Premelting at the ice--SiO2...

Premelting at the ice–SiO2 interface

A high-energy x-ray microbeam diffraction study

Von der Fakultat fur Mathematik und Physik der Universitat Stuttgartzur Erlangung der Wurde eines Doktors der

Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung

Vorgelegt von

Simon Christoph Engemann

aus Stuttgart

Hauptberichter: Prof. Dr. Helmut DoschMitberichter: Prof. Dr. Clemens Bechinger

Eingereicht am: 7. Oktober 2004Tag der mundlichen Prufung: 4. Februar 2005

Institut fur Theoretische und Angewandte Physikder Universitat Stuttgart,

Max-Planck-Institut fur Metallforschungin Stuttgart

2005

Bibliografische Information Der Deutschen Bibliothek: Die Deutsche Bibliothekverzeichnet diese Publikation in der Deutschen Nationalbibliografie; detailliertebibliografische Daten sind im Internet uber <http://dnb.ddb.de> abrufbar.

Bibliographic information published by Die Deutsche Bibliothek: Die DeutscheBibliothek lists this publication in the Deutsche Nationalbibliografie; detailedbibliographic data are available in the Internet at http://dnb.ddb.de.

Engemann, Simon:Premelting at the ice–SiO2 interface, a high-energy x-ray microbeam diffractionstudyDownload at http://www.ice-premelting.net/diss/.

c© 2005 Simon EngemannHerstellung und Verlag: Books on Demand GmbH, Norderstedt.

ISBN 3-8334-3980-7

Contents

Contents vi

Deutsche Zusammenfassung vii0.1 Eis und Wasser . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii0.2 Grenzflachenschmelzen von Eis . . . . . . . . . . . . . . . . . . . viii0.3 Messprinzip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix0.4 Probenpraparation und Probenumgebung . . . . . . . . . . . . . . x0.5 Ergebnisse und Diskussion . . . . . . . . . . . . . . . . . . . . . . xi

0.5.1 Morphologie der Substrate . . . . . . . . . . . . . . . . . . xi0.5.2 Wachstum der quasiflussigen Schicht . . . . . . . . . . . . xi0.5.3 Struktur der quasiflussigen Schicht . . . . . . . . . . . . . xii0.5.4 Weitere Experimente . . . . . . . . . . . . . . . . . . . . . xiii0.5.5 Bedeutung der Ergebnisse . . . . . . . . . . . . . . . . . . xiii

0.6 Ausblick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction 1

2 Ice and water 32.1 Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The H2O molecule and the hydrogen bond . . . . . . . . . . . . . 42.3 Anomalies and mysteries . . . . . . . . . . . . . . . . . . . . . . . 52.4 The quest for the water structure . . . . . . . . . . . . . . . . . . 62.5 Ice Ih . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Interface melting 113.1 The melting transition . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Theory of interface melting . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Phenomenological description . . . . . . . . . . . . . . . . 143.2.2 Landau-Ginzburg models . . . . . . . . . . . . . . . . . . . 163.2.3 Density functional theory . . . . . . . . . . . . . . . . . . 193.2.4 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.5 Other approaches . . . . . . . . . . . . . . . . . . . . . . . 193.2.6 Molecular dynamics simulations . . . . . . . . . . . . . . . 20

iii

iv CONTENTS

3.2.7 Interfacial melting and substrate roughness . . . . . . . . . 203.2.8 Further aspects . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Experimental evidence for interface melting . . . . . . . . . . . . 213.4 Interface melting of ice . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Theory and simulations . . . . . . . . . . . . . . . . . . . . 233.4.2 The free surface of ice . . . . . . . . . . . . . . . . . . . . 233.4.3 Ice in porous media . . . . . . . . . . . . . . . . . . . . . . 243.4.4 Ice–solid interfaces . . . . . . . . . . . . . . . . . . . . . . 263.4.5 Further aspects . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Consequences of ice premelting . . . . . . . . . . . . . . . . . . . 303.5.1 Permafrost . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.2 Glacier motion . . . . . . . . . . . . . . . . . . . . . . . . 303.5.3 Thunderstorms and atmospheric chemistry . . . . . . . . . 313.5.4 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . 31

4 Theory of x-ray reflectivity 334.1 Index of refraction for x-rays . . . . . . . . . . . . . . . . . . . . . 334.2 Reflection at an ideal interface . . . . . . . . . . . . . . . . . . . . 344.3 Reflection at multiple interfaces . . . . . . . . . . . . . . . . . . . 354.4 Arbitrary dispersion profiles . . . . . . . . . . . . . . . . . . . . . 374.5 The kinematical approximation . . . . . . . . . . . . . . . . . . . 374.6 Data analysis and phase inversion . . . . . . . . . . . . . . . . . . 384.7 Description of rough interfaces . . . . . . . . . . . . . . . . . . . . 394.8 Reflectivity from rough interfaces . . . . . . . . . . . . . . . . . . 40

4.8.1 Specular reflectivity . . . . . . . . . . . . . . . . . . . . . . 424.8.2 Integrated diffuse intensity . . . . . . . . . . . . . . . . . . 444.8.3 Off-specular reflectivity . . . . . . . . . . . . . . . . . . . . 44

4.9 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 High-energy x-ray-reflectivity experiments 475.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Application to the interface melting of ice . . . . . . . . . . . . . 505.3 Experimental and instrumental considerations . . . . . . . . . . . 51

5.3.1 Source and optics . . . . . . . . . . . . . . . . . . . . . . . 515.3.2 Sample stage . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.3 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3.4 Scattering geometry . . . . . . . . . . . . . . . . . . . . . 565.3.5 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.6 Integration by the detector . . . . . . . . . . . . . . . . . . 615.3.7 Illumination of the sample . . . . . . . . . . . . . . . . . . 615.3.8 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3.9 Data correction . . . . . . . . . . . . . . . . . . . . . . . . 68

CONTENTS v

6 Sample preparation and environment 71

6.1 The substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 The ice samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3 The cold room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.4 Interface preparation . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.5 The in situ chamber . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.6 Temperature stability and accuracy . . . . . . . . . . . . . . . . . 80

7 Results and discussion 85

7.1 Overview of the main experiments . . . . . . . . . . . . . . . . . . 86

7.2 Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.2.1 Raw data analysis . . . . . . . . . . . . . . . . . . . . . . . 92

7.2.2 Reconstruction of density profiles . . . . . . . . . . . . . . 93

7.2.3 Reliability of the fits . . . . . . . . . . . . . . . . . . . . . 99

7.3 Substrate morphology . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.1 Smooth substrate . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.2 Rough substrate . . . . . . . . . . . . . . . . . . . . . . . 105

7.3.3 AFM measurements . . . . . . . . . . . . . . . . . . . . . 112

7.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.4 Growth law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.4.1 What is expected from theory? . . . . . . . . . . . . . . . 114

7.4.2 Experimentally observed growth law . . . . . . . . . . . . 115

7.4.3 Onset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.4.4 Growth amplitude . . . . . . . . . . . . . . . . . . . . . . 120

7.4.5 Influence of roughness . . . . . . . . . . . . . . . . . . . . 121

7.4.6 Comparison with surface melting . . . . . . . . . . . . . . 122

7.4.7 Influence of temperature error . . . . . . . . . . . . . . . . 122

7.5 Density and structure of the quasiliquid . . . . . . . . . . . . . . . 125

7.5.1 Experimentally observed density . . . . . . . . . . . . . . . 125

7.5.2 Conclusions about the structure . . . . . . . . . . . . . . . 127

7.6 Si wafer as substrate . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.6.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.6.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 132

7.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.7 Neutron reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.7.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.7.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 136

7.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.8 Substrate termination and radiation effects . . . . . . . . . . . . . 138

7.9 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

vi CONTENTS

8 Outlook 1438.1 Influence of the substrate material and the confinement . . . . . . 1438.2 Surface melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.3 Influence of the substrate morphology . . . . . . . . . . . . . . . . 1458.4 Influence of impurities . . . . . . . . . . . . . . . . . . . . . . . . 1458.5 Growth law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.6 Structure of the quasiliquid . . . . . . . . . . . . . . . . . . . . . 1468.7 Influence of electric fields . . . . . . . . . . . . . . . . . . . . . . . 147

9 Summary 149

List of acronyms 155

List of figures 159

List of tables 161

Bibliography 163

Acknowledgements 179

Deutsche Zusammenfassung

Im Rahmen dieser Doktorarbeit wurde das Grenzflachenschmelzen von Eis mit-tels einer neuen Rontgenstreumethode basierend auf hochenergetischen Mikro-strahlen untersucht. Dieses Kapitel stellt eine deutsche Zusammenfassung derin Englisch verfassten Dissertation (folgende Kapitel) dar. Eine kurzer gefassteenglischsprachige Zusammenfassung findet sich in Kapitel 9.

0.1 Eis und Wasser

Eis und Wasser bedecken einen Großteil unseres Planeten und haben seine Ober-flache uber Jahrmillionen hinweg gepragt. Das Gleichgewicht von Eis, flussigemWasser und Wasserdampf ist entscheidend fur das Klima. Wasser bildet dieGrundlage unseres Lebens und ist in seinen verschiedenen ErscheinungsformenTeil unseres Alltags.

Obwohl der Aufbau des Wassermolekuls sehr einfach erscheint, weist Wassereine Reihe von außergewohnlichen Eigenschaften und Anomalien auf. Oftmalssind gerade diese Anomalien von entscheidender Bedeutung bei der Rolle, die dasWasser in der Natur und fur unser Leben spielt. Trotzdem sind diese Anomalienbis heute nicht vollstandig verstanden [1].

Anomalien treten u.a. in den Antwortfunktionen des Wassers, wie der iso-thermen Kompressibilitat, der isobaren Warmekapazitat und dem Warmeaus-dehnungskoefizienten auf. Der Betrag dieser Funktionen steigt beim Abkuhlenstark an, beim Unterkuhlen wird dieser Anstieg noch starker. Die Antwortfunk-tionen scheinen bei einer singularen Temperatur TS = 228 K zu divergieren [2].In einer normalen Flussigkeit wurden die genannten Großen mit sinkender Tem-peratur langsam abfallen.

Eine weitere Besonderheit ist die so genannte Dichteanomalie des Wassers.Wasser weist die großte Dichte bei +4C auf, bei weiterem Abkuhlen sinkt dieDichte wieder. Dies ist equivalent zu einem Vorzeichenwechsel des Warmeaus-dehnungskoefizienten bei +4C. Ferner ist die feste Phase Eis weniger dicht alsflussiges Wasser.

Auch beim Fest-Flussig-Ubergang, d.h. dem Schmelzen von Eis, treten Beson-derheiten zu Tage. So fuhrt der schon genannte Dichteunterschied zwischen Eisund Wasser dazu, dass sich Eis durch Druck verflussigen lasst. Außerdem zeigt

vii

viii DEUTSCHE ZUSAMMENFASSUNG

Eis eine ausgepragte Tendenz zum Oberflachenschmelzen [3], welches im nachstenAbschnitt erlautert werden soll.

0.2 Grenzflachenschmelzen von Eis

Befindet sich ein Festkorper s in Kontakt mit einem anderen Medium b, so kannsich an der Grenzflache s–b eine dunne Schicht des Festkorpers schon unter-halb der Schmelztemperatur Tm im Volumen des Materials verflussigen. DiesesPhanomen nennt man

”Grenzflachenschmelzen“. Handelt es sich bei dem Medi-

um b um Vakuum, Luft oder die Gasphase von s, so bezeichnet man die Grenz-flache ublicherweise als Oberflache von s und das eben genannte Phanomen als

”Oberflachenschmelzen“. Handelt es sich hingegen bei dem Medium b um einen

anderen Festkorper (im folgenden auch Substrat genannt), mithin um eine Fest–Fest-Grenzflache, so spricht man von Grenzflachenschmelzen im engeren Sinne.Man bezeichnet die geschmolzene Schicht an der Grenzflache normalerweise als

”quasiflussig“ (englisch abgekurzt mit qll), da im allgemeinen nicht zu erwarten

ist, dass sie die gleiche Struktur wie die Flussigkeit im Volumen hat. Man erwar-tet vielmehr eine durch den Kontakt mit dem darunter liegenden Festkorper sund dem daruber liegenden Medium b modifizierte Struktur.

Das Grenzflachenschmelzen wird vom Prinzip der Minimierung der freienEnergie getrieben. In einem phanomenologischen Modell (siehe z.B. [4]) lasst sichdie Anderung der freien Energie durch das Auftreten einer dunnen quasiflussigenSchicht der Dicke L folgendermaßen beschreiben:

∆F (L) = ρqllQmLTm − T

Tm

+ ∆γ (L) . (1)

Hierbei bezeichnet T die Temperatur, ρqll die Dichte der Quasiflussigkeit undQm die Schmelzwarme. ∆γ (L) ist der Unterschied in den Grenzflachenenergien,der durch das Auftreten der quasiflussigen Schicht erzeugt wird. Aufgrund derWechselwirkung zwischen den beiden Grenzflachen s–qll und qll–b hangt ∆γ vonderen Abstand, d.h. der Dicke der quasiflussigen Schicht, ab.

Abhangig vom Typ der Wechselwirkungen ergibt die Minimierung der frei-en Energie unterschiedliche Wachstumsgesetze fur die Dicke der quasiflussigenSchicht als Funktion der Temperatur.

Im Falle kurzreichweitiger exponentiell zerfallender Wechselwirkungen erhaltman ein logarithmisches Wachstumsgesetz der Form

L (T ) = L0 ln

(Tm − T0

Tm − T

). (2)

Das Grenzflachenschmelzen setzt bei der Temperatur T0 ein. Die Amplitude L0

kann mit der Korrelationslange der Quasiflussigkeit identifiziert werden.

0.3. MESSPRINZIP ix

Im Falle langreichweitiger Van-der-Waals-Wechselwirkungen ergibt sich einPotenzgesetz der From

L (T ) ∝(Tm − T

Tm

)−1/(n+1)

. (3)

Hierbei gilt n = 2 fur nicht-retardierte und n = 3 fur retardierte Wechselwirkun-gen.

Die bisherige Beschreibung trifft im Prinzip auf das Oberflachenschmelzen wieauch auf das Grenzflachenschmelzen an Fest–Fest-Grenzflachen zu. Im letzterenFall hangen die Grenzflachenenergien und damit das Schmelzverhalten naturlichvom Material des Substrats ab. Ferner konnte die Morphologie des Substrats einewichtige Rolle spielen, was in der bisherigen Betrachtung nicht berucksichtigt ist.

Das Grenzflachenschmelzen ist von fundamentaler Bedeutung fur den gesam-ten Schmelzprozess, da die quasiflussige Schicht an der Grenzflache als Nukleati-onskeim fur das Schmelzen im Volumen dienen kann. Außerdem hat es speziell imFall von Eis wichtige Konsesquenzen fur eine Reihe von Prozessen in Natur undTechnik [5], wie z.B. Gletscherbewegung, Stabilitat von Permafrost und Vereisungvon Flugzeugtragflachen.

Oberflachenschmelzen wurde an einer Vielzahl von Materialien beobachtet,besonders ausgepragt ist es bei Eis. Auch fur das Auftreten von Grenzflachen-schmelzen an Eis–Festkorper-Grenzflachen gibt es viele Hinweise. Allerdings feh-len hier schlussige mikroskopische Experimente an wohl definierten Grenzflachen.Dies liegt daran, dass es kaum geeignete Methoden gibt, um solche vergrabenenGrenzflachen zu untersuchen.

0.3 Reflektivitatsmessungen mit hochenergeti-

scher Rontgenstrahlung

Zur Untersuchung von Grenzflachen wurde in unserer Arbeitsgruppe ein neuesVerfahren entwickelt [6]. Der Aufbau dazu wurde am Strahlrohr ID15A der ESRF(European Synchrotron Radiation Facility) in Grenoble installiert. Das Verfahrenberuht auf der Verwendung brillianter und hochenergetischer (in dieser Arbeitca. 70 keV) Rontgenstrahlung, die an modernen Synchrotronstrahlungsquellenerzeugt werden kann.

Hochenergetische Rontgenstrahlung kann mehrere Millimeter oder gar Zen-timeter in Materialien eindringen und daher auch tief vergrabene Grenzflachenerreichen. Mit den herkommlichen Rontgenstreumethoden dagegen konnen nurGrenzflachen von dunnen Schichten untersucht werden. Dies fuhrt insbesonde-re bei Flussigkeiten zu der experimentellen Schwierigkeit, die Schichtdicke kon-stant zu halten. Zum anderen interferiert bei dunnen Schichten das Streusi-gnal der Grenzflache mit dem Streusignal der Oberflache. Mit hochenergetischer

x DEUTSCHE ZUSAMMENFASSUNG

Rontgenstrahlung lassen sich auch tief vergrabene Grenzflachen untersuchen unddamit die geschilderten Probleme vermeiden.

Aufgrund der kleinen Streuwinkel bei hochenergetischer Rontgenstrahlungwerden allerdings sehr hohe Anforderungen an die Genauigkeit und Stabilitatdes gesamten Aufbaus gestellt (besser als ±10 µrad bzw. ±1 µm). Ferner er-fordert der Einsatz von hochenergetischen Rontgenstrahlen sehr kleine Strahl-durchmesser am Probenort, da bei den kleinen Streuwinkeln die Projektion derGrenzflache senkrecht zum Strahl sehr klein ist. Dies wurde in den Experimen-ten fur diese Arbeit mittels Brechungslinsen fur Rontgenstrahlen erreicht, diees erlauben, Rontgenstrahlen auf einen Durchmesser von wenigen Mikrometernzu fokussieren. Dies hat zudem den Vorteil, dass sich durch entsprechend kleineBlendenoffnungen der Streuuntergrund weitgehend unterdrucken lasst. Dadurchergibt sich ein sehr großer dynamischer Bereich der Messungen von bis zu 10Großenordnungen.

Die in dieser Arbeit angewandten Rontgenreflektivitatsmessungen [7] erlaubendie Rekonstruktion des Dichteprofils senkrecht zur Grenzflache mit einer Auflo-sung bis in den atomaren Bereich. Dies ermoglicht, etwaiges Grenzflachenschmel-zen am Erscheinen einer zusatzlichen Schicht im Dichteprofil zu erkennen. Eslassen sich sowohl die Dicke als auch die Dichte der Schicht temperaturabhangigverfolgen. Die Experimente wurden in einem Temperaturbereich von −30 bis−0.022C durchgefuhrt.

0.4 Probenpraparation und Probenumgebung

In dieser Arbeit wurde das Grenzflachenschmelzen von Eis an Eis–SiO2–Si Grenz-flachen untersucht. Diese konnen als Modell fur Eis–Mineral-Grenzflachen, die inder Natur vorkommen, betrachtet werden.

Fur die SiO2–Si Substrate wurde einkristallines Silizium verwendet, das che-momechanisch poliert und anschließend aufwendig chemisch gereinigt wurde. AnLuft bildet Silizium dann ein natives amorphes Oxid von ca. 1–2 nm Dicke. DieSubstrate sind ursprunglich hydrophob, unter Bestrahlung mit hochenergetischerRontgenstrahlung und im Kontakt mit H2O bildet sich allerdings eine hydrophileTerminierung. Die Hauptexperimente wurden mit zwei Substraten durchgefuhrt,die eine unterschiedliche Oberflachenmorphologie aufweisen, ein

”glattes“ Sub-

strat und ein”raues“ Substrat (Details s.u.). Damit sollte der Einfluss der Rau-

igkeit auf das Grenzflachenschmelzen untersucht werden.Eis-Einkristalle wurden von Prof. Bilgram (ETH Zurich) aus hochreinem Was-

ser gezuchtet [8]. Mittels eines Zweikreis-Diffraktometers wurden die Eis-Kristallemit der c-Achse senkrecht zum Substrat ausgerichtet. Beim Kontaktieren mit demSubstrat wurde durch kurzzeitiges Erwarmen des Substrats die Eisoberflache auf-geschmolzen. Beim anschließenden Abkuhlen wurde das Substrat standig nach-gefahren, so dass die flussige Schicht immer sehr dunn war. Dadurch wurden et-

0.5. ERGEBNISSE UND DISKUSSION xi

waige Verunreinigungen ausgeschwemmt und Lufteinschlusse vermieden, so dassnach dem langsamen Rekristallisieren eine homogene und glatte Eis–SubstratGrenzflache vorlag. Die gesamte Probenpraparation wurde in einem begehbarenKuhlraum durchgefuhrt.

Fur die eigentlichen Rontgenstreuexperimente am Synchrotron wurde einemobile Probenumgebung konstruiert. Sie ermoglicht eine sehr stabile Tempera-turregelung uber Peltier-Elemente, die an zwei Seiten der Probe angebracht sind.

0.5 Ergebnisse und Diskussion

0.5.1 Morphologie der Substrate

Die Morphologie der Substrate wurde sowohl anhand der Rontgenstreudatenals auch erganzend mittels Rasterkraftmikroskopie untersucht. Der quadratischeMittelwert σ der Hohenabweichung fur das glatte Substrat betragt nach denRontgenmessungen (2.7±0.4) A. Die Rauigkeit ist lateral nur schwach korreliert.Beim rauen Substrat dagegen weist die Rauigkeit ein selbst-affines Verhaltenauf, wie es oft als Folge von Wachstums- oder Atzprozessen beobachtet wird.Die Messung der diffusen Rontgenreflektivitat ermoglicht es, die lateralen Kor-relationen der Rauigkeit zu bestimmen: g(R) = 0.11 · R2·0.34. In diesem Fall istσ unbestimmt. Die Analyse der Rasterkraftbilder bestatigt die Ergebnisse derRontgenmessungen.

0.5.2 Wachstum der quasiflussigen Schicht

Sowohl am rauen als auch am glatten Substrat konnte eindeutig das Auftretenvon Grenzflachenschmelzen nachgewiesen werden.

Das Wachstum der quasiflussigen Schicht kann uber einen weiten Bereich derTemperatur T durch ein logarithmisches Wachstumsgesetz beschrieben werden(s.o.). Im Falle des glatten Substrats ist die Ubereinstimmung mit einem loga-rithmischen Wachstum außerordentlich gut. Beim rauen Substrat dagegen zeigensich Abweichungen von einem logarithmischen Wachstum. Die Amplitude L0 desWachstumsgesetzes betragt (3.7±0.3) A beim glatten Substrat und (8.2±0.4) Abeim rauen Substrat. Diese Werte liegen im Bereich der Literaturwerte fur dieKorrelationslange von Wasser.

Beim Vergleich des glatten und des rauen Substrats ergibt sich eine guteUbereinstimmung im Bereich tiefer Temperaturen bis ca. −0.7C. Danach steigtdie Schichtdicke am rauen Substrat starker als am glatten Substrat. Die hochsteSchichtdicke betragt 55 A bei −0.036C am rauen Substrat gegenuber 27.5 A bei−0.022C am glatten Substrat. Das Schichtwachstum am rauen Substrat lasstsich auch gut durch ein Potenzgesetz beschreiben. Der Exponent liegt hierbeinahe bei einem Wert von −1/3, wie es fur nicht-retardierte Van-der-Waals-Wech-

xii DEUTSCHE ZUSAMMENFASSUNG

selwirkungen zu erwarten ist. Das Wachstum am glatten Substrat hingegen lasstsich nicht durch ein Potenzgesetz beschreiben.

Diese Ergebnisse lassen sich so interpretieren, dass es durch die Rauigkeit zueinem fruheren Ubergang von einem logarithmischen Gesetz zu einem Potenz-gesetz kommt. Fur sehr große Schichtdicken wurde man solch einen Ubergangin jedem Fall erwarten, da dann die langreichweitigen Van-der-Waals-Wechsel-wirkungen dominieren. Wenn das logarithmische Wachstumsgesetz fur das raueSubstrat nicht gultig ist, kann die daraus bestimmte Amplitude naturlich nichtmehr mit der Korrelationslange verglichen werden.

Ein besonderes Verhalten zeigt sich bei sehr dunnen Schichtdicken (d.h. beisehr tiefen Temperaturen). Hier scheint eine dunne Schicht auch noch bei tieferenTemperaturen flussig zu bleiben, als man aus der Extrapolation des logarithmi-schen Wachstumsgesetzes erwarten wurde. Dies ist insofern nicht verwunderlich,als im Bereich sehr dunner Schichtdicken das Kontinuumsmodell, das zur Herlei-tung der Wachstumsgesetze verwendet wurde, nicht mehr gultig ist.

Die beobachteten Schichtdicken der quasiflussigen Schicht liegen bei gleichenTemperaturen deutlich unter den beim Oberflachenschmelzen gemessenen Wer-ten.

0.5.3 Struktur der quasiflussigen Schicht

Die Rontgenreflektivitatsmessungen erlauben auch, die mittlere Dichte ρqll derquasiflussigen Schicht zu bestimmen. Die Werte von 1.20 g/cm3 am rauen und1.19 g/cm3 am glatten Substrat stimmen gut uberein und sind deutlich hoherals die Dichte ρl=1.0 g/cm3 von normalem Wasser. Es stellt sich also die Frage,wie die Struktur dieser hochdichten quasiflussigen Schicht aussehen konnte. BeimVergleich mit anderen Wasserphasen zeigt sich, dass die ermittelte Dichte nahebei der Dichte ρHDA=1.17–1.19 g/cm3 von hochdichtem amorphen Eis [9, 10](englisch abgekurzt mit HDA) bei Atmospharendruck liegt.

Dies legt eine strukturelle Verwandtschaft mit HDA nahe. Allerdings han-delt es sich beim Grenzflachenschmelzen um ein Gleichgewichtsphanomen, wasdas Auftreten einer metastabilen Struktur fraglich erscheinen lasst. In aktuel-len Theorien zur Struktur des Wassers wird jedoch eine der HDA entsprechendehochdichte flussige Form von Wasser (englisch abgekurzt mit HDL) postuliert [1].Diesen Theorien zufolge werden die (oftmals anomalen) Eigenschaften von Wasserdurch Fluktuationen dieser hochdichten und einer ebenfalls postulierten niedrig-dichten (englisch abgekurzt LDL) Form von Wasser bestimmt. Es konnten ander Grenzflache also Fluktuationen in die postulierte hochdichte Form von Was-ser stabilisiert werden. Ein derartiges Phanomen legen auch andere Experimentean Wasser-Grenzflachen nahe [11, 12].

0.6. AUSBLICK xiii

0.5.4 Weitere Experimente

Ein weiteres Experiment wurden mit einem sehr glatten Silizium-Wafer als Sub-strat durchgefuhrt. Bei diesem Experiment wurde ein anderes Reinigungsverfah-ren fur das Substrat verwendet, das direkt zu einer hydrophilen Terminierungmit einer dicken Oxidschicht fuhrt. Leider machte eine Krummung des nur etwa0.6 mm dicken Wafers die Messung vollstandiger Reflektivitatskurven unmoglich.Trotzdem konnte das Auftreten von Grenzflachenschmelzen auch bei diesem Sub-strat bestatigt werden.

Andere Experimente wurden mittels Neutronenreflektivitatsmessungen, derStandardmethode fur tief vergrabene Grenzflachen, durchgefuhrt. Aufgrund desim Vergleich zu Synchrotronstrahlungsquellen geringen Flusses der Neutronen-quellen sind allerdings nur kleine Impulsubertrage bei Reflektivitatsmessungenzuganglich. Dadurch ist die erreichbare Auflosung im Realraum zu sehr begrenzt.

0.5.5 Bedeutung der Ergebnisse

Die Ergebnisse dieser Arbeit haben Bedeutung fur das Verstandnis wichtigerPhanomene in der Natur (s.o.). Die Auswirkungen des beobachteten Grenzfla-chenschmelzens hangen von den bisher unbekannten Eigenschaften der quasi-flussigen Schicht ab. In diesem Zusammenhang ist auch die Beobachtung einerhochdichten Form von Wasser bedeutsam, da dies auch Unterschiede in den ande-ren Eigenschaften, wie z.B. der Viskositat oder der Loslichkeit von Verunreinigun-gen, nahe legt. Da in der Natur vorkommende Grenzflachen normalerweise rausind, ist der beobachtete Einfluss der Substratmorphologie wichtig fur Schluss-folgerungen uber

”reale“ Systeme. Außerdem konnen Grenzflachenexperimente

wie sie in dieser Arbeit durchgefuhrt wurden, neue Einblicke in die Struktur vonWasser liefern, insbesondere, wenn tatsachlich die postulierten Wasserformen anGrenzflachen stabilisiert werden konnen.

0.6 Ausblick

Aus dieser Arbeit ergeben sich weitere Fragen und mogliche Forschungsthemen.Darunter befinden sich die Frage nach dem Verhalten von Eis und Wasser imKontakt mit weiteren Materialien und die Frage nach dem Einfluss von Verun-reinigungen. Dazu gehort auch die Frage, ob die beobachtete hochdichte Formvon Wasser nur in dem dunnen Spalt zwischen Eis und dem verwendeten Sub-strat stabilisiert werden kann, oder ob dies auch an der Grenzflache zu flussigemWasser moglich ist.

Zukunftige Experimente konnten auch einen großeren Temperaturbereich ab-decken. Dann konnte das Verhalten nahe des Schmelzpunktes und ein moglicherWechsel des Schichtwachstums zu einem Potenzgesetz untersucht werden. Dieswurde allerdings eine Verbesserung der Temperaturstabilitat voraussetzen. Zum

xiv DEUTSCHE ZUSAMMENFASSUNG

anderen konnte das Einsetzen des Grenzflachenschmelzens bei tiefen Temperatu-ren genauer untersucht werden.

Die wohl spannendste aber zugleich schwierigste Aufgabe wird darin beste-hen, die Struktur der quasiflussigen Schicht zu bestimmen. Hierzu reichen Reflek-tivitatsmessungen nicht aus, da sie prinzipiell nur erlauben, die lateral gemittelteStruktur zu untersuchen. Eine Moglichkeit bietet die evaneszente Braggstreuung[13], allerdings ware in dem vorliegenden Fall eine Trennung der Streusignale vonder Quasiflussigkeit und der amorphen Oxidschicht kaum moglich. Hier mussteein kristallines Substrat verwendet werden. Selbst dann bleibt eine Messung mithochenergetischer Rontgenstreuung aufgrund der extrem kleinen Streuwinkel sehrschwierig.

Chapter 1

Introduction

Much of the current research in condensed matter physics is not devoted to thethe understanding of bulk materials, but matter in confinement and reduced di-mensions. This includes nanoparticles, thin films, and all kinds of interfaces. Inall of these situations, the properties of the materials involved can differ drasti-cally from the bulk.

The special case of the free surface has been studied in detail over the lastdecades, and x-ray scattering techniques have made a significant contribution tothe understanding of surface structures. The center of interest has now moved tosolid–solid, solid–liquid, and liquid–liquid interfaces. Such interfaces are of greattechnological interest (for example electrode–electrolyte interfaces) and play animportant role in other disciplines (for example biology).

Deeply buried interfaces, however, are difficult to probe experimentally. Theideal probe would allow non-destructive in situ measurements with a resolutionon the atomic scale. X-rays meet all these requirements, but usually lack thenecessary penetration depth. In this work, a recently developed scheme hasbeen used, which exploits the properties of high-energy x-ray microbeams. Itallows to apply the established surface-sensitive x-ray scattering techniques tostudy deeply buried interfaces. Only modern Synchrotron Radiation sources canprovide beams with the brilliance and stability required by this scheme.

Water and ice are not only of paramount importance for the biosphere, butalso exhibit a large number of surprising properties, which are still not fullyunderstood. At the free surface of ice, a phenomenon called ‘surface melting’ oc-curs. It is the formation of a liquid-like layer below the bulk melting temperature.There are many indications that an analogous effect exists at ice–solid interfaces.Because of the experimental difficulties in probing such interfaces, little is knownabout this ‘interface melting’, which might depend on the morphology of theinterface, for example.

In this work, model interfaces of ice in contact with SiO2 have been stud-ied. Measurements of the x-ray reflectivity reveal the density profile across theinterface. These measurements allow to observe premelting layers on nanoscopic

1

2 CHAPTER 1. INTRODUCTION

length scales, and to determine their thickness and density as a function of tem-perature. SiO2 substrates with different morphology have been used to determinethe influence of the roughness.

Other projects in the context of this thesis include ordering and segregationat CuPd-surfaces1 and Neutron Compton Scattering experiments on ice2. Theseprojects have some relation to the work on interface melting, but are beyond thescope of this dissertation.

For the bigger part, the organization of this dissertation should be self-ex-planatory, but sometimes a justification is given where I deemed it helpful. Eachchapter starts with a short overview of its content.

1in collaboration with H. Reichert, C. Mocuta, W. Schweika, and H. Dosch2in collaboration with H. Reichert, J. Mayers, G. Reiter, J. Bilgram, and H. Dosch

Chapter 2

Ice and water

The purpose of this chapter is to provide a short overview of the properties ofwater and ice. It will concentrate on aspects which are relevant for this work.

Sec. 2.1 explains the importance of water and ice, Sec. 2.2 introduces theH2O molecule as the building block of the water and ice structure, and Sec. 2.3highlights the anomalous properties of H2O. The complex phase behavior of H2Ois part of the mystery and in the focus of current water theories trying to explainthe anomalous properties of H2O. These issues will be discussed in Sec. 2.4;for more information see [1] and references therein. The chapter closes with adescription of the ice structure in Sec. 2.5. A more detailed description, includingcrystallographic data, can be found in [14].

For a general overview of ice physics and chemistry, the reader is referred tothe work of Petrenko and Whitworth [14], Hobbs [15], and Whalley [16]. Moreinformation about water can be found in the books of Franks [17] and Ball [18].

2.1 Importance

Water in its various forms is virtually omnipresent on the surface of the Earth,it touches nearly all aspects of our everyday life, and without water, life wouldnot even exist.

Ice and water shape the surface of our planet. 70% of the Earth is covered byoceans, 10% of the land mass is currently covered by ice (up to 30% were coveredduring the Earth’s history), and around 5% of the oceans are covered with ice,depending on the season. Retreating oceans have left their sediments, rivers cutdeep valleys and glaciers sculpt the landscape.

The climate crucially depends on the presence of ice, water, and vapor.Snow, ice, and cloud cover determine the balance of radiation received andreflected/emitted by the Earth. Evaporation of water, snowfall, and subsequentflow (see Sec. 3.5.2) of polar ice back into the oceans form another delicate equi-librium. Ice in the atmosphere is important for the production of rain and for the

3

4 CHAPTER 2. ICE AND WATER

0.9572 Å

104.52°

Figure 2.1: Free water molecule made up of one oxygen (large sphere) and twohydrogens (small spheres). Note the bent shape, which gives rise to an electricdipole moment. The H—O—H bond angle (104.52) is close to, but not exactlythe tetrahedral angle (109.47).

scavenging of atmospheric pollutants. It affects the chemistry of the atmosphere,as for example the reactions responsible for ozone depletion (see Sec. 3.5.3). Iceparticles also play a great role in the electrification of thunderstorms (see alsoSec. 3.5.3). Under certain conditions, atmospheric ice reaches the ground in theform of hail, often causing considerable damage.

Ice is of great relevance for buildings and infrastructure (frost heave, ice onpower lines, freezing pipes, avalanches), traffic (slippery and snow covered roads,ice on airplane wings, icebergs and pack ice disturbing shipping traffic), andagriculture (freeze damage). The effects on constructions and infrastructure es-pecially concern the permafrost regions (see Sec. 3.5.1), which cover some 20%of the land mass on the northern hemisphere.

Whereas the effects of ice often cause problems, we admire its beauty in theform of snowflakes, icicles, and frost patterns on windows. We enjoy skiing andice skating, and we use ice to preserve food and to cool drinks. Before the adventof refrigerators and freezers, ice was an important trading good [19, 20, 21].

Ice can also be found throughout the universe. Tiny particles in cold areas ofinterstellar space are covered by thin ice layers. In the solar system, ice is presenton moons and comets.

For many of the aspects mentioned here, interface melting of ice is essential(see Sec. 3.5).

2.2 The H2O molecule and the hydrogen bond

Water and ice are made up of H2O molecules bound together by hydrogen bonds.The arrangement of the oxygen and the two hydrogens is shown in Fig. 2.1. Fora free molecule, the O—H distance is (0.9572±0.003) A and the H—O—H angleis (104.52±0.05) The bent shape is a consequence of the ground state. It givesrise to an electric dipole moment of (6.186±0.001)×10−30 Cm. It also defines thepossible arrangements of molecules in the crystal structures.

Hydrogen bonds are a distinct type of chemical bond where a hydrogen atom

2.3. ANOMALIES AND MYSTERIES 5

sits between two highly electronegative atoms (F, O, N). If the highly electroneg-ative atoms are oxygens, the bond can be represented as O—H· · ·O. The hy-drogen atom stays covalently bound to one of the oxygen atoms (O—H), theproton ‘donor’. The distance to the proton ‘acceptor’ is much larger (O· · · ). Thestrength of the hydrogen bond is between that of covalent bonds and Van derWaals interactions. Hydrogen bonds are crucial for the properties of water, butdifficult to account for in calculations. As every H2O molecule can act as a protondonor for two hydrogen bonds and as a proton acceptor for two additional bonds,complex networks of hydrogen bonds can form. They also play an important rolein biochemistry, where nearly all processes take place in aqueous environments.

2.3 Anomalies and mysteries

Water in its various forms has always evoked interest and fascination in manyfields, and has been in the focus of scientific research since its beginnings. Despitethe importance of water and the amount of research dedicated to its understand-ing, water still holds unsolved mysteries [1].

The elusive simplicity of the water molecule (see Sec. 2.2) contrasts with thecomplex behavior of water, its unusual and all too often counterintuitive proper-ties, and its large number of solid phases (see Sec. 2.4) Therefore, the scientificinterest in water stems not only from its relevance in nature and technology, butalso from the fundamental questions it poses for condensed matter physics. Theanomalous properties of water are still not fully understood, but it is those veryanomalies that are responsible for the importance of water.

One of the anomalous characteristics of water is the behavior of its responsefunctions (see Fig. 2.2) like the isothermal compressibility KT , the isobaric heatcapacity CT , and the thermal expansion coefficient αT . Their magnitude increasessharply upon cooling (from a certain point on). When water is cooled furtherand eventually supercooled, the increase becomes even more pronounced. Whenextrapolated, the response functions seem to diverge at a singular temperatureTS = 228 K [2]. In a typical liquid, all the mentioned response functions woulddecrease slightly upon cooling (see Fig. 2.2). Adding to these oddities, the co-efficient of thermal expansion in water changes its sign at 4C, which expressesthe anomalous and well known density maximum at 4C. Furthermore, the vis-cosity of water decreases and its diffusivity increases upon compression, again incontrast to typical liquids. Water can only be supercooled down to about TH =231 K, the homogenous nucleation temperature, before it crystallizes to ice Ih.Therefore, its behavior at TS cannot be probed directly by experiments.


Tm 319 K

Iso

the

rma

lc

om

pre

ss

ibil

ity

KT

Tm 308 K

Temperature

Co

eff

icie

nt

of

the

rma

le

xp

an

sio

n

P

Tm

277 KTemperature

Iso

ba

ric

he

at

ca

pa

cit

yC

T

Temperature

Water normal liquid

Figure 2.2: The anomalous response functions of water. The response functions ofwater (solid lines) increase strongly in magnitude upon cooling and supercooling.The response functions for a ‘normal’ liquid are shown for comparison (dashedlines). The coefficient of thermal expansion of water changes its sign from negativeto positive at 277 K (4C), which is equivalent to a density maximum at thistemperature.

2.4 The quest for the water structure

There are at least 13 different crystalline forms of water of which 9 are stable(see Fig. 2.3). Only the ‘ordinary’ ice Ih is stable at atmospheric pressure. Inaddition, there are several amorphous forms, a behavior called polyamorphism.

Glassy water was first produced by depositing water vapor onto a cold metalplate [22]. Direct vitrification of the liquid by rapid cooling (hyperquenching) waslater achieved by Bruggeler and Mayer [23, 24]. After annealing, those forms ofglassy water relax to low-density amorphous ice (LDA). When ice is compressedto about 11 kbar at 77 K, high-density amorphous ice (HDA) is formed [9, 10].Both LDA and HDA can be recovered at atmospheric pressure with the densitiesρLDA = 0.94 g/cm3 and ρHDA = 1.17–1.19 g/cm3 respectively. LDA and HDAhave distinct structures manifested by their large density difference. What distin-guishes them from common glassy states, is a sharp and reversible transformationbetween the two forms at about 2 kbar and 135 K [25], which is characteristicfor a thermodynamic phase transition, as is the large change in density.

There are currently two different conjectures for a coherent theory of waterwhich could explain its anomalous properties and their relation to the amorphousstates: the liquid-liquid phase transition hypothesis [26] and the singularity-freescenarios [27]. Both share the idea that LDA and HDA are the vitreous counter-parts of two different forms of liquid water, a low-density liquid phase (LDL) anda high-density liquid phase (HDL). According to the liquid-liquid phase transitionhypothesis, the transition between the two liquids is of first order and terminatesat a critical point (Tc ≈ 220 K, pc ≈ 1 kbar, see Fig. 2.3a) below the bulk meltingpoint. At higher temperatures, the two phases become indistinguishable, and the

2.4. THE QUEST FOR THE WATER STRUCTURE 7

310

290

270

250

230

210

190

170

150

130

110

90

Liquid

Ih

(0.92)V VI

II

Tem

pera

ture

(K)

HDA(1.17)

C

LDA(0.94)

HDL

Pressure (MPa)200 400 600 8000

LDL

III

XI

VI

IhII

VII

VIII

Liquid

Pressure (GPa)

X

?

0.1 1.0 10 100

600

400

200

0

Tem

pera

ture

(K)

III V

a b

Figure 2.3: H2O phase diagram compiled from [14, 31]. (a) Phase diagram formoderate pressures showing the stable phases, the metastable amorphous formsHDA (high-density amorphous) ice and LDA (low-density amorphous) ice, as wellas the postulated corresponding liquid phases HDL (high-density liquid) waterand LDL (low-density liquid) water. HDL and LDL terminate at a second criticalpoint C (see the text). The numbers in brackets denote the density at ambientpressure in g/cm3. (b) Phase diagram for very high pressures showing the stablephases. The dotted green box marks the region shown in a.

characteristics of water are determined by the coexisting fluctuations of these twostates. In the singularity free scenarios, the transition between the two liquidsis continuous, and the response functions show strong maxima, but do not di-verge. Owing to the supercooling limit mentioned before, the two suggestions aredifficult to probe directly. There are experiments in support of the liquid-liquidphase transition hypothesis [28, 29] as well as the singularity-free theories [30],and no consensus has emerged so far. Up to now, the LDL and HDL phase ofwater have not been observed experimentally.

It has also been suggested that there is more than one liquid-liquid transitionin water [32] and that HDA is not even a glass in the sense of a quenched liquid,but rather a poorly crystalline form of ice, so the quenched liquid might lookdifferent [33]. The role of the recently discovered very-high-density amorphousice (VHDA, ρVHDA = 1.25 g/cm3 at atmospheric pressure) [34, 35] is not yet cleareither.

Although the possibility of different forms of liquid water may sound ratherspeculative, liquid-liquid transitions have been experimentally observed in othermaterials, for example phosphorus [36]. Recent theoretical work indicates thatliquid-liquid transitions might actually be a rather generic phenomenon [37].


[ 100]1

[11 0]2

[0001]

Figure 2.4: The structure of ice Ih. View along the [0001]-direction (c-axis).The oxygen atoms are represented by large spheres, the hydrogen atoms by smallspheres. The connecting lines represent the hydrogen bonds. The oxygen atomsfollow the ‘wurtzite’ structure. Note the puckering of the hexagonal rings and theopen channels formed along the c-axis. Note also the disorder in the distributionof the hydrogen atoms, which follows the ice rules (see the text).

It is commonly accepted that the highly directional (tetrahedral) hydrogenbonding is to a large extent responsible for the behavior of water. (It causesthe tendency of the ordered states to have a higher specific volume, e.g.) Modelcalculations were in fact able to exhibit both scenarios, the liquid-liquid transitionand the singularity-free scenario, by altering the geometrical constraints of thebonding [38].

2.5 Ice Ih

The structure of ice Ih is illustrated in Fig. 2.4. The oxygen atoms are arrangedon a hexagonal lattice. The lattice parameters are a = 4.519 A and c = 7.357 Aat −20C [39]. The arrangement of the oxygen atoms follows the ‘wurtzite’ struc-ture. Each oxygen atom is tetragonally (O—O—O angle of 109.47) surroundedby 4 nearest neighbors. They form layers of puckered hexagonal rings perpen-dicular to the c-axis. The stacking of these layers has the sequence ABABAB...known from hexagonal close-packed metals.

The arrangement of the hydrogen atoms follows the model proposed by Paul-ing [40]. There are two possible hydrogen sites on each line between neighboringoxygen atoms, and the distribution of the hydrogen atoms satisfies the two icerules :

1. There are two hydrogens adjacent to each oxygen.

2.5. ICE IH 9

2. There is only one hydrogen between two neighboring oxygens.

The crucial point in Pauling’s model is the lack of long-range order in the occu-pation of the two possible hydrogen sites on each bond, therefore, the structure ofice Ih has an intrinsic disorder (see Fig. 2.4). This hydrogen disorder leads to anexcess entropy [41]. On average, each hydrogen site is occupied by half a hydro-gen atom. The space group for this average structure is P63/mmc. The oxygenatoms are covalently bonded to their two adjacent hydrogen atoms forming H2Omolecules (see rule 1). These molecules are connected via hydrogen bonds. Eachline between two oxygen atoms represents such a hydrogen bond, and either ofthe two H2O molecules can provide the hydrogen (see rule 2). The disorder inthe occupation of the hydrogen sites can thus also be seen as a disorder in theorientation of the H2O molecules.

The lattice parameters were first determined correctly by Dennison [42] usingx-ray diffraction. Based on theoretical considerations, Bragg [43] then proposed astructure with the correct position of the oxygen atoms and the hydrogen atomshalfway between the oxygen atoms. But already the question was raised whetherthe hydrogen atoms were shifted from the center (thus destroying the symmetryof the system). At this time, however, x-ray scattering experiments were notable to resolve the position of the hydrogen atoms due to their weak contributionto the scattering signal. Single crystal diffraction experiments by Barnes [44]affirmed the arrangement of the oxygen atoms proposed by Bragg.

Bernal and Fowler [45] suggested that the water molecule as shown in Fig. 2.1would stay intact in ice, which was backed by the similarity of the Raman spectraof water, ice, and vapor. Such molecules, however, cannot be arranged on thesites of the observed unit cell without destroying the hexagonal symmetry. Thesimplest structure retaining the hexagonal symmetry requires a unit cell 3 timeslarger as the one proposed by Barnes. Such a structure would be polar, but thiswas deemed more probable than the smallest non-polar structure, which wouldhave an extremely complicated and large (96 molecules) unit cell.

After Giauque and Ashley [41] found out by experiments that ice had anexcess entropy, Pauling finally proposed his model (presented above) supposingthat no particular ordering of the H2O molecules was stabilized (at least at ordi-nary temperatures). Pauling also calculated the entropy of its proposed structurewhich is in good agreement with the experimental value of Giauque and Ashley.Neutron diffraction allowed the first crystallographic study of the hydrogen posi-tions. Powder diffraction experiments by Wollan et al. [46] agreed with Pauling’smodel and ruled out several others. Final confirmation came from a single crys-tal neutron diffraction experiment by Peterson and Levy [47]. An equivalentx-ray experiment, which was later performed by Goto et al. [48], agreed with theneutron experiment.

X-rays, however, are sensitive to the electron density distribution, whereasneutrons are sensitive to the distribution of the nuclei. It was thus possible to


detect the deviation between the position of the hydrogen nuclei and the centerof the electron distribution, which is shifted towards the oxygen atoms.

Chapter 3

Interface melting

Several scenarios can lead to premelting, i.e. the formation of a (quasi)liquidequilibrium phase in the solid region of the bulk phase diagram. Among thosescenarios are interfaces (as the surface, solid–solid interfaces, and grain bound-aries) and more complex confinement situations (like in small particles and porousmedia). The effect of an interface may depend on its chemical composition, cur-vature, and roughness. It is difficult to separate the contribution of these variousmechanisms in the more complicated situations like porous media.

Supercooled liquids do not fall in the category of premelting, since they are ina metastable state. Also, premelting does not include the reduction of the bulkmelting temperature due to dissolved impurities or change of pressure.

In the case of premelting caused by the influence of an interface, the effect iscalled interface melting, or also interfacial melting. If this interface is the solid–vapor, solid–vacuum or solid–air interface, the effect is called surface melting (seeFig. 3.1a). Surface melting is thus a special case of interface melting and theunderlying theory is the same. Heterogenous solid–solid interfaces (see Fig. 3.1b),which are the focus of this work, represent another class of interfaces. The terminterface melting in the narrower sense refers to these interfaces. Experimentson interface melting at well-defined solid–solid interfaces are very rare due to thedifficulties in probing deeply-buried interfaces.

Premelting occurs in all types of materials and is quite pronounced in thecase of ice, where it has also important implications for many environmental andtechnical processes (see Sec. 3.5).

A short section (3.1) in this chapter deals with the melting process in general.The following sections present the theory (3.2) and experiments (3.3) related tointerface melting of various materials, whereas a separate section (3.4) is dedi-cated to the interface melting of ice. The vast majority of the literature concernssurface melting, and only a part of the work can be presented here. The emphasiswill be on the much smaller number of studies on interface melting at solid–solidinterfaces. The last section in this chapter (3.5) deals with the consequences ofice premelting.

11

12 CHAPTER 3. INTERFACE MELTING

vapor

ice

quasiliquid qll

solid

ice

quasiliquid qll qll

a b csurface melting interface melting permafrost

Figure 3.1: Interface melting scenarios for ice. (a) Surface melting of ice. (b)Interface melting of ice at a heterogenous ice–solid interface (interface meltingin the narrower sense). (c) Permafrost shows an abundance of such ice–solidinterfaces.

This chapter cannot cover all aspects of interface melting and the related lit-erature. For a review and more details, see for example [49, 4] (focus on surfacemelting experiments), [50], [51], [52, 53] (focus on theory, metal surfaces and in-terplay with other surface phenomena), [54, 55] (surface melting and roughening),[5] (premelting of ice and environmental consequences).

3.1 The melting transition

Melting and the reverse process of freezing are among the most prominent anddramatic phase transitions. The melting of ice may be the most important phasetransition on Earth.

Thermodynamics provides a description in terms of the Gibbs free energyG(p, T ). It is a continuous function of p and T during the transition, whereasother thermodynamic quantities such as the volume V or the entropy S undergodiscontinuous changes. Nearly all materials expand upon melting (∆V > 0)with a few exceptions, among them Sb, Bi, Ga, silica—and ice. With the onlyexception of He, the entropy increases upon melting (∆S > 0), melting is thus adisordering transition.

The relevant order parameters for the melting transition are the Fourier com-ponents of the density, measured directly by the Bragg scattering intensities.Melting is a first-order transition, i.e. the order parameter changes discontinu-ously at the transition. It is characterized by latent heat and coexistence of thesolid and the liquid phase at the transition.

Thermodynamics provides little information about the mechanism of meltingand its kinetics. Several theoretical approaches have been developed to gain amicroscopic understanding of the melting process. Most of them start from eitherthe liquid, or the solid phase, but melting involves both.

In the liquid-based approach, density functional theory is used to describe

3.2. THEORY OF INTERFACE MELTING 13

freezing as a condensation of liquid density modes. Whereas this approach hasallowed to gain microscopic insight into the freezing process, the construction ofthe functional is ad hoc and the positional order of the solid not a result of thecalculation, but an input. The properties of the solid are not exactly reproduced,either.

Solid-based theories focusing on lattice stability provide some useful phe-nomenological criteria for melting. The Lindemann criterion [56] states thatmelting sets in, when the root-mean-square displacement

√〈(∆r)2〉 reaches about

15% of the interparticle distance. The underlying model describes melting interms of individual atomic properties and ignores the cooperative character ofthis phase transition. Nevertheless, it provides a quasi-universal empirical esti-mate for the melting transition. It has later been generalized by Ross [57].

The Born criterion [58] links the melting transition to the decrease of theshear elastic moduli, which finally leads to a mechanical instability of the solidstructure. It was later modified to reach better agreement with experimental data[59] and to incorporate contributions from external stress [60]. Another class ofsolid-based theories concentrates on structural defects like vacancies [61, 62, 63]or dislocations [64, 65], but today it seems clear that defect generation is not themechanism for bulk melting.

Despite its ubiquity, the microscopic mechanism of melting is still not fullyunderstood and subject of current research (for example [66]). It is now commonlyaccepted that surfaces and interfaces play a great role for the melting process, aswill be explained in the next section.

3.2 Theory of interface melting

We consider the case of an interface between a solid s and another medium b.When the temperature of the system approaches the bulk melting point Tm ofthe solid s, a thin premelting layer of thickness L can intervene between s andb (see Fig. 3.2). This phenomenon is called interface melting. The structure ofthe strongly confined premelting layer may differ from the bulk liquid phase of s,therefore, it is usually referred to as the quasiliquid layer (qll).

This ‘liquid embryo’ may serve as a nucleation site for the bulk melting. Theinterface would act as a large natural defect initiating the melting process. It hasbeen argued that this could be the reason for the difficulties in superheating solids,while most liquids can be supercooled (every real solid has at least one interface,its surface). This idea is supported by the fact that solids can be superheatedunder certain circumstances, namely when special coatings are applied to changethe surface properties.

Interface melting starts at a certain onset temperature T0. Further increaseof the temperature leads to the growth of the quasiliquid layer thickness L. Asthe bulk melting point is reached, L diverges: L → ∞ for T → Tm. Interface


solid s

medium b medium b

solid s

quasiliquid qllL

TTm TTma b

Figure 3.2: Interface melting of a solid s in contact with another medium b.(a) Shows the situation far below the bulk melting temperature Tm. (b) Closeto, but still below the bulk melting temperature Tm, a liquid-like (quasiliquid)layer qll might intervene at the interface between the solid s and the mediumb. The thickness L(T ) of this layer is determined by the competition betweenthe possible reduction of the interfacial free energies γ and the energy needed totransform the layer from the solid to the quasiliquid state.

melting can be considered as a special case of a wetting transition, where a solidis wetted by its own melt.

There are, however, cases of incomplete wetting or blocked interface melting,where L remains finite up to the melting point: L→ Lm for T → Tm.

The driving force for interface melting is the minimization of the free energyof the system.

3.2.1 Phenomenological description

The simplest approach to interface melting is a phenomenological thermodynamicmodel. The description presented in this section is based on a continuum model.It is not applicable for very thin quasiliquid films of less than a few molecularlayers.

We calculate the free energy per unit area for the system shown in Fig. 3.2with an intervening quasiliquid layer of thickness L at the temperature T :

F (L) =

ρqllQmL

Tm − T

Tm

+ γs−qll + γqll−b + P (L) for L > 0,

γs−b for L = 0.

(3.1)

Here, ρqll is the density of the quasiliquid and Qm denotes the latent heatof melting. The interfacial energies γs−qll, γqll−b, and γs−b are not known, ingeneral, and difficult to determine experimentally. P (L) represents the inter-action between the two interfaces (s–qll and qll–b) and can be considered as athickness-dependent correction to the interfacial energies.


Interfacial melting occurs, if the free energy F has its minimum at a finitethickness L > 0 at a temperature T < Tm. Minimizing F with respect to Lyields the growth law L (T ) of the quasiliquid layer. In order to minimize F , theterm P (L), which depends on the nature of the molecular interactions, has to beknown.

If we assume that exponentially decaying short-range forces are dominating,the phenomenological expression

P (L) = −Ae−L/L0 (3.2)

can be used, where L0 is a correlation length of the quasiliquid, and A definesthe strength of the interactions.1

Minimization of the free energy(

∂(∆F )∂L

= 0)

yields the equilibrium thickness

of the quasiliquid layer

L (T ) = L0 ln

(TmA

ρqllQmL0 (Tm − T )

), (3.3)

which can be rewritten as

L (T ) = L0 ln

(Tm − T0

Tm − T

)(3.4)

with the onset temperature

T0 = Tm

(1− A

ρqllQmL0

). (3.5)

For T < T0, where the argument of the logarithm in Eq. 3.4 is negative, thequasiliquid layer is unstable and L = 0.

The logarithmic growth law (Eq. 3.4) is characteristic for short-range forces.For dominating long-range Van der Waals type dispersion forces, P (L) has theform

W

Ln(3.6)

with n = 2 for non-retarded and n = 3 for retarded Van der Waals forces. W isthe Hamaker constant. Minimization of the free energy then yields an algebraicgrowth law (power law) for W > 0:

L (T ) =

nWTm

ρqllQm (Tm − T )

1/(n+1)

∝ (Tm − T )p (3.7)

1Assuming F to be continuous at L = 0 implies A = −γs−qll−γqll−b +γs−b, see for example[49]. But for L → 0, i.e. layers with a thickness of about the molecular diameter, the continuumapproach presented in this section is not valid anyway.


with the exponent p = −1/(n + 1). If W is negative, the Van der Waals inter-actions lead to blocked melting, since long-range forces dominate from a certainlayer thickness on.

The Hamaker constant W is a measure for the strength of the Van der Waalsinteraction. In the case of surface melting (at the free surface), the Hamakerconstant can be approximated by [67]

W =π

12ελ6 (ρs − ρl) ρl (3.8)

for non-retarded pair interactions decaying as −ε (r/λ)−6. The densities of thesolid and liquid phase are ρs and ρl, respectively. The Hamaker constant at thefree surface has the same sign as the density difference between the solid and theliquid phase. This implies that blocked surface melting should occur when theliquid phase has a higher density then the solid phase.

It should be noted that Eq. 3.6 is no longer valid for small L, i.e. W is nota constant anymore. A detailed discussion of dispersion forces and the Hamakerconstant can be found in [68].

As the thickness of the quasiliquid layer increases with temperature, short-range forces get damped, while Van der Waals forces become more important.This can lead to a cross-over from a logarithmic to an algebraic growth law (asobserved for the premelting of Ne films [69]). For even larger values of L, therecan be another cross-over from non-retarded to retarded Van der Waals forces.

Fig. 3.3 shows model calculations of the free energy for different types ofinteractions and the associated growth laws. The various contributions to thefree energy are illustrated in Fig. 3.4.

Interfaces can also induce layering in liquids. Such layering has been observedat liquid surfaces (see for example [70]) as well as at solid-liquid interfaces [71, 72],and is expected to play a role in interfacial melting, since the quasiliquid isstrongly confined between two solids. In order to include such layering effects,P (L) has to be complemented by terms of the form

b cos (k1L) e−L/a. (3.9)

This expresses the preference for layer thicknesses which are a multiple of thenearest-neighbor distance 2π/k1 of the particles in the quasiliquid.

3.2.2 Landau-Ginzburg models

The expressions for the interfacial free energy as presented in Sec. 3.2.1 can alsobe derived by considering Landau-Ginzburg models. These models are still phe-nomenological in the sense that they do not provide a microscopic descriptionwhich would allow to calculate the interfacial free energies in Eq. 3.1. In theframework of Landau theory, interface melting is a special case of an interface-induced disordering transition in a semi-infinite system with a first-order bulk


10-4

10-3

10-2

10-1

100

101

102

0

20

40

60

80

100 A: =1, =0A W

B: =0, =2A W

C: =1, =2A W

D: =1, =-1A W

D

C

B

A

L(Å

)

Tm-T (K)

a growth-laws b free energy

0 10 20 30 40 5010

-2

100

101

4x101

A

BD

CA short-range forcesB VdW forcesC cross-overD blocked melting

F(a

rb.units

)

L (Å)

Figure 3.3: Free energy calculations and growth laws for different types ofinteractions with ρqllQm = 1 (energies in arbitrary units), Tm = 273.15 K,γs−qll + γqll−b = 0, L0 = 5 A. Several cases are considered here. A: only short-range forces, A = 1, W = 0, which leads to a logarithmic growth law. B: onlyVan der Waals forces, A = 0, W = 2, which leads to a power law growth. C:both short-range and Van der Waals forces, A = 1, W = 2, which leads to across-over from a logarithmic to a power law. D: both short-range and Van derWaals forces, but a negative Hamaker constant, A = 1, W = −1, which leads toblocked melting. (a) Growth law L (T ). (b) Free energy F (L) for Tm−T = 1 K.In case D the calculated free energy goes to −∞ for L → 0, which does notcorrespond to reality, of course. The approach is simply not valid for very thinfilms (see the text).


0 10 20 30 40 5010

-4

10-3

10-2

10-1

100

101

102

meltingshort-range forcesVdW forces

F(a

rb.units

)

L (Å)

a

0 10 20 30 40 500.0

0.1

0.2

0.3

0.4

0.5

meltingshort-range forcesVdW forces

F(a

rb.units

)

L (Å)

b

Figure 3.4: Contributions to the free energy from melting (solid line), short-range forces (short dashed line), and Van der Waals forces (long dashed line).The parameters for this calculation are the same as in Fig. 3.3, case C. (a)Logarithmic plot, the Van der Waals forces can be seen to dominate over theexponentially decaying short-range forces for large L. (b) Linear plot.

transition. This theoretical approach was introduced by Lipowsky [73] and fur-ther developed in numerous papers [74, 75, 67, 76, 77, 78, 79, 80]. Most of themodels were initially developed for explaining surface phenomena, this is whythe established terms refer to surfaces, like ‘surface induced order’ for example.Although the models are here applied to interfaces in general, the establishedterms will be used.

The presence of an interface strongly influences the phase behavior of a phys-ical system. A system which undergoes a first-order bulk transition can showseveral types of interface transitions, in particular ‘surface induced order’ (SIO)and ‘surface induced disorder’ (SID). In the case of SIO, the interface remainsordered up to an interface transition temperature TSIO>Tbulk.

In the case of SID, the order parameter at the interface vanishes continuouslyon approaching Tbulk. Interface melting is a special case of SID, which should inprinciple be described by a multi-component order parameter [77, 79]. A layer ofthe (nearly) disordered phase grows from the interface into the ordered bulk. Thethickness of this layer follows the growth laws presented in the previous section.In the case of dominating short-range forces this is the logarithmic growth lawfrom Eq. 3.4. The prefactor L0 in this equation is also called ‘growth amplitude’and related to a decay length a of the system. This can be [78, 81]

• a decay length aOP of an order parameter (OP) density, in this case

L0 =aOP

2, (3.10)


• a decay length aNO of a non-ordering (NO) density, in this case

L0 = aNO (without the factor1

2). (3.11)

This decay length can then be compared with a corresponding correlation lengthof the system.

3.2.3 Density functional theory

For a Lennard-Jones (LJ) system near the triple point, surface melting could beobserved with density functional theory (DFT) [82]. Such a LJ system may serveas a model for rare gases. The density functional used in this first study, however,does not provide a phase diagram in agreement with simulations. A subsequentstudy applied a better weighted-density approximation (WDA) functional [83].It showed complete surface melting for the fcc-crystal–gas interface of LJ sys-tems. The thickness of the premelting layer varied with surface orientation. Itstemperature dependence followed a logarithmic growth law.

3.2.4 Lattice theory

Trayanov and Tosatti have developed a microscopic lattice theory of surface melt-ing [84, 85, 86]. It is based on the minimization of the free energy with respect todensity and ‘crystallinity’ as spatially varying order parameters. By introducinga discrete reference lattice and applying mean-field and free-volume approxima-tions, their approach allows to calculate the partition function of the system. Itwas applied to the case of the (100) and (110) LJ crystal surfaces. The densityand crystallinity profiles show a rather abrupt jump at the interface betweenthe solid and the quasiliquid. It might, however, be due to the mean-field (MF)approximations which suppress fluctuations. The layer thickness of the quasiliq-uid was calculated from the density profiles and follows an algebraic growth lawwith the exponent −1/3. A change from long-range to short-range interactionschanges the growth law from algebraic to logarithmic. Switching the long-rangetail of the interactions from attractive to repulsive leads to blocked surface melt-ing. The difference between different surface orientations (anisotropy) was shownto diminish with temperature.

3.2.5 Other approaches

Other approaches include phonon theory [87]. In this approach, it was found thatthe surface becomes unstable with respect to melting before the bulk, and thatthis instability then proceeds into the bulk. The approach was applied to thecase of copper [88, 89], where it was also used to explain the dependence on thesurface orientation.


3.2.6 Molecular dynamics simulations

Molecular dynamics simulations were used extensively to study surface and inter-face melting, as well as melting in other confinement situations like nano-particles.The first simulations were performed for LJ systems [90], for a review see [91].They were then extended to metals, where a lot of experimental data was avail-able. In this case, additional difficulties for the simulations arise due to thepresence of many-body interactions. The simulations on metals include Au [92],Cu [93], Ni [94, 95] and Al [96, 97]. Semiconductors (for example Si [98]) andoxides (for example Cr2O3 [99]) have been investigated as well.

3.2.7 Interfacial melting and substrate roughness

The influence of substrate roughness and curvature on wetting phenomena hasbeen studied in detail, see for example [100, 101, 102, 103, 104, 105] and referencestherein. The question is not yet completely solved, which is illustrated by thenumber of recent publications. The effect of roughness can be dramatic and evenchange a non-wetting to a wetting scenario (roughness-induced wetting [103]).

In general, roughness affects the wetting behavior in quite different ways. Itsfirst consequence is the increase of the effective interface area. This obvious ef-fect can be expected to amplify the general tendency with respect to interfacialmelting. If the interfacial energy is lowered by an intervening quasiliquid layer,it is so even more when the interface area is larger. However, the roughnessof the substrate is not necessarily replicated at the interfaces between differentlayers on top of the substrate. In the case of interfacial melting of ice, this isthe ice–quasiliquid interface. The morphology of this interface is itself a conse-quence of the minimization of the free energy. This does not only complicate thecalculations, but also influences the net roughness effect.

The second consequence of the roughness concerns the interface potentialdenoted with P (L) in Eq. 3.1, which depends on the molecular interactions.Since the roughness changes the distances between particles in the various layers,it modifies P (L) (see for example [103]). This effect could be called a change ofthe effective layer thickness, although it does not appear as a simple correctionto the layer thickness in the calculations.

A third effect concerns only solid wetting films. In this case, the bendingenergy of the solid film picking up the substrate roughness must be taken intoaccount [104, 105].

The effect of roughness on wetting phenomena thus depends on the specific sit-uation. This includes the type of roughness (self-affine roughness, mound rough-ness, periodic structures, ...) and its length scales, the type of interactions (Vander Waals interactions, exponentially decaying interactions, ...), and the wettinglayers involved (solid or liquid).

3.3. EXPERIMENTAL EVIDENCE FOR INTERFACE MELTING 21

3.2.8 Further aspects

At surfaces, other phenomena can occur, which can either be superimposed topremelting, or lead to an interplay with premelting. Such phenomena are rough-ening, pre-roughening and faceting. Pre-roughening [106, 107] starts at a tem-perature Tpr where the free energy cost for step formation vanishes. It producesa roughly half-filled outermost layer by the formation of surface vacancies andadatoms, which associate to form islands and holes. When the step free energyvanishes with an essential singularity, this gives rise to a roughening [108, 109]transition at the temperature Tr. Islands form on top of other islands and holesinside other holes, which causes the surface width to diverge. As the surface freeenergy depends on the crystal orientation, the macroscopic orientation of crys-tals can be unstable with respect to faceting. Such crystals form large low-energyfacets while retaining the average orientation. The surface phenomena describedhere can also appear at solid–(quasi)liquid interfaces. They might thus also playa role at solid–solid interfaces once premelting has set in.

As the surface free energy varies with crystal orientation, the surface meltingbehavior can also be anisotropic. An example is the surface behavior of aluminum.Whereas the relatively open Al(110) surface shows surface melting, the close-packed Al(111) surface remains stable up to the bulk melting point [110].

The description in the sections 3.2.1 and 3.2.2 obtained in the framework ofLandau theory underestimates the effect of fluctuations [73]. Fluctuations of theinterface, capillary waves at surfaces, e.g., can modify the behavior.

3.3 Experimental evidence for interface melting

Surface melting has been studied with a large variety of techniques and has beenobserved in many classes of materials. Only examples will be cited here. Thereader is referred to the reviews mentioned above (page 12).

The first microscopic studies have been performed on metals including Pb[111], Al [110], Au [112], Ni [113], and Ga [114, 115]. The general tendencyof metals is to show complete surface melting on the relatively open faces suchas fcc(110), no surface melting for the densely packed faces like fcc(111), and in-complete surface melting on faces with intermediate packing density like fcc(100).The growth law usually shows a cross-over from a logarithmic to a power law.Other studies have been performed on the rare gases Ne and Ar [116, 69], whereagain a cross-over in the growth law was observed. Experimental evidence for (atleast incomplete) surface melting has also been found for the semiconductors Ge[117] and Si [118]. Finally, surface melting can also occur in organic substances,as has been shown for caprolactam (C6H11ON) [119] and methane (CH4) [120].For the surface melting of ice see Sec. 3.4.

Surface melting is a well-established phenomenon that occurs in a wide range


of materials. For solid–solid interfaces however, few experimental results areavailable. Most studies on interface melting were done on ice (see Sec. 3.4).

However, some experimental observations attributed to surface melting mightactually be due to interface melting. An example is the work of Zhu et al. [116, 69]cited above. They attributed heat capacity anomalies of adsorbed Ne and Arfilms to surface melting and roughening. As their measurements do not provideany spatial information, they could be explained by surface melting, or interfacemelting, or a combination of both. In another case, Chernov and Yakovlev [121]observed premelting of biphenyl in contact with glass by ellipsometry. Theycalled their observation surface melting, although the premelting occurred at asolid–solid interface. The distinction is very important, as no premelting wasobserved on the free surface of biphenyl in a study from another group usingx-ray reflectivity [122].

Other evidence for interface melting was found for Ar in porous Vycor glass byheat capacity and vapor pressure measurements [123]. High-resolution transmis-sion microscopy observations of the interface between Al and amorphous Al2O3

in nanoparticles were also attributed to interface melting of Al [124].

For nano-crystals embedded in the matrix of another material, superheatinginstead of premelting has been observed (for example Pb embedded in Zn [125]).The experimental results suggest that in these cases, the size and shape of theparticles play a crucial role.

3.4 Interface melting of ice

The melting of ice is part of our everyday life. The melting point of ice atstandard atmospheric pressure, 273.15 K, is the zero of the Celsius scale, 0C.However, the anomalous properties of water (see Sec. 2.3) also show up in themelting of ice. Two particular features are linked to its melting behavior. One ispressure melting, caused by the anomalous density increase upon melting. Thesolid phase ice has a lower density (ρs = 0.92 g/cm3) than the liquid phase water(ρl = 1.0 g/cm3), in contrast to most other materials. Clausius-Clapeyron’srelation then implies a negative slope of the melting curve Tm(p). The otherfeature is surface melting (see below), which also occurs in other materials, butis especially pronounced in the case of ice. This is striking, as the negativedensity difference ∆ρ = ρs − ρl between the solid and the liquid phase rendersthe Hamaker constant negative and surface melting energetically unfavorable (seeSec. 3.2.1). The quasiliquid, however, may have a different density than the bulkliquid, thus modifying the value of the Hamaker constant. In the case of interfacemelting, the Hamaker argument does no longer hold in its original form and theHamaker constant depends on the specific interface (see below).

3.4. INTERFACE MELTING OF ICE 23

3.4.1 Theory and simulations

In principle, the theoretical considerations presented in Sec. 3.2 apply to iceinterfaces as well. Interface melting is driven by the minimization of the freeenergy. In the case of ice with its network forming hydrogen bonds, this is morecomplicated than in other materials.

In an early theory Fletcher [126] (see also [127, 128]) evaluated the free energyat the free surface of ice. Taking into account electrostatic effects and the dipoleand quadrupole moment of the water molecule, he concluded that the free energygain due to surface polarization is sufficient to induce surface melting at about−5C with a quasiliquid layer thickness reaching 10 to 40 A close to 0C.

Elbaum and Schick [129] applied the theory of dispersion forces to the surfaceof ice. They found that electromagnetic interactions result in incomplete surfacemelting with a maximum layer thickness of about 30 A. According to the authors,fluctuations could lead to larger layer thicknesses but would not lead to completemelting.

In a subsequent study Wilen et al. [130] evaluated the contribution of Vander Waals forces to interface melting at various ice–solid interfaces. Taken alone,Van der Waals interactions can lead to complete or incomplete interface melt-ing, depending on the substrate. However, additional (short-range) interactionscan change the overall behavior. An estimate shows that electrical interactionscan indeed be dominant if present. Layer thicknesses for various substrates areestimated to be of the order of 10 A at −0.1C.

A recent theory of Ryzhkin and Petrenko [131] links surface melting to pro-ton disorder and supports the idea of two overlapping surface regions, a protondisordered region and a second region where the oxygen lattice breaks down.

Several molecular dynamics studies have been performed on surface meltingof ice. In an early work by Weber and Stillinger [132], surface melting was seenin a simulation of ice crystallite melting. Later studies by Kroes [133] and Nadaand Furukawa [134] (among others) directly addressed the problem of meltingat ice surfaces. Both used the TIP4P potential and observed the formation ofquasiliquid structures at the surface. Kroes only investigated the basal face,whereas Nada and Furukawa studied basal and prismatic faces and observedanisotropic behavior with the basal face exhibiting thicker quasiliquid layers.

Wettlaufer [135] has investigated the effect of impurities on surface and in-terface melting of ice by calculating Van der Waals and Coulombic interactionsin contaminated interfacial films. His results suggest that impurities can have adramatic influence on surface and interface melting.

3.4.2 The free surface of ice

Surface melting of ice has been studied in a large number of experiments with var-ious techniques, among them photoemission (Nason and Fletcher [136]), proton


backscattering (Golecki and Jaccard [137]), ellipsometry (Beaglehole and Na-son [138], Furukawa et al. [139] ), laser reflection (Elbaum et al. [140]), sum-frequency vibrational spectroscopy (Wei et al. [141]), and photoelectron spec-troscopy (Bluhm et al. [142]). While practically all experiments confirm thepresence of a quasiliquid layer at the surface of ice, there are large discrepancies inthe layer thicknesses and onset temperatures reported, which cannot be discussedin detail here. Part of the discrepancies might be due to the fact that smoothand clean ice surfaces are difficult to prepare and even more difficult to maintainin the same state for the duration of the experiment. The high vapor pressureof ice causes problems for many standard surface techniques which require UHV(ultra-high vacuum). The vapor pressure has to be controlled very precisely, oth-erwise the surface morphology changes rapidly due to sublimation/resublimation.Another problem might be to avoid contamination of the surface by impurities,which can have a significant influence on the melting process (see above). Fur-thermore, care has to be taken when comparing different experiments, as thesurfaces of ice against air and against pure water vapor as well as different crys-tal orientations do not behave in the same way. Some experiments were actuallyperformed on thin films, where the strong confinement as well as the interfacewith the underlying substrate may significantly influence the experimental re-sults. Another part of the discrepancies might stem from the fact that differentphysical properties are probed by the various experimental techniques. Experi-mental support for this idea comes from experiments by Dosch et al. [143], whoobserved that a loss in long-range coherence of the hydrogen network occurs in adeep-ranging surface layer prior to actual surface melting.

The relevant order parameters for the solid–liquid transition are the Fouriercomponents of the solid density-density correlation function measured as Braggscattering intensities. Lied et al. [144, 3, 145, 146, 143] performed a series ofsurface-sensitive x-ray-diffraction experiments in order to directly probe this orderparameter. They studied several surfaces with high-symmetry orientations andfound onset temperatures between−13.5 and−12.5C. The layer thickness couldbe best fitted with a logarithmic growth law, with deviations towards a higherthickness for temperatures above −1C. The amplitude of the growth law variedbetween 37 A and 84 A. These findings will later be discussed in comparison withthis work (see Chapt. 7.4).

3.4.3 Ice in porous media

While the experimental results which will be described in this section all giveevidence of premelting, they have two important shortcomings: First, the inter-pretation of the physical origin is difficult, as powders, porous media, and soilsare not well characterized with respect to size distribution of grains and pores,curvature, roughness, surface termination, impurities, ice crystallinity and orien-tation, etc. Second, these experiments only indicate the existence of a premelting


liquid. Neither do they allow to locate this liquid, nor do they permit to observevariations of the properties within the layer. It might be possible that premeltingonly occurs in the smallest pores, for example, but the experiments presented inthis section only allow to estimate an average layer thickness. In order to addressthe first problem, experiments on better defined solid–solid interfaces have beenperformed, which will be presented in the next section.

The advantage of porous media is the abundance of interfaces. If interfacialmelting occurs, this leads to a macroscopic quantity of quasiliquid material. Asignal from this quasiliquid can then be detected with common bulk methods.Porous media can also serve as a more realistic model for permafrost soil.

Maruyama et al. [147] have studied H2O-saturated powders of graphitizedcarbon black and talc by quasi-elastic neutron scattering (QENS). They observedunfrozen water down to temperatures below−30C. The temperature dependenceof the calculated liquid fraction is linked to size effects. The translational diffusioncoefficient of the liquid fraction differs from supercooled water. A later analysisof the results by Cahn et al. [148] incorporated separate terms for interfacial andcurvature melting and found good agreement with the measurements.

A later study by Gay et al. [149] was aimed at reducing the curvature effectsin fine powders. Therefore, melting of D2O in exfoliated graphite with a laminarstructure was studied by neutron diffraction. The liquid fraction grows accordingto a power law with exponent −0.54, significantly larger than the value −1/3expected for long-range forces. The discrepancy is again attributed to size effects.

Bellissent-Funel and Lai [150] performed neutron scattering experiments ofD2O confined in porous Vycor glass. They observed the formation of cubic iceand the persistence of a small liquid fraction down to −40C.

Ishizaki et al. [151] used pulsed NMR to study ice in porous silica in a temper-ature range of −30 to 0C. They deduced from their measurements a (average!)thickness for the quasiliquid layer at the ice–silica interface as a function of tem-perature. At −30C, this layer is still ≈10 A thick. It seems to diverge at adepressed melting point, which the authors explain by effects from the pore cur-vature. The behavior strongly depends on the pore size. The growth of thequasiliquid layer follows a power law with exponent −0.60.

Ordered porous silica materials with cylindrical pores of uniform (and tunable)size have become more readily available. They enable studies of the premeltingas a function of the pore diameter (in the range of a few nanometers). Theseexperiments (see [152] and references therein) agree with the predicted lineardependence of the melting point depression on the inverse pore radius. They alsohint to a thin (≈4 A) layer of interfacial water remaining liquid down to very lowtemperatures.


3.4.4 Ice–solid interfaces

Ellipsometry

Furukawa and Ishikawa [153] have studied ice single crystals of unspecified orien-tation in contact with BK 7 optical glass by ellipsometry. Ellipsometry exploitsthe change of polarization when light is reflected. It can be characterized by theellipticity

ρ =Rp

Rs

= tanψ exp (i∆) , (3.12)

where Rp and Rs are the Fresnel coefficients for the p- and s-componentsof the polarized light. The polarization change is sensitive to the profile of therefractive index n across the interface. Monochromatic ellipsometry employed byFurukawa and Ishikawa yields only 2 numbers (tanψ and ∆) which can be used toreconstruct the profile of the refractive index. Therefore, strong assumptions haveto be made. Furukawa and Ishikawa used the measured ellipticity values to deducethe refractive index and the thickness of a quasiliquid premelting layer betweenice and glass. The values were different from sample to sample, but a systematicchange with temperature could be observed. The 10 nm microroughness of theglass sample produces a smearing of the observed profile of the refractive index.For this reason, an apparent layer with 10 nm thickness and a refractive indexbetween water and glass is already observed at −5C. This layer has no physicalmeaning but limits the resolution to about 10 nm. At temperatures above −1,the authors see a change in the ellipticity which they attribute to a quasiliquidlayer with the refractive index of bulk water reaching a thickness of more than100 nm. It is not clear how the microroughness of the glass is included in theirmodel in this regime. Optical anisotropy is not considered in their analysis.

In a subsequent study, Beaglehole and Wilson [154] applied ellipsometry to icein contact with different glass surfaces: smooth and clean glass, roughened glass,glass with surface impurities, and roughened glass with a hydrophobic coating.They tried to take into account as a sort of background the ellipticity changeinduced by the roughness. Therefore, they performed reference measurements onthe various glass substrates in contact with water. For ice in contact with theclean and smooth, or with the hydrophobic glass, they saw no significant changein the ellipticity up to about −0.1 to −0.05C. For the roughened glass, theyobtained water thicknesses reaching up to about 200 nm at −0.2C. They didnot evaluate the refractive index (density) of this layer. In the case of the glasssubstrate with impurities, significant influence of the impurities on the refractiveindex are expected. Rather than measuring the refractive index, the authors per-formed an estimate based on literature values and calculated the layer thicknessusing these assumed values of the refractive index. Their plot of the tempera-ture dependence of the layer thickness contains three data points. The deducedexponent for the temperature variation is −1. In conclusion, Beaglehole and Wil-


son show clear differences for the different glass substrates. However, the samelimitations as for the measurements of Furukawa and Ishikawa apply:

• The profile of the refractive index has to be deduced from only 2 numbers,therefore strong assumptions have to be made.

• It is difficult to distinguish apparent layers due to substrate roughness fromactual layers of a different material.

• The spatial resolution in the measurement of the layer thickness appears tobe on the order of 10 nm.

For the study of interfaces by ellipsometry, one of the two materials has to betransparent for the wavelength used (typically in the optical regime). In princi-ple, this could be the ice. But due to the high vapor pressure of ice, it is difficultto prepare and preserve a smooth ice surface through which the light beam couldpenetrate. This limits the range of other ice–solid interfaces accessible to ellip-sometry measurements.

Sum-frequency vibrational spectroscopy

Wei, Shen et al. [155, 156, 157, 158, 159] reported results from sum-frequencyvibrational spectroscopy (SFVS) measurements at various ice and water inter-faces. This method yields information about the various bond modes at theinterface which could not be obtained with other experimental techniques. Butthe method is only sensitive to the topmost layer. It does not allow any depthprofiling across the quasiliquid layer, nor does it provide information about itsthickness [159]. The most detailed analysis has been performed for the ice andwater surface (ice–air, water–air). The measurements allow to deduce the ori-entational order and maximum tilt angle for the free OH bonds at the surface.Surface disordering of ice sets in at 200 K. This temperature should not be iden-tified with the onset temperature T0 from Sec. 3.2. The onset temperature T0

presumes that a quasiliquid layer intervenes at the interface replacing it by twonew interfaces. The surface disordering here refers to a partial disordering of thefirst monolayer, which might set in well below the actual onset of surface meltingas described in Sec. 3.2. Therefore, this finding is not in disagreement with otherstudies reporting much higher onset temperatures. Another interesting featureof the results from Wei et al. is the observation of strong differences between thequasiliquid surface of ice and the surface of water (even if supercooled) whichshows that the structure of the quasiliquid indeed differs from bulk water.

Further measurements have been performed on hydrophilic ice–silica inter-faces and hydrophobic ice–OTS–silica interfaces (silica coated with a hydropho-bic octadecyltrichlorosilane monolayer). These measurements are more difficultto interpret than those for the free surface [159]. The authors conclude that at


the hydrophobic interface, the dangling OH bonds are highly disordered in ori-entation, regardless of the temperature (this refers to the first monolayer at theinterface, the method is not sensitive to deeper layers). As for the free surface,the hydrogen-bonded OH-peak decreases with temperature. At the hydrophilicinterface, no peak for dangling OH bonds is observed due to hydrogen bonding tothe SiOH (silanol) groups of the silica surface. The hydrogen-bonded OH-peakremains quite strong up to −1C. Measurements have been performed across themelting transition, and the authors conclude from these measurements that thenet orientation of water molecules at the hydrophilic interface flips upon melting.They also suggest that interface melting at the hydrophilic interface only occursvery close to the bulk melting temperature.

Atomic force microscopy

A common method in surface analysis is atomic force microscopy (AFM), whichhas also been applied to study ice surfaces. Where the microscope tip is incontact with the surface, however, there is no surface in the strict sense anymore,but an interface between the sample and the tip. This is important, as thepresence of the tip might have an effect on the behavior of the sample. In thecase of surface melting studies with AFM, the question is whether an eventuallyobserved quasiliquid layer is to be attributed to surface melting, or interfacemelting induced by the contact with the microscope tip. Petrenko [160] comes tothe conclusion that thermal equilibrium between tip and ice always occurs, butBluhm et al. [161] conclude that in their lateral force microscopy experiments,the quasiliquid layer between tip and ice is squeezed out.

The AFM measurements are presented in this subsection about ice–solid in-terfaces, as an influence of the tip cannot be excluded a priori . The interfaceice–tip, however, is not as well defined as the interfaces studied in the other ex-periments presented in this subsection (based on methods like ellipsometry andsum-frequency vibrational spectroscopy). First of all, the tip obviously is notflat, but has a radius of curvature in the range of 10 to 100 nm. Then, it also hasa (unknown) microscopic roughness. Furthermore, there might be heating due tothe laser beam used to detect the deflection of the cantilever. Eastman and Zhu[162] estimated this effect to be significant, therefore special precautions had beentaken in the studies presented here to minimize the heating of the cantilever. Ifon the other hand the temperature of the cantilever is too low, condensation ofwater vapor can also disturb the measurements. This is why Pittenger et al. [163]keep the tip at a temperature about 0.1–0.3C above the sample temperature.

Several groups have conducted AFM studies of ice surfaces, among them Pe-trenko [160], Pittenger, Fain, Slaughterbeck et al. [164, 165, 163], Doppenschmidt,Butt et al. [166, 167, 168], and Bluhm et al. [161]. In general, such experimentsseem to be quite tricky. The results of the various studies show large differences,and even results obtained in the same study under identical experimental con-


0.1 1 1010

0

101

102

103

L(Å

)

Tm-T (K)

Figure 3.5: Growth laws for interfacial melting of ice from the literature. From[154], determined with ellipsometry: ice against roughened float glass (squares),ice against float glass with impurities (triangles). From [151], determined withNMR: ice in porous silica with 500 A pore diameter (circles). From [163], de-termined with AFM: ice against uncoated AFM tip (diamonds), ice against hy-drophobic AFM tip (crosses).

ditions scatter strongly [160, 167]. Most of the studies agree on the presenceof a quasiliquid layer. The thickness of this layer is calculated from measuredforce curves. The interpretation of such force curves is not trivial, as adhesion,capillary fores, elastic and plastic deformation of the ice, flow of the quasiliq-uid under the tip, as well as electrostatic and Van der Waals forces can playa role. The models describing these contributions often have to include strongassumptions, as for example for the viscosity of the quasiliquid [163]. The mostrecent and most convincing study from Pittenger et al. [163] covers temperaturesbetween −1 and −10C, where the calculated thickness of the quasiliquid layergrows like L ≈ 1.1 nm (Tm − T )−0.68 for a silicon tip. This leads to a layer thick-ness of about 1 nm at −1C. When the tip has a hydrophobic coating, the layerthicknesses are slightly smaller. Petrenko [160] obtains 2–16 nm at −10.7C,Doppenschmidt and Butt [167] about 30–50 nm at −1C and about 5–15 nm at−10C. In a later analysis, Butt et al. [168] explained their measurements withplastic deformation of the ice rather than a quasiliquid layer.

3.4.5 Further aspects

Premelting of ice is a vast topic and cannot be presented here in its full scopeand depth. One of the aspects that should not be omitted is the premelting atgrain boundaries (see [169, 170]). The macroscopic manifestations of premeltingare another issue. Some of them are discussed in Sec. 3.5. An example is the


work of Jellinek [171] on the adhesive properties of ice. For snow-ice sandwichedbetween stainless steel, he observed a breakdown of the adhesive strength startingat about −15C, the temperature where interface and surface melting typicallyset in.

In the interface melting scenario, the quasiliquid layer is strongly confinedbetween two solids. Therefore, other studies of liquids in confinement might berelevant. The structure of the quasiliquid may have similarities with the structureof water in nanopores, for example.

3.5 Consequences of ice premelting

The environmental consequences of ice premelting have been reviewed by Dashet al. [5] (see also [172]). Another good source is [173].

A few issues will be presented in this section.

3.5.1 Permafrost

Permafrost is a composite structure of rock or soil remaining at or below 0C fortwo or more years. It contains in many cases over 30% ice, and hence abundantice–mineral interfaces (see Fig. 3.1c). Permafrost covers about 20% of the landmass on the northern hemisphere and can be up to several hundred meters thick.Permafrost can be an ideal terrain to build on, but any internal melting processcan turn it into a slurry-like material with disastrous consequences for buildingsand infrastructure. Damage can also occur through frost heave (expansion of thesoil). Permafrost is characterized by massive transport of water (and solutes),which—due to premelting phenomena—is present in soils even at temperatureswell below 0C (see for example Williams [174]).

3.5.2 Glacier motion

There are three mechanisms of glacier motion: ice deformation, bed deformation,and basal sliding [175]. The first process is usually dominant, but rather slow(some 10 meters per year). Basal sliding spans a much greater range of velocitiesand can change over periods of hours. It strongly depends on the presence ofwater at the base of the ice sheet. One possible mechanism for a reduction ofthe melting temperature at the glacier bed is pressure melting (see Sec. 3.4). Asthe negative slope of the melting curve is very small, dTm

dp= −7.15 mK/atm, the

ice melting temperature at the base of a 1000 m thick ice sheet is only loweredto −0.7C (assuming uniform distribution of the weight). Any premelting at theice–rock interface significantly below the bulk melting point would therefore bea dominant factor for the basal sliding of glaciers. The observed sliding of polarglaciers has been attributed to interfacial melting of ice [176].

3.6. SUMMARY AND CONCLUSIONS 31

3.5.3 Thunderstorms and atmospheric chemistry

Much is known about the structure of thunderstorms and their formation, butthe microscopic mechanism responsible for the charging remains unclear. It isgenerally agreed that charging occurs in collisions of tiny ice particles. The liquidgenerated by surface melting of these particles was suggested to play a key rolein the charge transfer [177, 178].

Ice particles play a similarly important role for atmospheric chemistry. Itis known, for example, that polar stratospheric clouds are relevant for ozonedepletion. Surface melting may be involved in these processes. The quasiliquidlayer on ice particles can serve as a reservoir and reaction site for the chemicals(HCl, e.g.) implicated in atmospheric reactions [179].

3.5.4 Friction

The coefficient of friction of ice can be very low compared to other common ma-terials, which makes ice skating and skiing possible. Several effects contributeto the friction of ice and snow: dry friction, lubricated friction, ploughing, andcapillary forces. Lubrication comes from a thin liquid film, the debate is aboutthe origin of this film. Three mechanisms have been proposed: pressure melting,frictional heating, and interface melting. In many physics textbooks (for exam-ple [180]), ice skating is still explained by pressure melting, although a simplecalculation shows that its contribution is negligible for ice skaters with a masssmaller than a few tons. For ice skating, frictional heating seems to be a domi-nant factor [181, 182], but interface melting also contributes to lubrication [183].The respective contributions depend on factors like temperature, skating speed,and the properties of the skate.

3.6 Summary and conclusions

Surface melting is a well-established phenomenon and occurs in a wide range ofmaterials (see Secs. 3.2 and 3.3). There is a lot of experimental evidence forinterface melting of ice in contact with other solids, but microscopic observa-tions at well-defined interfaces are scarce, owing to the difficulties in probingdeeply buried interfaces (see Sec. 3.4). Interface melting of ice has importantconsequences for processes in nature and technology (see Sec. 3.5) and is also ofgreat interest from a purely scientific point of view. It might be related to theanomalies of water and the unsolved puzzle of the water structure (see Chapt. 2).Although the relevant order parameters for the solid–liquid transition are theFourier components of the solid density-density correlation function, the averagedensity plays an important role for the melting of ice (see Sec. 3.4).

Chapter 4

Theory of x-ray reflectivity

The aim of this chapter is to acquaint the reader with the basic principles ofx-ray reflectivity needed for the understanding of this work. An extensive reviewof x-ray reflectivity techniques can be found in [7], the basic principles are alsocovered in [184]. Other references are given in the text.

4.1 Index of refraction for x-rays

We consider a homogeneous material consisting of N different atom types withthe respective number densities ηj. For x-rays, the index of refraction of thismaterial can be written as

n = 1− δ + iβ (4.1)

with the dispersion and absorption terms

δ =λ2

2πre

N∑j=1

ηj

(f 0

j + f ′j (λ))

(4.2)

and β =λ2

2πre

N∑j=1

ηjf′′j (λ) =

λ

4πµ . (4.3)

Here λ denotes the x-ray wavelength, re the classical electron radius, and µthe linear absorption coefficient. The x-ray form factor is

fj = f 0j + f ′j (λ) + if ′′j (λ) , (4.4)

where f ′j (λ) and f ′′j (λ) account for dispersion and absorption corrections, re-spectively. f 0

j depends on the momentum transfer q, but can be considered to beconstant over the q-range typically covered by x-ray reflectivity measurements. Agood approximation (far from absorption edges) is f 0

j ≈ Zj (Zj being the atomicnumber), and thus

δ =λ2

2πreσe, (4.5)

33

34 CHAPTER 4. THEORY OF X-RAY REFLECTIVITY

ki

kf

kt

interface

f=

i

t

i

n1

n2

Figure 4.1: Reflection and refraction of a plane electromagnetic wave. An incidentplane wave with wave vector ki hits an interface at an incident angle αi. It splitsinto a reflected (αf = αi) wave with wave vector kf and a refracted wave withwave vector kt transmitted at an angle αt. For incident angles αi < αc, totalreflection occurs (see the text).

where σe denotes the electron density. Tabulated values for the form factor canbe found in [185].

4.2 Reflection at an ideal interface

A plane electromagnetic wave Ei (r) = E0i exp (iki · r) with wave vector ki and

amplitude E0i impinging on an interface at an incident angle αi splits into a

reflected wave (αf = αi, amplitude E0f ) and a refracted wave transmitted at the

angle αt (see Fig. 4.1). The angle of the refracted wave is linked to the angle ofthe incident wave by Snell’s law:

cosαi

cosαt

=n2

n1

, (4.6)

where n1, n2 denote the respective indices of refraction for the media of the in-coming wave and the transmitted wave. If the incident angle is smaller than thecritical angle

αc = arccos

(1− δ21− δ1

)≈√

2 (δ2 − δ1) , (4.7)

total external reflection occurs: No transmitted wave is created, and only anevanescent wave field is induced in the second medium. Apart from small lossesdue to absorption, all incoming radiation is reflected. The critical angle is onlydefined for δ2 > δ1. This means that total reflection can only occur, when themedium on the side of the incoming wave is optically denser (Re(n1)−Re(n2) =δ2 − δ1 > 0).

4.3. REFLECTION AT MULTIPLE INTERFACES 35

The reflection coefficients r for the components of the electric field parallel ‖and perpendicular ⊥ to the interface are1

r‖ =E0

f‖

E0i‖, (4.8)

r⊥ =E0

f⊥

E0i⊥. (4.9)

They can be calculated using the fact that the tangential components of theelectric and magnetic field have to be continuous at the interface. In the case ofx-rays, where n is close to unity, there is practically no difference between thedifferent polarizations and

r =ki,z − kt,z

ki,z + kt,z

=sinαi −

n2

n1

sinαt

sinαi +n2

n1

sinαt

, (4.10)

where ki,z = k sinαi and kt,z = n2/n1

ksinαt are the z-components of the wave

vector of the incident and transmitted wave, respectively.The intensity of the reflected wave, the so-called Fresnel reflectivity, is

RF = |r|2 . (4.11)

For αi & 3αc, the Fresnel reflectivity can be well approximated by

RF ≈(αc

2αi

)4

≈

(k√

2∆δ

qz

)4

. (4.12)

For the connection between angles and the momentum transfer, see Sec. 5.3.4.

4.3 Reflection at multiple interfaces

Now we consider the case of a multilayer system, where the reflections fromall interfaces contribute to the total reflection. A sketch of a system consistingof N + 1 layers is shown in Fig. 4.2. The layer j (j = 1 . . . N + 1) has therefractive index nj and the thickness dj. The layers 1 and N +1 are semi-infinite(d1 = d∞ = ∞). The system has N interfaces at the positions zj.

Two waves are created at each interface: a ‘reflected’ wave (Rj) propagatingin the layer j and a ‘transmitted’ wave (Tj+1) propagating in the layer j+1. Butunlike at a single interface, there are also two incoming waves at each interface:the transmitted wave from the interface j−1 (Tj) and the reflected wave from the

1The components parallel and perpendicular to the interface are independent, see [184].


T =11 R1

f=ii

n2

x

T2R2

T2

R2

R3T3

Tj-1 Rj-1

RjTj

TjRj

Rj+1

Tj+1

TN-1

RN-1

RN TN

TN RN

TN+1

z

n1

n

nj-1

nj

nj+1

nN-1

nN

nN+1

dj

dN

z1=0

z2

zj-1

zj

zN-1

zN

...

...

layer 1

layer 2

layer j

...

layer N

...

layer +1N

d2

Figure 4.2: Reflection and refraction of a plane wave at a system of multipleinterfaces. The system shown in this figure consists of N+1 layers with refractiveindices nj and thicknesses dj separated by N interfaces. A recursive approachallows to calculate the reflectivity (see the text).

4.4. ARBITRARY DISPERSION PROFILES 37

interface j + 1 (Rj+1). The amplitude of the incoming wave in the semi-infinitelayer 1 is normalized to unity, T1 = 1, and no reflected wave is propagatingthrough the last layer, RN+1 = 0. The Parratt formalism [186] connects the Rj

and Tj:

Xj =Rj

Tj

= exp (−2ikz,jzj)rj,j+1 +Xj+1 exp (2ikz,j+1zj)

1 + rj,j+1Xj+1 exp (2ikz,j+1zj), (4.13)

where

rj,j+1 =kz,j − kz,j+1

kz,j + kz,j+1

(4.14)

is the Fresnel coefficient of the interface between layer j and layer j + 1, and kz,j

denotes the z-component of the wave vector in layer j.Recursive application of Eq. 4.13 with RN+1 = XN+1 = 0 as the start of the

recursion yields after N iterations the reflectivity

R = |X1|2 = |R1|2 . (4.15)

4.4 Arbitrary dispersion profiles

Arbitrary (continuous) dispersion profiles can be treated with the Parratt for-malism by slicing the profile into a large number of thin layers of thickness εwith constant δ in each layer and sharp interfaces. If the dispersion profile isdescribed with subatomic resolution (ε 1 A, taking into account the electrondistribution of the atoms), the Parratt formalism allows to calculate the intensitydistribution on the specular rod over the whole momentum transfer range fromq = 0 A−1 up to the Bragg peaks. Such a scheme has recently been used bySchweika et al. [187] in a study of surface segregation and ordering in CuAu.

Arbitrary dispersion profiles can also be calculated in the kinematical approx-imation (see the next section).

4.5 The kinematical approximation

In the kinematical approximation, multiple scattering effects are neglected. Itis only valid, when the scattering cross section is small (‘weak scattering’). Forx-rays, the kinematical approximation can often be used and allows to analyzereflectivity data in a straight-forward way (in contrast to electron diffraction, e.g.,where a dynamical scattering theory is always necessary). In the kinematicalapproximation, the reflectivity is the Fourier transform of the derivative of thedispersion profile δ (z) multiplied by the Fresnel reflectivity RF (qz) (a derivationcan be found in Sec. 4.8.1):

R (qz) = RF (qz)

∣∣∣∣ 1

δ−∞ − δ+∞

∫dδ (z)

dzexp (iqzz) dz

∣∣∣∣2 , (4.16)


where δ−∞ and δ+∞ denote the dispersion at z = −∞ and z = +∞, respectively.

This so-called ‘Master formula’ is in very good agreement with the exacttreatment except for the region around and below the critical angle. A betteragreement in the vicinity of the critical angle can be reached by replacing qz

in the Fourier transform of Eq. 4.16 by q′z = 2k sin(√

α2i − α2

c

), where αc =√

2 (δ−∞ − δ+∞) (see Eq. 4.7).

The advantage of the kinematical approximation is the closed form expres-sion for the reflectivity. This provides good insight into the relation between thedispersion profile and the reflectivity, and allows to some extent a qualitativeinterpretation of reflectivity curves. Since it permits an effective numerical cal-culation of the reflectivity, the kinematical approximation is also often the basisfor advanced data analysis techniques like phase inversion.

4.6 Data analysis and phase inversion

A basic problem, not only for reflectivity measurements, but also for other diffrac-tion techniques, is the loss of the information about the phase of the scatteredwave, as only the square modulus (intensity) of the complex wave amplitude canbe measured.

Consequently, a direct Fourier back transformation of Eq. 4.16 is not possible.The usual approach for analyzing reflectivity data is thus to assume a model ofthe dispersion profile (incorporating all knowledge about the system) and fit thefree parameters of the model to the measured data.

Several techniques have been used to overcome ambiguities in the analysis ofdiffraction data, among them

• adding a known reference layer to an unknown system (see for example[188]),

• exploiting different polarizations of the incoming and reflected beam (neu-tron reflectivity),

• using anomalous scattering.

While these methods have specific experimental requirements, other tech-niques only concern the data analysis. For x-ray reflectivity measurements, phase-guessing methods have been used with success. They are based on the fact thatthe phase, although unknown, is not arbitrary. For more information about thesemethods, see [7, 189] and references therein.

4.7. DESCRIPTION OF ROUGH INTERFACES 39

R=( , )x y

z( )0

z( )R

z

xy

Figure 4.3: Sketch of an interface contour z (R).

4.7 Description of rough interfaces

A single interface without overhangs can be defined by a contour function (seeFig. 4.3)

z (R) with R = (x, y) . (4.17)

We introduce the mean height

z = 〈z (R′)〉R′ , (4.18)

where the angle brackets denote a spatial average. Assuming ergodicity, thiscorresponds to an ensemble average. The interface can also be described by theheight fluctuations h (R) around the mean height z:

h (R) = z (R)− z. (4.19)

The interface can be characterized by statistical properties, such as:

• the root mean square (rms) roughness σ,

σ2 =⟨[h (R′)]

2⟩

R′, (4.20)

• the height-height correlation function

C (R) = 〈h (R′)h (R′ + R)〉R′ , (4.21)


• the height-difference correlation function

g (R) =⟨[h (R′)− h (R′ + R)]

2⟩

R′= 2σ2 − 2C (R) . (4.22)

From Eq. 4.21, it also follows that

σ2 = C (0) . (4.23)

For isotropic surfaces g and C only depend on R = |R|.Many isotropic solid surfaces have a self-affine character and can be described

by

g (R) = 2σ2

1− exp

[−(R

ξ

)2h]

, or equivalently by

C (R) = σ2 exp

[−(R

ξ

)2h],

(4.24)

where ξ is the cutoff length and h the ‘Hurst parameter’ with 0 < h ≤ 1. Smallvalues of h lead to jagged surfaces, while values close to 1 correspond to a surfacewith smaller gradients. For R ξ, the surface is self-affine rough, g (R) ∼ R2h,while for R ξ, the surface appears to be smooth, and g(R) saturates at 2σ2.

For an ideally self-affine surface with no cut-off, g(R) does not saturate forR→∞, and

g (R) = BR2h. (4.25)

4.8 Reflectivity from rough interfaces

in the kinematical approximation

This section is based on work from Sinha et al. [190] and Rauscher et al. [191].In the kinematical approximation, an incident plane wave

Ei (rE) = exp (iki · rE) (4.26)

creates a scattered wave

E (rE) = −exp (ikrE)

4πrE

∫d3r exp (−iq · r) ρ (r) , (4.27)

where q denotes the momentum transfer kf − ki and r = (x, y, z). The densityρ used here is defined as

ρ = k2(1− n2

), (4.28)

which can be written using Eq. 4.1 and δ = bNλ2/2π as

ρ ≈ k22δ = 4πNb, (4.29)

4.8. REFLECTIVITY FROM ROUGH INTERFACES 41

x

z

interface contour

Figure 4.4: Illustration of an interface contour with a fixed (conformal) densityprofile. The density is represented by the color.

where bN is the scattering length density.We now consider the case of a rough interface described by an interface contour

h (R) (see Sec. 4.7) with a density profile across the interface which only dependson the distance z − h (R) from the interface along the z-direction (see Fig. 4.4):

ρ (r) = ρ (z − h (R)) . (4.30)

For such a system, Eq. 4.27 can be written as


4πrE

∫d2R exp (−iQ ·R)

∫ ∞

−∞dz exp (−iqzz) ρ (z − h (R)) ,

(4.31)where Q = (qx, qy) is the momentum transfer parallel to the interface and R =(x, y). By substituting z′ = z − h (R), one obtains


4πrE

∫d2R exp (−iQ ·R− iqzh (R))

∫ ∞

−∞dz′ exp (−iqzz

′) ρ (z′)

= −exp (ikrE)

4πrE

∫d2R exp [−iQ ·R− iqzh (R)] ρ (qz)

(4.32)

with

ρ (qz) =

∫ ∞

−∞dz′ exp (−iqzz

′) ρ (z′) (4.33)

being the Fourier transform of the density profile.


From this, the differential scattering cross section is obtained:

dσ

dΩ= r2

E |E|2

=1

(4π)2

∫∫d2Rd2R′ exp [−iQ · (R−R′)] exp [−iqz (h (R)− h (R′))] |ρ (qz)|2 .

(4.34)

Assuming that [h (R)− h (R′)] is a Gaussian random variable and that the x-raycoherence length is large compared to the correlation length of h(R′′), this yieldswith the substitution R′′ = R−R′:

dσ

dΩ=A (q)

(4π)2

∫d2R′′ exp (−iQ ·R′′) exp

[−q

2z

2g (R′′)

]|ρ (qz)|2 , (4.35)

where A (Q) is the illuminated interface area.Usually, g (R′′) saturates at 2σ2 for R′′ →∞ (see Eq. 4.24). The differential

cross section then splits into a specular and a diffuse (off-specular) part (seeFig. 4.5):(

dσ

dΩ

)spec

=A (q)

(4π)2exp

(−q2

zσ2)δ (Q) |ρ (qz)|2 , (4.36)(

dσ

dΩ

)diff

=A (q)

(4π)2exp

(−q2

zσ2) ∫

d2R′′ exp[C (R′′) q2

z

]− 1

exp (Q ·R′′) |ρ (qz)|2 .

(4.37)

The measured intensity I is the integral of the differential cross section overthe solid angle covered by the detector,

I =

∫ (dσ

dΩ

)dΩdetector, (4.38)

which is calculated in Sec. 5.3.6.The illuminated interface area A (q) is calculated in Sec. 5.3.7. It depends on

the vertical and the horizontal size of the beam, wz and wy, respectively.In the following, we assume that the intensity has been measured by a so-

called ‘rocking scan’ (see Sec. 5.3.6) and that the full incident beam illuminatesthe sample (see Fig. 5.12c+d).

4.8.1 Specular reflectivity

With Eqs. 4.36, 5.15, and 5.22, the specular part of the reflected intensity is∫ (dσ

dΩ

)spec

dΩ =wzwy

2

exp (−σ2q2z)

q2z

|ρ (qz)|2 . (4.39)

4.8. REFLECTIVITY FROM ROUGH INTERFACES 43

0.000 0.001 0.002 0.003 0.004

10-4

10-3

10-2

10-1

100

-3.0x10-4 0.0 3.0x10

-4

qx(Å

-1)

Inte

nsi

ty(a

rb.units

)

i(rad)

C

A

BC B

Figure 4.5: Schematic graph of (A) Specular and (B+C) diffuse reflectivity ina so-called ‘rocking scan’ (αi + αf = const). The ‘Yoneda wings’ (C) can onlybe explained by dynamical scattering, whereas the regions (A+B) are covered bythe kinematical scattering theory described in this chapter.

The reflectivity is defined as the reflected intensity divided by the incidentintensity (measured in the same way), which leads to

Rspec =exp (−σ2q2

z)

q2z

|ρ (qz)|2

=exp (−σ2q2

z)

q2z

1

q2z

∣∣∣∣∣ ˜(

dρ

dz

)∣∣∣∣∣2

,

(4.40)

where(

dρdz

)denotes the Fourier transform of the derivative of the density profile.

One can see that the specular reflectivity is the same, if the interface is treatedas smooth, but the laterally averaged density profile is taken. For Gaussianroughness, this corresponds to a convolution of ρ (z′) with an error function ofwidth σ. In comparison with the reflectivity from a perfectly smooth (σ = 0)interface, the reflectivity from the rough interface decays much faster with qz dueto the damping factor exp (−σ2q2

z) in Eq. 4.40.

If we use Eqs. 4.28 and 4.12, the specular reflectivity can be rewritten as

Rspec =(k2∆δ)2

q4z

∣∣∣∣ 1

∆δ

∫ ∞

−∞

dδ

dz′exp (−iqzz

′) dz′∣∣∣∣2 (4.41)

with ∆δ = δ−∞ − δ+∞.


The prefactor (k2∆δ)2

q4z

is ≈ RF for qz k. We can replace it with RF to

obtain a better approximation (refraction phenomena have been neglected in thederivation so far):

Rspec = RF

∣∣∣∣ 1

∆δ

∫ ∞

−∞

dδ

dz′exp (−iqzz

′) dz′∣∣∣∣2 . (4.42)

This is the Master formula (Eq. 4.16) already presented in Sec. 4.5. The sameresult is obtained for the specular reflectivity measured with a finite detector slitopening instead of a rocking scan.

4.8.2 Integrated diffuse intensity

In this section, we calculate the measured intensity if the contributions fromboth the specular and the diffuse part of the reflected intensity are completelyintegrated in a rocking scan. We start from Eq. 4.35 containing specular anddiffuse contributions and use Eqs. 5.14 and 5.22 as in Sec. 4.8.1, which yields2

∫dσ

dΩdΩ =

wzwy

2q2z

∫d2Q

∫d2R′′ exp (−iQ ·R′′) exp

[−q

2z

2g (R′′)

]|ρ (qz)|2

=wzwy

2q2z

|ρ (qz)|2 .

(4.43)

The same expression as for the specular part of the reflected intensity (Eq. 4.39)is found, but without the damping factor exp (−σ2q2

z). This can be understoodintuitively, as the damping factor accounts for intensity ‘lost’ due to scatteringfrom roughness in off-specular directions. This intensity is regained via the inte-gration.

4.8.3 Off-specular reflectivity

Here, only isotropic rough interfaces will be considered and two specific casesdiscussed:

• ideally self-affine rough surfaces with no cutoff described by Eq. 4.25,

• self-affine rough surfaces with cutoff described by Eq. 4.24.

2Eq. 5.22 can only be used if dσdΩ (qx) ≈ 0 for |qx| >

q2z

2k , i.e. if the scattered intensity isrestricted to the vicinity of the specular rod. Otherwise, the exact expression for the illuminatedarea has to be used to correct variations of the illuminated area during a rocking scan.

4.9. FURTHER REMARKS 45

Self-affine rough surfaces with no cutoff

In this case, the scattering does not split into specular and diffuse parts. We haveto start from Eq. 4.35, which may now be expressed using Eq. 4.25 as

dσ

dΩ=

A

8π

∫dR′′R′′ exp

[−q

2z

2g (R′′)

]J0 (QR′′) |ρ (qz)|2 , (4.44)

where J0 denotes the Bessel function.

Self-affine rough surfaces with cutoff

In this case, the scattering splits into specular and diffuse parts. The specularpart is given by Eq. 4.36. The diffuse part given by Eq. 4.37 may be expressedwith Eq. 4.24 as

(dσ

dΩ

)diff

=A

8πexp

(−q2

zσ2) ∫

dR′′

R′′

exp[q2zσ

2 exp(− (R′′/ξ)

2h)]− 1J0 (QR′′) |ρ (qz)|2 .

(4.45)

4.9 Further remarks

Reflectivity from a rough interface can also be calculated in the Distorted-WaveBorn Approximation (DWBA), see for example [190, 192]. The main differencesin the formula for the diffuse scattering are

• the transmission functions, which lead to an enhancement of the densityfor αi or αf close to αc (the so-called ‘Yoneda-wings’), and

• the appearance of qtz as the wave vector transfer in the medium of the

transmitted wave instead of qz.

The formulae derived in this section refer to a rough interface with a fixeddensity profile. If this density profile contains several layers, the interfaces be-tween the layers completely replicate the roughness (conformal roughness). Thisassumption is not always fulfilled. This is probably not a problem, if just uncor-related roughness is added between the other layers, as this will simply lead toan additional broad diffuse background. The situation will be more complicatedfor partial correlations between layers. For a discussion of these questions, see [7]and references therein.

Chapter 5

High-energy x-ray-reflectivityexperiments

A number of surface-sensitive x-ray diffraction techniques, such as x-ray reflec-tivity, evanescent x-ray diffraction, and crystal truncation rod diffraction havebeen developed in the past. They allow detailed and non-destructive structuredetermination and have been applied with great success to the study of surfaces,interfaces, and multilayers. Much of our current understanding of surface struc-tures is due to surface-sensitive x-ray scattering experiments.

The free surface is a special (and idealized) case of an interface. Many phe-nomena in nature and technology occur at more complex, deeply-buried solid–solid, solid–liquid or liquid–liquid interfaces. Examples of such phenomena arelubrication, electrochemistry, processes at membranes—and interfacial melting.Unfortunately, the surface sensitive x-ray scattering techniques mentioned beforeare not well suited for the study of such interfaces.

A new x-ray transmission-reflection scheme for the study of deeply-buriedinterfaces using high-energy microbeams has been recently developed in our group[6]. The inherent limits of the conventional techniques and the principle of ournew x-ray scattering scheme will be discussed in Sec. 5.1. The use of this methodfor the study of melting at ice–solid interfaces will be shortly presented in Sec. 5.2.The experimental and instrumental details will be explained in Sec. 5.3.

5.1 Principle

Typical x-ray energies for conventional x-ray scattering techniques are in therange of 10–20 keV. Because of the high attenuation coefficient for x-rays in thisenergy range (see Fig. 5.1), the penetration depth is usually in the µm-regime.Therefore, deeply-buried interfaces are generally not accessible to these methods.

For the study of interfaces, this limitation is usually circumvented by placingthe interface as close as possible to the surface, so that it is covered with a film

47

48 CHAPTER 5. HIGH-ENERGY X-RAY-REFLECTIVITY EXPERIMENTS

20 40 60 80 100 120 140

10-1

100

101

102

103

µ(1

/cm

)

E (keV)

water

Si

Cu

Pb

Figure 5.1: The linear attenuation coefficient µ decreases rapidly with the x-rayenergy E (data from [185]).

thin enough to be penetrated by the x-ray beam (see Fig. 5.2a, [7]). This methodhas several drawbacks.

• The in situ study of real devices with deeply-buried interfaces is not possi-ble.

• The interface beneath the thin film might not behave like the deeply-buriedinterface.

• The measured x-ray signal is a complicated interference pattern created bycontribution from all interfaces passed by the x-ray beam, including thesurface (see Fig. 5.2a).

• The thin film covering the interface of interest has to be stable throughoutthe experiment. This is not trivial in the case of liquid films, where aslight change in the temperature and thus the vapor pressure can have adramatic influence on the film thickness. The same holds true for ice films.In addition, sublimation and condensation can also change the morphologyof the ice surface.

• Due to background scattering, the dynamic range is usually limited to 8–9orders of magnitude, especially if the x-ray beam has to penetrate liquidsor amorphous materials.

A new x-ray transmission reflection scheme for studying deeply-buried in-terfaces [6] has recently been developed in our group. It is based on the useof modern Synchrotron Radiation (SR) sources and compound refractive lenses(CRL), which allow to produce high-energy (≥70 keV) x-ray microbeams. As

5.1. PRINCIPLE 49

up to cm

<µm

up to cm

buriedinterface

R1

R2

R1

conventional scheme

a b

transmission-reflection schemewith high-energy microbeam

Figure 5.2: Comparison of (a) conventional surface scattering scheme and (b)transmission-reflection scheme with high-energy microbeam. In the conventionalscheme waves reflected from all interfaces create a complicated interference signaland the interface of interest can only be covered with a thin film.

the attenuation coefficient decreases rapidly with the x-ray energy (see Fig. 5.1),high-energy x-rays can penetrate materials with a macroscopic thickness of up toseveral centimeters.

In the transmission-reflection scheme, a high-energy x-ray microbeam is usedto penetrate the sample from the side (thereby avoiding reflections from the sur-face and other interfaces) and probe the buried interface of interest (see Fig. 5.2b).In principle, the whole spectrum of x-ray techniques used at surfaces (see above)can be applied at buried interfaces if high-energy x-rays are used. In this workx-ray reflectivity (see Chapt. 4 and Sec. 5.2), has been used. This technique issensitive to the (electron) density profile perpendicular to the interface. In an-other work, we have already used evanescent x-ray diffraction to study the lateralstructure of liquid lead at the interface with silicon [193].

Compound refractive lenses made of aluminum are used to produce highly-collimated microbeams with a small divergence (≈30 µrad) and a small beamsize (≈10 µm). There are two reasons why this is necessary:

1. On its long path through the material the x-ray beam produces diffusescattering background, which would normally limit the measurement to asmall dynamic range. The small beam size and divergence allow to stronglylimit the x-ray phase space covered by the detector and thereby minimizethe diffuse background.

2. The incident angles for measurements at high x-ray energies are very small(the critical angle for an ice–Si interface at 70 keV is ≈0.3 mrad, seeTab. 7.1). Consequently, the projection of the interface perpendicular to


the beam is very small (≈10 µm at the critical angle for a sample lengthof 30 mm). The small beam size ensures that even at these small incidentangles the entire beam hits the interface (see Sec. 5.3). This maximizes thescattering signal for reflectivity measurements and is absolutely necessaryfor evanescent scattering to avoid bulk scattering.

The small incident angles, and, as a consequence, the small projection of theinterface perpendicular to the beam are the reasons for the major difficultiesassociated with this method. A high angular resolution (< 10 µrad) is needed fordefining such small angles. A high resolution (< µm) in the height adjustment isneeded to place the interface into the beam (see Fig. 5.12). Similar requirementsconcern the beam and sample stability, as well as the sample quality (curvature).

The possibility of radiation damage is always a concern with experimentsmaking use of highly brilliant Synchrotron Radiation. High-energy x-rays interactless strongly with matter (see Fig. 5.1) than x-rays with lower energies , but thesmall beam size used in these experiments can be a problem (see Sec. 7.8).

5.2 Application to the interface melting of ice

The main reason why experimental data on interface melting of ice is quite scarceis that adequate methods have been lacking so far (see Sec. 3.4). Conventionalsurface-sensitive x-ray scattering techniques in conjunction with thin films of ice(see Sec. 5.1) are not suitable because of the high vapor pressure of ice and theimportant changes in the surface morphology due to sublimation/resublimation.The high-energy transmission-reflection scheme presented in this chapter is theideal probe for investigating ice–solid interfaces. It has already been successfullyapplied to study metal(liquid)–semiconductor(solid) interfaces [193, 194].

A schematical sketch of the setup for the study of an ice–SiO2–Si interface(see Chapt. 6) is shown in Fig. 5.3. A high-energy (≈70 keV) x-ray beam froma Synchrotron Radiation (SR) source is focused by a compound refractive lense(CRL). It penetrates the sample through the ice and hits the interface at a (small)incident angle. As can be seen from Fig. 5.1, the attenuation coefficient for wateris very low at these energies. The reflected intensity is measured as a function ofthe momentum transfer qz perpendicular to the interface. The reflectivity profilesobtained in this way allow to deduce the profile of the (laterally averaged) densityacross the interface (see Chapt. 4). If a quasiliquid layer (qll) with differentdensity emerges due to interfacial melting, interference fringes will appear on thereflectivity curve (see inset of Fig. 5.3). The distance of these fringes is relatedto the thickness L of the layer. Although the relevant order parameters for themelting transition are the Fourier components of the density, the average densityprobed by the reflectivity measurements plays an important role for the meltingof ice (see Sec. 3.4).

5.3. EXPERIMENTAL AND INSTRUMENTAL CONSIDERATIONS 51

CRLice

SiO2

Si

qz

sample cell

qllSR source

70 keV

T T m

qc

2/Lqll

Inte

nsi

ty

qz

2 /L

T Tm

Figure 5.3: Schematical sketch of the setup for the study of ice–solid interfaceswith high-energy x-ray microbeams (see the text).

5.3 Experimental and instrumental considera-

tions

Results from several experiments are presented in this dissertation (see Chapt. 7).The two main experiments of this work were carried out using the high-energy x-ray microbeam transmission-reflection scheme. The basic concept of this schemehas been presented in Sec. 5.1 of this chapter. In this section the experimentaldetails of this scheme as used in the two main experiments are discussed. Theexperimental parameters of these two experiments were almost identical. Addi-tional experiments have been performed, some of them with completely differentmethods (neutron reflectivity). Since these experiments are of minor importancefor this work, their experimental aspects are not discussed in the same detail. Forthe sake of clarity, the experimental aspects of these experiments are presentedtogether with their results in separate sections of Chapt. 7.

A sketch of the setup for the high-energy microbeam transmission-reflectionscheme is shown in Fig. 5.5. The principal components are the source, themonochromator, the focusing device, the sample stage, and the detector. Thesecomponents are presented in the following subsections. Some important aspectslike resolution, coherence, and correction factors will also be discussed.

The setup presented here has been installed at the high-energy beamlineID15A of the ESRF (a schematic layout of the beamline is shown in Fig. 5.4).For each experiment, the whole setup had to be reinstalled. A permanent setupwill be available in the near future.

5.3.1 Source and optics

The source of ID15A is a seven period permanent magnet asymmetric wigglerlocated at the center of the straight section. It has a critical energy of 44.1 keV


0 m

AMPW

26

BL A PS

38 42

MP MS

ID15 Bhutch S

ID15 Ahutch P

SS ID15 Ahutch B

42

MB

experimental setup

SW

Figure 5.4: Schematic layout of the high-energy beamlines ID15A with hutchesP (port) and B (bow), and ID15B with hutch S (starboard) [195]. AMPW:Asymmetric multipole wiggler, SW: shield wall, BL: beam limiter, A: attenuator,PS: primary slits, SS: secondary slits, MP: monochromator for experimental hutchP (Laue), MS: monochromator for hutch S (Bragg/Laue), MB: monochromatorfor hutch B (for this experiment: double Laue)

at a gap of 20.3 mm. The first element of the beamline is a 4 mm thick aluminumabsorber which filters out x-ray energies below 40 keV (see Fig. 5.4). The experi-ments were carried out at an x-ray energy of around 71 keV (wavelength 0.175 A,see Tab. 7.1). The energy was calibrated by measuring the scattering angle ofa specific Si or ice Bragg reflection. For monochromatization of the beam, twoasymmetrically cut and bent Si crystals in a fixed exit Laue geometry were used(see Fig. 5.5) [196, 197]. The first crystal creates a virtual source for the secondcrystal, which allows to compensate the divergence of the primary beam in onedirection (horizontal). The energy bandpass of the monochromator crystals isdetermined by the asymmetric cut to the (111) reflection. A 37.74 cut for thefirst crystal and a 37.76 cut for the second crystal were used, which leads toa relatively large energy band pass of 165 eV, corresponding to an energy reso-lution ∆E/E ≈ 2 × 10−3. As the monochromator crystals are only exposed tothe high-energy part of the x-ray spectrum, direct cooling of the first crystal inan In-Ga eutectic is sufficient and the crystals can be kept in air. Due to thesimplicity of the setup and the low thermal load on the monochromator crystals,a high stability of the monochromatized beam is achieved.

The beam is focused on the sample position (alternatively to the detectorposition) by a compound refractive lense1 [198, 199, 200] with an effective apertureof about 0.35 mm (at the x-ray energy of 71 keV). It consists of around 230 singlealuminum lenses and produces a spot size of about 5 µm (vertical) × 20 µm(horizontal) at a focal distance of about 4.5 m (for exact numbers see Tab. 7.1).The beam profiles at the sample position are shown in Fig. 5.6. The long focal

1provided by A. Snigirev (ESRF)


230

CRL

mono1

mono2

virtualsource

2

sampletower

detectortable

detector

monochromatorMB

0.8 m 2 m 4 m 1.3 m

a top view

b side view

1

2

zy

xi

2

Figure 5.5: Sketch of the setup for the high-energy microbeam transmission-reflection scheme. (a) Top view. (b) Side view. Two asymmetrically cut and bentSi crystals in a fixed exit Laue geometry are used for monochromatization (71 keV,bandwidth 165 eV). A compound refractive lense (CRL) serves to focus the beamon the sample position (spot size 5×20 µm2). A specially built diffractometercarries the sample chamber and defines the sample position and sample tilt. Thedetector is placed on a table which allows to set the horizontal and verticalscattering angle.


-20

-10

0

10

20

10 0 -10 -20 -30 -40 -50 -60

dI/dz(arb. units)

z(µ

m)

-50 0 50

0

-5

-10

-15

dI/dx

(arb

.units

)

x (µm)

a vertical b horizontal

Figure 5.6: Beam profiles at the sample position. The measured intensity profile(derivative of a ‘knife-edge’ scan, open circles) is shown together with Gaussianfits (solid line). (a) Vertical profile, FWHM = 5.6 µm. (b) Horizontal profile,FWHM = 19.2 µm.

distance leads to a small divergence of about 30 µrad. A flux of 2×1010 photons/sat a storage ring current of 200 mA can be reached in the focused beam. Incomparison with the previously used [193] microbeam collimator, the lenses leadto a strongly enhanced stability and a flux which is about 7 times higher (at thesame vertical spot size and a horizontal spot size of 20 µm for the lenses and 350µm for the collimator).

5.3.2 Sample stage

The sample chamber (see Chapt. 6.5) is mounted on a specially designed diffrac-tometer in vertical axis geometry (see Fig. 5.7), which defines the sample positionand tilt. It allows to control the vertical position z of the sample with an ac-curacy of 0.2 µm and the incident angle αi with an accuracy of 5 µrad. Thesurface normal of the sample is aligned with a double tilt station (χ1 and χ2)and the lateral position of the sample (x and y) can be adjusted with a doubletranslation stage. The sample can also be rotated around the surface normal(important for measurements of anisotropies in the lateral structure as in [193]).The diffractometer can carry a load of up to 120 kg which allows to use heavyUHV chambers. For stability reasons, the whole diffractometer is mounted on a2 ton granite block.


Figure 5.7: Diffractometer with sample chamber.

5.3.3 Detector

The scattering angle is defined by a set of detector slits made of tungsten. Anidentical slit system is used as scatter slits which define the field of view forthe detector. Even when focusing on the sample position, the beam size atthe detector position is as small as 50 µm vertical × 100 µ horizontal. Thisallows to work with small slit gaps leading to a very small solid angle coveredby the detector on the order of 10−9 (10−10 when the beam is focused ontothe detector). Consequently, the diffuse scattering background is reduced toa minimum. Together with careful shielding, this leads to a background levelof only about 1 count/s in the detector at a primary beam flux of the order of1010 photons/s, which allows to access a dynamic range of up to 10 orders ofmagnitude.

As x-ray detectors, a Si-PIN-diode was used for the highest intensities and ascintillation counter for lower intensities. Due to the limited dynamic range of thescintillation counter (≈2×104), beam attenuators had to be used. An additionalPIN-diode placed behind the CRL was used for monitoring the incident beamflux.

The detector and slit system were mounted on a separate table decoupled fromthe sample tower. The vertical scattering angle (2θ⊥) is set by two independenttranslations integrated in the detector table, the horizontal scattering angle (2θ‖)by an additional translation and rotation.


5.3.4 Scattering geometry

The scattering geometry is illustrated in Fig. 5.8. The momentum transfer iscontrolled by varying the angles αi, αf and β. Its components are given by

qx = k (cosαf cos β − cosαi) ,

qy = k cosαi sin β,

qz = k (sinαi + sinαf ) .

(5.1)

In the case of in-plane scattering (used to measure specular reflectivity, forexample) β = 0 and

qx = k (cosαf − cosαi) ,

qy = 0,

qz = k (sinαi + sinαf ) .

(5.2)

In the case of specular reflectivity αf = αi and

qx = 0,

qy = 0,

qz = k2α.

(5.3)

with α = αi = αf .

In this work, a so-called ‘rocking-scan refers to a rotation of the sample aroundthe y-axis in the in-plane scattering geometry (see Fig. 5.9). During a rockingscan 2α = αi + αf = const. Such a rocking scan corresponds to a transversemomentum scan along qx. The perpendicular momentum transfer qz stays prac-tically constant during the scan.

All the angles are usually quite small in reflectivity experiments, especially inexperiments with high-energy x-rays. In this work, the largest angles are about20 mrad. Therefore, the following approximations can often be used:

sin η ≈ η ≈ 0,

cos η ≈ 1− η2/2 ≈ 1,(5.4)

where η is a small angle.

5.3.5 Resolution

Calculating the total differentials from Eq. 5.2 and using the approximations inEq. 5.4 yields the momentum transfer resolution for in-plane scattering geometry(see also Fig. 5.8):


iqx

qy

f

b top view

x

y

ki

kf

h

i

i

ki

kf

f

q

qz

qx

f

v

interface

detector

detector slits

a side view

x

z

ki

kf

q

interfacec steric view

x

z

y

fi

Figure 5.8: Scattering geometry. The incident and scattered wave vectors areki and kf , the incident and exit angles αi and αf , β. The momentum transferis q = kf − ki with the components qz perpendicular and qx, qy parallel to theinterface. The resolution is determined by the divergence (∆αi vertical, ∆βi

horizontal), the acceptance (∆αf vertical, ∆βf horizontal) set by the detectorslit openings (∆v, ∆h), and the wavelength spread. The latter leads to a spreadof the wave vector length not shown in this illustration.


i

kikf

f

q

qz

interfacex

z

qx

y

Figure 5.9: Illustration of a rocking scan. The sample is rotated around the y-axis. 2α = αi + αf is constant. The momentum transfer is fixed in the referenceframe of the laboratory, but its components with respect to the interface change.

δqx =∆λ

λqx + k (αi∆αi + αf∆αf ) ,

δqy = k∆β = k (∆βi + ∆βf ) ,

δqz =∆λ

λqz + k (∆αi + ∆αf ) .

(5.5)

The resolution of the momentum transfer qz perpendicular to the interfaceis most relevant for reflectivity measurements. The resolution in qy is usuallyrelaxed, leading to an effective integration over qy.

The wavelength spread ∆λ/λ can be calculated from the energy resolution(bandwidth) ∆E/E:

λ =hc

E, (5.6)

∆λ

λ=

∆E

E. (5.7)

∆αi and ∆βi are the angular beam divergence in the vertical and horizontaldirection, respectively.

∆αf and ∆βf are the acceptance of the detector in the vertical and horizontaldirection, respectively. They are defined by the opening of the detector slits, ∆vin the vertical and ∆h in the horizontal direction (see Fig. 5.8):

∆αf ≈ ∆vl, (5.8)

∆βf ≈ ∆hl, (5.9)

where l is the distance between the detector slits and the sample. Additionalcontributions from the finite size of the illuminated area can be neglected. The


0.0 0.2 0.4 0.6 0.8 1.00.000

0.005

0.010

0.015

0.020total

q

z(Å

-1)

qz(Å

-1)

0.0 0.2 0.4 0.6 0.8 1.00.000

0.005

0.010

0.015

0.020

total

q

z(Å

-1)

qz(Å

-1)

a rough substrate b smooth substrate

ii

f

f

Figure 5.10: Momentum transfer resolution δqz perpendicular to the interface asa function of qz. The contributions from the divergence ∆αi, acceptance ∆αf ,and wavelength spread ∆λ/λ are plotted. (a) Experiment with rough substrate.The jump in the momentum resolution is due to a change of the slit settings. (b)Experiment with smooth substrate. The accessible qz-range is larger, since thesmall roughness leads to a slower decay of the reflected intensity.

same applies to corrections due to the tilt of the detector slit gap and the changeof the projected distance between detector and slits.

The horizontal detector slit opening used in this work leads to an effectiveintegration over qy (see above). The vertical detector slit opening leads to amoderate resolution in qz.

In the experiments presented in this work, we are dealing with thin layers (≈x nm), which lead to rather well separated (≈(0.6/x) A−1) and broad features onthe reflectivity curve with modest requirements for the resolution. Several otherconsiderations favor a larger resolution element:

• increased intensity,

• less sensitivity to misalignment,

• less sensitivity to sample imperfections (e.g., curvature), especially at smallincident angles.

Therefore, the choice of the resolution (i.e. the vertical detector slit opening) isa compromise and the smallest possible resolution element δqz is not necessarilythe best option.

The resolution is shown Figs. 5.10, 5.11 and summarized in Tab. 5.3.5. Thevalues have been calculated with the experimental parameters from Tab. 7.1.


0.0 0.2 0.4 0.6 0.8 1.00.0

5.0x10

1.0x10

1.5x10

-5

-4

-4

qx= 0 Å

-1

rough substrate

smooth substrate

q

x( Å

-1)

qz(Å

-1)

a on specular path b during rockings scan

-1.0x10-3 0.0 1.0x10

-30.0

-5

-4

-4

qz= 0.3 Å

-1

smooth substrate

rough substrate

q

x(

5.0x10

1.0x10

1.5x10

Å-1)

qx(Å

-1)

Figure 5.11: Momentum transfer resolution δqx parallel to the interface. (a) Asa function of qz (at specular condition). (b) As a function of qx (at fixed qz andqy = 0 A−1).

Table 5.1: Momentum transfer resolution (mean values). For q-dependence, seeFigs. 5.10 and 5.11.

substrate δqx δqy δqz(A−1) (A−1) (A−1)

rough substrate 7× 10−5 0.06 0.02smooth substrate 5.5× 10−5 0.01 0.01


5.3.6 Integration by the detector

For the determination of the density profile from diffuse reflectivity, integratedrocking scans are used in this work (see Sec. 4.8.2). Integration of a rockingscan corresponds to the integration over a certain solid angle with the conditionαi + αf = 2α = const.: ∫

dαfdβ

∣∣∣∣αi+αf=2α=const

.

But even without explicit integration, the finite size of the detector slit open-ing always leads to an integration: ∫

dαfdβ.

The integration in angular space leads to an effective integration in momentumspace. In this section, the connection between the integration over the solid angle∫

dαfdβ and the integration over the lateral momentum transfer∫

dqxdqy willbe calculated for the case of a rocking scan and for the case of a finite detectorslit opening (without rocking the sample).

The partial derivatives are calculated from Eq. 5.1 (using the approximationsfrom Eq. 5.4):

Jrocking =∂ (qx, qy)

∂ (β, αf )

∣∣∣∣αi+αf=2α=const.

≈ k

(−β −2α1 −βαf

), (5.10)

Jdetector =∂ (qx, qy)

∂ (β, αf )≈ k

(−β −αf

1 −βαf

). (5.11)

From this, we get

det (Jrocking) = k2(β2αf + 2α

)≈ k22α ≈ kqz, (5.12)

det (Jdetector) = k2(β2αf + αf

)≈ k2αf≈

kqz2. (5.13)

The last approximation in Eq. 5.13 is only valid close to the specular condition.We finally obtain ∫

dαfdβ

∣∣∣∣rocking

=1

kqz

∫dqxdqy, (5.14)∫

dαfdβ

∣∣∣∣detector

=2

kqz

∫dqxdqy. (5.15)

5.3.7 Illumination of the sample

The illuminated interface area and the proportion of the beam hitting the in-terface depend on the incident angle αi. The different situations are depicted in


Fig. 5.12. For αi < α0 = arcsin (wz/a), only a proportion fbeam = w′z/wz of the

beam hits the interface, and thus the reflected intensity is reduced by the sameratio. This geometric effect has to be accounted for, either by correcting themeasured data, or by including it in the calculation of the reflected intensities:

Rcorrected = f−1beamRmeasured (5.16)

orRcalculated = fbeamRideal, (5.17)

where

fbeam =

a sin (αi)

wz

for αi < α0 = arcsin (wz/a) ,

1 otherwise.

(5.18)

The angle of total illumination, α0, decreases with the sample length a. Therefore,a smaller part of the reflectivity curve is affected by the illumination correction forlonger samples. With the experimental parameters for this work (see Tab. 7.1),α0 ≈ 0.24 mrad.

Eq. 5.18 is valid when the beam has a box profile (in the vertical direction).The measured beam profile, however, is a Gaussian (see Fig. 5.6). For an arbitrarybeam profile I(z), the incident intensity I ′ calculates as

I ′ =

∫ z+

z−

I(z)dz, (5.19)

where z± = ±a sin (αi) denotes the extension of the interface projected perpen-dicular to the beam (see Fig. 5.13). The correction factor is then

fbeam =I ′

I0=

∫ z+

z−I(z)dz∫ +∞

−∞ I(z)dz. (5.20)

Graphs of the correction factor for a box profile and a Gaussian profile, as wellas corrected and uncorrected reflectivity curves are shown in Fig. 5.14. However,the area correction often does not work as well as in the example shown in thegraph. This is in part caused by the extreme sensitivity on the alignment. Ifthe beam does not exactly hit the center of the sample, the formulae for thearea correction are not strictly valid anymore. Furthermore, the effective samplelength is in most cases smaller than the substrate length. One reason is thecurvature of the sample (see Sec. 6.1). Another reason is that over the durationof a beamtime, some of the ice evaporates preferentially at the edges of the sample.

Once the whole beam illuminates the interface (αi = α0), its footprint a′ onthe interface (see Fig. 5.12) rapidly decreases with αi (qz), see Fig. 5.15:

a′ =

wz

sin (αi)for αi > α0 = arcsin (wz/a) ,

a otherwise.

(5.21)


a i=0

i

wz

w ’<z

a

a’=a

i

w ’z=

a’<a

i

a’=a

b i<

0

c i=

0d

i>

0

interface beamw

z

wz

w ’z= w

z

Figure 5.12: Illumination of the interface. αi incident angle, a length of thesample, a′ illuminated part of the interface, wz beam height, w′

z part of the beamheight which illuminates the interface. (a) αi = 0. (b) αi < α0. The whole lengthof the interface is illuminated (a′ = a), but not all of the beam hits the interface(h′ < h), thus the reflected intensity is reduced. (c) αi = α0. At this angle, theprojection of the beam exactly covers the sample length (a′ = a, w′

z = wz). (d)αi > α0. At larger angles, the whole beam hits the interface (w′

z = wz), but itsfootprint is shorter than the sample length (a′ < a).


i

z

I(z)

z+

z-

I’

beam profile

a

Figure 5.13: Illumination of the interface with an arbitrary beam profile. Theincident intensity I ′ is the integral of the intensity profile I(z) over the projectionof the interface onto the z-axis.

0.02 0.04 0.06 0.08

10-2

10-1

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

measured datacorrected data:box profileGaussian profile

0.00 0.02 0.04 0.060.0

0.2

0.4

0.6

0.8

1.0

i>

0

i<

0

box profileGaussian profile

1/f

beam

qz(Å

-1)

a correction factor b measured and corrected data

Figure 5.14: Illumination correction. (a) Correction factor as a function of qzfor a box-profile (Eq. 5.18 ) and a Gaussian profile (Eq. 5.20). (b) Open cir-cles: measured (uncorrected) reflectivity data , solid line: corrected with boxprofile, dashed line: corrected with Gaussian profile. The Gaussian profile allowsa slightly better correction (note the small dip on the edge of total reflection forthe correction with the box profile). Renormalization of the data after correctionleads to a small difference even at large qz, where the correction factor is 1 forboth beam profiles.


0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

experiment 1a = 24 mm, w

z= 5.6 µm

experiment 2a = 30 mm, w

z= 7.1 µm

Footp

rinta'(

mm

)

qz(Å

-1)

Figure 5.15: Length a′ of the beam footprint on the interface for the roughsubstrate (solid line) and the smooth substrate (dashed line) calculated fromEq. 5.21. a: sample length, wz vertical beam size (FWHM).

For a Gaussian profile, wz denotes the FWHM of the beam profile and a′ theFWHM of the beam projected onto the interface.

The small footprint for larger values of qz has several advantages:

• less sensitivity to sample curvature,

• less sensitivity to sample inhomogeneities (in this case: temperature gradi-ents),

• no influence from the edges of the sample.

For small values of qz, however, the footprint is large. While temperature gradi-ents are nonetheless small (see Sec. 6.6), scattering from the edges of the samplecan be a problem.

According to Eq. 5.21 the illuminated area A is for αi > α0

A =wzwy

sinαi

≈ 2kwzwy

qzclose to the specular condition,

(5.22)

wy denotes the horizontal width of the beam profile. This equation is used inChapt. 4 where the diffuse reflectivity is discussed. Unlike for specular reflectivity,the intensity of the diffuse reflectivity is proportional to the illuminated area.


l

t

a

ki

kf

a

b

i

f

r

x

z

c

t

i

x

t

t

z

b

wz

Figure 5.16: Coherence in x-ray scattering. (a) Illustration of longitudinal (ξl)and transverse (ξt) coherence. (b) Projections of the transverse coherence. (c)Path length difference a− b for scattering from two points separated by r.

5.3.8 Coherence

A wave field is called coherent, when it is able to produce interference phenom-ena. This requires a spatial and temporal relation between the phases of thewave field. The degree of coherence is usually characterized by a longitudinaland a transverse coherence length, ξl and ξt, respectively (see Fig. 5.16a). Thelongitudinal coherence is a measure of the monochromaticity of the source,

ξl =λ2

∆λ, (5.23)

where λ denotes the wavelength and ∆λ/λ the wavelength spread. The longitu-dinal coherence length is associated with a coherence time

τ =ξlc, (5.24)

c being the speed of light.The transverse coherence length is related to the finite size s of the source,

ξt =λd

s, (5.25)

where d denotes the distance from the source.Note that the definitions for the longitudinal and transverse coherence length

that can be found in the literature sometimes contain an additional factor of 1/2or 1/π.

In the case of high-energy x-rays the wavelength is small, which leads to alow degree of coherence. In this work, only a moderate energy resolution ∆λ/λhas been used, which further decreases the longitudinal coherence length (for thevalues see Tab. 5.2).


The degree of coherence is crucial for the interpretation of the x-ray scatteringsignal. In most cases, an intermediate degree of coherence is desired. If the scat-tering were completely incoherent, no interference would be possible. If, on theother hand, the scattering is completely coherent (the effective coherence lengthreaches the dimensions of the sample), the scattering signal does not correspondto an average structure anymore. In this case, a complicated speckle pattern isproduced, which contains information about the whole sample. Some techniquesexploit the coherent regime, but in most cases it is better to avoid this situationin order to obtain an averaged information.

The longitudinal and transverse coherence length can be used to calculatewhether scattered waves from two points of the sample separated by the vectorr can interfere. First, the lateral coherence length ξl has to be larger than thepath length difference l for a given momentum transfer q (see Fig. 5.16c),

ξl > l = b− a = rx (cosαf − cosαi) + rz (sinαf + sinαi) =r · qk

. (5.26)

If we define

ξxl =

ξlk

qx,

ξzl =

ξlk

qz,

(5.27)

the condition expressed by Eq. 5.26 is fulfilled for

(ξxl > rx and rz = 0) or

(ξzl > rz and rx = 0).

(5.28)

The second condition for interference is that the transverse coherence lengthprojected on r is larger than r. This can also be written as (see Fig. 5.16b)

ξxt =

ξtsinαi

> rx and

ξzt =

ξtcosαi

> rz.

(5.29)

If the limiting coherence length is smaller than the linear dimensions of thescattering volume, the scattering signal corresponds to an averaging over thisvolume.

In the experiments of this work, the coherence is large enough to produceinterference from the density profile perpendicular to the interface and from theroughness of the interface (see Tab. 5.2). But at the same time, the transversecoherence length is smaller than the beam size, and thus the transverse coher-ence length projected on the interface smaller than the illuminated length. This


Table 5.2: Coherence parameters.rough smooth

substrate substrate

λ (A) wavelength 0.1740 0.1736∆λλ

(%) wavelength spread 0.23 0.23d (mm) distance source (lenses) to sample 4380 4530s (µm) source size (effective aperture) 350 350ξl (A) longitudinal coherence length 76 75ξt (A) transverse coherence length 2200 2200qmaxx (A−1) max. parallel 0.003 10−4

momentum transferqmaxz (A−1) max. perpendicular 0.57 1.0

momentum transferξxl (A) lower limit (at qx = qmax

x ) 9.1×105 2.7×107

ξzl (A) lower limit (at qz = qmax

z ) 4800 2700ξxt (A) lower limit (at qz = qmax

z ) 2.8×104 1.6×104

ξzt (A) lower limit (at αi = 0) 2200 2200wz (µm) beam size vertical 5.6 7.1wy (µm) beam size horizontal 19.2 ≈24

means that the scattering signal corresponds to an (incoherent) average over theilluminated sample area as assumed in Sec. 4.8.

The reader is referred to the literature for more details on coherence in x-ray physics in general [201], and coherence in x-ray scattering experiments inparticular [202].

5.3.9 Data correction

The correction due to different illumination of the interface has already beendiscussed in detail in the last section.

As mentioned above, a PIN diode has been used to monitor the intensityof the incident beam. This intensity varies due to the decrease of the storagering current between injections, temperature changes of the monochromators andother reasons. Therefore, all measured intensities have been normalized to theincident beam intensity.

Several types of background show up in the data. One part is due to scatteringfrom the air, optical elements, and the bulk of the sample. These contributionscan be strongly reduced with shielding and small slit openings. When specularreflectivity is measured, the diffuse reflectivity from the roughness of the inter-face represents another type of ‘background’. Separation and analysis of thebackground will be discussed in Sec. 7.3.


As mentioned before, absorbers had to be used due to the limited dynamicrange of the detectors. Different parts of the reflectivity curves were thus mea-sured with different absorber settings and had to be adjusted (see Sec. 7.2.1).Finally, the data had to be scaled appropriately (see Chapt. 7) as the measure-ments do not yield directly the absolute reflectivity, but only a scattered intensityproportional to the reflectivity.

Chapter 6

Sample preparation andenvironment

In this chapter, the samples, preparation techniques, and the sample environmentare described. The chapter includes sections about the substrates and their sur-face preparation (Sec. 6.1), the ice single crystals (Sec. 6.2), and the preparationof well-defined ice–substrate interfaces (Sec. 6.4). Another section describes thein situ chamber designed for the ice experiments and the temperature control sys-tem (Sec. 6.5). The last section discusses the temperature stability and accuracy(Sec. 6.6).

6.1 The substrates

The aim of this work was to investigate the melting behavior at the heterogeneousinterface between ice and a solid ‘substrate’. There were two main considerationsfor the choice of SiO2 as the substrate material:

1. the relevance of this particular ice–substrate interface,

2. the suitability of the substrate for the sample preparation and the reflec-tivity measurements.

A number of interfaces could serve as model systems for important ice inter-faces in nature and technology. The ice–SiO2 interface can be regarded as anidealized model system for the situations where ice is in contact with a minerallike at glacier beds or in permafrost.

As discussed before, the high-energy x-ray transmission-reflection scheme (seeChapt. 5) used for the experiments puts high demands on the sample quality. Thesample has to be smooth on an atomic scale, must have a very low waviness andcurvature (which has been characterized by interferometry1, see Fig. 6.1), andshould be long in the direction of the beam.

1in collaboration with A. Weißhardt (MPI fur Metallforschung)

71

72 CHAPTER 6. SAMPLE PREPARATION AND ENVIRONMENT

-15 -10 -5 0 5 10 15-1.5

-1.0

-0.5

0.0

0.5x (long) direction,

y=0

y (short) direction,x=0

z(

m)

x (mm),y (mm)-10 -5 0 5 10

-2.0

-1.5

-1.0

-0.5

0.0

0.5

y direction,x=0

x direction,y=0

z(

m)

x (mm),y (mm)

a rough substrate b smooth substrate

1 m1 m

Figure 6.1: Figure error of the substrates. (a) Rough substrate. (b) Smoothsubstrate. In both cases, the figure error is about 1 µm over the whole sample,which corresponds to an angular spread of about 50-200 µrad.

A SiO2 substrate which fulfills these requirements can be prepared from a sil-icon block. Large single crystals of silicon are readily available and can be easilypolished. After cleaning, a native amorphous oxide forms at the surface whensilicon is kept in air. Compared to quartz glass, an oxidized silicon block alsohas the advantage of a high thermal conductivity (about two orders of magnitudelarger), which is important for the sample preparation (see Sec. 6.4). Moreover,a substrate with an amorphous termination comes closer to an ideal hard wall.Unlike a crystalline substrate, it does not impose a specific structure in the adja-cent quasiliquid and, therefore, is thought to reveal more intrinsic properties ofwater.

Several parameters can have an influence on interface melting (see Chapt. 8),like

• the substrate material,

• substrate morphology,

• crystal orientation,

• impurities.

In this work, interface melting at the ice–SiO2 interface was studied. In addition,the influence of one of the aforementioned parameters, namely the substratemorphology, was tested. Therefore, two substrates with different roughness wereprepared, a ‘rough’ substrate and a ‘smooth’ substrate. See Sec. 7.1 for anoverview of the experiments and the substrates used. For a detailed discussionof the substrate morphology, see Sec. 7.3.

6.1. THE SUBSTRATES 73

Figure 6.2: Photograph of a polished Si block used as substrate (here: the smoothsubstrate). The top face is contacted with the ice single crystal after cleaning.

The dimensions l × w × h of the substrates in mm (with the length l alongthe beam and the height h perpendicular to the interface) were 24 × 24 × 10for the rough substrate and 30 × 24 × 8 for the smooth substrate. The surfacecut was (111) for the rough and (100) for the smooth substrate. The chemo-mechanical polishing of the substrates was performed by H. Wendel (MPI furFestkorperforschung) for the rough sample and by the crystal laboratory groupof the ESRF for the smooth sample. A photograph of a substrate is shown inFig. 6.2 and a typical reflectivity measurement on a substrate in Fig. 6.3.

After polishing the substrates were thoroughly cleaned in several steps usingsome standard recipes:

1. ‘Piranha clean’

• H2SO4 (80%) and H2O2 (30%), volume ratio 3:1 (gets hot2), 5 minutes,removes organic contaminants and chemically oxidizes the surface

• rinsing with H2O, 3 minutes

• HF dip (5%), 2 minutes, removes the oxide layer

• rinsing with H2O

2. RCA clean

• H2O, NH3 (25%) and H2O2 (30%), 4:1:1, 70C, 10 minutes, also re-moves organic contaminants


• HF dip (5%), 1 minute

2Contact with residues of organic solvents must be avoided, explosive reactions!


-2.0x10-4 0.0 2.0x10

-4

0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

Inte

nsi

ty(a

rb.units

)

qx(Å

-1)

0.0 0.1 0.2 0.3 0.4 0.5

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

a reflectivity b transverse momentum scan

qz=0.171 Å

-1

see b

Figure 6.3: Reflectivity measurement of a substrate with the diffractometer ofthe cold room (Cu-Kα x-ray radiation). (a) Reflectivity curve. Open circles:measured points. Filled squares: integrated transverse momentum scans. Solidcurve: fit with oxide layer thickness of 15 A and rms roughness of 3.9 A. (b)Example of a transverse momentum scan (filled squares) and fit (solid line) witha Gaussian (representing the resolution function).


• H2O, HCl (30%) and H2O2 (30%), 8:1:1, 70C, 10 minutes, removesmetallic contaminants, oxidizes the surface


• HF dip (5%), 1 minute, removes the oxide

The silicon sample was then kept in air which leads to the formation of a thinnative amorphous oxide layer. The thickness of this oxide layer is 20 A for therough substrate and 16 A for the smooth substrate.

The contact angle of the substrates depends on the chemical treatment usedfor the surface preparation (see for example [203]. The method used in this workinitially leads to a hydrophobic surface (due to the HF dip at the end). Theoxide-covered surface is usually hydrophilic, but can become hydrophobic dueto contaminations with hydrocarbons from the air. When exposed to intensivehigh-energy x-ray radiation in the presence of water, the surface becomes stronglyhydrophilic again (see Sec. 7.8).

An additional experiment was performed with Si wafers and a different chemi-cal treatment was used. For details see Sec. 7.6. For more information on Si/SiO2

surfaces and preparation techniques, see [204, 205, 203, 206, 207].

6.2. THE ICE SAMPLES 75

I

hot wire

ice

a b

Figure 6.4: Cutting of ice crystals. (a) Schematic view: A resistance wire grid ispulled through the ice. (b) Photograph after cutting: A short disk has just beencut from the large cylindrical ice single crystal.

6.2 The ice samples

Ice single crystals were provided by J. Bilgram (ETH Zurich) [8]. The singlecrystals were grown with a zone-melting technique from high-purity water (con-ductance < 10−7 S/cm). They are of very high quality with a dislocation densityof about 103/cm2 (determined from Lang topographs) and a mosaicity of around0.3 mrad (determined from x-ray rocking scans). The crystals have a cylindricalshape with a diameter of about 50 mm and a length of about 300 mm. For thecrystals used in this work, the cylinder axis was oriented along the [0001] direc-tion (c-axis), but other orientations are possible. Smaller pieces with a lengthof about 15 mm were ‘thermally’ cut from the large crystal using hot wires thatwere pulled through the ice (see Fig. 6.4). In principle, mechanical sawing is alsopossible below ≈−15C, but leads to more crystal defects [208].

A walk-in cold room was installed and operated for handling the ice sam-ples (see Sec. 6.3). Due to the high vapor pressure of ice, the samples cannotbe kept in air for long periods of time. Instead, they were kept in heptane (n-heptane C7H16 ‘for analysis’, density 0.68 g/cm3, melting temperature −90.6,vapor pressure 48 hPa), which is practically insoluble in ice. It prevents sublima-tion/resublimation of ice that would otherwise lead to a visible deterioration ofthe ice single crystals [208, 14]. When taken from the heptane, the ice sampleswere kept in air for some time before further use. This allowed the remainingtraces of the highly volatile heptane to evaporate. For the transport of ice sam-ples, a mobile freezer was used, which can also be operated in a car using the


on-board electrical system.

6.3 The cold room

A walk-in cold room (≈20 m3)was installed for handling the ice samples (storage,characterization, preparation, and mounting). It consists of a main room (about−17C) and an entry room (about −14C), which is the only access to the mainroom. This reduces air exchange between the main room and the exterior andleads to low humidity levels in the main room despite of its small size.

For stability reasons, a rigid metal frame, which all the experimental tablesare bolted to, has been mounted in the cold room. Besides the ice samples,the cold room contains all the sample preparation tools (see Secs. 6.2 and 6.4)and a two-circle diffractometer for orienting ice samples prior to contacting. Thediffractometer also allows to check the crystallinity of the samples and to measurethe reflectivity of the substrates. The x-ray tube (Cu target, operational up to 60kV and 1.5 kW), high-voltage generator, motor controls, and control computersare placed outside the cold room. The x-ray beam is fed into the cold room viaa beam tube, and the cold room also serves as radiation shielding. The distancebetween source and sample is about 0.6 m. A focusing Ge(111) monochromatorand two slit systems are used, a pair (horizontal and vertical) of entry slits and apair of detector slits. A NaI scintillation counter is used as x-ray detector. Sev-eral feedthroughs, cameras, a temperature monitoring system, and an intercomsystem have also been installed.

6.4 Interface preparation

Samples which had been cut from the large crystals were then mounted on agoniometer head. Using the two-circle diffractometer, a specific crystal axis (herethe [0001]-direction) could be aligned parallel to the axis of the goniometer. Thegoniometer head was then mounted on an optical bench preserving the orientationof the ice sample. The substrate was mounted on the same optical bench withits surface normal aligned along the crystal axis of the ice sample (see Fig. 6.5).The subsequent contacting of the two surfaces (see below) lead to an orientedice–SiO2 interface. The interface normal deviated from the desired crystal axisby less than 0.1 (miscut). For the first experiment (rough sample), however,the two-circle diffractometer was not yet operational, therefore the miscut wasabout 1.

The temperature of the substrate during the preparation process was con-trolled via a Peltier element. The substrate was heated to about +4C and thenbrought into contact with the ice sample leading to the formation of a moltenzone between the ice and the substrate. The temperature was then slowly re-

6.4. INTERFACE PREPARATION 77

c

-5°C

+4°CSi

ice

goniometerhead

Peltier

optical bench

a b

Figure 6.5: Interface preparation. (a) Sketch of the setup for contacting ice withthe substrate (side view). The desired ice crystal axis (here the c-axis) is alignedwith a goniometer head. The Si substrate is heated with a Peltier element andmoved against the ice creating a thin molten layer. Subsequent cooling leads torecrystallization, if the right temperature gradient is maintained. (b) Photographof the sample after contacting (still mounted on the preparation stage). Theprotruding parts of the ice still need to be molten away.


a b c

Figure 6.6: Photographs of the sample. (a) Protruding parts of the ice have beenmolten away. (b) A temperature sensor has been molten into the ice. (c) Thewhole sample has been wrapped in aluminum foil.

duced while continuously moving the substrate against the ice (with a microme-ter screw) and thus keeping the molten layer very thin. This prevents inclusionof air bubbles in the molten layer and flushes out contaminants (which are muchmore soluble in water than in ice). The temperature at the substrate was alwayskept higher than the temperature at the ice, which leads to epitaxial recrystal-lization of the molten layer at the ice–water interface instead of the formation ofpolycristalline ice at the water–substrate interface.

The ice part of the sample was then brought into the right form by carefullymelting away protruding parts, so that the whole sample finally had the shapeof a parallelepiped (see Fig. 6.6a). In the next step, a calibrated Pt100 resis-tive temperature sensor was molten into the ice (see Fig. 6.6b). The sample wasthereafter sealed in high-purity aluminum foil to prevent contamination and sub-limation/resublimation of the ice (see Fig. 6.6c). The sample was then ready tobe mounted in the sample chamber used for the actual experiments (see Sec. 6.5).

6.5 The in situ chamber

A special sample chamber (see Fig. 6.7) has been designed for experiments withice samples and high-energy x-ray radiation. Only a small fraction of the high-energy x-ray radiation is absorbed by the aluminum windows of 0.1 mm thickness.Since the windows are also practically transparent for neutrons, the chamber issuited for neutron experiments as well. The inner part of the chamber can beexchanged. A different inner setup has been designed for Neutron ComptonScattering experiments on ice (see Chapt. 8).

For the interface experiments, the sample was mounted on a copper holder.The temperature was controlled via two Peltier elements (lapped and sealed)situated on two sides of the sample holder, thereby minimizing temperature gra-

6.5. THE IN SITU CHAMBER 79

1

2

3

4

5

X-ray beamba

dc

1

fe

6

100 mm

Figure 6.7: In situ chamber for x-ray and neutron scattering experiments withice providing stable control of the sample temperature (see the text). (a)+(b)Top view of open chamber. (c)+(d) Side view of open chamber, photograph).(e) Side view of closed chamber. (f) Look into open chamber (rendered image).(1) Sample (sealed with aluminum foil, calibrated sensor frozen into the ice).(2) Peltier element. (3) Heat exchanger. (4) Tube for the cooling liquid. (5)Electrical feedthrough for sensor and Peltier leads. (6) Aluminum window.


dients in the sample. A cooling liquid (about −15 to −5C) was used to removeheat from the backside of the Peltier elements via heat exchangers. A refrigeratedbath/circulator (Haake DC10-K15 [209], temperature accuracy ±0.02 K) servedfor keeping the temperature of the cooling liquid constant. For measurementsclose to the melting point, instead of raising the temperature of the cooling liq-uid, the polarity of the current for the Peltier elements was reversed. In thismode, they were heating ‘against’ the cooling liquid. This has two advantages:First, the cooling liquid prevents the sample from melting, even if the Peltierelements fail. Second, the temperature stability is better when the temperatureof the cooling liquid remains constant.

On each side of the sample, a Pt100 temperature sensor (Heraeus M-FK 222,DIN 1/3 B [210]) was placed on the sample holder. Together with the two Peltierelements, they formed two independent control loops. A factory calibrated Pt100sensor (Lakeshore Model Pt-111 [211]) was molten into the ice for measuring thesample temperature. Additional Pt100 sensors were installed to monitor thetemperature in other parts of the chamber (on the heat exchangers, e.g.). Thesesensors had been calibrated using the factory calibrated sensor which was latermolten into the ice. All sensor measurements were performed with a 4-leadtechnique to eliminate the effect of lead resistance. Thin manganin wires withlow thermal conductivity were used for the wiring in order to minimize heattransfer along the leads. A Lakeshore Model 340 temperature controller [211]was used for temperature measurement and control. The sensor excitation was1 mA. Two DC power supplies (HP/Agilent 6553A [212]) delivered the currentfor the Peltier elements. They were programmed by the Lakeshore controller andhad to be connected to the latter via a voltage converter specifically designed forthis purpose. A small programme was written for logging the temperatures on apersonal computer connected to the Lakeshore controller via a serial connection.A schematic view of the control setup is shown in Fig. 6.8.

6.6 Temperature stability and accuracy

The temperature stability was in the mK regime as can be seen from Fig. 6.9 show-ing part of a temperature log. Temperature stabilities of ±1 mK were reached,which is better than the specification (at 300 K) of the Lakeshore 340 temperaturecontroller (±5 mK).

One has to distinguish between temperature stability and accuracy though.The specified accuracy of the controller is ±30 mK. For the measurement of thesample temperature, a factory calibrated Lakeshore Model Pt-111 was used (seeabove). This sensor was also used for calibrating the remaining sensors. Theerror of this calibration was estimated by performing two calibrations, one beforethe experiment and one after the experiment. The differences were on the orderof ±10 mK.

6.6. TEMPERATURE STABILITY AND ACCURACY 81

loop 1

loop 2

Tsample

heatexchanger

Si

icetemperature

controller

signal converter

sensor

Peltier

sampleholder

cooling liquid

bathpower

supplies

\b\n\r\f\b

\n

\f

\

\f\

\f

\\b

\f\

\ \b\b

\

\n\b\n

\b\

!\b\b

\n\b

!\b\b

\n\b

\b!\b

%&\r\n

\f

\n

computer

Figure 6.8: Schematic view of the control setup. The temperature at two sidesof the sample is controlled via independent control loops. A temperature con-troller reads out the Pt100 sensors and programs the power supplies providing thecurrent for the Peltier elements, which can be either used as cooling or heatingdevices. The temperature on the backside of the Peltier elements is defined by acooling liquid circuit. The temperature measurements are logged on a computer.

00:00 02:00 04:00 06:00

-0.040

-0.038

-0.036

-0.034

-0.032

-0.030

Tem

pera

ture

(°C

)

Time (h)

Figure 6.9: Part of temperature log (from first experiment with the rough sub-strate). The sample temperature −0.036C varies by only ±1 mK over the du-ration of a measurement (≈ 6 hours).


Another source of errors are temperature gradients. The effect of these gra-dients is two-fold:

1. The temperature of the sensor is not exactly the temperature of the interfacearea probed by the x-ray beam.

2. Owing to the finite size of the x-ray beam the measured signal correspondsto an average over a certain temperature range.

In order to estimate effects of temperature gradients, measurements of thetemperature distribution have been performed. Therefore 8 sensors were used (2were used for the control loops and placed on the sample holder, 6 were molteninto the ice, see Fig. 6.10a). The variation ∆T of the temperature over the wholesample gives an upper limit of the error caused by temperature gradients. It isplotted in Fig. 6.10b as a function of temperature. It decreases on approach-ing 0C, as the temperature difference between the sample and the surroundinggets smaller. As this is exactly where the measurements are most temperaturesensitive (due to the divergence of the quasiliquid layer thickness), the followingconsiderations refer to ≈0C. At this temperature the ∆T is as small as ±15 mK.Temperature gradients along the different directions are plotted in Fig. 6.10c.

The gradient along the z-direction (perpendicular to the interface) is largest(1.9 mK/mm). As the sensor used for measuring the sample temperature duringthe experiments is placed approximately 7 mm above the interface, it may yield atemperature which is ≈13 mK higher than the actual interface temperature (fora discussion of the effect on the results, see Sec. 7.4).

The gradients in the x and y-direction (parallel to the interface) cause anaveraging over a certain temperature range. The gradient in the y-direction (per-pendicular to the copper holder) is 0.92 mK/mm. This direction is perpendicularto the beam and the temperature variation over the width (≈0.02 mm) of thebeam is negligible. However, the beam might not be exactly perpendicular tothe y-direction, and therefore cover a larger y-range. But even in the case of a 3

tilt, the temperature range covered by the beam is only ±0.5 mK at full illumi-nation of the sample length. The temperature distribution in the y-direction isnot symmetric which indicates that the thermal contact is not the same on bothsides of the sample.

The assessment of the effect of gradients in the x-direction (along the beam)is more complicated, as the beam footprint on the interface changes with the mo-mentum transfer qz (see Sec. 5.3.7). Fig. 6.10d shows the temperature range alongthe x-direction covered by the beam footprint as a function of the momentumtransfer qz. It starts with a maximum value of 15 mK and rapidly decreases withqz. At qz = 0.1 A−1, where the first features appear on the reflectivity curves,it is as small as 2.6 mK. The temperature distribution along the x-direction issymmetric as would be expected.

To summarize this subsection:

6.6. TEMPERATURE STABILITY AND ACCURACY 83

-20 -15 -10 -5 0

0

5

10

15

20

gra

d(T

)(m

k/m

m)

Temperature (°C)

dT/dxdT/dydT/dz

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

14

16

18

10-9

10-7

10-5

10-3

10-1

Tx

T

x(m

K)

qz(Å

-1)

Reflectivity

Inte

nsi

ty(a

rb.units

)

S2 S3

S1

S4

S5

S6 S7 S8

ice

Si

x

y

z

a

b

-20 -15 -10 -5 00

50

100

150

200

250

T

(mK

)

Temperature (°C)

c d

on sample holder

X-rays

Figure 6.10: Temperature distribution in the sample. (a) Setup for measuringthe temperature distribution. Sn: Pt100 temperature sensors. S1: factory cali-brated sensor. S2, S3: control loop sensors on sample holder, S4-S8: sensors formeasuring the temperature distribution (together with S1). The x-ray beam isroughly parallel to the x-direction. (b) Variation ∆T of the temperature overthe whole sample as a function of the nominal temperature. With increasingtemperature, the temperature difference between the sample and the surround-ing decreases. This results in a more uniform temperature distribution over thesample. (c) Temperature gradients along different directions. Gradients parallelto the interface (x and y) lead to an averaging of the temperature ‘seen’ by thex-ray beam (see d). The gradient perpendicular to the interface (z) may inducean offset between the measured temperature and the true interface temperature.(d) Temperature range ∆Tx covered by the footprint of the x-ray beam alongx (solid line, experimental parameters from the smooth substrate). A measuredreflectivity curve at T = −0.034C is shown in the same graph (dashed line). Atqz = 0.1 A−1, where the first features appear on the reflectivity curve, ∆Tx is assmall as 2.6 mK.


• The temperature stability is of the order of ±1 mK (see Fig. 6.9).

• The temperature error is between ≈ +40 and ≈ −20 mK (specified accuracyplus offset between sensor and interface).

• The temperature range, over which the beam averages, is <3 mK for therelevant part of the reflectivity curve and never larger than 15 mK.

Suggestions for improving the temperature accuracy are discussed in Sec. 8.5.

Chapter 7

Results and discussion

The results obtained in this work will be presented and discussed in this chapter.The main results were obtained from two high-energy x-ray reflectivity experi-ments. The theory (Chapt. 4) and experimental details (Chapt. 5) of high-energyx-ray reflectivity have been presented in previous chapters. In the two main ex-periments, well-defined ice–solid model interfaces (see Chapt. 6) have been stud-ied as a function of temperature, and interfacial melting has been observed. Twosubstrates with different morphology were used, which are referred to as the roughand the smooth substrate. The morphology of the substrates is characterized inSec. 7.3.

An overview and a qualitative interpretation of the reflectivity measurementsfrom the two main experiments is given in Sec. 7.1. Quantitative results can beobtained by reconstructing density profiles from the reflectivity measurementsusing the methods presented in Chapt. 4. This is done in Sec. 7.2 and thereliability of the results is discussed. The reconstruction of the density profilesyields the growth law of the quasiliquid layer, which is presented and discussedin Sec. 7.4. An intriguing aspect of this work is the extraordinarily high densityobserved in the quasiliquid. Sec. 7.5 highlights this point and offers a possibleexplanation. Sec. 7.8 deals with the effect of high-energy x-rays on the substratetermination and the consequences for interfacial melting. Sec. 7.9 finally discussesimplications of this work and its relation to other studies.

Further evidence for interfacial melting of ice has been obtained from two otherexperiments, one with high-energy x-ray reflectivity (using a slightly modifiedsetup) and a silicon wafer as the substrate (Sec. 7.6), and the other with neutronreflectivity (Sec. 7.7). The results from these two experiments are presentedin separate sections of this chapter together with short subsections about therespective samples and methods. This division has been chosen because thesetwo experiments represent only a minor contribution to the overall results ofthis work, but use a different experimental approach (at least in the case of theneutron experiment). Considering the significance of the results, a more detaileddiscussion of these experiments and the methods does not seem appropriate.

85

86 CHAPTER 7. RESULTS AND DISCUSSION

Table 7.1: Overview of experimental parameters.rough substrate smooth substrate

beamtime 08/2002 and 08/200312/2002

x-ray energy (keV) 71.26 71.44bandwidth (%) 0.23 0.23x-ray wavelength (A) 0.1740 0.1736critical angle αc (mrad) 0.3266 0.3257corresponding qc (A−1) 0.02359 0.02358no. of lenses in CRL 232 223focal distance of CRL (mm) 4380 4530beam size∗ vertical (µm) 5.6 7.1beam size∗ horizontal (µm) 19.2 ≈24detector slit vert. (mm) 0.06–0.6 0.25detector slit horiz. (mm) 2 0.3divergence vertical (µrad) 30 30divergence horizontal (µrad) 54 54resolution δqz (A−1) 4.3×10−3–2.1×10−2 9.6×10−3–1.2×10−2

temperature range (C) −30 to −0.036 −30 to −0.02max. momentum transfer (A−1) 0.57 1.0size of substrate (mm) 24 × 24 × 10 30 × 24 × 8orientation of substrate (111) (001)orientation of ice (0001) (0001)miscut of ice 1 0.1

rms roughness of substrate (A) -† (2.7± 0.4) A∗ FWHM† self-affine roughness, see Sec. 7.3

7.1 Overview of the main experiments

The main results of this work have been obtained from two experiments using twosubstrates of different morphology. An overview of the experimental parametersof these two experiments is given in Tab. 7.1.

On each substrate, reflectivity measurements have been performed at varioustemperatures, ranging from far below (−30C) up to very close (≈ −0.022C) tothe bulk melting temperature. In order to test the reversibility with temperature,measurements have been done during alternating cooling and heating cycles, andrepeated measurements have been performed for some temperatures. Figs. 7.1and 7.2 show ‘time lines’ for the experiments indicating the temperature andchronology of the individual reflectivity measurements. For the rough substrate,some additional measurements have been performed during another beamtime inorder to verify the observations. These measurements are also included in Fig. 7.1.

7.1. OVERVIEW OF THE MAIN EXPERIMENTS 87

1beamtime

st 2beamtime

nd

60

10

1

0.1

0.01

1E-3

1E-4

T01

-12.7

T02

-0.5

T03

-30.0

T04

-14.7

T05

-7.7

T06

-3.1

T07

-1.7

T08

-1.0

T09

-0.5

T10

-0.2

5

T11

-0.1

14

T12

-0.0

36

T13

-25.0

T14

-0.1

75

T15

subst

rate

Tm-T

(K)

number of measurement

y(m

m)

Figure 7.1: Time line of the experiments with the rough substrate. The tem-perature (left axis, bottom curve) and the position (right axis, top curve, onlyschematical) are plotted against the time (represented by the number of the mea-surement). Each measurement is consecutively numbered (Tnn, where nn denotesthe number) and labelled with the temperature in C. Measurements with therough substrate were performed during two different beamtimes. During the sec-ond beamtime, a measurement on the bare substrate (denoted T15 substrate) wasalso performed after removing the ice.

In the case of the smooth substrate, measurements have also been performed atdifferent positions on the interface. This is also indicated in Fig. 7.2.

The measured reflectivity curves together with a fit to the data (see Sec. 7.2.2)are shown in Figs. 7.3 and 7.4. With increasing temperature, an additionalsignal appears at high momentum transfers qz and gets gradually shifted towardssmaller qz. With a further increase of the temperature, this signal develops intointerference fringes, causing a modulation of the reflectivity curve. The distancebetween interference fringes decreases with temperature.

This observation unambiguously shows that a layer with different densityemerges (giving rise to interference fringes) and that the layer thickness (inverselyrelated to the distance between the interference fringes) grows with temperature.The qualitative result from this observation is: Interfacial melting indeed occursat the ice–solid interface investigated.


5 10 15 20 25 30

60

10

1

0.1

0.01

1E-3

1E-4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

T01

-24.7

T02

-1.0

0T03

-0.2

53

T04

-0.1

37

T05

-0.5

18

T06

-0.8

19

T07

-1.0

2T08

-1.7

2T09

-3.1

2T10

-14.7

T11

-25.0

T12

-2.6

2T13

-2.1

2T14

-1.3

7T15

-1.0

2 T16

-0.1

26

T17

-0.1

25

T18

-0.1

26

T19

-0.0

72

T20

-0.0

34

T21

-0.0

35

T22

-0.0

35

T23

-0.0

98

T24

-0.3

68

T25

-0.6

57

T26

-3.1

2T27

-6.2

3T28

-12.4

T29

-20.0

T30

-30.0

T31

-0.0

52

T32

-0.0

2

Tm-T

(K)

number of measurementy

(mm

)

Figure 7.2: Time line of the experiment with the smooth substrate. The tem-perature (left axis, bottom curve) and the position (right axis, top curve) areplotted against the time (represented by the number of the measurement). Eachmeasurement is consecutively numbered (Tnn, where nn denotes the number) andlabelled with the temperature in C. The measurements which are not performedon the initial position y = 0 are highlighted with a dashed pattern.


0.0 0.1 0.2 0.3 0.4 0.5 0.610

-32

10-29

10-26

10-23

10-20

10-17

10-14

10-11

10-8

10-5

10-2

101

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

T12 -0.036°C

T11 -0.114°C

T14 -0.175°C

T10 -0.25°C

T09 -0.5°C

T02 -0.5°C

T08 -1.0°C

T07 -1.7°C

T06 -3.1°C

T05 -7.7°C

T01 -12.7°C

T04 -14.7°C

T13 -25°C

T03 -30°C

T15 substrate

Figure 7.3: Reflectivity measurements for ice in contact with the rough substrate.The measurements show that an additional layer appears and grows in thicknesswith temperature. The solid lines are fits to the data. The curves are displacedby factors of 30.


0.0 0.2 0.4 0.6 0.8 1.0 1.210

-43

10-40

10-37

10-34

10-31

10-28

10-25

10-22

10-19

10-16

10-13

10-10

10-7

10-4

10-1

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

T32 -0.022°CT20 -0.034°CT31 -0.052°CT19 -0.072°CT23 -0.098°CT16 -0.126°CT04 -0.137°CT03 -0.25°CT24 -0.37°CT05 -0.52°CT25 -0.66°CT06 -0.82°CT02 -1.00°CT07 -1.02°CT08 -1.72°CT26 -3.12°CT27 -6.23°CT28 -12.4°CT29 -20.0°CT01 -24.7°CT30 -30.0°C

Figure 7.4: Reflectivity measurements for ice in contact with the smooth sub-strate. The measurements show that an additional layer appears and grows inthickness with temperature. The solid lines are fits to the data. The curves aredisplaced by factors of 45.


0.0 0.2 0.4 0.6 0.8 1.010

-27

10-24

10-21

10-18

10-15

10-12

10-9

10-6

10-3

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

T09 -3.12°C

T10 -14.7°C

T11 -25.0°C

T12 -2.62°C

T13 -2.12°C

T14 -1.37°C

T15 -1.02°C

T17 -0.125°C

T18 -0.126°C

T21 -0.035°C

T22 -0.035°C

Figure 7.5: Reflectivity measurements on different positions of the sample(smooth substrate, compare Fig. 7.2) that were not irradiated long enough tochange the termination from hydrophobic to hydrophilic. This is thought to bethe reason why the interfacial melting is not pronounced in this case. The curvesare displaced by factors of 45.


The data in Figs. 7.3 and 7.4 also demonstrate that this behavior is reversible(compare the timeline in Figs. 7.1 and 7.2), with the exception of the lowesttemperatures.

The reflectivity measurements shown in Fig. 7.5, however, do not display thisevolution with temperature. They have been measured during the experimentwith the smooth substrate like those from Fig. 7.4, but at another position on thesample. Only later did the reason for this observation become clear: Under theinfluence of the high-energy x-ray microbeam and in the presence of water, thesubstrate termination changes from hydrophobic to hydrophilic, which stronglyinfluences the interface melting behavior. This observation is discussed in Sec. 7.8.This radiation-induced change of the substrate termination requires some time.When the measurements shown in Fig. 7.5 where performed, the substrate wasstill hydrophobic at this position, since it had not been illuminated long enough.The measurements shown in Figs. 7.3 and 7.4, in contrast, have been performedafter significant irradiation (setup and alignment time), rendering that positionhydrophilic.

The effect of the radiation might also explain why the low-temperature mea-surements are not fully reversible. For both experiments, the low-temperaturecurves measured at the beginning of a beamtime do not match those measuredlater.

The reflectivity of the bare substrates (without ice) has also been measured inair. In the case of the rough substrate, it has been measured at the end of a beam-time after removing the ice (lowest curve in Fig. 7.3). For the smooth substrate,only measurements with a Cu-Kα sealed tube are available (see Fig. 6.3).

7.2 Density profiles

7.2.1 Raw data analysis

As described in Sec. 5.3.9, all measured intensities have been normalized to themonitored primary intensity. Parts of reflectivity curves measured with differentabsorbers have been adjusted. Rocking scans (transverse momentum scans, seeSec. 5.3.4) have been measured in order to determine the background and evaluatethe morphology of the interface (see Sec. 7.3). In the case of the rough substrate,integrated intensities have been obtained from rocking scans (Secs. 7.3 and 4.8).The illumination correction has been performed on the data from the smoothsubstrate. For the data from the rough substrate, the illumination correction hasbeen included in the fitting routine.

7.2. DENSITY PROFILES 93

a

-40 -20 0 20 40

8.0x10-6

1.2x10-5

1.6x10-5

2.0x10-5

0.3

0.4

0.5

0.6

0.7

dis

pers

ion

z (Å)

ele

ctro

ndensi

ty

e(e

/Å3)

-40 -20 0 20 40

8.0x10-6

1.2x10-5

1.6x10-5

2.0x10-5

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

dis

pers

ion

z (Å)

mass

densi

ty

(g/c

m3)b

H O

(ice)2 H O

(qll)2 SiO2

Si

Figure 7.6: Dispersion and density profiles. (a) Dispersion/electron density pro-file. (b) Dispersion profile (solid line) and corresponding mass density profile(dashed line) assuming the composition indicated on the graph.

7.2.2 Reconstruction of density profiles

The (specular) x-ray reflectivity is determined by the laterally averaged dispersionprofile (see Sec. 4). For a large x-ray energy range, the dispersion is proportionalto the electron density (see Eq. 4.5). The mass density profile can be calculatedfrom the dispersion or electron density profile if the the chemical composition isknown (see Fig. 7.6).

The dispersion profiles perpendicular to the interface have been reconstructedby fitting a model to the measured data. For this purpose, the program ‘Winfit’written by A. Ruhm has been used [213]. It is based on the Parratt formalismdescribed in Sec. 4.3.

The density/dispersion profile is described with a model consisting of 4 lay-ers (see Fig. 7.7). The first layer represents the ice, it is semi-infinite1 and hasthe dispersion of ice. The second layer represents the premelting quasiliquid.Its thickness and density are the parameters which should be determined as afunction of temperature. The third and fourth layer represent a thin SiO2 layercovering a semi-infinite2 Si block (taking literature values for the dispersion).Each interface between two layers is characterized by a parameter for the rmsroughness. The thickness of the SiO2 layer and the associated roughness param-eters are in principle free parameters. They should, however, be consistent forall measurements at different temperatures, since the SiO2 layer is not supposedto change with temperature. In the case of the lowest temperatures, where theinterfacial melting has not yet set in, the second layer describing the quasiliquidcan be omitted.

This is the simplest model which can be used to adequately describe the

1The incoming wave enters the ice through its side and no reflection occurs at the top ofthe ice. It can therefore be treated as semi-infinite.

2Due to the small incident angles and the large thickness of the Si–SiO2 substrate, thetransmitted wave leaves the substrate through its side and does not get reflected from thebottom. The substrate therefore appears to be semi-infinite.


SiO

ice

quasiliquidL

SiO2

Si

LSiO2

ice

qll

SiO2

Si

ice

Si

2

Figure 7.7: Model for fitting the reflectivity data. The densities are denotedwith ρ, the rms roughness parameters with σ. The densities ρice, ρSiO2 and ρSi

are fixed to the nominal values. The thickness LSiO2 of the SiO2 layer and theroughness parameters σSiO2 , σSi of the substrate have to be consistent over alltemperatures. This leaves the thickness L of the quasiliquid layer, its density ρqll,and the roughness σice of the ice–quasiliquid interface as temperature dependentparameters.

interface. It contains the lowest possible number of layers and free parameters.It does not account for a variation of the density within the quasiliquid layer.

Reflectivity curves calculated from the fitted model are shown in Figs. 7.3 and7.4 (solid lines). The simple model allows to reproduce quite well the measureddata. The remaining differences might indeed be due to an internal structure ofthe quasiliquid layer. But since the differences are small, they cannot be used tounambiguously deduce the internal profile of the quasiliquid layer.

In the case of the rough substrate, the reflectivity has been obtained fromintegrated intensities of rocking scans (see Sec. 4.8.2). By doing so, the informa-tion about the roughness gets lost. The roughness parameters obtained from thefitting only describe a ‘local’ roughness on small length scales (see Sec. 7.3, wherethe morphology of the rough substrate is discussed).

The reconstructed density profiles are shown in Figs. 7.8 and 7.9, the corre-sponding fit parameters are listed in Tabs. 7.2 and 7.3. Not listed in the table area scaling factor, a constant background, and the angle of complete illumination inthe case of the rough substrate (for the smooth substrate, the correction for theilluminated area has been applied to the data, not to the calculated reflectivity).

For the lowest temperatures the density profile simply shows ice in contactwith the substrate consisting of a thin SiO2 layer and the underlying siliconblock. At higher temperatures, a layer with a different density intervenes andincreases in thickness with temperature. An atomic scale illustration representingthe interface model is shown in Fig. 7.10.

The data has been analyzed with the dynamical Parratt formalism. However,the kinematical approximation shows practically no differences for typical reflec-tivity curves from this work, as can be seen from Fig. 7.11. Differences would


-50 0 50

1

2

3

4

5

6

7

(g

/cm

3 )

z (Å)

T12 -0.036°C

T11 -0.114°C

T14 -0.175°C

T10 -0.25°C

T09 -0.5°C

T02 -0.5°C

T08 -1.0°C

T07 -1.7°C

T06 -3.1°C

T05 -7.7°C

T04 -14.7°C

T13 -25°C

T03 -30°C

Figure 7.8: Reconstructed density profiles for the rough substrate. The dashedlines indicate the positions of the layers. The curves are displaced by verticaloffsets of 0.4 (g/cm3).


-50 0 50

1

2

3

4

5

6

7

8

9

10

(g

/cm

3 )

z (Å)

T32 -0.022°CT20 -0.034°CT31 -0.052°CT19 -0.072°CT23 -0.098°CT16 -0.126°CT04 -0.137°CT03 -0.25°CT24 -0.37°CT05 -0.52°CT25 -0.66°CT06 -0.82°CT02 -1.00°CT07 -1.02°CT08 -1.72°CT26 -3.12°CT27 -6.23°CT28 -12.4°CT29 -20.0°CT01 -24.7°CT30 -30.0°C

Figure 7.9: Reconstructed density profiles for the smooth substrate. The dashedlines indicate the positions of the layers. The curves are displaced by verticaloffsets of 0.4 (g/cm3).


Table 7.2: Fit parameters for the rough substrate.label Tm − T L ρqll LSiO2 σice

∗ σSiO2∗ σSi

∗

(K) (A) (g/cm3) (A) (A) (A) (A)T12 0.036 55.0 1.12 18.5 0 3.46 0T11 0.114 45.5 1.13 20.1 0 3.89 0T14† 0.175 35.9 1.13 19.6 0 5.20 0T10 0.25 36.3 1.13 21.5 0 4.43 0T09 0.5 26.7 1.25 18.9 0 6.45 0T02 0.5 23.7 1.22 19.7 0 4.80 3.73T08 1.0 17.2 1.18 22.1 0 4.25 0T07 1.7 14.1 1.17 22.0 0 4.51 0T06 3.1 11.6 1.26 23.3 0 3.73 0T05 7.7 9.93 1.26 21.5 0 3.68 0T01 12.7 −‡ −‡ −‡ −‡ −‡ −‡

T04 14.7 7.46 1.21 22.9 0 4.53 5.33T13† 25.0 0 − 21.4 0 5.93 0T03 30.0 7.96 1.25 19.1 0 4.23 3.15T15 -§ (10.6)‖ (1.00)‖ 15.2 0 6.92 0average 1.20 20.4 0 4.7 0.9sd¶ 0.05 2.1 0 1.1 1.8∗ fit parameters do not reflect the real roughness (see the text)† second beamtime‡ no significant fit possible due to small q-range of measurements§ bare substrate‖ the bare substrate (measured in air) is covered with an adsorbed water layer¶ standard deviation


Table 7.3: Fit parameters for the smooth substrate.label Tm − T L ρqll LSiO2 σice σSiO2 σSi

(K) (A) (g/cm3) (A) (A) (A) (A)T32 0.022 27.5 1.16 19.1 1.75 3.00 0T20 0.034 26.5 1.20 17.0 0.85 3.02 0T31 0.052 26.3 1.13 17.9 2.01 2.62 0T19 0.072 23.8 1.18 16.6 2.18 3.00 0T23 0.098 23.9 1.23 16.4 1.56 2.99 0T16 0.126 23.1 1.18 15.7 3.42 2.43 0T04 0.137 20.8 1.31 14.5 2.79 2.31 0T03 0.25 18.5 1.24 14.4 2.11 1.86 0T24 0.37 19.3 1.24 15.2 1.94 2.77 0T05 0.52 14.6 1.19 13.5 0 3.11 2.12T25 0.66 17.3 1.18 13.8 1.63 2.51 0T06 0.82 13.2 1.16 13.5 0 3.01 0.44T02 1.00 13.2 1.08 12.3 0 2.32 0T07 1.02 12.6 1.12 13.0 0 3.13 0.90T08 1.72 10.9 1.10 13.7 0 3.17 0T26 3.12 8.84 1.24 15.7 2.88 2.53 0T27 6.23 7.66 1.27 15.4 2.27 2.50 0T28 12.4 6.97 1.27 15.1 1.82 2.79 0T29 20.0 4.19 1.12 20.2 3.65 2.96 0T01 24.6 0 − 13.5 0 2.35 0T30 30.0 4.51 1.27 19.0 2.23 2.23 0average 1.19 15.5 1.58 2.70 0.22sd∗ 0.06 2.2 1.19 0.4 0.43∗ standard deviation


ice hI

quasiliquidlayer

amorphousSiO2

Si

T = -1°C

2.1 nm

1.7 nm

Figure 7.10: Illustration of the model for the ice–SiO2–Si interface. The lengthscales and the densities correspond to the experimentally determined values forthe rough substrate at −1C. For clarity, the atomic radii are not on scale.

only become apparent for large layer thicknesses, when features on the reflectivitycurve appear in the vicinity of the critical angle.

7.2.3 Reliability of the fits

The density profiles are reconstructed by fitting a model to the measured data.This raises two question:

• Are other density profiles capable of explaining the measurements?

• How accurate can the model parameters be determined by fitting?

In order to address these issues, several tests have been performed.Fig. 7.12 shows calculated curves for ice–SiO2–Si interfaces with intervening

layers of different thickness and density. It can be seen that the layer thicknessdetermines the distance between interference fringes (compare Eq. 4.16) whereasthe density is linked to the amplitude of the modulation. The layer thickness is,therefore, a very robust parameter, whereas the density cannot be determinedwith the same precision, which is why the fitted density values scatter morestrongly (compare Figs. 7.33 and 7.29). Nevertheless, the measured reflectivityis sensitive to both the thickness and the density (as well as the roughness).

In order to verify the reliability of the deduced parameter values, fitting runshave been performed, where one of the parameters (the density of the quasiliquid,


0.0 0.2 0.4 0.6 0.8 1.0

10-9

10-7

10-5

10-3

10-1

Inte

nsi

ty(a

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)

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-1)

dynamicalkinematical

Figure 7.11: Comparison between kinematical and dynamical calculation of re-flectivity. Practically no differences can be seen for typical reflectivity curvesfrom this work.

for example) was fixed and the other parameters were free. These series of fittingruns show that the quality of the fit effectively depends on the value of the fixedparameter (see Fig. 7.13).

Such tests show that the fits are not very sensitive to the roughness at theSiO2–Si interface (see Fig. 7.14). In most cases, the best fit is obtained withσSi = 0 A, but a finite value on the order of σSiO2 is much more realistic andmatches the measured data almost as well (see Fig. 7.14). Due to the special wayof measuring the reflectivity in the case of the rough substrate, the roughnessparameters in this case are not very meaningful anyhow (see Sec. 7.3). Thisexplains why the roughness obtained at the ice–quasiliquid interface is zero inthe case of the rough substrate.

The sensitivity of the reflectivity on the model parameters and the degree towhich different fits of the same reflectivity curve scatter are used to estimate theerror bars of the fit parameters as plotted in Figs. 7.29 and 7.33. The errorsare not necessarily the same for all measurements. Large layer thicknesses canbe determined with better accuracy than small ones because they are fixed by ahigher number of interference fringes.

For individual reflectivity curves, several simple density profiles may existwhich fit well the measured data. But the model should allow to explain all reflec-tivity curves measured at different temperatures simultaneously. This eliminatesmost ambiguities for a single reflectivity measurement, since many parameters arenot completely free, but need to be consistent when comparing measurements atdifferent temperatures.

The results have also been verified with phase inversion (see Sec. 4.6).


0.0 0.2 0.4 0.6 0.8 1.010

-11

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10-1

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= 1.5 Å = 2.0 Å = 2.5 Å = 3.0 Å = 3.5 Å

0.0 0.2 0.4 0.6 0.8 1.0

10-10

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100

= 0.98 g/cm³ = 1.08 g/cm³ = 1.18 g/cm³ = 1.28 g/cm³ = 1.38 g/cm³

Inte

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a b

0.0 0.2 0.4 0.6 0.8 1.0

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Inte

nsi

ty(a

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-1)

L = 20.3 ÅL = 18.8 ÅL = 17.3 ÅL = 15.8 ÅL = 14.3 Å

c

Figure 7.12: Calculated reflectivity curves showing the influence of model pa-rameters. The black line corresponds to the fit results for the smooth substrateat −0.657C. The other lines are calculated with different values for (a) thequasiliquid layer thickness L, (b) the density ρ of the quasiliquid, and (c) the rmsroughness σ of the substrate.


0.8 1.0 1.2 1.4 1.60

10

20

30

40

50

60

70

T = -0.036°CT = -1°C

L(Å

)

qll

(g/cm³)0.8 1.0 1.2 1.4 1.6

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2

qll

(g/cm³)

T = -1°CT = -0.036°C

a b

Figure 7.13: Reliability of the fits. (a) Quality of the fit (measured by the meansquare deviation χ2) as a function of the model parameter ρ denoting the densityof the quasiliquid (rough substrate). The other parameters were free except forthe dispersion of ice and silicon, which were fixed to the nominal values. It can beseen that χ2 strongly depends on the density. In particular, good fits cannot beachieved with the density of water (1 g/cm3). The open boxes mark the resultsthat were finally obtained from the fitting. These are not necessarily the fits withthe smallest χ2, as the retained fits have to satisfy certain constraints: The resultsneed to be physical and consistent over all temperatures. (b) Fit parameter Ldenoting the thickness of the quasiliquid layer as a function of the density (samefitting runs as in (b)). This plot shows that the thickness and the density whichresult from the fits are practically uncorrelated. Only a strongly different densityleads to a significant change of the thickness.

0.0 0.2 0.4 0.6 0.8 1.01E-15

1E-13

1E-11

1E-9

1E-7

1E-5

1E-3

0.1

Si

(Å)

00.51.01.52.0

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

Figure 7.14: Reliability of the roughness parameter σSi. Reflectivity curves cal-culated with different values for this parameter show that the measurements arenot very sensitive to the roughness at the SiO2–Si interface.

7.3. SUBSTRATE MORPHOLOGY 103

-1.0x10-5 0.0 1.0x10

-5

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15

20

Inte

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FWHM = 87 µrad= 0.25 mm v = 0.1 mm FWHM = 60 µrada bv

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-5

0

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6

8

10

Inte

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-1)

Figure 7.15: Rocking scans at qz = 0.063 A−1 for the smooth substrate withdifferent resolutions defined by the vertical detector slit gap. The width of thespecular peak is limited by the resolution.

7.3 Substrate morphology

Two substrates with different morphology have been mainly used in this work.They are referred to as the smooth substrate and the rough substrate. Themorphology does not only have an influence on the feasibility of reflectivity mea-surements and their interpretation (see Chapts. 4 and 5), but is also one of theparameters determining the interface melting behavior (see Chapt. 3). Therefore,a characterization of the morphology is needed. This has been done by analyzingthe reflectivity data itself, in particular the off-specular scattering. As a com-plementary approach, AFM images of the bare substrate have been taken3 andanalyzed.

7.3.1 Smooth substrate

Rocking scans (see Sec. 5.3.4) on the smooth substrate show a narrow resolution-limited peak. It is the convolution of the specular reflectivity, characterized bya δ function (see Sec. 4.8), with the instrumental resolution and the curvatureof the sample. Consequently, the peak width decreases when the resolution isimproved by choosing a smaller vertical detector slit gap (see Fig. 7.15).

The peak width changes with qz like the resolution function and hence in-creases linearly with qz in momentum space (see Fig. 7.16). In angular space, itremains constant. The width is slightly smaller than the calculated resolution,which indicates that the effective detector slit gap and/or beam divergence aresomewhat smaller than the nominal values used for calculating the resolution. Infact, the peak width of the specular reflectivity can be regarded as a measurement

3in collaboration with U. Taffner (MPI fur Metallforschung)


0.0 0.1 0.2 0.3 0.4 0.50.0

2.0x10-5

4.0x10-5

6.0x10-5 measurement

linear fitresolution(nominal)

w(Å

-1)

qz(Å

-1)

Figure 7.16: Peak width in a rocking scan as a function of qz (smooth substrate).The resolution is shown as a dashed line.

qz= Å

-10.19 q

z= Å

-10.32 q

z= 0.50 Å

-1

back-ground

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1.6

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0.0

0.2

0.4

0.6

0.8In

tensi

ty(a

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)

qx(Å

-1)

Figure 7.17: Rocking scans with the smooth substrate for different values of qz.The peak always has a Gaussian line shape.

of the resolution.

The line shape of the rocking scans does not change with qz (see Fig. 7.17)and can be well described by a Gaussian. The background contains the diffusenon-specular reflectivity (and some other contributions like air scattering), andis orders of magnitude smaller. This shows that the roughness must be ratherweakly correlated.

The background has been subtracted from the peak intensity measured atqx = 0 A in order to obtain the true specular reflectivity. The specular reflectivityallows to reconstruct the laterally averaged dispersion profile (see Sec. 7.2.2).These profiles contain the rms roughness of the substrate. Its value is (2.7±0.4) A(see Tab. 7.3).


7.3.2 Rough substrate

In the case of the rough substrate, a specular component cannot be recognizedin the rocking scans (see Figs. 7.18 and 7.23). In Sec. 4.8, it has been shown,that the density profile can nevertheless be reconstructed from the integrated in-tensity of the rocking scans. Such measurements are very time-consuming, sincea complete rocking scan profile has to be measured for each value of qz (eachpoint on the reflectivity curve). If the line shape of the rocking scans does notchange with qz, however, the peak intensity at qx = 0 A is a measure of theintegrated intensity. It only needs to be multiplied by a qz-dependent correctionfactor accounting for the change of the peak width (see Fig. 7.18). However, theintegrated intensity does not contain any information about the roughness any-more. The reflectivity corresponds to that of a perfectly smooth interface. Evenso, fitting the reflectivity curves obtained in this way does not always yield zeroroughness (see Tab. 7.3). The reason is the finite range of integration in momen-tum space, which allows only partial integration of the diffuse reflectivity. Theroughness on small lateral length scales, therefore, still shows up in the reflectiv-ity profiles. This ‘local roughness’ also cannot be expected to be fully conformal.The rms value of this local roughness obtained from fitting the reflectivity curves(integrated intensities) is (4.7± 1.1) A.

The fact that even at small qz no specular component can be seen in therocking scans shows that the correlation length of the roughness is very large. Inthe case of self-affine roughness, there is no cutoff of the self-affine behavior up tothe length scales probed by the x-rays. The height-difference correlation functiong (R) is then given by Eq. 4.25:

g (R) = BR2h.

The information about correlations of the roughness is contained in the dif-fuse reflectivity, this means the transverse momentum scans (rocking scans). Thetheory for this diffuse reflectivity is discussed in Sec. 4.8. For self-affine roughnesswithout cutoff, it is described by Eq. 4.44. In general, this cannot be calculatedanalytically, except for the special cases h = 0.5 and h = 1. Since the resolu-tion in qy is usually relaxed (see Sec. 5.3.5), the measurement corresponds to anintegration of Eq. 4.44 over qy.

The line shape of the rocking scans depends on the Hurst parameter h, whilethe amplitude B only influences the peak width. Fig. 7.19 shows calculatedrocking scans for different h. For the special cases h = 0.5 and h = 1 the lineshape is Lorentzian and Gaussian, respectively. For values of h close to 0.5, theline shape is not very sensitive to h. The qz-dependence of its FWHM, however,strongly depends on h and B. A fit of the measured width w as a function of qzyields h =0.34± 0.01 and B =0.107± 0.006 (see Fig. 7.20).

Fig. 7.22 compares calculated rocking scan profiles with the measurement.The measured data has been corrected for the variation of the illuminated area


0.0 0.1 0.2 0.3 0.4 0.510

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integratedcorrecteduncorrected

0.000 0.002 0.0040

1

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Inte

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ty(a

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i(rad)

qz = 0.16 Å-1

a rocking scan b reflectivity

Figure 7.18: Measurement of integrated intensity (rough substrate). (a) Rockingscan. The central peak is diffuse reflectivity, the other two are the so-calledYoneda wings (compare Fig. 4.5). The asymmetry is explained in the text. (b)Reflectivity. The solid circles are the integrated intensity obtained from rockingscans (corresponding to the shaded area in a). The peak intensity has beenmeasured at qx = 0 A−1 (solid square in a). The background has been obtainedfrom two points (open squares in a) between the diffuse reflectivity peak andthe Yoneda wings. Substraction of the background yields the lower curve in b(triangles). This curve represents the peak height, not the integrated intensity.Therefore, it has to be corrected using the intensities obtained from integratingrocking scans, yielding the upper curve in b (open circles).


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01234567

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a = 0.1h b = 0.25h c h = 0.34

d = 0.5 (Lorentzian)h e = 1 (Gaussian)h

Figure 7.19: Calculated rocking scans at qz = 0.16 A−1 for different values ofthe Hurst parameter h. The amplitude B has been obtained from a fit of theqz-dependence of the peak width (see Fig. 7.20). This fit yields h = 0.34, thecorresponding profile is shown in c. For intermediate values of h (b-d) the rockingscans are not very sensitive to h.

(see Sec. 5.3.7). This removes most of the asymmetry present in the raw data(compare to Fig. 7.18a). Since the calculation is based on a kinematical theory,the Yoneda wings cannot be reproduced. In Fig. 7.22a, where h and B have beenfixed to the values determined from fitting w(qz), only a constant backgroundand a scaling factor have been used to adjust the calculation to the measurement.Fig. 7.22b+c show fits of the measurement with a Lorentzian and a Gaussian lineshape, corresponding to h = 0.5 and h = 1, respectively. For these two cases,the width of the peak has been fitted to the data. This corresponds to a fit ofB. While a Gaussian line shape (h = 1) can clearly be excluded, the distinctionbetween h = 0.34 and h = 0.5 cannot be made from a single rocking scan (hencethe fit of w(qz)).

With the roughness parameters h and B obtained from fitting w(qz), all rock-ing scan profiles (at different qz) can be calculated. They fit very well to the mea-surements, illustrated by Fig. 7.23. Again, only a scaling factor and a constantbackground have been used to adjust the calculated profiles to the measurements.The remaining asymmetry in the profiles measured at large qz (Fig. 7.23e) is dueto the variation of the resolution with qx (compare Fig. 5.11). The effect is morepronounced at large qz, since here the peak width, and hence the qx-range ofthe profile, is larger. The variation of the resolution is also demonstrated by thefact that the width of the two Yoneda wings is different (Fig. 7.23e). At very


0.0 0.1 0.2 0.3 0.4 0.5 0.6

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a different fit of (linear plot)h, B b log-log plot)different fit of (h, B

h=0.1

h=1

h=0.5

h=0.25

h=0.34

h=0.1h=0.25h=0.34

h=1

h=0.5

d comparision with resolution

resolution

c different ,B h=0.34

B=0.11

B=0.05

B=0.2

0.05 0.1 0.510

10

10

10

10

-6

-5

-4

-3

-2

w(Å

-1)

qz(Å

-1)

Figure 7.20: Width w of the diffuse reflectivity peak (FWHM) in a rocking scanas a function of qz. (a) Measurement (solid squares) and fits (linear plot). Thebest fit (bold line) is obtained with h =0.34 ± 0.01 and B =0.107 ± 0.006. Fitswith fixed h are shown for comparison (thin lines). (b) Same as a (log-log plot).(c) Best fit (solid line) and calculations for different values of B (fixed h = 0.34).(d) Comparison with the resolution function. The measured peak width is alwayslarger than the resolution function.


substrate

quasiliquid

ice

Figure 7.21: Roughness replication at substrate–quasiliquid–ice interface. Theroughness at the substrate–quasiliquid interface is replicated at the quasiliquid–ice interface (conformal roughness), except for small length scales.

small qz (Fig. 7.23a), the calculated peak profile has a smaller width than themeasured one, even if convoluted with the resolution function. The discrepancycan be explained with broadening due to sample curvature. At small qz, thebeam footprint is approximately as long as the interface (24 mm), therefore, themeasurement is very sensitive to the curvature. With a figure error of around1 µm (see Sec. 6.1) a broadening of around 40–80 µrad can be expected. Theangular width of the measured peak shown in Fig. 7.23a is 65 µrad. At larger qz,the curvature has no visible effect, since the intrinsic width of the peak is largerand at the same time the beam footprint is smaller.

The rocking scan profiles are virtually identical (apart from the height, ofcourse) for all temperatures, and for measurements with the bare substrate.This indicates that the roughness is basically conformal. The morphology atthe substrate–quasiliquid interface must be replicated on the quasiliquid–ice in-terface, except for roughness on very small length scales (see Fig. 7.21). Thisseems quite plausible, as long as the interface is not too jagged. It justifies theanalysis of the integrated diffuse reflectivity as presented in Sec. 4.8.2.


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b = 0.5(Lorentzian)

h c = 1(Gaussian)

h

a = 0.34h

backgroundYoneda

wing

Figure 7.22: Comparison of calculated (open circles) and measured (solid lines)rocking scans for different h. The measured data has been corrected for theilluminated area. In (a), h and B from the fit of w(qz) (see Fig. 7.20) havebeen used, only a scaling factor and a constant background have been fitted tothe rocking scan. In (b) and (c) h has been fixed, but B fitted to the rockingscans. While h = 1 can clearly be excluded, the distinction between h = 0.34and h = 0.5 cannot be made from a single rocking scan.


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a qz= 0.03 Å

-1

b qz= 0.07 Å

-1

qz= 0.25 Å

-1q

z= 0.16 Å

-1

dc

qz= 0.35 Å

-1

e

Figure 7.23: Comparison of calculated (solid lines) and measured (open circles)rocking scans for different qz. The values for h and B have been obtained fromthe fit of w(qz). Only a scaling factor and constant offset (background) have beenfitted to the individual rocking scans. The calculated curves fit very well to themeasurement, except for very small qz (see a). In this case, the calculation yieldsa much smaller peak width (dashed line). A better match is found when theresolution is included (solid line). For a more detailed discussion, see the text.


-32 Å

+32 Å

1 m 1 m

ba

Figure 7.24: AFM images of (a) the smooth substrate and (b) the rough substrate.The same scale and color coding have been used. The AFM picture of the roughsubstrate shows larger height variations and a stronger ‘texture’. More detailsare revealed by a statistical analysis (see the text and Fig. 7.25).

7.3.3 AFM measurements

AFM measurements of the substrates provide real-space images of the topogra-phy and thereby allow a ‘visual’ comparison of the substrates (see Fig. 7.24).The AFM images directly show that the rough substrate has larger and morestructured height variations.

Unlike x-ray diffraction, AFM does not directly yield statistical informationabout the substrate. Such information has to be calculated from the measuredheight values. This has been done for the height-difference correlation functiong(R) (see Fig. 7.25). This information is still local in the sense that it is calcu-lated from a single AFM image taken from one spot on the substrate. A morerepresentative result could be obtained by averaging images from different spotson the substrate.

For intermediate length scales, the substrate morphology shows the charac-teristics of self-affine roughness. In this regime, g(R) appears as a straight line ina log-log plot (see Fig. 7.25). A fit yields h = 0.38 and B = 0.098 for the roughsubstrate, which agrees well with the x-ray data. For the smooth substrate4,h = 0.56 and B = 6.5× 10−4 is obtained. Compared to the rough substrate, h islarger, which corresponds to a less jagged surface. The parameter B setting the

4In order to preserve the ice–substrate interface, the AFM measurements were performed onanother identical substrate. The substrate belongs to the same batch as the smooth substrateused for the x-ray measurements. Both substrates have been polished and cleaned together.


103

104

105

106

100

1000

smooth substraterough substratefits

g(R

)(Å

2)

R (Å)

Figure 7.25: Height difference correlation functions g(R) determined from AFMmeasurements. The solid lines are fits assuming self-affine roughness (see thetext).

scale of the self-affine roughness is about two orders of magnitude smaller.Owing to the finite size of the image, the number of points entering the

calculation of g(R) decreases with R. Therefore, the calculated g(R) is not validfor large R. Similarly, the calculation is affected by the size of the AFM tip(here around 100–500 A) at small length scales [214]. A number of other factorscan possibly lead to errors in the interpretation of the AFM images. Since themeasurements were performed in air, small dust particles are found on some partsof the images. Sometimes, these particles are dragged by the AFM tip. Areasaffected by dust particles were excluded from the analysis. As the sample surfacedoes not coincide with the z = 0 plane of the instrument, the measured heightshave to be corrected for the sample tilt. Otherwise, the tilt of the sample leads toan artificial increase of g(R). Furthermore, consecutive lines of the AFM imagecan have a large offset in the height. Therefore, the images shown in Fig. 7.24 havebeen ‘flattened’ to remove this offset. In order to avoid any artifacts caused bythe offset or the flattening, only points within the same line have been consideredfor the calculation of g(R).

7.3.4 Conclusion

The morphology of the rough and the smooth substrate differs strongly. Thisapplies to the amplitude of the roughness as well as its lateral correlations. Theroughness has been characterized using the x-ray data and AFM measurements,which are in good agreement. The characteristics of the two substrates are sum-marized in Tab. 7.4.


Table 7.4: Morphology of the substrates.‘rough’ substrate ‘smooth’ substrate

Si orientation (111) (001)SiO2 layer thickness∗ 20 A 16 Afigure error† ≈ 1 µm ≈ 1 µm

concave convexrms roughness∗ not defined‡ (2.7± 0.4) Alateral correlations g(R) = 0.107 ·R2·0.34 weakAFM g(R) = 0.098 ·R2·0.38 g(R) = 6.5× 10−4 ·R2·0.56

∗ see Tabs. 7.2 and 7.3† see Sec. 6.1‡ no cut-off of self-affine roughness (see the text), ‘local’ rms roughness (4.7± 1.1) A

In the case of the smooth substrate, the correlations of the roughness are weak(low level of diffuse reflectivity). Detailed measurements of the diffuse reflectivitywould allow to better characterize these correlations, like in the case of the roughsubstrate, but would require much more time.

In the case of the rough substrate, no cut-off of the self-affine behavior canbe detected on the length scales probed by the x-rays. This means that the rmsroughness does not saturate, and hence the ‘amplitude’ of the roughness cannotbe determined from the x-ray data (besides a ‘local’ rms roughness, see above).

Various growth mechanisms can lead to self-affine roughness [215]. It has alsobeen observed on etched Si surfaces [216, 217]. It is thus no surprise to findself-affine roughness on the substrates used in this work, which have undergoneseveral cycles of etching and oxide growth during the sample preparation (seeSec. 6.1). The chemical treatment was the same for both substrates, but theinitial chemomechanical polishing was performed in different laboratories. Thedifference in the Si orientation might also play a role, since the etching ratedepends on the orientation.

7.4 Growth law

7.4.1 What is expected from theory?

The theory for interface melting has been discussed in Sec. 3.2. In the case ofdominating short-range interactions, a logarithmic growth law is expected fromLandau theory. For large layer thicknesses L, long-range interactions will domi-nate and cause a cross-over to a power-law (the cross-over thickness depends onthe ratio of the Hamaker constant and the strength of the exponentially decayinginteractions). For very small layer thicknesses on the order of a few molecu-lar layers, the mean-field approach is not valid anymore. Therefore, deviations

7.4. GROWTH LAW 115

0.1 1 100

1

2

3

4

5

6

L(n

m)

Tm-T (K)

powerlaw

logarithmiclaw

layering

Figure 7.26: Growth law for interfacial melting as expected by theory. For largelayer thicknesses, long-range forces dominate and the growth follows a power law.Short-range interactions at smaller layer thicknesses lead to a logarithmic growthlaw. For very small layer thicknesses, a layer-by-layer growth may occur.

from the continuous logarithmic law may occur, as for example layer-by-layergrowth. It can also be expected that the very first layer in contact with thesubstrate shows a specific behavior like premelting at much lower temperaturesor substrate-induced ordering (as for example observed for water films adsorbedon certain metal substrates, see [218] and references therein). The theoreticallyexpected growth law can thus be divided into three regions: power-law for largeL, logarithmic law for intermediate L, deviations like layer-by-layer growth forsmall L. This is illustrated in Fig. 7.26.

7.4.2 Experimentally observed growth law

In this work, the thickness of the quasiliquid layer has been obtained (see Sec. 7.2)from reflectivity measurements. The values are plotted as a function of temper-ature in Figs. 7.27 and 7.28. The individual points are labelled with the corre-sponding reflectivity measurement in Figs. 7.27a and 7.28a.

The data can be fitted with a logarithmic growth law appearing as a straightline in a plot where the scale of the relative temperature Tm − T is logarithmic(see Figs. 7.27b and 7.28b). The fit is much better in the case of the smoothsubstrate, whereas the growth law in the case of the rough substrate seems to bemore complicated and might have to be separated into several regimes (discussedin Sec. 7.4.5). Due to the limited number of data points for the rough substrate,one could only speculate about deviations from a pure logarithmic growth lawat first. Such deviations became only clear once it was able to compare the datafrom the rough and the smooth substrate (see Fig. 7.29). In the case of the


smooth substrate, a logarithmic growth law fits the experimental data extremelywell over the whole temperature range of the experiment.

Deviations from a logarithmic growth law can be observed for both substratesin the low-temperature region, which will be discussed in the next section.

Fits with a power law have also been performed, appearing as straight lines ina log-log plot (see Figs. 7.27c and 7.28c). The power law is clearly not valid in thecase of the smooth substrate, where it strongly deviates from the measurementat high temperatures. In the case of the rough substrate, however, the powerlaw matches quite well the data, especially in the high temperature regime (seeFig. 7.27f). This might indicate a cross-over from a logarithmic to a power law(see Sec. 7.4.5). The fits with a power law yield an exponent of −(0.33 ± 0.03)for the rough substrate, which is very close to the exponent −1/3 expected fornon-retarded Van der Waals forces. The Hamaker constant W (defined as inSec. 3.2.1) can also be determined from the fit with a power law, which gives (6.6±1.3)× 10−21 J for the rough substrate. This is the typical order of magnitude forHamaker constants (see for example [130]). In the case of the smooth substrate,only the low temperature data can be fitted to a power law, which yields anexponent of −(0.29±0.03) (corresponding to n = 2.4, i.e. more strongly retardedVan der Waals interactions) and a small value of (2.6 ± 0.4) × 10−25 J for theHamaker constant.

7.4.3 Onset

Extrapolating the logarithmic growth law to zero layer thickness yields the on-set temperature T0 with the values (−19 ± 3)C for the rough substrate and(−47± 16)C for the smooth substrate. However, the fit is not very sensitive tothe value of the onset temperature. Furthermore, the mean-field approximationbehind the logarithmic growth law is not valid anymore in the regime of smalllayer thicknesses. The real on-set for interfacial melting might be different. Theexperimental values for the layer thickness shown in Fig. 7.29 do not seem togive a coherent picture. Some values are actually zero and would be consistentwith T0 as determined from the fit with a logarithmic growth law. These arevalues from measurements performed at the beginning of a beam-time (compareFigs. 7.1 and 7.2). Only later, we realized that after a certain time, a quasiliquidlayer with finite thickness seems to persist down to temperatures that are evenlower than the initial temperature. This can be explained with the observedradiation-induced change of the substrate termination from hydrophobic to hy-drophilic. This means that the data points showing a finite layer thickness atlow temperatures are the ones corresponding to the same substrate terminationas the rest of the data. For the hydrophilic termination, the ‘real’ onset of inter-face melting is thus considerably lower than the T0 deduced from extrapolatingthe logarithmic growth law. This is consistent with other studies showing thepresence of a quasiliquid at very low temperatures (see Sec. 3.4).

7.4. GROWTH LAW 117

0.1 1 100

10

20

30

40

50

60

L(Å

)

Tm-T (K)

0.1 1 100

10

20

30

40

50

60

L(Å

)

Tm-T (K)

d

0.1 1 100

10

20

30

40

50

60

L(Å

)

Tm-T (K)

c

0.1 1 10

10

100

L(Å

)

Tm-T (K)

a

0.1 1 100

10

20

30

40

50

60

L(Å

)

Tm-T (K)

T08T07

T05T04

T12

T11

T10

T09

T02

T06T03

T14

T13

b

0.1 1 100

10

20

30

40

50

60

L(Å

)

Tm-T (K)

fe

T13c

Figure 7.27: Growth law for the rough substrate. (a) Data from first experiment(filled symbols) and second experiment (open symbols) and labels. (b) Fit withlogarithmic growth law, which yields L0=(8.2± 0.4) A and T0=(−19± 3)C. (c)Fit with a power law yielding an exponent of −(0.33±0.03). (d) Possible layeringat low temperatures. The solid line indicates the c lattice unit of ice. (e) Fit withseparate logarithmic growth laws for the high temperature (solid squares) andlow temperature (crosses) parts, see the text. (f) Fit with a power law for thehigh temperature (solid squares) part and a logarithmic growth law for the lowtemperature (crosses) part, see the text.


0.1 1 100

10

20

30

L(Å

)

Tm-T (K)

a b

dc

0.1 1 10

0

10

20

30 T32 T31T23

T04T24

T25T02

T08T27

T29T30

T20T19

T16T03

T05

T06T07

T26T28

T01

L(Å

)

Tm-T (K)

0.1 1 10

10

L(Å

)

Tm-T (K)

0.1 1 100

10

20

30

L(Å

)

Tm-T (K)

T01

c

Figure 7.28: Growth law for the smooth substrate. (a) Data points and labels.(b) Fit with logarithmic growth law, which yields L0=(3.7±0.3) A and T0=(−47±16)C. (c) Inadequate fit with a power law yielding an exponent of −(0.29±0.03).(d) Possible layering at low temperatures. The solid line indicates the c latticeunit of ice.

7.4. GROWTH LAW 119

0.1 1 100

10

20

30

40

50

60 rough substratesmooth substrate

L(Å

)

Tm-T (K)

c

Figure 7.29: Influence of roughness on the growth law. Comparison of the roughand the smooth substrate. The solid lines indicate fits with a logarithmic growthlaw and the c lattice unit of ice. The logarithmic growth law fits very well the dataof the smooth substrate. The growth behavior at the rough substrate correspondswell to the one at the smooth substrate up to about −0.7C. From this point onthe layer thickness at the rough substrate increases much faster (discussed in thetext).

As the mean-field approach is not valid for very thin layers, it is not surprisingto observe indications for some sort of layering (see Figs. 7.27d and 7.28d) insteadof a continuous growth at low temperatures. In the case of the smooth substrate,two jumps in the layer thickness can be recognized. The two jumps togethercorrespond approximately to the lattice unit c of the ice crystal perpendicularto the interface and thus to about two molecular layers. In the case of therough substrate, the observed interfacial melting directly sets in with this layerthickness, but this is probably only due to the limited spatial resolution. (Inthe case of the rough substrate, the qz range of the reflectivity measurements issmaller, thereby limiting the spatial resolution perpendicular to the interface.)Further experiments could address the question of the onset, but it might bedifficult to answer, as the reflectivity has to be measured up to high momentumtransfers in order to reach atomic resolution. Moreover, interfacial roughnessmight mask a very thin layer by smearing the density profile.

It has to be noted that apart from the data points at very low temperatures,the observations are completely reversible (see the ‘timeline’ in Figs. 7.1 and 7.2for the data shown in Fig. 7.29). The reversibility is also illustrated by Fig. 7.30showing reflectivity curves measured at approximately the same temperature, butduring different heating/cooling cycles.


0.0 0.2 0.410

-8

10-6

10-4

10-2

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

a rough substrate

b smooth substrate

T04 -14.7°CT02 -0.5°C T09 -0.5°C

0.0 0.2 0.410

-8

10-6

10-4

10-2

100

Inte

nsi

ty(a

rb.units

)q

z(Å

-1)

0.0 0.2 0.410

-8

10-6

10-4

10-2

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

0.0 0.2 0.4 0.6 0.8 1.010

-10

10-8

10-6

10-4

10-2

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

0.0 0.2 0.4 0.6 0.8 1.010

-10

10-8

10-6

10-4

10-2

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

0.0 0.2 0.4 0.6 0.8 1.010

-10

10-8

10-6

10-4

10-2

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

T02 -1.0°C T04 -0.137°C T07 -1.0°C

range of T02

Figure 7.30: Reversibility of the quasiliquid layer formation. Illustrated by reflec-tivity measurements for (a) the rough substrate and (b) the smooth substrate.

7.4.4 Growth amplitude

The fit with a logarithmic growth law yields the growth amplitude5 L0 =(8.2 ±0.4) A for the rough substrate and L0 =(3.7 ± 0.3) A for the smooth substrate.The fit is extremely good for the smooth substrate, but does not match too wellthe data for the rough substrate. As discussed in Sec. 3.2, the growth amplitudecan be compared to the correlation length of the quasiliquid. The experimentallydetermined values for the correlation length in bulk water range from 4.5 A [219]to 8 A [220]. This signifies that the correlation length of the quasiliquid layer isof the same order as in bulk water. In the case of the smooth substrate, the valuelies at the lower end of the water data, the correlations in the quasiliquid seemsto be less pronounced. In the case of the rough substrate, the conclusion wouldbe that the quasiliquid has stronger correlations. But as a single logarithmicgrowth low does not fit too well to the data, the reason for the faster growth ofthe layer thickness is more likely a difference in the growth law (see Sec. 7.4.5).This interpretation is supported by the fact that the growth law for the roughsubstrate in the region of smaller layer thickness follows quite well the one for thesmooth substrate and deviates only at higher temperatures. Otherwise, it wouldremain to explain why the quasiliquid on a rough substrate should have a larger

5the amplitude defines the slope of the logarithmic growth law in a log-linear plot as forexample in Fig. 7.29

7.4. GROWTH LAW 121

correlation length. This might not be be impossible. Effects of the substratemorphology on the density of an adjacent liquid have been discussed recently[221]. But in our case, the observed density of the quasiliquid in contact withboth substrates is the same (see Sec. 7.5), which does not support the idea of astrong structural difference.

7.4.5 Influence of roughness

Comparing the layer thickness for the rough and for the smooth substrate, onerealizes that there is a good match for low temperatures up to about −0.7C (ora corresponding layer thickness of about 16 A). The layer thickness for the roughsubstrate is systematically higher, but the difference is within the error bars. Thethickness might also appear larger because the density profile is more stronglysmeared out by the larger roughness. Up to this point, and with the exception ofthe very low temperatures, the interface melting at the rough and at the smoothsubstrate follows a logarithmic growth law with a growth amplitude on the orderof the bulk correlation length of water. Fitting this part of the growth law forthe rough substrate with a logarithmic law yields a growth amplitude of 3.5 A(see Fig. 7.27e), which is close to the value obtained for the smooth substrate.For temperatures above −0.7C, the quasiliquid layer at the rough substrategrows much faster with temperature. At the highest temperatures covered by theexperiment, its thickness is about twice as large at the rough substrate than at thesmooth substrate (55 A compared to 27 A). This is approximately the ratio of thevalues for the rms roughness (‘local’ roughness in the case of the rough substrate,see Sec. 7.3) of the two substrates. The high temperature part of the growthlaw for the rough substrate can be fitted with a different logarithmic growthlaw (see Fig. 7.27e), but this would be difficult to explain theoretically. Fittingthis part with a power law (see Fig. 7.27f) yields the exponent −(0.31 ± 0.03),still very close to the value expected for non-retarded Van der Waals forces. Apossible interpretation is that the roughness shifts the cross-over from logarithmicto power law growth. For the smooth substrate, the necessary layer thickness islarger than the values reached in the experiment, therefore only a logarithmicgrowth law is observed. For the rough substrate, in contrast, the cross-overthickness is lowered by the morphology. This leads to the observation of anapparent cross-over in the growth law.

An alternative interpretation could be that the interface melting, at both therough and the smooth substrate, in principle follows a power law. In the case ofthe smooth substrate, the interfacial melting is blocked, therefore the growth lawdeviates from the power law at high temperatures. The roughness switches thebehavior to complete melting, therefore a power law fits the data of the roughsubstrate over the whole temperature range.

Roughness effects have been discussed in the context of other wetting phe-nomena (see Sec. 3.2.7). Netz and Andelman [103] investigated theoretically the


influence of roughness for the case of interfacial melting. They consider the caseof Van der Waals type interactions and a simple sinusoidal roughness. In contrast,the substrates used in this work exhibit self-affine roughness (see Sec. 7.3), andthe interfacial melting appears to be governed by exponentially decaying inter-actions for small L. It should be possible, however, to use Netz and Andelman’sapproach with the type of roughness and interactions found in this work.

Other experiments have also shown that curvature effects as they occur forexample in porous media have an influence on ice premelting (see Sec. 3.4.3).Beaglehole and Wilson [154] also concluded from their ellipsometry measurementsthat the roughness of a glass substrate favors the interface melting of ice (seeSec. 3.4.4).

7.4.6 Comparison with surface melting

There is a large number of experiments on the surface melting of ice whose resultsscatter quite strongly (see Sec. 3.4.2). For comparison, the work of Lied et al.(see references in Sec. 3.4.2) is presented. They studied the surface melting of iceusing evanescent x-ray diffraction [13]. This allows to directly probe the relevantorder parameter of the melting transition contained in the Bragg intensities.Lied et al. have investigated three different high-symmetry faces of ice, amongthem the basal orientation used in this work. The observed layer thickness asa function of temperature for various orientations is shown in Fig. 7.31c. Thetemperature range covered by those experiments is smaller than in this work. Allcurves can be approximated by logarithmic growth laws. Deviations appear athigh temperatures, but unlike in this work, no deviations are observed for lowtemperatures.

The onset temperature obtained from the surface melting experiments is be-tween −12.4C and −13.5C, and thus higher than the T0 obtained from extrap-olating the logarithmic growth law at the interface. A difference in the onsettemperature is no surprise, as T0 is determined by the interfacial free energiesaccording to mean-field theory (see Eq. 3.5). The interfacial energies for an icesurface and an ice–solid interface are different, of course.

More striking is the large growth amplitude for surface melting between 36 Aand 84 A, and a quasiliquid layer thickness of up to 500 A at around −0.3C. Thisimplies that either the quasiliquid at the surface of ice displays much strongercorrelations, or there is another effect enhancing the surface melting, which is notaccounted for in the model predicting the logarithmic growth law.

7.4.7 Influence of temperature error

Due to the logarithmic or power law growth of the quasiliquid layer, errors in thetemperature measurement are of minor importance for low-temperatures, butcan have a strong influence on the observed growth law for temperatures close

7.4. GROWTH LAW 123

0.1 1 10

1

10

100

1000

surfacerough interfacesmooth interface

L(Å

)

Tm-T (K)

0.1 1 100

100

200

300

400

500

600

L(Å

)

Tm-T (K)

(0001) basal(1100)(1000) prism

a surface/interface melting, basal face

c surface melting, different faces

b surface/interface melting, basal face

0.1 1 100

100

200

300

400

500

600

surfaceroughinterfacesmoothinterface

L(Å

)

Tm-T (K)

Figure 7.31: Growth law for surface melting of ice. (a) Comparison of surface[144] and interface melting (this work), basal orientation. (b) Same in a log-logplot. (c) Surface melting of ice on different faces, from [144].


0.01 0.1 1 100

20

40

60

80

100

T = +20 mKT = 0 mKT = -13 mKT = -40 mK

L(Å

)

Tm-T

measured(K)

Tsmooth

Trough

Figure 7.32: Effect of temperature error ∆T = Treal−Tmeasured on observed growthlaw. The long dashed lines correspond to the maximum errors (see Sec. 6.6). Thegreen short dashed line corresponds to the error expected from the measuredtemperature distribution in the sample. The effect on the growth law is largerfor positive ∆T , as in this case the real temperature is closer to the bulk meltingpoint Tm where the layer thickness diverges. The layer thickness then alreadydiverges at temperatures that are nominally below Tm (upper dashed line), butsuch a behavior is not observed in the experiments. The maximum temperaturereached in the experiments with the rough and the smooth substrate (Trough andTsmooth, respectively) is indicated.

to the melting point. Fig. 7.32 shows the calculated temperature dependence ofthe layer thickness that would be observed for various offsets in the temperaturemeasurement (the real growth law is always the same). The effect is strongestfor a positive offset ∆T = Treal − Tmeasured, which causes a positive curvature ofthe observed L(T ) close to Tm. Such a behavior is not found in this work, andaccording to Sec. 6.6, ∆T is expected to be about −13 mK (which correspondsto the green short dashed line in Fig. 7.32). The real layer thickness is, therefore,expected to be slightly larger than the observed one, but with no significantinfluence on the type of growth law observed. For measurements even closer tothe bulk melting point, however, the absolute temperature accuracy has to beimproved (see Sec. 8.5).

7.5. DENSITY AND STRUCTURE OF THE QUASILIQUID 125

0.1 1 100.8

1.0

1.2

1.4

1.6

1.8

qll(g

/cm

³)

Tm-T (K)

0.1 1 100.8

1.0

1.2

1.4

1.6

1.8

ice Ihwater

HDA

rough substratesmooth substrate

qll(g

/cm

³)

Tm-T (K)

0 10 20 30 40 50 600.8

1.0

1.2

1.4

1.6

1.8

ice Ihwater

HDA


qll(g

/cm

³)

L (Å)

0.1 1 100.8

1.0

1.2

1.4

1.6

1.8

qll(g

/cm

³)

Tm-T (K)

b smooth substrate

c both substrates

a rough substrate

d both substrates (as a function of )L

Figure 7.33: Density of quasiliquid layer. The solid lines indicate the trend. (a)Rough substrate. (b) Smooth substrate. (c) Comparison of the rough and thesmooth substrate. The densities of ice Ih, bulk water, and HDA are indicatedby dashed lines. (d) Density as a function of the layer thickness. The solid lineis a linear fit (in this log-lin plot) and indicates the decay of the average densitytowards the bulk value as the layer thickness increases.

7.5 Density and structure of the quasiliquid

7.5.1 Experimentally observed density

The density of the quasiliquid layer is part of the model describing the dispersionprofile of the interface used to fit the reflectivity data. In the beginning, itwas fixed to the density ρwater = 1.0 g/cm3 of bulk water. But it soon becameapparent that a layer with the density of water would not be able to reproducethe measured data. The density of the quasiliquid has then be used as a freeparameter. The values obtained from the fitting are shown in Fig. 7.33 as afunction of temperature for both the rough and the smooth substrate.

The values scatter quite strongly, but are distinctly higher than the densityof bulk water. The error bars are upper bound uncertainties as they include


systematic errors due to the nature of the fitting process used. The mean valuesare 1.20 g/cm3 for the rough substrate and 1.19 g/cm3 for the smooth substrate.The density of the quasiliquid is thus about 20% higher than in bulk water. Thiscorresponds to a compression by a factor of 3

√1.2 ≈ 1.063.

The agreement of the observed densities at the smooth substrate and therough substrate shows that the density enhancment is not a roughness-inducedartifact from the fitting procedure. It also shows that the enhanced density isnot caused by the specific substrate morphology.

The strongly enhanced density of the quasiliquid layer is an experimental fact.The reliability of the model has been extensively verified (see Sec. 7.2.3). Onlya strongly enhanced density can consistently explain the measured reflectivitycurves (see for example Fig. 7.13). In the following, the origin of this experimen-tally observed density enhancement will be discussed.

In principle, impurities dissolved in the quasiliquid layer can lead to an in-creased density. Impurities were found to be the origin of the anomalous prop-erties of the so-called ‘polywater’ reported in the 1960s, see for example [222].There are several arguments against impurities being the reason of the densityincrease observed in this work:

• The sample preparation process renders significant contaminations very un-likely: The ice single crystals are grown from high-purity water, and mostcontaminations have a very low solubility in ice. The substrates have beenthoroughly cleaned (see Sec. 6.1). When contacting the ice with the sub-strate, a molten layer is created and squeezed out, which flushes away im-purities (see Sec. 6.4).

• A strong degree of contamination would be needed to obtain the observeddensity increase.

• If contaminations were responsible for the density increase, the densitywould decrease strongly with the thickness of the quasiliquid layer due todilution.

• It would be unlikely to find the same degree of contamination in two differ-ent samples (rough substrate and smooth substrate).

Dissolution of the substrate also has to be considered as a source for theincreased density. The solubility of SiO2 in water is practically zero [223]; foramorphous silica, the reported values are around 100 ppm [204]. The solubilityincreases with temperature, pressure, and pH (for pH values larger than 8). Ata pH of 11, the solubility is as high as 1000 ppm [204]. The radiation-inducedproduction of OH− could have a similar effect, but the amount of dissolved ma-terial would still be far too low. Moreover, the dissolution of SiO2 would lead toa decrease of the SiO2 layer thickness. The data analysis, which also yields thethickness of the SiO2 layer, does not show such a decrease (see Fig. 7.34).


0 10 200

5

10

15

20

25


dS

iO2(Å

)

number of measurement

Figure 7.34: Thickness of the SiO2 layer as a function of the temperature. Sincethe measurements are not very sensitive to the thickness of the SiO2 layer, thevalues scatter considerably, but do not indicate a significant decrease.

7.5.2 Conclusions about the structure

One is forced to conclude that the high density of the quasiliquid comes fromstructural differences compared to bulk water, i.e. a different local arrangementof the H2O molecules. Structural differences are not unexpected. They are thereason why the premelting layer is called quasiliquid instead of simply liquid. Inthe experiments reported in this work, the structural differences must be verypronounced.

The main question is how the structure of the quasiliquid may be related tothe other known structures of H2O. Are there actually any possible local arrange-ments of H2O molecules yielding such a high density? In order to answer thesequestions, the experimentally determined density of the quasiliquid has beencompared with the density of other forms of H2O (see Tab. 7.5) . Many of theseforms can only be produced under high pressure, but can also be recovered atatmospheric pressure, if the temperature is low enough (usually, the temperatureof liquid nitrogen, 77 K, is sufficient). Tab. 7.5 includes the densities measuredat atmospheric pressure, which were compared to the density of the quasiliquid.For most of the structures listed in Tab. 7.5, temperature dependent data is notavailable. Regular ice Ih expands by 1.7% upon heating from 85 K to 265 K.This can be used as an estimate for the other H2O structures. This relativelysmall effect due to thermal expansion is neglected in the following.

Comparing the densities one realizes that the density of the quasiliquid isvery close to the one of high-density amorphous (HDA) ice, which points to aclose structural relationship. There are also crystalline H2O structures with sim-ilar densities (ice II, V, and IX, see Tab. 7.5), but the density of HDA matches


Table 7.5: Solid forms of H2O. Densities are given for the temperature T andpressure p stated.

Ice Crystal Proton T p Density Referencesystem order (K) (GPa) (g/cm3)

Ih Hexagonal No 250 0 0.920 [14]Ic Cubic No 78 0 0.931 [14]II Rhombohedral Yes 123 0∗ 1.170 [14]III Tetragonal No 250 0.28 1.165 [14]

0† 1.14 [224]IV Rhombohedral No 260 0.5 1.292 [14]

110 0∗ 1.272 [14]V Monoclinic No 223 0.53 1.283 [14]

98 0∗ 1.231 [14]VI Tetragonal No 225 1.1 1.373 [14]

0† 1.31 [224]VII Cubic No 295 2.4 1.599 [14]

0† 1.50 [224]VIII Tetragonal Yes 10 2.4 1.628 [14]

0† 1.46 [224]IX Tetragonal Yes 165 0.28 1.194 [14]

0† 1.16 [224]X Cubic Symmetric 300 62 2.79 [14]

0† 2.51 [224]XI Orthorhombic Yes 5 0 0.934 [14]XII Tetragonal No 260 0.50 1.292 [14]

0† 1.29 [224]LDA Amorphous n/a 77 0∗ 0.94 [9]HDA Amorphous n/a 77 1.0 1.31 [9]

77 0∗ 1.17-1.19 [9, 10]VHDA Amorphous n/a 77 0∗ 1.25 [34]

∗samples recovered at low temperatures†see [224]


best. Several arguments can be used to argue against the formation of a dif-ferent crystalline structure at the interface upon approaching the bulk meltingtemperature.

• Interfacial melting is considered to be a disordering transition (like surface-induced disorder), with

• the quasiliquid having liquid properties as shown by measurements of ad-hesion [171] and friction, as well as experiments in porous media (seeSec. 3.4.3) showing the presence of a liquid fraction.

• The peculiar features of the possible crystalline phases: Ice II and ice IXshow proton ordering, ice IX has no region of stability and ice V an ex-tremely complicated structure with 28 molecules in the unit cell. The largeunit cell makes a continuous growth of the layer difficult.

As the observed interfacial melting is an equilibrium phenomenon, the forma-tion of the metastable HDA would be quite odd. It could not fully account for theliquid-like properties either. In current theories of the water structure HDA iceis the glassy counterpart of a postulated high-density liquid (HDL) form of water(see Sec. 2.4). The idea that the quasiliquid might be governed by fluctuationsinto the hypothetical HDL phase is therefore very appealing. According to thehypothesis of a liquid-liquid transition, bulk water consist of fluctuations of HDLand LDL (low-density liquid), see Fig. 7.35a. Interfaces might then display apreference for one of the two liquid phases leading to an effective pinning of partof the density fluctuation spectrum present in water, in this case the high-densityfluctuations (see Fig. 7.35b). This may lead to an effective stabilization of a thinHDL (or LDL) layer at interfaces and explain the observation of an enhanceddensity in the quasiliquid. The stabilization of one of the liquid forms might alsobe influenced by the nano-confinement between the two solids SiO2 and ice (seeFig. 7.35c).

This picture is supported by a number of other experiments aimed at under-standing the interface between liquid water and a hydrophobic substrate [11, 12].These experiments consistently show an interfacial water layer with a density de-pletion of around 10%. Although these results have been interpreted differentlyso far, they fit well to the idea that different water fluctuations can be stabilizedat interfaces. They also give a hint to what leads to the preference of either HDLor LDL fluctuations. It seems that hydrophobic interface potentials stabilize alow-density form of water, whereas hydrophilic interface potentials created by theparticular interface used in this work stabilize a high-density form of water.

In other experimental studies of the Au(111)–electrolyte interface [225] andthe Ag(111)–electrolyte interface [226], a strong density increase in the interfacialwater has been observed as well. The interface effect is restricted to the firstmolecular layers. In the case of [226], a layering of the water molecules leading


a bulk b interface c confinement

Figure 7.35: Schematic illustration of fluctuations in water. Black: High-densityfluctuations. White: Low-density fluctuations. (a) In bulk water, both high-density and low-density fluctuations are present and lead to the average densityof 1 g/cm3. (b) Interface potentials may lead to an effective stabilization of part ofthe fluctuation spectrum. This effect is restricted to an interfacial layer, while thebulk situation is found further away from the interface. (c) A strongly confinedlayer can be completely governed by one part of the fluctuation spectrum.

to an oscillation of the density profile has been found. The ‘areal density’ withinthe first layer is increased by a factor of four. Although the density oscillates,there is a net density excess in the interface region (see Fig. 7.36).

The experiments that we carried out in this work lack the resolution to resolveunambiguously individual layers of water molecules (possible layering within thequasiliquid). Instead, they provide an average density and the overall thicknessof the quasiliquid.

7.6 Si wafer as substrate

As already mentioned in Sec. 7.1, an additional experiment has been performedusing a thin and ultra-smooth silicon wafer as substrate. The cleaning procedurefor the wafer (see Sec. 7.6.1) was different from the one used for the other ex-periments (compare Sec. 6.1). The same experimental technique was used (seeChapt. 5), but the setup was slightly different (see Sec. 7.6.2).

7.6.1 Sample

Si(001) wafers with a thickness of ≈ 0.6 mm were kindly provided by P. Dreier(Siltronic AG). The rms roughness of these wafers was extremely small (σ ≈ 2 A).The experiment with one of these wafers aimed at obtaining more information

7.6. SI WAFER AS SUBSTRATE 131

0 5 10 15 200

1

2

3

4

5

Rela

tive

densi

ty

z (Å)

datarunning average (3 Å)running average (6 Å)

Figure 7.36: Water density at Ag(111)/electrolyte interface at +0.52 V of thepotential of zero charge (normalized to the value far from the interface). Datafrom Toney et al. [226] (solid line) and running averages (dashed lines).

max

max

max

t

b

wafer

Figure 7.37: Reflection from the backside of thin samples can occur for incidentangles larger than αmax.

about the influence of roughness on interface melting. Furthermore, a low rough-ness leads to a slow decay of the reflectivity curve, which allows measurementsup to high momentum transfers, corresponding to a high resolution in real space.Despite the small thickness of the wafers, reflection of the beam from the bottomside of the wafer does not occur, because the incident angles are sufficiently small.The maximum angle αmax without reflection from the back side of the wafer is(see Fig. 7.37)

αmax = arctant

l= 24 mrad, (7.1)

where t denotes the thickness of the wafer and l its length. This formula appliesto the case of full illumination of the interface. In practice, the footprint at themaximum angle is very small and located in the center of the sample. Then lcan be replaced by l/2 and the maximum angle is even larger. But already thesmaller value corresponds to a momentum transfer of 2 A−1 at the x-ray energyused.

The wafer was first cleaned in concentrated HNO3 at ≈ 70 C for several


hours. This also leads to the formation of a thick oxide layer with a stronglyhydrophilic termination. This step was succeeded by the RCA standard clean1 (H2O2-NH4OH-H2O, 1:1:5). After cleaning, the wafer was placed on a cleanblock of pure aluminum and contacted with an ice single crystal as described inSec. 6.4.

7.6.2 Experimental setup

The wafer experiment was carried out at beamline ID15B of the ESRF, unlikethe other x-ray experiments described in this work. For monochromatization, abent Laue-type Si(511) monochromator was used at an energy of 78.2 keV (cor-responding to a wavelength of 0.159 A). The energy bandwidth was 0.3%, definedby the secondary slits in this setup. The monochromator also demagnifies thebeam in the horizontal direction to about 50 µm. For focusing in the verticaldirection (which is more relevant for this experiment, see Chapt. 5) we commis-sioned a new bent multilayer (W/B4C, 150 layers, period 23.73 A, 5.5% d-spacinggradient along the length of 240 mm, slope error ≈ 1 µrad). It allows to focus thex-ray beam to a vertical beam size of less than 4 µm at a distance of 2200 mm.For a comparable focal size, a focal distance of ≈ 4000 mm is needed with theCRL used in the other experiments. For smaller focal distances, the numberof single lenses in the CRL has to be increased, which considerably reduces thetransmission. Therefore, the setup with the multilayer is also of great interest forother experiments where limited space is an issue. Like the optics setup presentedin Chapt. 5, it allows a good control of the various beam parameters such as focalsize, x-ray energy, and bandwidth, at a comparable or even higher flux. It is moresensitive to beam instabilities, but can nevertheless be used for high-energy x-rayreflectivity experiments. The sample tower and detector setup is identical to theone described in Chapt. 5. A schematic view of the setup is shown in Fig. 7.38.

7.6.3 Results

Unfortunately, complete reflectivity curves could not be measured for this sam-ple. The reason is that the thin wafer is also rather flexible, which causes a slightbending of the wafer (probably during the preparation of the ice–substrate in-terface). As a consequence, the incident angle is not well defined (see Fig. 7.39).The local incident angle actually varies on the interface. This can be seen bymoving the interface across the beam (see Fig. 7.40).

This leads to

• a change of the incident angle by minor movements of the beam relative tothe sample, since this also considerably moves the beam footprint on theinterface (compare Fig. 7.40),

• an averaging over a range of incident angles by the beam footprint.

7.6. SI WAFER AS SUBSTRATE 133

2

sampletower

detectortable

detector

monochromator

42 m 5 m 4.4 m 1.3 m

a top view

b side view

1

2

zy

xi

2

multilayer

secondaryslits

2.2 m

2ML

Figure 7.38: Sketch of the high-energy setup with focusing multilayer. (a) Topview. (b) Side view. The focusing in the horizontal direction is achieved by abent monochromator and decoupled from the focusing in the vertical directionobtained by the bent multilayer. The reflection angle 2θML of the multilayer islargely exaggerated in the sketch. The real angle is close to 35 mrad .


bent substrate

X-ray beam

1

2

z

x

Figure 7.39: Reflection from bent substrate. The incident angle depends on wherethe beam hits the substrate.

-60 -40 -20 0 20 40 60

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0-15 -10 -5 0 5 10 15

(mra

d)

z (µm)

x (mm)

Figure 7.40: Bending of the wafer. Measured by the difference between actualand nominal (3.5 mrad) incident angle as a function of the sample position z.The movement of the sample along z causes a movement of the beam footprintalong x (see also Fig. 7.39).

7.7. NEUTRON REFLECTIVITY 135

0.2 0.4 0.6 0.8 1.010

-6

10-4

10-2

100 -0.32°C

-0.79°C-15.60°C

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

Figure 7.41: Reflectivity measurements for ice in contact with a Si wafer. Theappearance of interference fringes with increasing temperature indicates interfa-cial melting. Unfortunately, measurement of complete reflectivity curves was notpossible due to bending of the wafer.

As the size of the beam footprint on the interface changes with the momentumtransfer (see Sec. 5.3), the averaging range also changes. The range is smallerfor large momentum transfers (as are the movements of the footprint along theinterface). Therefore, parts of the reflectivity curves could still be measured atlarge momentum transfers. A few measurements for different temperatures areshown in Fig. 7.41. It can be seen that the measurements qualitatively behave likethe ones performed with the silicon blocks as substrates (see Sec. 7.1): With in-creasing temperature, interference fringes with decreasing intervals appear. Thismeans that a layer with different density and increasing thickness forms. Thequalitative conclusion from this observation is that interfacial melting occurs forthe wafer substrate as well. This observation is important, since the substrate inthis case has undergone a different chemical treatment leading to a thick oxidelayer that is hydrophilic from the beginning (see also Sec. 7.8).

7.7 Neutron reflectivity

First attempts to study interface melting with neutron reflectivity have beenundertaken by Lied [144]. In the first experiment of this work, neutron reflectivitywas used as well, and the results of this experiment will be presented here.

The formalism for neutron reflectivity is analogous to x-ray reflectivity (de-scribed in Chapt. 4) when the materials involved are not magnetic. However, thescattering length density for neutrons is not connected to the electron density.The neutrons interact mainly with the nuclei, and the scattering length den-


sity depends crucially on the isotopes involved. As the typical extinction lengthfor neutrons is large, they are often the method of choice for studying buriedinterfaces. There are some drawbacks though:

• the low flux compared to Synchrotron radiation sources, which limits thedynamic range of reflectivity measurements to about 6 orders of magnitude,

• the low resolution in q-space,

• the low spatial resolution in real space (large beam size and limited flux)

• the necessity to use D2O, since H2O produces strong background scatteringdue to the large incoherent neutron scattering cross section of H (see [227]).

7.7.1 Sample

As in most of the x-ray experiments, a silicon block covered with a native oxidelayer was used as substrate. The chemical preparation of this substrate and thepreparation of the interface were as described in Sec. 6.1. The ice single crystalswere made of D2O instead of H2O.

7.7.2 Experimental setup

The neutron reflectivity experiments were carried out at the evanescent-wavereflecto-diffractometer EVA [228] of the Institut Laue-Langevin (ILL) in Greno-ble. The instrument uses cold neutrons delivered from a reactor source. A py-rolytic graphite (002) monochromator provides a neutron beam with a wavelengthof 5.5 A (corresponding to an energy of 2.7 meV) and a bandwidth of 1.2%. Acooled Be filter suppresses higher harmonics. The neutron flux at the sample po-sition is about 1× 106 n/s. The EVA diffractometer resembles the one describedin Sec. 5.3, but does not have the same level of mechanical accuracy (which isnot needed for the neutron reflectivity experiments, as the typical angles of inci-dence are much larger). The same sample chamber as in the x-ray experiments(see Sec. 6.5) was used. The aluminum windows are suitable for experimentswith neutrons and high-energy x-rays as well. For the neutron reflectivity ex-periment, however, the beam penetrates the sample through the silicon side andgets reflected from the ice interface (see Fig. 7.43), as D2O has a higher neu-tron scattering length density than silicon. A 3He linear wire counter serves asa 1-dimensional position-sensitive detector (PSD) for measuring the scatteredintensity (see Fig. 7.42).

7.7.3 Results

Measured neutron reflectivity curves for two different temperatures are shown inFig. 7.44a in comparison with measurements from Lied (Fig. 7.44b) and calcula-

7.7. NEUTRON REFLECTIVITY 137

100 200 300 400 500

0

400

800

1200

measurement

Inte

nsi

ty(a

rb.units

)

channel number

Gaussian fit

primarybeam

reflectedbeam

i= 10 mrad

Figure 7.42: Example of PSD recording showing the reflected and primary neu-tron beam.

ice (D O)2

SiO2

Si

qz

sample cell

qll

n

PSD

Figure 7.43: Schematic geometry for neutron reflectivity measurements at ice–SiO2–Si interfaces.


tions (Fig. 7.44c). At high momentum transfers, the reflectivity curves measuredclose to the bulk melting point (Tm = 3.8C for D2O [229]) lie all above the curvesmeasured far below. The effect is small, but consistent with the measurementsperformed by Lied. It indicates that there is actually a change in the densityprofile as the temperature approaches the melting point. The calculations showthat the formation of a layer with increased density can explain the change ofthe reflectivity. Unfortunately, the density profile cannot be resolved due to thelimited q-range accessible in the neutron reflectivity measurements.

7.8 Substrate termination and radiation effects

As discussed in Sec. 7.1, long exposure to high-energy x-rays apparently causesthe substrate to change from hydrophobic to hydrophilic. Further experimentshave been undertaken in order to investigate this effect.

First, the assumption of a radiation-induced change of the substrate termi-nation has been verified by controlled irradiation. Therefore, a substrate wasprepared as described in Sec. 6.1, which was initially hydrophobic. One sam-ple was placed in a small plastic container half-filled with milli-Q water. Thesample was then exposed to high-energy x-ray radiation from an x-ray tube witha W target (Kα1 59.3 keV). In order to maximize the flux, no monochromatorwas used, but the low-energy part of the spectrum was filtered out by a 2 mmthick aluminum absorber. The part of the sample which was immersed in waterchanged from hydrophobic to hydrophilic where it was exposed to radiation (seeFig. 7.45). The part of the sample which had stayed in air showed this changeonly on tiny isolated spots (where water droplets may have condensed duringthe exposure). This shows that the radiation damage is apparently mediated byradicals created by photolysis of H2O molecules (H2O → H+ + OH−). This fitsto the fact that these radicals are also used in Si cleaning procedures leading toa hydrophilic termination (for example treatment with KOH solution or HNO3).

In order to characterize the change of the substrate termination, XPS mea-surements have been performed6 on irradiated hydrophilic and non-irradiatedhydrophobic parts of the substrate. Fig. 7.46 shows a comparison of the spectrafor silicon and carbon. The carbon peak comes from hydrocarbon contamina-tions which cannot be completely avoided when the sample is kept in air for anextended period of time. For the Si spectrum, an additional component can bedistinguished on the hydrophobic part when compared to the hydrophilic one.A scan across the irradiated part shows that this feature also occurs at a pointabout 3 mm from the hydrophilic zone, which is probably one of the hydrophilicspots outside the water-immersed area. The difference in the silicon spectrumcoincides with an energy shift of the carbon peak indicating a different bondingof the hydrocarbons in the hydrophilic and the hydrophobic region.

6in collaboration with L. Jeurgens and M. Wieland (both MPI fur Metallforschung)

7.8. SUBSTRATE TERMINATION AND RADIATION EFFECTS 139

0.00 0.05 0.10 0.1510

-6

10-5

10-4

10-3

10-2

10-1

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

3.5°C-15.0°C

0.00 0.05 0.10 0.1510

-6

10-5

10-4

10-3

10-2

10-1

100

L = 32 ÅL = 0 Å

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

a this work b previous measurements (Lied)

b simulation

0.00 0.05 0.10 0.1510

-6

10-5

10-4

10-3

10-2

10-1

100

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

3.5°C-22.2°C

Figure 7.44: Neutron reflectivity from D2O(ice)–SiO2–Si interfaces. (a) Measure-ments from this work. (b) Measurements from Lied [144]. (c) Calculated neutronreflectivity curves for different values of the quasiliquid layer thickness L (thick-ness of SiO2 layer 20 A, rms roughness 5 A). The measurements close the meltingpoint (Tm=3.8C) show increased reflectivity (see a+b), which can be explainedby the formation of a quasiliquid layer (see c).


hydrophilic

Figure 7.45: Photograph of substrate showing radiation induced change of thehydrophilicity. The central part of the sample has been exposed to high-energyx-ray radiation for several days. It was half-way immersed in water during thistime. The irradiation has produced a strongly hydrophilic termination where thesample was in contact with water (arrow). The irradiated area exposed to aironly shows a few hydrophilic spots (above).

As mentioned in Sec. 7.1, the reflectivity measurements on briefly irradiatedparts of the sample do not clearly indicate interfacial melting. This means thateither the hydrophobic termination of the substrate does not lead to interfacialmelting, or it leads to a low-density quasiliquid as suggested by other experiments(see Sec. 7.5.2). In the latter case, the density contrast between ice and thequasiliquid might be too small for detecting the quasiliquid layer (see Fig. 8.1).

7.9 Implications

The main conclusion from this work is that intrinsic interface melting of ice isindeed possible, which has been controversially debated in literature. Interfacialmelting has been observed at the interface between ice and SiO2. The growth lawis in good agreement with expectations from theory and other experiments. Theeffect strongly depends on the hydrophilicity of the SiO2. Further experimentsare necessary to investigate the influence of the substrate chemistry. A distinctinfluence of the substrate morphology was also observed, as suggested by previousexperiments and theory.

An intriguing property of the observed quasiliquid layer is its surprisingly largedensity, which is about 20% higher than the density of bulk water. This suggeststhat the quasiliquid resembles the postulated HDL form of water. Therefore,the results of this work do not only have implications for the theory of interfacemelting, but may also have implications for our understanding of water. The

7.9. IMPLICATIONS 141

a b

dc

e

98 100 102 104 106

0

2

4

6

8

Si 2p extra

Si 2p oxide

Si 2p metallic

Inte

nsi

ty(a

rb.units

)

Energy (eV)

98 100 102 104 106

0

2

4

6

8

Si 2p oxide

Si 2p metallic

Inte

nsi

ty(a

rb.units

)

Energy (eV)

y = 7 mm

y = 0 mm

-2 0 2 4 6 8 10 12 140

5

10

15

20

25

30

35 Si 2p extra

Peak

are

a(a

rb.units

)

Position y (mm)

280 285 290 2951

2

3

4

5

C 1s peak 1

Inte

nsi

ty(a

rb.units

)

Energy (eV)

280 285 290 2951

2

3

4

5

C 1s peak 1

Inte

nsi

ty(a

rb.units

)

Energy (eV)

f C 1s peak 1

-2 0 2 4 6 8 10 12 14

285.4

285.6

285.8

286.0

286.2

Energ

y(e

V)

Position y (mm)

Figure 7.46: XPS spectra of partially irradiated substrate. First column: siliconpeaks. Second column: carbon peaks. (a)+(b) Non-irradiated part showing anadditional component in the silicon spectrum. (c)+(d) Irradiated part. (e)+(f)Scan across the irradiated part showing the additional component in the siliconspectra and a shift of the carbon peaks.


question is how the observation of a dense form of water depends on the chemistryof the interface and on the nano-confinement between ice and SiO2. If thesequestions are answered, systems might be tailored to stabilize thin films of eitherHDL or LDL and study its properties.

The results of this work also have important ramifications for processes in na-ture and technology, some of which have been mentioned in Sec. 2.1. They mightoffer a better understanding of phenomena like glacier motion and permafrost,for example. It is also conceivable that further research could be technologicallyexploited, for example for the development of coatings capable of preventing theicing of airplane wings.

The observation of a dense form of water with hitherto unknown properties isimportant in this context. The physical and chemical properties of the quasiliquiddetermine the impact of interfacial melting in the aforementioned situations. Thepostulated HDL form of water is expected to have a lower viscosity and a largersolubility for impurities than bulk water. A quasiliquid with similar propertieswould promote glacier motion better than normal bulk water. The high solubilityfor impurities could imply that enrichment of impurities in the quasiliquid leadto a self-amplifying process, where the impurities cause a true reduction of themelting point, which further increases the thickness of the premelting layer.

In a wider context, the results of this work affect all situations where wateris in confinement or at interfaces. If the modification of the water structure insuch situations is indeed a generic phenomenon, it will be important for a muchwider range of issues, from electrochemistry to biological systems.

A more detailed discussion of the questions raised by this work follows inChapt. 8.

Chapter 8

Outlook

This chapter summarizes open questions and prospects for further research.

8.1 Influence of the substrate material and the

confinement

Measurements with different substrates, especially substrates with different hy-drophilicty/hydrophobicity, could address some important open questions:

1. How does interface melting depend on the substrate termination? (In thecase of SiO2, our experiments suggest that interface melting might onlyoccur when the termination is hydrophilic.)

2. How does the observation of a high-density liquid depend on the sub-strate? (Our observation in conjunction with other experiments suggeststhat hydrophilic substrates favor the high-density liquid and hydrophobicsubstrates favor the low-density liquid, see Sec. 7.5.)

Another question is how the formation of the high-density and low-densityliquid depends on the nano-confinement between ice and the substrate. Therefore,experiments should be performed for ice in contact with a substrate (interfacemelting scenario), and for bulk liquid water in contact with the same substrate.

A problem for future experiments is that if the LDL form of water is stabilizedupon interfacial melting of ice, the contrast between ice (ρice = 0.92 g/cm3) andLDL (≈ ρLDA = 0.94 g/cm3) would be very low. In this case, it might be difficultto observe a LDL layer in reflectivity measurements (see Fig. 8.1). If the ice isreplaced by liquid water, the density contrast is larger, and it is thus easier toobserve a LDL layer at the water–substrate interface than in the interface meltingscenario.

From our observations, we have concluded that interfacial melting occurs whenice is in contact with hydrophilic SiO2. But unfortunately we only have complete

143

144 CHAPTER 8. OUTLOOK

0.0 0.2 0.4 0.6 0.8 1.01E-17

1E-15

1E-13

1E-11

1E-9

1E-7

1E-5

1E-3

0.1

Inte

nsi

ty(a

rb.units

)

qz(Å

-1)

qll=1.00 g/cm (water)3

qll=0.94 g/cm (LDL)3

without quasiliquid layer

qll=1.18 g/cm (HDL)3

Figure 8.1: Calculated reflectivity profiles for various densities of the quasiliq-uid. The curve for a quasiliquid layer with the density of LDL is practicallyindistinguishable from the curve without a quasiliquid layer. Parameters for thiscalculation: L = 25 A, LSiO2 = 15 A, σice = 1.5 A, σSiO2 = 3 A, σSi = 1 A, x-rayenergy 71 keV.

data sets for the substrates that were rendered hydrophilic by irradiation. Forthe thin wafers, which were hydrophilic from the beginning, complete reflectivitycurves could not be measured due to bending of the wafers (see Sec. 7.6). It would,therefore, be interesting to repeat these measurements with thick substrates.

Silicon samples with a hydrophobic coating are a good choice for a hydropho-bic substrate, since smooth samples can be easily prepared. The problem here isto find a coating which is stable against radiation damage. We have performedtest experiments with coatings of trimethylchlorosilane (TMCS), but the coatingwas removed after exposure times of just a few seconds. If possible, substrateswith a hydrophilic and a hydrophobic part should be prepared, or even withparts having different degrees of hydrophilicity. In this way, the influence of thehydrophilicity could be tested without the need for exchanging the sample andwithout altering the experimental conditions.

For a more fundamental understanding, experiments with well-controlled hy-drophobic and hydrophilic substrates are most interesting. But one can also lookfor substrates that are of specific interest for processes in nature and technology.The SiO2 substrates used in this work may serve as a model for ice–mineral inter-faces as they occur in nature. The interface between ice and stainless steel mayrepresent the contact of an ice skate with the rink, and the interface between iceand rubber might be investigated in order to improve the contact between tiresand icy roads. Many other materials are of interest, however, the experimentaltechnique used in this work requires that smooth surfaces can be prepared fromthe substrate material. Also for some substrates, the large density compared to

8.2. SURFACE MELTING 145

ice, or an internal density profile, can make it difficult to resolve a thin quasiliquidlayer in the density profile. Further problems may arise for the preparation ofthe ice–substrate interface, if the heat conductivity of the substrate is small.

Another question is the difference between amorphous and crystalline sub-strates (with the same composition). For the substrates used in this work, thismeans replacing the amorphous SiO2 with a quartz single crystal. A crystallinesubstrate is more likely to induce a specific structure in the interfacial water, asfor example seen in the adsorption of water on Pt(111) [218]. The amorphoussubstrates used in this work come closer to an ideal hard wall and, therefore,are thought to reveal intrinsic water properties like the fluctuations in HDL andLDL. An advantage of crystalline substrates is the possibility to compare theexperimental results to theoretical calculations, whereas amorphous substratesare still very difficult to calculate. For the interface between quartz and water,results from calculations can already be found in the literature [230, 231].

8.2 Surface melting

Some of the experiments on surface melting of ice are not sensitive to the averagedensity. In the analysis of other experiments, the density of the quasiliquid is oftenassumed to be the same as in bulk water. It might be worthwhile reexaminingsome of these results to see whether there are any indications for a change in theaverage density at the free surface as well.

8.3 Influence of the substrate morphology

This work has shown that the substrate morphology can have a pronouncedinfluence on interface melting. Further experiments could help to understandthis influence. In any case, this influence has to be taken into account whencomparing results from different substrates.

8.4 Influence of impurities

Impurities can cause a drastic reduction of the ice melting temperature. Accord-ing to theory [135] and experiments [154], impurities should also have a greatinfluence on surface and interface melting. This influence is of great interest for‘real’ interfaces in nature which are practically never free of impurities. It is ofspecial interest for atmospheric chemistry (see Sec. 3.5.3), since ice particles inthe atmosphere serve as a reservoir and reaction site for chemicals.

The experimental challenge consists in controlling the concentration of impu-rities at the interface. One way might be by doping ice crystals. The advantageof this method is that the doping with certain chemicals (like HF or NH3) is


possible after contacting with the substrate1. Otherwise, the distribution of im-purities will most probably change during the interface preparation (see Sec. 6.4),as it involves the melting of the ice surface.

8.5 Growth law

As discussed in Sec. 7.4, deviations from a logarithmic growth law are observed atlow temperatures, where a thin quasiliquid layer seems to persist below the onsettemperature obtained by extrapolating the logarithmic growth law. This behaviorshould be explored by measurements at low temperatures. Several factors definethe lower temperature limit that can be reached with the current setup:

• the maximum temperature difference between back side and front side ofthe Peltier elements (about 65 K),

• the cooling power of the Peltier elements,

• the cooling power of the circulator for the cooling liquid,

• the admissible temperature range of the hoses used for the cooling liquid(currently −20C).

With the current setup, sample temperatures down to about−50 or−60C shouldbe possible.

However, reflectivity measurements up to very high momentum transfers ofabout 1.8 A−1 are required to achieve the real space resolution necessary toobserve molecular monolayers.

The second question concerning the growth law is the behavior very close tothe melting point. From theory, a cross-over to a power-law would be expected asalready suggested by the experiment with the rough substrate. In order to con-duct reliable measurements at temperatures even closer to the bulk melting point,the temperature accuracy and uniformity has to be improved. This presumablyrequires replacing the sample holder by an additional inner chamber thermallydecoupled from the exterior by an insulating vacuum in the outer chamber.

8.6 Structure of the quasiliquid

While the density of the interfacial quasiliquid hints to a close structural relation-ship with the HDA ice (and might actually be identified with the postulated HDLwater), definite proof is missing. Therefore, diffraction experiments are necessaryto resolve the structure of the quasiliquid.

1However, the use of such chemicals in a cold-room would certainly be a safety issue.

8.7. INFLUENCE OF ELECTRIC FIELDS 147

This can in principle be achieved with evanescent x-ray diffraction, success-fully applied to the study of Pb(liq.)–Si interfaces [193]. Unfortunately, thismethod cannot be applied to the system studied in this work, since the signalfrom the amorphous SiO2 layer cannot be effectively separated from the signalof the quasiliquid. One would need a crystalline substrate to do that. Also, asubstrate with a lower electron density than ice would be a great advantage. Inthis case, the evanescent wave can be induced in the quasiliquid instead of thesubstrate, which leads to an enhancement of the signal. Due to the low elec-tron density of ice (ρel

ice = 0.31 A−1), a suitable substrate with lower electrondensity is difficult to find. Beryllium, for example, has an electron density ofρel

Be = 0.49 A−3, Kapton (polyamide) ρelKapton = 0.44 A−3.

Another possibility for probing the structure of the quasiliquid would be touse a sample containing a large number of parallel ice–substrate interfaces Ifthe fraction of the quasiliquid in the sample is large enough, bulk scatteringtechniques could be used.

8.7 Influence of electric fields

One possible microscopic mechanism for interfacial melting is the preference of acertain orientation of the water molecules with respect to the interface (‘interfacepolarization’, see the work of Fletcher mentioned in Sec. 3.4.1). Such an alignmentwould violate the ice rules and could in this way break the hydrogen bond networkin ice. Due to the dipole moment of the water molecules, a similar effect couldbe achieved with electric fields. It would, therefore, be interesting to study theeffect of electric fields on interfacial melting. However, large electric fields areneeded to induce an appreciable alignment of the water molecules. The ratio rof molecules with parallel and antiparallel alignment with respect to an electricfield E behaves as r = exp

(µEkT

), where µ = 6.186 × 10−30 Cm is the dipole

moment of a water molecule (see Sec. 2.2), k = 1.381×10−23 J/K the Boltzmannconstant, and T the temperature. For a 10% orientation (r =55/45) and at thebulk melting point T = 273.15 K, an electric field of about 1.2× 108 V/m wouldthen be required. A possibility might be to use a tip-shaped electrode, whichallows to produce locally very strong fields.

Chapter 9

Summary

In this work the interfacial melting of ice has been investigated with a novelhigh-energy x-ray diffraction scheme.

Ice and water are abundant on Earth, of paramount importance for the bio-sphere, and part of our everyday life. Despite the apparent simplicity of thewater molecule, H2O has astonishing and often anomalous properties of whichmany are still not completely understood. (See Chapt. 2.)

Among these phenomena is the melting of ice which exhibits two special fea-tures. The first is pressure melting, i.e. the possibility to melt ice by applyingpressure. The second is surface melting, i.e. the appearance of a thin (quasi)liquidlayer at the surface of ice at temperatures below the bulk melting point.

Surface melting is observed in other materials, but appears to be particularlystrong in the case of ice. The phenomenological explanation of this effect is theminimization of the free energy of the system by introducing a quasiliquid layer.The quasiliquid at the surface may serve as a nucleation site for the bulk melting,and is thus of fundamental importance for understanding the melting process ingeneral. (See Chapt. 3.)

Surface melting is well established and in the case of ice, it has importantconsequences for environmental processes. In most cases, however, ice is in con-tact with other materials, and the question arises whether a similar effect alsooccurs at ice–solid interfaces. There are many indications that such interfacemelting is possible. But it may depend on many parameters like the materialof the solid substrate and its surface morphology. So far, investigations of well-defined ice–solid interfaces on nanoscopic length scales have been missing. (SeeChapt. 3.)

This is mainly due to the lack of suitable experimental techniques for prob-ing in situ deeply buried interfaces with adequate resolution. A recently devel-oped high-energy x-ray transmission-reflection scheme is an ideal probe for suchcases. This technique is based on the use of high-energy x-ray (here ≈70 keV) mi-crobeams, which can only be produced at modern Synchrotron Radiation sources.At the energies used in this scheme, x-rays can penetrate up to several centimeters

149

150 CHAPTER 9. SUMMARY

of material. Compound refractive lenses (CRLs) allow to focus the x-ray beamdown to a spot size of a few microns. Among the advantages of this scheme arethe large dynamic range and the low background level for diffraction experiments.However, very high angular and positional accuracy are required. Therefore, aspecial diffractometer with selected components was used for this work. Thewhole setup was installed at beamline ID15A of the ESRF. (See Chapt. 5.)

The high-energy x-ray scheme allows to apply established x-ray surface tech-niques at buried interfaces. In this work, x-ray reflectivity measurements havebeen used. These measurements are sensitive to the (electron) density profile per-pendicular to the interface. If a quasiliquid layer forms due to interfacial melting,it gives rise to interference fringes in the measured reflectivity profiles. Completedensity profiles can be reconstructed with the Parratt formalism. (See Chapt. 4.)

In this work, interfacial melting of ice at ice–SiO2–Si interfaces has been stud-ied. This particular interface might serve as a model for ice–mineral interfacesas they occur in nature. The SiO2–Si substrates were prepared from polished Sisingle crystals. After cleaning, a native amorphous oxide of 1–2 nm thicknessforms at the Si surface in air. The substrates become strongly hydrophilic whenexposed to high-energy x-ray radiation in the presence of water. Single crystalsof ice were provided by J. Bilgram (ETH Zurich). Smooth and homogeneousinterfaces were prepared by melting and subsequent recrystallization of the icesurface in contact with the substrate. The ice crystals were oriented with theirc-axis [0001] perpendicular to the interface. The sample preparation was carriedout in a walk-in cold room. For the Synchrotron experiments, a mobile samplechamber was constructed. It allows a precise control of the sample temperature,which is necessary for measurements close to the melting point. (See Chapt. 6.)

Two substrates with different morphology have been used in the experiments.The roughness has been analyzed with x-ray and AFM measurements. The‘smooth’ substrate exhibits an rms roughness of ≈ (2.7 ± 0.4) A and only weaklateral correlations. The ‘rough’ substrate can be described by means of a height-difference correlation function g(R) = 0.11 ·R2·0.34 (from the analysis of the x-raydata). (See Sec. 7.3.)

The x-ray reflectivity measurements clearly show that interfacial melting ofice occurs in contact with both the rough and the smooth substrate. Growthof the interfacial quasiliquid layer sets in at about −20C, but a very thin layerseems to persist down to lower temperatures. The thickness of the quasiliquidlayer increases with temperature and reaches 55 A at −0.036C for the roughsubstrate and 27 A at −0.022C for the smooth substrate.

Theory predicts a logarithmic growth of the quasiliquid layer thickness Las a function of the temperature T when exponentially decaying (short-ranged)interactions dominate:

L (T ) = L0 ln

(Tm − T0

Tm − T

), (9.1)

151

where Tm is the bulk melting temperature. A fit with a logarithmic growth lawmatches extremely well the measured data for the smooth substrate. It yields thegrowth amplitude L0 =(3.7± 0.3) A, which is slightly smaller than the reportedbulk correlation length of water (4.5 A to 8 A). In the case of the rough substrate,the fit with a logarithmic growth law does not match the measured data too well,although the growth amplitude L0 =(8.2 ± 0.4) A is still close to the reportedcorrelation length of water. (See Sec. 7.4.)

The quasiliquid layer thickness at the rough substrate agrees with the smoothsubstrate at low temperatures up to about −0.7C, corresponding to a layerthickness of about 16 A. From then on, the quasiliquid layer at the rough substrategrows much faster. The growth law for the rough substrate can also be describedby a power law

L (T ) ∝ (Tm − T )p (9.2)

with an exponent p close to −1/3 expected from theory for dominating (non-retarded) Van der Waals type dispersion forces (see Tab. 9). Since these in-teractions are long-ranged, a cross-over from a logarithmic to a power law canoccur. The observed difference in the growth laws for the rough and the smoothsubstrate may indicate a shift of the cross-over thickness due to the roughness.The mechanism for the roughness effect in this particular situation is not yetfully understood, but studies of roughness effects in other wetting scenarios offerpromising routes for a theoretical study of this aspect. (See Sec. 7.4.5.)

For both the rough and the smooth substrate, deviations from the logarithmicgrowth law are also visible at very low temperatures (corresponding to very smalllayer thicknesses). A thin layer seems to remain molten at temperatures belowthe onset expected from the fit of the layer thickness to a logarithmic growth law.In this regime, however, the continuum model leading to the logarithmic growthlaw is no longer valid. On the molecular scale, one might rather expect some sortof layering as suggested by the experimental observations. (See Sec. 7.4.3.)

The x-ray-reflectivity measurements also reveal the density ρqll of the inter-facial quasiliquid, 1.20 g/cm3 for the rough substrate, and 1.19 g/cm3 for thesmooth substrate. This is much higher than the density ρl=1.0 g/cm3 of bulkwater, but close to the density ρHDA=1.17–1.19 g/cm3 of high-density amorphous(HDA) ice at atmospheric pressure. This points to a close structural relationshipbetween the quasiliquid and HDA. (See Sec. 7.5.)

In the postulated two-phase scenario (see Sec. 2.4), the properties of waterare explained by fluctuations into a high-density liquid (HDL) and a low-densityliquid (LDL) form of water, terminating at a second critical point around 220 K(below the supercooling limit). The high-density and low-density amorphous ice(HDA and LDA) are the vitreous counterparts of the postulated liquid forms.

The fact that interfacial melting is an equilibrium phenomenon suggests thatthe quasiliquid is governed by fluctuations into the postulated (liquid) HDL in-stead of the metastable (solid) HDA. The fluctuations into HDL might be sta-


Table 9.1: Summary of the results.‘rough’ substrate ‘smooth’ substrate

rms roughness not defined∗ (2.7± 0.4) Alateral correlations g(R) = 0.11 ·R2·0.34 weaklogarithmic growth law

fit good for low T goodamplitude (8.2± 0.4) A (3.7± 0.3) A

power lawfit good strong deviations for high Texponent −(0.33± 0.03) −(0.29± 0.03)

average density 1.20 g/cm3 1.19 g/cm3

largest layer thickness 55 A at −0.036C 27.5 A at −0.022C∗ ‘local’ rms roughness (4.7± 1.1) A

bilized by the particular (hydrophilic) interface and by the confinement betweenthe ice and the substrate. A similar effect has recently been reported for waterin contact with hydrophobic substrates. In this case, a low-density form of waterhas been observed at the interface. (See Sec. 7.5.)

In this work, additional experiments were performed using an ultra-smoothSi wafer as substrate. An alternative chemical cleaning procedure has been usedin this case. It leads to a thick oxide layer with a hydrophilic termination (evenwithout irradiation). Unfortunately, measurements of complete reflectivity curveswere not possible due to bending of the thin wafer. The measurements never-theless show that interfacial melting occurs with this substrate as well. (SeeSec. 7.6.)

Other experiments have initially been performed with neutron reflectivity,the standard technique for probing buried interfaces. However, due to the smallflux of neutron sources, the range in momentum space accessible to reflectivitymeasurements and hence the resolution in real space are too limited. This demon-strates the necessity for and the potential of the new high-energy x-ray scheme.(See Sec. 7.7.)

The results of this work have important implications. The presence of aquasiliquid layer at ice–SiO2 interfaces has ramifications for many phenomenain nature, like the motion of glaciers and the stability of permafrost. Theseconsequences depend on the so far unknown properties of the quasiliquid. In thiscontext, the observation of a high-density form of water is important, since itsuggests differences in other properties, like the viscosity and the solubility forimpurities. As ice interfaces in nature are usually rough, the observed influence ofthe substrate morphology is relevant for conclusions about ‘real’ systems. Finally,interfaces or confinement situations as provided by interface melting can revealnew information about the intrinsic properties of water and may eventually allow

153

to study the postulated forms of water. (See Sec. 7.9.)A number of important questions is raised by this work. This includes the

influence of the specific substrate material, the role of the confinement, the effectof the substrate morphology, and the influence of impurities. Future experimentscould also cover a larger temperature range to study the transition from inter-face melting to bulk melting (which requires a better control of the temperature)and the initial stage of interface melting at low temperatures. The most intrigu-ing (and most challenging) task will certainly be to resolve the structure of theinterfacial quasiliquid. (See Chapt. 8.)

List of acronyms

AFM atomic force microscopy/microscopeCRL compound refractive lenseDFT density functional theoryDWBA Distorted-Wave Born ApproximationESRF European Synchrotron Radiation FacilityEVA evanescent wave diffractometer (instrument at the ILL)fcc face-centered cubicFWHM full width at half maximumHDA high-density amorphous iceHDL high-density liquid waterILL Institut Laue-LangevinLDA low-density amorphous iceLDL low-density liquid waterLJ Lennard-JonesMF mean-fieldNCS Neutron Compton ScatteringNMR nuclear magnetic resonancePSD position-sensitive detectorQENS quasi-elastic neutron sscatteringqll quasiliquid layerrms root mean squareSID surface induced disorderSIO surface induced orderSFVS sum-frequency vibrational spectroscopySR Synchrotron RadiationUHV ultra-high vacuumXPS x-ray photoelectron spectroscopy

155

156 LIST OF ACRONYMS

List of figures

2.1 Free water molecule. . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Response functions of water. . . . . . . . . . . . . . . . . . . . . . 6

2.3 H2O phase diagram for moderate and high pressures. . . . . . . . 7

2.4 The structure of ice Ih. . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Interface melting scenarios for ice. . . . . . . . . . . . . . . . . . . 12

3.2 Interface melting of a solid s in contact with another medium b. . 14

3.3 Free energy calculations and growth laws for different types ofinteractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Contributions to the free energy. . . . . . . . . . . . . . . . . . . . 18

3.5 Growth laws for interfacial melting of ice from the literature . . . 29

4.1 Reflection and refraction of a plane wave. . . . . . . . . . . . . . . 34

4.2 Reflection and refraction of a plane wave at multiple interfaces. . 36

4.3 Sketch of an interface contour z (R). . . . . . . . . . . . . . . . . 39

4.4 Illustration of an interface contour with a fixed (conformal) densityprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Schematic graph of specular and diffuse reflectivity. . . . . . . . . 43

5.1 Linear attenuation coefficient µ as a function of the x-ray energy. 48

5.2 Comparison of conventional x-ray surface scattering scheme andthe transmission-reflection scheme. . . . . . . . . . . . . . . . . . 49

5.3 Schematical sketch of the setup for the study of ice–solid interfaceswith high-energy x-ray microbeams. . . . . . . . . . . . . . . . . . 51

5.4 Schematic layout of the high-energy beamlines ID15A/B. . . . . . 52

5.5 Sketch of the setup for the high-energy microbeam transmission-reflection scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.6 Beam profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.7 Diffractometer with sample chamber. . . . . . . . . . . . . . . . . 55

5.8 Scattering geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.9 Illustration of a rocking scan. . . . . . . . . . . . . . . . . . . . . 58

5.10 Momentum transfer resolution δqz. . . . . . . . . . . . . . . . . . 59

5.11 Momentum transfer resolution δqx. . . . . . . . . . . . . . . . . . 60

157

158 LIST OF FIGURES

5.12 Illumination of the interface. . . . . . . . . . . . . . . . . . . . . . 635.13 Illumination of the interface with an arbitrary beam profile. . . . 645.14 Illumination correction. . . . . . . . . . . . . . . . . . . . . . . . . 645.15 Footprint of the beam on the interface. . . . . . . . . . . . . . . . 655.16 Coherence in x-ray scattering. . . . . . . . . . . . . . . . . . . . . 66

6.1 Figure error of the substrates. . . . . . . . . . . . . . . . . . . . . 726.2 Photograph of a Si block used as substrate. . . . . . . . . . . . . . 736.3 Reflectivity measurement of a substrate. . . . . . . . . . . . . . . 746.4 Cutting of ice crystals. . . . . . . . . . . . . . . . . . . . . . . . . 756.5 Interface preparation. . . . . . . . . . . . . . . . . . . . . . . . . . 776.6 Photographs of the sample. . . . . . . . . . . . . . . . . . . . . . 786.7 In situ chamber for x-ray and neutron scattering experiments with

ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.8 Schematic view of the control setup. . . . . . . . . . . . . . . . . . 816.9 Part of temperature log. . . . . . . . . . . . . . . . . . . . . . . . 816.10 Temperature distribution in the sample. . . . . . . . . . . . . . . 83

7.1 Time line of the experiment with the rough substrate. . . . . . . . 877.2 Time line of the experiment with the smooth substrate. . . . . . . 887.3 Reflectivity measurements for ice in contact with the rough substrate. 897.4 Reflectivity measurements for ice in contact with the smooth sub-

strate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.5 Reflectivity measurements on different positions of the sample. . . 917.6 Dispersion and density profiles. . . . . . . . . . . . . . . . . . . . 937.7 Model for fitting the reflectivity data. . . . . . . . . . . . . . . . . 947.8 Reconstructed density profiles for the rough substrate. . . . . . . 957.9 Reconstructed density profiles for the smooth substrate. . . . . . . 967.10 Illustration of the model for the ice–SiO2–Si interface. . . . . . . . 997.11 Comparison between kinematical and dynamical calculation of re-

flectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.12 Calculated reflectivity curves showing the influence of model pa-

rameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.13 Reliability of the fits. . . . . . . . . . . . . . . . . . . . . . . . . . 1027.14 Reliability of the roughness parameter σSi. . . . . . . . . . . . . . 1027.15 Rocking scans for the smooth substrate with different resolutions. 1037.16 Peak width in a rocking scan for the smooth substrate. . . . . . . 1047.17 Rocking scans with the smooth substrate for different qz. . . . . . 1047.18 Measurement of integrated intensity. . . . . . . . . . . . . . . . . 1067.19 Calculated rocking scan profiles for different values of the Hurst

parameter h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.20 Width of the diffuse reflectivity peak as a function of qz. . . . . . 1087.21 Roughness replication at substrate–quasiliquid–ice interface. . . . 109

LIST OF FIGURES 159

7.22 Comparison of calculated and measured rocking scans for differenth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.23 Comparison of calculated and measured rocking scans for differentqz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.24 AFM images of the substrates. . . . . . . . . . . . . . . . . . . . . 1127.25 Height-difference correlation functions g(R) determined from AFM

measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.26 Growth law for interfacial melting as expected by theory. . . . . . 1157.27 Growth law for the rough substrate. . . . . . . . . . . . . . . . . . 1177.28 Growth law for the smooth substrate. . . . . . . . . . . . . . . . . 1187.29 Influence of roughness on the growth law. . . . . . . . . . . . . . . 1197.30 Reversibility of the quasiliquid layer formation. . . . . . . . . . . 1207.31 Growth law for surface melting of ice. . . . . . . . . . . . . . . . . 1237.32 Effect of temperature error on observed growth law. . . . . . . . . 1247.33 Density of the quasiliquid layer. . . . . . . . . . . . . . . . . . . . 1257.34 Thickness of the SiO2 layer. . . . . . . . . . . . . . . . . . . . . . 1277.35 Schematic illustration of fluctuations in water. . . . . . . . . . . . 1307.36 Water density at Ag(111)/electrolyte interface. . . . . . . . . . . . 1317.37 Reflection from the backside of thin samples. . . . . . . . . . . . . 1317.38 Sketch of the high-energy setup with focusing multilayer. . . . . . 1337.39 Reflection from bent substrate. . . . . . . . . . . . . . . . . . . . 1347.40 Bending of the wafer. . . . . . . . . . . . . . . . . . . . . . . . . . 1347.41 Reflectivity measurements for ice in contact with a Si wafer. . . . 1357.42 Example of PSD recording for neutron reflectivity. . . . . . . . . . 1377.43 Scattering geometry for neutron reflectivity . . . . . . . . . . . . . 1377.44 Neutron reflectivity from D2O(ice)–SiO2–Si interfaces. . . . . . . . 1397.45 Photograph of substrate showing radiation induced change of the

hydrophilicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.46 XPS spectra of irradiated substrate. . . . . . . . . . . . . . . . . . 141

8.1 Calculated reflectivity profiles for various densities of the quasiliquid.144

160 LIST OF FIGURES

List of tables

5.1 Momentum transfer resolution. . . . . . . . . . . . . . . . . . . . 605.2 Coherence parameters. . . . . . . . . . . . . . . . . . . . . . . . . 68

7.1 Overview of experimental parameters. . . . . . . . . . . . . . . . . 867.2 Fit parameters for the rough substrate. . . . . . . . . . . . . . . . 977.3 Fit parameters for the smooth substrate. . . . . . . . . . . . . . . 987.4 Morphology of the substrates. . . . . . . . . . . . . . . . . . . . . 1147.5 Solid forms of H2O. . . . . . . . . . . . . . . . . . . . . . . . . . . 128

9.1 Summary of the results. . . . . . . . . . . . . . . . . . . . . . . . 152

161

162 LIST OF TABLES

Bibliography

[1] P. G. Debenedetti and H. E. Stanley. Supercooled and Glassy Water. Phys.Today, 56(6), 40 (2003).

[2] R. J. Speedy and C. A. Angell. Isothermal compressibility of supercooledwater and evidence for a thermodynamic singularity at -45C. J. Chem.Phys., 65, 851 (1976).

[3] A. Lied, H. Dosch, and J. H. Bilgram. Surface Melting of Ice Ih SingleCrystals Revealed by Glancing Angle X-Ray Scattering. Phys. Rev. Lett.,72, 3554 (1994).

[4] J. W. M. Frenken and H. M. van Pinxteren. Surface melting: an experi-mental overview. In D. A. King and D. P. Woodruff (eds.), Phase Transi-tions and Adsorbate Restructuring at Metal Surfaces (Elsevier, Amsterdam,1994).

[5] J. G. Dash, H. Y. Fu, and J. S. Wettlaufer. The premelting of ice and itsenvironmental consequences. Rep. Prog. Phys., 58, 115 (1995).

[6] H. Reichert, V. Honkimaki, A. Snigirev, S. Engemann, and H. Dosch. Anew X-ray transmission-reflection scheme for the study of deeply buriedinterfaces using high-energy microbeams. Physica B, 336, 46 (2003).

[7] M. Tolan. X-ray Scattering from Soft-Matter Thin Films (Springer, Berlin,1999).

[8] J. Bilgram, H. Wenzl, and G. Mair. Perfection of zone refined ice singlecrystals. J. Cryst. Growth, 20, 319 (1973).

[9] O. Mishima, L. D. Calvert, and E. Whalley. ”Melting ice”I at 77 K and 10kbar: a new method of making amorphous solids. Nature, 310, 393 (1984).

[10] O. Mishima, L. D. Calvert, and E. Whalley. An apparently first-ordertransition between two amorphous phases of ice induced by pressure. Nature,314, 76 (1985).

163

164 BIBLIOGRAPHY

[11] R. Steitz et al. Nanobubbles and Their Precursor Layer at the Interface ofWater Against a Hydrophobic Substrate. Langmuir, 19, 2409 (2003).

[12] T. R. Jensen et al. Water in Contact With Extended Hydrophobic Surfaces:Direct Evidence of Weak Dewetting. Phys. Rev. Lett., 90, 086101 (2003).

[13] H. Dosch. Critical Phenomena at Surfaces and Interfaces: Evanescent X-Ray and Neutron Scattering (Springer, Heidelberg, 1992).

[14] V. F. Petrenko and R. W. Whitworth. Physics of Ice (Oxford UniversityPress, Oxford, 1999).

[15] P. V. Hobbs. Ice Physics (Clarendon Press, Oxford, 1974).

[16] E. Whalley. Physics and chemistry of ice (Royal Soc. of Canada, Ottawa,1973).

[17] F. Franks (ed.). Water - A Comprehensive Treatise (Plenum Press, NewYork, London, 1972).

[18] P. Ball. H2O: a biography of water (Weidenfeld & Nicholson, London, 1999).

[19] U. Hellmann. Kunstliche Kalte - Die Geschichte der Kuhlung im Haushalt(Anabas, Giessen, 1990).

[20] T. Shachtmanuth. Minusgrade - Auf der Suche nach dem absolutenNullpunkt: Eine Chronik der Kalte (Rowohlt, Reinbek, 2001).

[21] C. Reinke-Kunze. Die PackEisWaffel. Von Gletschern, Schnee undSpeiseeis (Birkhauser, Basel, 1996).

[22] E. F. Burton and W. F. Oliver. The Crystal Structure of Ice at Low Tem-peratures. Proc. R. Soc. London, Ser. A, 153, 166 (1935).

[23] P. Bruggeller and E. Mayer. Complete vitrification in pure liquid water anddilute aqueous solutions. Nature, 288, 569 (1980).

[24] E. Mayer and P. Bruggeller. Vitrification of pure liquid water by high pres-sure jet freezing. Nature, 298, 715 (1982).

[25] O. Mishima. Reversible first-order transition between two H2O amorphs at∼0.2 GPa and ∼135 K. J. Chem. Phys., 100, 5910 (1994).

[26] P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley. Phase behaviourof metastable water. Nature, 360, 324 (1992).

[27] S. Sastry, P. G. Debenedetti, F. Sciortino, and H. E. Stanley. Singularity-free interpretation of the thermodynamics of supercooled water. Phys. Rev.E, 53, 6144 (1996).

BIBLIOGRAPHY 165

[28] O. Mishima and H. E. Stanley. Decompression-induced melting of ice IVand the liquid-liquid transition in water. Nature, 392, 164 (1998).

[29] O. Mishima and Y. Suzuki. Propagation of the polyamorphic transition ofice and the liquid-liquid critical point. Nature, 419, 599 (2002).

[30] C. A. Tulk et al. Structural Studies of Several Distinct Metastable Formsof Amorphous Ice. Science, 297, 1320 (2002).

[31] O. Mishima and H. E. Stanley. The relationship between liquid, supercooledand glassy water. Nature, 396, 329 (1998).

[32] I. Brovchenko, A. Geiger, and A. Oleinikova. Multiple liquid-liquid transi-tions in supercooled water. J. Chem. Phys., 118, 9473 (2003).

[33] C. A. Tulk et al. Another Look at Water and Ice - Response. Science, 299,45 (2003).

[34] T. Loerting, C. Salzmann, I. Kohl, E. Mayer, and A. Hallbrucker. A seconddistinct structural ”state”of high-density amorphous ice at 77 K and 1 bar.Phys. Chem. Chem. Phys., 3, 5355 (2001).

[35] J. L. Finney et al. Structure of a New Dense Amorphous Ice. Phys. Rev.Lett., 89, 205503 (2002).

[36] Y. Katayama et al. A first-order liquid-liquid phase transition in phospho-rus. Nature, 403, 170 (2000).

[37] G. Franzese, G. Malescio, A. Skibinsky, S. V. Buldyrev, and H. E. Stanley.Generic mechanism for generating a liquid-liquid phase transition. Nature,409, 692 (2001).

[38] T. M. Truskett, P. G. Debenedetti, S. Sastry, and S. Torquato. A single-bond approach to orientation-dependent interactions and its implicationsfor liquid water. J. Chem. Phys., 111, 2647 (1999).

[39] K. Rottger, A. Endriss, J. Ihringer, S. Doyle, and W. F. Kuhs. LatticeConstants and Thermal Expansion of H2O and D2O Ice Ih Between 10 and265 K. Acta Crystallogr. B, 50, 644 (1994).

[40] L. Pauling. The Structure and Entropy of Ice and of Other Crystals withSome Randomness of Atomic Arrangement. J. Am. Chem. Soc., 57, 2680(1935).

[41] W. F. Giauque and M. F. Ashley. Molecular Rotation in Ice at 10K. FreeEnergy of Formation and Entropy of Water. Phys. Rev., 43, 81 (1933).

166 BIBLIOGRAPHY

[42] D. M. Dennison. The Crystal Structure of Ice. Phys. Rev., 17, 20 (1921).

[43] S. W. H. Bragg. The Crystal Structure of Ice. Proc. Phys. Soc. London,34, 98 (1922).

[44] W. H. Barnes. The Crystal Structure of Ice between 0C and -183C. Proc.R. Soc. London, Ser. A, 125, 670 (1929).

[45] J. D. Bernal and R. H. Fowler. A Theory of Water and Ionic Solution, withParticular Reference to Hydrogen and Hydroxyl Ions. J. Chem. Phys., 1,515 (1933).

[46] E. O. Wollan, W. L. Davidson, and C. G. Shull. Neutron Diffraction Studyof the Structure of Ice. Phys. Rev., 75, 1348 (1949).

[47] S. W. Peterson and H. A. Levy. A Single-Crystal Neutron Diffraction Studyof Heavy Ice. Acta Crystallogr., 10, 70 (1957).

[48] A. Goto, T. Hondoh, and S. Mae. The electron density distribution in iceIh determined by single-crystal x-ray diffractometry. J. Chem. Phys., 93,1412 (1990).

[49] J. F. van der Veen, B. Pluis, and A. W. D. van der Gon. Surface Melting.In R. Vanselow and R. F. Howe (eds.), Chemistry and Physics of SolidSurfaces VII (Springer, Berlin, 1988).

[50] J. G. Dash. Surface Melting. Contemp. Phys., 30, 89 (1989).

[51] H. Lowen. Melting, freezing and colloidal suspensions. Phys. Rep., 237,249 (1994).

[52] A. C. Levi. Roughening, wetting and surface melting: theoretical consider-ations. In D. A. King and D. P. Woodruff (eds.), Phase Transitions andAdsorbate Restructuring at Metal Surfaces (Elsevier, Amsterdam, 1994).

[53] F. D. D. Tolla, E. Tosatti, and F. Ercolessi. Interplay of melting, wet-ting, overheating and faceting on metal surfaces: theory and simulation. InK. Binder and G. Ciccotti (eds.), Monte Carlo and Molecular Dynamcis ofCondensed Matter Systems (SIF, Bologna, 1996).

[54] A. A. Chernov. Roughening and Melting of Crystalline Surfaces. Prog.Cryst. Growth Ch., 26, 195 (1993).

[55] M. Bienfait. Roughening and surface melting transitions: consequences oncrystal growth. Surf. Sci., 272, 1 (1992).

[56] F. A. Lindemann. Uber die Berechnung molekularer Eigenfrequenzen. Phys.Z., 11, 609 (1910).

BIBLIOGRAPHY 167

[57] M. Ross. Generalized Lindemann Melting Law. Phys. Rev., 184, 233 (1969).

[58] M. Born. Thermodynamics of Crystals and Melting. J. Chem. Phys., 7,591 (1939).

[59] J. L. Tallon. The volume dependence of elastic moduli and the Born-Durandmelting hypothesis. Philos. Mag. A, 39, 151 (1979).

[60] J. H. Wang, S. Yip, S. R. Phillpot, and D. Wolf. Crystal Instabilities atFinite Strain. Phys. Rev. Lett., 71, 4182 (1993).

[61] J. Frenkel. Kinetic Theory of Liquids (Clarendon Press, Oxford, 1946).

[62] T. Gorecki. Vacancies of Changes of the First Coordination Sphere Radiusof Metals at the Melting Point. Z. Metallkd., 67, 269 (1976).

[63] F. H. Stillinger and T. A. Weber. Point defects in bcc crystals: Structures,transition kinetics, and melting implications. J. Chem. Phys., 81, 5095(1984).

[64] R. M. Cotterill, E. J. Jensen, and W. D. Kristensen. Melting: Theoriesand Recent Computer Simulations. In T. Riste (ed.), Anharmonic lattices,structural transitions and melting (Noordhoff, Leiden, 1974).

[65] F. Lund. Instability Driven by Dislocation Loops in Bulk Elastic Solids:Melting and Superheating. Phys. Rev. Lett., 69, 3084 (1992).

[66] B. J. Siwick, J. R. Dwyer, R. E. Jordan, and R. J. D. Miller. An Atomic-Level View of Melting Using Femtosecond Electron Diffraction. Science,302, 1382 (2003).

[67] R. Lipowsky. Surface critical phenomena at first-order phase transitions.Ferroelectrics, 73, 69 (1987).

[68] R. H. French. Origins and Applications of London Dispersion Forces andHamaker Constants in Ceramics. J. Am. Ceram. Soc., 83, 2117 (2000).

[69] D. M. Zhu and J. G. Dash. Surface Melting of Neon and Argon Films -Profile of the Crystal-Melt Interface. Phys. Rev. Lett., 60, 432 (1988).

[70] O. M. Magnussen et al. X-Ray Reflectivity Measurements of Surface Lay-ering in Liquid Mercury. Phys. Rev. Lett., 74, 4444 (1995).

[71] W. J. Huisman et al. Layering of a liquid metal in contact with a hard wall.Nature, 390, 379 (1997).

[72] W. J. Huisman and J. F. van der Veen. Modelling the atomic density acrossa solid-liquid interface. Surf. Sci., 404, 866 (1998).

168 BIBLIOGRAPHY

[73] R. Lipowsky. Critical Surface Phenomena at First-Order Bulk Transitions.Phys. Rev. Lett., 49, 1575 (1982).

[74] R. Lipowsky and W. Speth. Semi-infinite systems with first-order bulktransitions. Phys. Rev. B, 28, 3983 (1983).

[75] R. Lipowsky. Surface-induced order and disorder: Critical phenomena atfirst-order phase-transitions (invited). J. Appl. Phys., 55, 2485 (1984).

[76] G. Gompper and D. M. Kroll. Surface melting and surface-induced-disordertransitions in thin films: The effect of hidden variables. Phys. Rev. B, 40,7221 (1989).

[77] R. Lipowsky, U. Breuer, K. C. Prince, and H. P. Bonzel. MulticomponentOrder Parameter for Surface Melting. Phys. Rev. Lett., 62, 913 (1989).

[78] R. Lipowsky. Surface-Induced Disorder and Surface Melting. In L. M. Fal-icov, F. Mejıa-Lira, and J. L. Moran-Lopez (eds.), Magnetic Properties ofLow-Dimensional Systems II, Proceedings of the 2nd Workshop (Springer,Berlin, Heidelberg, 1990).

[79] R. Lipowsky, U. Breuer, K. C. Prince, and H. P. Bonzel. MulticomponentOrder Parameter for Surface Melting - Reply. Phys. Rev. Lett., 64, 2105(1990).

[80] B. Pluis, D. Frenkel, and J. F. van der Veen. Surface-Induced meltingand freezing. 2. A semi-empirical Landau-type model. Surf. Sci., 239, 282(1990).

[81] R. Lipowsky. Private communication.

[82] T. A. Cherepanova and A. V. Stekolnikov. Statistical Models of CrystalSurfaces - First-Principle Approaches. J. Cryst. Growth, 99, 88 (1990).

[83] R. Ohnesorge, H. Lowen, and H. Wagner. Density functional theory ofcrystal-fluid interfaces and surface melting. Phys. Rev. E, 50, 4801 (1994).

[84] A. Trayanov and E. Tosatti. Lattice Theory of Crystal Surface Melting.Phys. Rev. Lett., 59, 2207 (1987).

[85] A. Trayanov and E. Tosatti. Lattice theory of surface melting. Phys. Rev.B, 38, 6961 (1988).

[86] A. Trayanov and E. Tosatti. Surface Supermelting. Solid State Commun.,84, 177 (1992).

[87] L. Pietronero and E. Tosatti. Surface Theory of Melting. Solid State Com-mun., 32, 255 (1979).

BIBLIOGRAPHY 169

[88] C. S. Jayanthi, E. Tosatti, and L. Pietronero. Surface melting of copper.Phys. Rev. B, 31, 3456 (1985).

[89] C. S. Jayanthi, E. Tosatti, A. Fasolino, and L. Pietronero. Multilayer Re-laxation and Melting of a Metal Surface. Surf. Sci., 152, 155 (1985).

[90] J. Q. Broughton and L. V. Woodcock. A molecular dynamics study ofsurface melting. J. Phys. C: Solid State, 11, 2743 (1978).

[91] V. Pontikis and P. Sindzingre. Surface Melting and Roughening Transition.Phys. Scr., T19B, 375 (1987).

[92] P. Carnevali, F. Ercolessi, and E. Tosatti. Melting and nonmelting behaviorof the Au(111) surface. Phys. Rev. B, 36, 6701 (1987).

[93] R. N. Barnett and U. Landman. Surface premelting of Cu(110). Phys. Rev.B, 44, 3226 (1991).

[94] E. T. Chen, R. N. Barnett, and U. Landman. Crystal-melt and melt-vaporinterfaces of nickel. Phys. Rev. B, 40, 924 (1989).

[95] E. T. Chen, R. N. Barnett, and U. Landman. Surface melting of Ni(110).Phys. Rev. B, 41, 439 (1990).

[96] P. Stoltze, J. K. Norskov, and U. Landman. Disordering and Melting ofAluminum Surfaces. Phys. Rev. Lett., 61, 440 (1988).

[97] P. Stoltze. Simulations of the Premelting of Al(110). J. Chem. Phys., 92,6306 (1990).

[98] K. Stokbro et al. Incomplete melting of the Si(100) surface from molecular-dynamics simulations using the effective-medium tight-binding model. Surf.Sci., 360, 221 (1996).

[99] M. A. S. Barrera, J. F. Sanz, L. J. Alvarez, and J. A. Odriozola. Molecular-dynamics simulations of premelting processes in Cr2O3. Phys. Rev. B, 58,6057 (1998).

[100] P. Pfeifer, Y. J. Wu, M. W. Cole, and J. Krim. Multilayer Adsorption ona Fractally Rough Surface. Phys. Rev. Lett., 62, 1997 (1989).

[101] M. Napiorkowski, W. Koch, and S. Dietrich. Wedge wetting by van derWaals fluids. Phys. Rev. A, 45, 5760 (1992).

[102] G. Palasantzas. Wetting on rough self-affine surfaces. Phys. Rev. B, 51,14612 (1995).

170 BIBLIOGRAPHY

[103] R. R. Netz and D. Andelman. Roughness-induced wetting. Phys. Rev. E,55, 687 (1997).

[104] A. Esztermann et al. Triple-Point Wetting on Rough Substrates. Phys.Rev. Lett., 88, 055702 (2002).

[105] G. Palasantzas and G. M. E. A. Backx. Wetting of van der Waals solidfilms on self-affine rough surfaces. Phys. Rev. B, 68, 035412 (2003).

[106] K. Rommelse and M. Dennijs. Preroughening Transitions in Surfaces. Phys.Rev. Lett., 59, 2578 (1987).

[107] M. Bernasconi and E. Tosatti. Reconstruction, disordering and rougheningof metal surfaces. Surf. Sci. Rep., 17, 363 (1993).

[108] W. K. Burton, N. Cabrera, and F. C. Frank. The Growth of Crystals andthe Equilibrium Structure of their Surfaces. Philos. Trans. R. Soc. London,Ser. A, 243, 299 (1951).

[109] H. van Beijeren and I. Nolden. The Roughening Transition. In W. Schom-mers and P. V. Blanckenhagen (eds.), Structure and Dynamics of SurfacesII (Springer, Berlin, 1987).

[110] A. W. D. van der Gon, R. J. Smith, J. M. Gay, D. J. Oconnor, and J. F.van der Veen. Melting of Al Surfaces. Surf. Sci., 227, 143 (1990).

[111] J. W. M. Frenken and J. F. van der Veen. Observation of Surface Melting.Phys. Rev. Lett., 54, 134 (1985).

[112] S. G. J. Mochrie, D. M. Zehner, B. M. Ocko, and D. Gibbs. Structure andPhases of the Au(001) Surface - X-Ray-Scattering Measurements. Phys.Rev. Lett., 64, 2925 (1990).

[113] Y. J. Cao and E. H. Conrad. Approach to Thermal Roughening of Ni(110):A Study by High-Resolution Low-Energy Electron Diffraction. Phys. Rev.Lett., 64, 447 (1990).

[114] R. Trittibach, C. Grutter, and J. H. Bilgram. Surface Melting of GalliumSingle-Crystals. Phys. Rev. B, 50, 2529 (1994).

[115] A. Ruhm et al. Bulk and surface premelting phenomena in alpha-gallium.Phys. Rev. B, 68, 224110 (2003).

[116] D. M. Zhu and J. G. Dash. Surface Melting and Roughening of AdsorbedArgon Films. Phys. Rev. Lett., 57, 2959 (1986).

BIBLIOGRAPHY 171

[117] E. G. Mcrae and R. A. Malic. A New Phase-Transition at the Ge(111)Surface Observed by Low-Energy Electron-Diffraction. Phys. Rev. Lett.,58, 1437 (1987).

[118] K. Sumitomo, H. Hibino, Y. Homma, and T. Ogino. Observation of In-complete Surface Melting of Si Using Medium-Energy Ion Scattering Spec-troscopy. Jpn. J. Appl. Phys. 1, 39, 4421 (2000).

[119] S. Chandavarkar, R. M. Geertman, and W. H. Dejeu. Observation of aPrewetting Transition During Surface Melting of Caprolactam. Phys. Rev.Lett., 69, 2384 (1992).

[120] M. Bienfait. Surface Premelting of CH4 Thin Films. Europhys. Lett., 4,79 (1987).

[121] A. A. Chernov and V. A. Yakovlev. Thin Boundary Layers of the Melt ofa Biphenyl Single Crystal and Its Premelting. Langmuir, 3, 635 (1987).

[122] W. H. D. Jeu, J. D. Shindler, S. Chandavarkar, R. M. Geertman, andK. Liang. On the (absence of) surface melting in biphenyl. Surf. Sci., 342,341 (1995).

[123] D. Wallacher and K. Knorr. Melting and freezing of Ar in nanopores. Phys.Rev. B, 6310, 104202 (2001).

[124] Z. Q. Yang, L. L. He, S. J. Zhao, and H. Q. Ye. Experimental evidence ofstructural transition at the crystal-amorphous interphase boundary betweenAl and Al2O3. J. Phys.: Condens. Matter, 14, 1887 (2002).

[125] R. Goswami and K. Chattopadhyay. The superheating of Pb embedded ina Zn matrix - The role of interface melting. Philos. Mag. Lett., 68, 215(1993).

[126] N. H. Fletcher. Surface Structure of Water and Ice. II. A Revised Model.Philos. Mag., 18, 1287 (1968).

[127] N. H. Fletcher. Surface Structure of Water and Ice. Philos. Mag., 7, 255(1962).

[128] N. H. Fletcher. Surface Structure of Water and Ice - a Reply and a Cor-rection. Philos. Mag., 8, 1425 (1963).

[129] M. Elbaum and M. Schick. Application of the theory of dispersion forces tothe surface melting of ice. Phys. Rev. Lett., 66, 1713 (1991).

[130] L. A. Wilen, J. S. Wettlaufer, M. Elbaum, and M. Schick. Dispersion-forceeffects in interfacial premelting of ice. Phys. Rev. B, 52, 12426 (1995).

172 BIBLIOGRAPHY

[131] I. A. Ryzhkin and V. F. Petrenko. Violation of Ice Rules Near the Surface:a Theory for the Quasiliquid Layer. Phys. Rev. B, 65, 012205 (2002).

[132] T. A. Weber and F. H. Stillinger. Molecular Dynamics Study of Ice Crys-tallite Melting. J. Phys. Chem., 87, 4277 (1983).

[133] G. J. Kroes. Surface melting of the (0001) face of TIP4P ice. Surf. Sci.,275, 365 (1992).

[134] H. Nada and Y. Furukawa. Anisotropy in Structural Transitions BetweenBasal and Prismatic Faces of Ice Studied by Molecular Dynamics Simula-tion. Surf. Sci., 446, 1 (2000).

[135] J. S. Wettlaufer. Impurity Effects in the Premelting of Ice. Phys. Rev.Lett., 82, 2516 (1999).

[136] D. Nason and N. H. Fletcher. Photoemission from ice and water surfaces:Quasiliquid layer effect. J. Chem. Phys., 62, 4444 (1975).

[137] I. Golecki and C. Jaccard. Intrinsic Surface Disorder in Ice Near Melting-Point. J. Phys. C: Solid State, 11, 4229 (1978).

[138] D. Beaglehole and D. Nason. Transition Layer on the Surface on Ice. Surf.Sci., 96, 357 (1980).

[139] Y. Furukawa, M. Yamamoto, and T. Kuroda. Ellipsometric study of thetransition layer on the surface of an ice crystal. J. Cryst. Growth, 82, 665(1987).

[140] M. Elbaum, S. G. Lipson, and J. G. Dash. Optical Study of Surface Meltingon Ice. J. Cryst. Growth, 129, 491 (1993).

[141] X. Wei, P. B. Miranda, and Y. R. Shen. Surface Vibrational SpectroscopicStudy of Surface Melting of Ice. Phys. Rev. Lett., 86, 1554 (2001).

[142] H. Bluhm, D. F. Ogletree, C. S. Fadley, Z. Hussain, and N. Salmeron. Thepremelting of ice studied with photoelectron spectroscopy. J. Phys.: Condens.Matter, 14, L227 (2002).

[143] H. Dosch, A. Lied, and J. H. Bilgram. Disruption of the hydrogen-bondingnetwork at the surface of Ih ice near surface premelting. Surf. Sci., 366, 43(1996).

[144] A. Lied. Eine Untersuchung des Oberflachenschmelzens von Eis mittelsoberflachensensitiver Rontgenstreuung. Ph.D. thesis, Ludwig-Maximilians-Universitat Munchen (1993).

BIBLIOGRAPHY 173

[145] A. Lied, H. Dosch, and J. H. Bilgram. Glancing angle X-ray scattering fromsingle crystal ice surfaces. Physica B, 198, 92 (1994).

[146] H. Dosch, A. Lied, and J. H. Bilgram. Glancing-angle X-ray scatteringstudies of the premelting of ice surfaces. Surf. Sci., 327, 145 (1995).

[147] M. Maruyama, M. Bienfait, J. G. Dash, and G. Coddens. Interfacial Melt-ing of Ice in Graphite and Talc Powders. J. Cryst. Growth, 118, 33 (1992).

[148] J. W. Cahn, J. G. Dash, and H. Y. Fu. Theory of Ice Premelting in Mono-sized Powders. J. Cryst. Growth, 123, 101 (1992).

[149] J. M. Gay, J. Suzanne, J. G. Dash, and H. Y. Fu. Premelting of ice inexfoliated graphite; a neutron diffraction study. J. Cryst. Growth, 125, 33(1992).

[150] M.-C. Bellissent-Funel, J. Lal, and L. Bosio. Structural study of waterconfined in porous glass by neutron scattering. J. Chem. Phys., 98, 4246(1993).

[151] T. Ishizaki, M. Maruyama, Y. Furukawa, and J. G. Dash. Premelting ofice in porous silica glass. J. Cryst. Growth, 163, 455 (1996).

[152] A. Schreiber, I. Ketelsen, and G. H. Findenegg. Melting and freezing ofwater in ordered mesoporous silica materials. Phys. Chem. Chem. Phys.,3, 1185 (2001).

[153] Y. Furukawa and I. Ishikawa. Direct evidence for melting transition atinterface between ice crystal and glass substrate. J. Cryst. Growth, 128,1137 (1993).

[154] D. Beaglehole and P. Wilson. Extrinsic Premelting at the Ice-Glass Inter-face. J. Phys. Chem., 98, 8096 (1994).

[155] Q. Du, E. Freysz, and Y. R. Shen. Vibrational Spectra of Water Moleculesat Quartz/Water Interfaces. Phys. Rev. Lett., 72, 238 (1994).

[156] Y. R. Shen. 1998 Frank Isakson Prize Address - Sum Frequency Generationfor Vibrational Spectroscopy: Applications to Water Interfaces and Filmsof Water and Ice. Solid State Commun., 108, 399 (1998).

[157] X. Wei. Sum-Frequency Spectroscopic Studies I. Surface Melting of Ice II.Surface Alignment of Polymers. Ph.D. thesis, University of California atBerkeley (2000).

[158] X. Wei and Y. R. Shen. Vibrational spectroscopy of ice interfaces. Appl.Phys. B, 74, 617 (2002).

174 BIBLIOGRAPHY

[159] X. Wei, P. B. Miranda, C. Zhang, and Y. R. Shen. Sum-frequency spectro-scopic studies of ice interfaces. Phys. Rev. B, 66, 085401 (2002).

[160] V. F. Petrenko. Study of the Surface of Ice, Ice/Solid and Ice/Liquid In-terfaces with Scanning Force Microscopy. J. Phys. Chem. B, 101, 6276(1997).

[161] H. Bluhm, T. Inoue, and M. Salmeron. Friction of ice measured usinglateral force microscopy. Phys. Rev. B, 61, 7760 (2000).

[162] T. Eastman and D. M. Zhu. Influence of an AFM Tip on Interfacial Meltingon Ice. J. Colloid Interface Sci., 172, 297 (1995).

[163] B. Pittenger et al. Premelting at ice-solid interfaces studied via velocity-dependent indentation with force microscope tips. Phys. Rev. B, 63, 134102(2001).

[164] C. R. Slaughterbeck et al. Electric field effects on force curves for oxi-dized silicon tips and ice surfaces in a controlled environment. J. Vac. Sci.Technol., A, 14, 1213 (1996).

[165] B. Pittenger, D. J. Cook, C. R. Slaughterbeck, and S. C. J. Fain. Investi-gation of ice-solid interfaces by force microscopy: Plastic flow and adhesiveforces. Journal of Vacuum Science & Technology A, 16, 1832 (1998).

[166] A. Doppenschmidt, M. Kappl, and H.-J. Butt. Surface Properties of IceStudied by Atomic Force Microscopy. J. Phys. Chem. B, 102, 7813 (1998).

[167] A. Doppenschmidt and H.-J. Butt. Measuring the Thickness of the Liquid-like Layer on Ice Surfaces with Atomic Force Microscopy. Langmuir, 16,6709 (2000).

[168] H.-J. Butt, A. Doppenschmidt, G. Huttl, E. Muller, and O. I. Vinogradova.Analysis of plastic deformation in atomic force microscopy: Application toice. J. Chem. Phys., 113, 1194 (2000).

[169] J. Q. Broughton and G. H. Gilmer. Thermodynamic Criteria for Grain-Boundary Melting: A Molecular-Dynamics Study. Phys. Rev. Lett., 56,2692 (1986).

[170] G. P. Johari, W. Pascheto, and S. J. Jones. Intergranular liquid in solidsand premelting of ice. J. Chem. Phys., 100, 4548 (1994).

[171] H. H. G. Jellinek. Adhesive properties of ice. Journal of Colloid Science,14, 268 (1959).

BIBLIOGRAPHY 175

[172] J. S. Wettlaufer and J. G. Dash. Melting Below Zero. Sci. Am., 282(2),50 (2000).

[173] J. S. Wettlaufer, G. J. Dash, and N. Untersteiner. Ice Physics and theNatural Environment (Springer, Berlin, Heidelberg, 1999).

[174] P. J. Williams. The Freezing of Soils: Ice in a Porous Medium and ItsEnvironmental Significance. In J. S. Wettlaufer, G. J. Dash, and N. Unter-steiner (eds.), Ice Physics and the Natural Environment (Springer, Berlin,Heidelberg, 1999).

[175] W. S. B. Paterson. Some Aspects of the Physics of Glaciers. In J. S.Wettlaufer, G. J. Dash, and N. Untersteiner (eds.), Ice Physics and theNatural Environment (Springer, Berlin, Heidelberg, 1999).

[176] K. M. Cuffey, H. Conway, B. Hallet, A. M. Gades, and C. F. Raymond.Interfacial water in polar glaciers and glacier sliding at -17 C. Geophys.Res. Lett., 26, 751 (1999).

[177] G. J. Turner and C. D. Stow. The quasi-liquid film on ice. Evidence from,and implications for contact charging events. Philos. Mag. A, 49, L25(1984).

[178] M. B. Baker and J. G. Dash. Mechanism of charge transfer between collidingice particles in thunderstorms. J. Geophys. Res., [Atmos.], 99, 10621 (1994).

[179] J. P. D. Abbatt et al. Interaction of HCl Vapor With Water-ice: Implica-tions for the Stratosphere. J. Geophys. Res., [Atmos.], 97, 15819 (1992).

[180] D. Meschede and C. Gerthsen. Gerthsen Physik (Springer, Berlin, 2002).

[181] S. C. Colbeck. Pressure Melting and Ice Skating. Am. J. Phys., 63, 888(1995).

[182] S. C. Colbeck, L. Najarian, and H. B. Smith. Sliding Temperatures of IceSkates. Am. J. Phys., 65, 488 (1997).

[183] J. J. de Koning, G. de Groot, and G. J. van Ingen Schenau. Ice FrictionDuring Speed Skating. J. Biomech., 25, 565 (1992).

[184] M. Born and E. Wolf. Principles of Optics: Electromagnetic Theory ofPropagation, Interference and Diffraction of Light (Cambridge UniversityPress, Cambridge, 1999).

[185] C. T. Chantler et al. X-Ray Form Factor, Attenuation and Scattering Tables(version 2.0). http://physics.nist.gov/ffast.

176 BIBLIOGRAPHY

[186] L. G. Parratt. Surface Studies of Solids by Total Reflection of X-Rays.Phys. Rev., 95, 359 (1954).

[187] W. Schweika, H. Reichert, W. Babik, O. Klein, and S. Engemann. Strain-induced incomplete wetting at CuAu(001) surfaces. Phys. Rev. B, 70,041401(R) (2004).

[188] B. Nickel, W. Donner, H. Dosch, C. Detlefs, and G. Grubel. Critical Ad-sorption and Dimensional Crossover in Epitaxial FeCo Films. Phys. Rev.Lett. (85).

[189] A. K. Doerr. Untersuchung der strukturellen Eigenschaften dunner Benet-zungsfilme. Ph.D. thesis, Christian-Albrechts-Universitat (1999).

[190] S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley. X-ray and neutronscattering from rough surfaces. Phys. Rev. B, 38, 2297 (1988).

[191] M. Rauscher et al. (unpublished).

[192] M. Rauscher. Rontgenstreuung an rauhen Oberflachen. Ph.D. thesis,Ludwig-Maximilians-Universitat (1995).

[193] H. Reichert et al. Observation of five-fold local symmetry in liquid lead.Nature, 408, 839 (2000).

[194] M. Denk. Structural Investigation of Solid-Liquid Interfaces - Metal-Semiconductor Interface. Ph.D. thesis, Universitat Stuttgart (to be pub-lished).

[195] P. Suortti and T. Tschentscher. High-energy scattering beamlines at Euro-pean Synchrotron Radiation Facility. Rev. Sci. Instrum., 66, 1798 (1995).

[196] P. Suortti and C. Schulze. Fixed-Exit Monochromators for High-EnergySynchrotron Radiation. J. Synchrotron Radiat., 2, 6 (1995).

[197] P. Suortti et al. Monochromators for High-Energy Synchrotron Radiation.Z. Phys. Chem., 215, 1419 (2001).

[198] A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler. A compound refractivelens for focusing high-energy X-rays. Nature, 384, 49 (1996).

[199] B. Lengeler et al. A microscope for hard x rays based on parabolic compoundrefractive lenses. Appl. Phys. Lett., 74, 3924 (1999).

[200] C. G. Schroer et al. Parabolic Compound Refractive Lenses for Hard X-Rays. In A. K. Freund et al. (eds.), Advances in X-Ray Optics (Bellingham,Washington, 2001).

BIBLIOGRAPHY 177

[201] B. Lengeler. Coherence in X-ray physics. Naturwissenschaften, 88, 249(2001).

[202] S. K. Sinha, M. Tolan, and A. Gibaud. Effects of partial coherence on thescattering of x rays by matter. Phys. Rev. B, 57, 2740 (1998).

[203] Q.-Y. Tong and U. Gosele. Semiconductor wafer bonding: science andtechnology (Wiley, New York, 1999).

[204] R. K. Iler. The Chemistry of Silica: Solubility, Polymerization, Colloid andSurface Properties, and Biochemistry (Wiley, New York, 1979).

[205] A. P. Legrand (ed.). The surface properties of silicas (Wiley, Chichester,1998).

[206] R. A. B. Devine, J.-P. Duraud, and E. Dooryhee. Structure and Imper-fections in Amorphous and Crystalline Silicon Dioxide (Wiley, Chichester,England, 2000).

[207] V. Lehmann. Electrochemistry of Silicon: Instrumentation, Science, Mate-rials and Applications (Wiley, 2002).

[208] J. Bilgram. Private communication.

[209] http://www.thermo.com/.

[210] http://www.heraeus-sensor-nite.de/.

[211] http://www.lakeshore.com/.

[212] http://www.agilent.com/.

[213] A. Ruhm. http://dxray.mpi-stuttgart.mpg.de/en/abteilungen/dosch/software/mfintro en.shtml.

[214] Q. M. Hudspeth et al. How does a multiwalled carbon nanotube atomic forcemicroscopy probe affect the determination of surface roughness statistics?Surf. Sci., 515, 453 (2002).

[215] A.-L. Barabasi and H. E. Stanley. Fractal concepts in surface growth (Cam-bridge University Press, Cambridge, 1995).

[216] M. E. R. Dotto and M. U. Kleinke. Kinetic roughening in etched Si. PhysicaA, 295, 149 (2001).

[217] M. E. R. Dotto and M. U. Kleinke. Scaling laws in etched Si surfaces. Phys.Rev. B, 65, 245323 (2002).

178 BIBLIOGRAPHY

[218] H. Ogasawara et al. Structure and Bonding of Water on Pt(111). Phys.Rev. Lett., 89, 276102 (2002).

[219] Y. L. Xie, K. F. Ludwig, Jr., G. Morales, D. E. Hare, and C. M. Sorensen.Noncritical Behavior of Density Fluctuations in Supercooled Water. Phys.Rev. Lett., 71, 2050 (1993).

[220] L. Bosio, J. Teixeira, and H. E. Stanley. Enhanced Density Fluctuationsin Supercooled H2O, D2O, and Ethanol-Water Solutions: Evidence FromSmall-Angle X-Ray Scattering. Phys. Rev. Lett., 46, 597 (1981).

[221] S. I. Mamatkulov, P. K. Khabibullaev, and R. R. Netz. Water at Hydropho-bic Substrates: Curvature, Pressure, and Temperature Effects. Langmuir,20, 4756 (2004).

[222] D. L. Rousseau and S. P. S. Porto. Polywater: Polymer or Artifact? Sci-ence, 167, 1715 (1970).

[223] R. C. Weast (ed.). CRC Handbook of chemistry and physics (CRC Press,Boca Raton, 1989).

[224] M. Chaplin. Water Structure and Behavior.http://www.lsbu.ac.uk/water/index.html.

[225] J. Wang, B. M. Ocko, A. J. Davenport, and H. S. Isaacs. In situ x-ray-diffraction and -reflectivity studies of the Au(111)/electrolyte interface: Re-construction and anion adsorption. Phys. Rev. B, 46, 10321 (1992).

[226] M. F. Toney et al. Voltage-dependent ordering of water molecules at anelectrode-electrolyte interface. Nature, 368, 444 (1994).

[227] V. F. Sears. Neutron scattering lengths and cross sections. Neutron News,3, 26 (1992).

[228] H. Dosch, K. A. Usta, A. Lied, W. Drexel, and J. Peisl. The evanescentneutron wave diffractometer: On the way to surface sensitive neutron scat-tering. Rev. Sci. Instrum. (63).

[229] I. Kirshenbaum. Physical properties and analysis of heavy water (McGraw-Hill, New York, 1951).

[230] S. Iarlori, D. Ceresoli, M. Bernasconi, D. Donadio, and M. Parrinello. De-hydroxylation and Silanization of the Surfaces of beta-Cristobalite Silica:An ab Initio Simulation. J. Phys. Chem. B, 105, 8007 (2001).

[231] G. M. Rignanese, J. C. Charlier, and X. Gonze. First-principles molecular-dynamics investigation of the hydration mechanisms of the (0001) alpha-quartz surface. Phys. Chem. Chem. Phys., 6, 1920 (2004).

Acknowledgements

Hauptberichter: Helmut DoschMitberichter: Clemens BechingerBetreuer: Harald Reichert

Special thanks to

Jorg Bilgram, Matthias Denk, Veijo Honkimaki, Oliver Klein, Markus Rauscher,Sebastian Schoder

and

Frank Adams, Stephanie Adelhelm, Esther Barrena, Jean-Francois Chemin, P.Dreier, Alwin Engemann, Christa Engemann, Felix Engemann, Helen Engemann,Dimas Garcia de Oteyza Feldermann, Robert Fendt, Ulrich Gebhardt, ErnstGunther, Christian Gutt, Rolf Henes, Lars Jeurgens, Peter Keppler, Anne-CecileLacroix, Klaus Mecke, Bert Nickel, Ben Ocko, Walter Plenert, Craig Priest, IngoRamsteiner, Adrian Ruhm, Werner Schweika, Andreas Schops, Udo Seifert, Ana-toly Snigirev, David Snoswell, Michael Sprung, Eugene Stanley, Reinhard Streitel,Ulrike Taffner, Metin Tolan, Philippe Villermet, Alexei Vorobiev, Peter Weiss,Annette Weißhardt, Helmut Wendel, Gunther Wiederoder, Michaela Wieland,Martin Zimmermann, the MPI machine shops, the OZ-team, the low-temperatureservice group, those I forgot.

This work has been funded by the Deutsche Forschungsgemeinschaft in the pri-ority program on ”wetting and structure formation at interfaces”.

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