Hysteresis Phenomena in Me So Porous Materials

Hysteresis Phenomena in

Mesoporous Materials

Von der Fakultät für Physik und Geowissenschaften

der Universität Leipzig

genehmigte

DISSERTATION

zur Erlangung des akademischen Grades

Doctor rerum naturalium

(Dr. rer. nat.)

vorgelegt von

Dipl.-Phys. Sergej Naumov

geboren am 31.10.1980 in Pskov

Gutachter: Prof. Dr. Jörg Kärger (Universität Leipzig)

Prof. Dr. Keith E. Gubbins (NC State University, USA)

Tag der Verleihung: 20.07.2009

Abstract

Sergej NaumovHysteresis Phenomena in Mesoporous Materials

Universitat Leipzig, Dissertation, 200995 pages, 112 references, 37 figures, 7 tables

This thesis deals with the recent efforts to elucidate the origin of the adsorptionhysteresis phenomenon typical for mesoporous materials. Utilizing the capabilitiesof pulsed field gradient nuclear magnetic resonance, the macroscopic information,accessible by transient sorption experiments, and the microscopic information, pro-vided by the effective self-diffusivities, have been correlated and thus shown to yieldfurther insight into the adsorption dynamics and the equilibrium properties of guestmolecules in mesopores. In particular, two mechanisms of molecular transport,namely self-diffusion and activated redistribution of the fluid in the pores, havebeen elucidated.

Basing on this finding, an explanation for the slowing down of the transientuptake with the onset of capillary condensation, observed in experiments, has beengiven. The activated nature of nucleation, growth and redistribution of the fluidphase inside the pores prevents equilibration on an experimental time scale.

Adsorption behavior in electrochemically etched porous silicon with linear poreshas been studied by means of Mean Field Theory. It has been shown that the direct-ing feature of many puzzling observations is the existence of a mesoscalic disorder,exceeding the disorder on an atomistic level. Thus, the linear, non-interconnectedchannels in mesoporous silicon turn out to exhibit all effects commonly associatedwith three-dimensional disordered networks.

i

Contents

1 Introduction 11.1 Hysteresis Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aims of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Basics 92.1 Liquid-Gas Phase Transition . . . . . . . . . . . . . . . . . . . . . . . 92.2 Capillary Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Adsorption Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Adsorption Hysteresis . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Sorption Scanning Curves . . . . . . . . . . . . . . . . . . . . 14

2.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Diffusion in Pores . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Materials and Methods 193.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Porous Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Porous Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Pulsed Field Gradient NMR . . . . . . . . . . . . . . . . . . . . . . . 223.3 Adsorption Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Adsorption from Vapour Phase . . . . . . . . . . . . . . . . . 263.3.2 BelSorp Mini II . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Mean Field Theory Approach . . . . . . . . . . . . . . . . . . . . . . 29

4 Random Pore Network 314.1 Adsorption and Diffusion Hysteresis . . . . . . . . . . . . . . . . . . . 324.2 Sorption Kinetics: Strong Surface Field . . . . . . . . . . . . . . . . . 434.3 Sorption Kinetics: Weak Surface Field . . . . . . . . . . . . . . . . . 484.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 One-Dimensional Channels 555.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Effect of Mesoscopic Disorder and

Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 Chemical Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 67

iii

5.4 Role of External Surface . . . . . . . . . . . . . . . . . . . . . . . . . 675.5 Effect of Pore Openings . . . . . . . . . . . . . . . . . . . . . . . . . 685.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Summary 75

Bibliography 77

Acknowledgements 85

Appendix 87VaporControl Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

List of Publications 91

Curriculum Vitae 95

iv

Chapter 1

Introduction

1.1 Hysteresis Phenomena

Systems exhibiting hysteresis behavior may be summarised as systems where the

output depends not only on the input parameters but also on the history how the

current state has been attained. In the case of a deterministic system, one can

predict the output at some instant of time only knowing the current state of the

system. The numerous examples of such systems include magnetisation hysteresis

in ferromagnetic materials, elastic hysteresis, temperature behavior in thermostats,

disposal development/unemployment hysteresis in economy, reaction hysteresis upon

neuron stimulus in neuroscience, and many others.

In general, complex systems composed of subdomains with interactions between

them can exhibit various hysteresis phenomena. It is also intuitively clear that it

is typical of non-equilibrium systems and, thus, is a sign of the departure from

the global equilibrium. The inability to attain a global equilibrium state, i.e. the

most favourable state with the lowest free energy, leads to the hysteretic behavior.

Consequently, hysteresis points out the existence of quasi-equilibrium, metastable

states, representing the local minima in the free energy, where the system becomes

trapped for relatively long periods of time.

Confined fluids in mesoporous materials ([1]), with typical pore sizes from 2 to 50

nm, usually exhibit the so-called adsorption hysteresis upon variation of the exter-

nal parameters like pressure, chemical potential or temperature in the temperature

range below the critical temperature ([2, 3, 4, 5]). The phenomenon of adsorption

hysteresis with many accompanying features, especially related to dynamical prop-

1

CHAPTER 1. INTRODUCTION

erties of the system in the hysteresis regime, will be in the focus of the present work.

The main aim of this chapter is to give a very general introduction to the present

state of art in this field.

Adsorption from gas phase may be understood as the enrichment of molecules

in an interfacial layer adjacent to a solid wall. The solid is typically called the

absorbent and the fluid which is adsorbed is called the adsorbate ([1]). There are

two ways how a molecule can be adsorbed on the surface: physisorbed molecules are

kept on the surface by the weak van der Waals forces, while chemisorbed molecules

become part of the solid. Thus, physisorption, which will be considered throughout

this work, is typically a reversible process. The equilibrium amount adsorbed on

a surface is determined by the involved intermolecular (fluid-solid and fluid-fluid)

interactions and the external parameters, such as temperature and gas pressure.

However, if one considers not flat but curved surfaces, which one finds, e.g., in

mesoporous materials, the physisorption processes may occur irreversibly, i.e., the

amounts adsorbed upon increasing or decreasing of gas pressure do not coincide

over a certain interval of pressures. This phenomenon is associated with liquid-gas

phase transitions under the porous confinement, which is referred to as capillary

condensation. In 1871, Sir William Thomson (also known as Lord Kelvin) first

described the change in vapour pressure due to a curved liquid/vapour interface.

This finding further may be used to understand the phenomenon that in a capillary

a fluid condenses at a vapour pressure below the bulk saturated vapour pressure.

The corresponding equation, relating the curvature of the liquid-gas interface and the

transition vapour pressure, called the Kelvin equation, will be discussed in Chap. 2.

There is a big diversity of studies of adsorption behavior and capillary conden-

sation in mesopores reported since more than one century. Besides the fundamental

interest, there is a big concern about the adsorption hysteresis in porous materials,

since the adsorption experiments are considered being one of the most powerful tools

to study the material structural properties [6].

In 1907, Zsigmondy ([7]) postulated that hysteresis may arise from the difference

in contact angles between the fluid and the wall during pore filling and emptying.

McBain ([8]) modified this theory in 1935 by proposing that the narrowing at the

pore openings can deteriorate the access of the external gas phase to the pore inte-

rior on the desorption. Nowadays, this phenomenon is referred to as pore-blocking

or ”ink-bottle” effect. Taking account of this effect, it is assumed that during ad-

sorption the pores are filled by the liquid in the order of increasing pore radii (as

2

1.1. HYSTERESIS PHENOMENA

predicted by the Kelvin equation (see Chap. 2)), while on desorption the dimension

of the pore necks (narrowest parts of the pore structure) controls the emptying of

the pores. This phenomenon has been a subject of intense research and it was shown

that it strongly depends on particular details of the pore geometry and the fluid-

wall interactions ([9]). Later on, in 1938, Cohan ([10]) proposed a still generally

accepted theory of hysteresis which relates the hysteresis to the different geometry

of the liquid-gas interface during adsorption (cylindrically concave) and desorption

(semi-spherically concave).

In order to account for a complex structure and distribution of pore sizes typ-

ical of most porous materials, Everett et al. introduced the so-called independent

domain model ([11, 12, 2]). The main assumption of this theory is based on the

representation of the porous network as an ensemble of independent pores, whose

behaviours during capillary condensation and evaporation do not depend on each

other. Considering the critical conditions at which vapour-liquid and liquid-vapour

transitions occur, the phenomenon of the adsorption hysteresis was then understood

at the level of one pore and brought about to the level of the entire pore network.

This model may well capture the basic features for a system of independent paral-

lel channels with negligible diameter variation along the pores, as can be found in

MCM-41 ([13]) or some conformations of SBA-15 materials ([14, 15]).

Importantly, this model may also explain the so-called scanning experiment,

where one records the amount adsorbed during incomplete filling and emptying

cycles. Such scanning curves show a whole hierarchy of subloops inside the major

hysteresis loop. In general, the behavior of the scanning curves is very important for

the validation of a developed theoretical model ([16]). Although the independent

domain model was a step forward and made a substantial advance in the field,

Everett himself pointed out that a more general theory of adsorption is necessary,

which especially considers the correlations that arise among the voids in a pore

network in order to adequately interpret sorption experiments ([17]).

Studying adsorption scanning curves on MCM-41, McNall et al. concluded that

no single model can account for all details of the measured adsorption hysteresis

data ([18]). Even in the case of seemingly ideal channels of MCM-41, there are

impurities, roughness and defects which may have effect upon adsorption behavior

([19, 20]).

In porous materials with random pore structure, found, e.g., in porous glasses

or silica gels, network effects seem to have a very strong influence and are a very

3


plausible argument for the adsorption hysteresis phenomenon. Mason introduced

a pore network model in the early 80’s ([21, 22]). He considered the effects of the

interconnections on the capillary condensation processes in interconnected pores.

For pores with a single internal cavity and four constrictions connecting this cavity to

the adjacent ones, full (containing capillary-condensed liquid) and empty (containing

gas phase in the pore interior) pores may have various configurations of such full

and empty neighbors. These different configurations have been used in the analysis

of the behavior of the material during adsorption and desorption. The model for

adsorption predicted a simple relation between adsorption scanning curves inside the

major hysteresis loop irrespective of the number of interconnections of the pores.

The last two decades are certainly marked by the rapid development of the

information technology in any domain of science. The enormous number of publi-

cations in this field show the particular importance of the methods developed, such

as Monte Carlo (MC), Molecular Dynamics (MD) simulations, and lattice based

calculations. Implementation of the microscopic properties of real materials in the

computer simulations confirmed the earlier suggestions that no single theory may

thoroughly describe all experimental data on adsorption.

In a comprehensive work of Evans, Marconi and Tarazona [23] calculations of

phase transitions in confined geometries by means of the Density Functional Theory

(DFT) have been presented. They confirmed the necessity to revise the Kelvin

equation for the analysis of adsorption experiments. The authors associated the

steep capillary-condensation transition with the limit of stability of the adsorbed

liquid layer on the pore walls. The steep knee on the desorption branch was related

to the limit of mechanical stability of the capillary-condensed liquid in the pores.

In [24], Marconi and van Swall reconsidered the role of the meniscus development

in the adsorption hysteresis by means of Mean Field DFT. They used a lattice model

of a slit pore to study capillary condensation processes in a finite and in an infinite

pore. It has been shown that the adsorption behavior in a slit pore with finite

length placed in contact with bulk fluid is very different from an infinite (boundary

conditions). The vapour liquid interface, thus, seem to have a strong influence on

the adsorption behavior.

In [9], Sarkisov and Monson presented a study of adsorption and desorption

in well-defined pore geometries implicitly taking account of diffusive mass transfer

using Molecular Dynamics (MD) simulations. The most significant result of their

work was the absence of the pore-blocking in an ”ink-bottle” configuration. The au-

4

1.1. HYSTERESIS PHENOMENA

thors have shown that evaporation from a larger cavity can occur even though the

neck of the pore remains filled with liquid. Notably, a variety of hysteresis models

(”ink-bottle”, interconnected network) are based on the assumption of pore-blocking

giving a simple explanation of the hysteresis phenomenon. The authors suggest that

the development of more realistic models describing real materials shall lead to the

confirmation of the fluid behavior in pore as observed in experiments ([25, 26]).

Later on, Woo at al. ([27]) studied the desorption from disordered mesoporous ma-

terials such as Vycor glass ([28]) by means of dynamic Monte Carlo simulations

with Kawasaki dynamics ([29]). They have observed the development of the fluid

configurations along the desorption isotherms due to the advancement of macro-

scopic front interfaces towards the interior. Importantly, the interface progress was

preceded and, possibly, initiated by a bubble nucleation (or cavitation) mechanism

on a length scale determined by the pore size and fluid-wall interaction ratio.

The transition between the cavitation and the pore-blocking regimes of the evap-

oration from the ink-bottle type pores has been observed by Ravikovitch et al.

studying the temperature dependence of the hysteresis loop ([30]). The analysis of

the hysteresis loops and the scanning isotherms revealed that evaporation from the

blocked cavities controlled by the size of connecting pores (classical ink-bottle or

pore blocking effect), but also spontaneous evaporation caused by cavitation of the

stretched metastable liquid may occur. The authors have found a near-equilibrium

evaporation in the region of hysteresis from unblocked cavities that have access to

the external vapour phase. In [31], the same authors studied the adsorption in

spherical cavities by means of the Nonlocal DFT. The method shows that for small

cavities with pore diameter ranging from 3 to 6 nm, the capillary condensation oc-

curs reversibly, while in bigger cavities the adsorption step corresponds to the limit

of thermodynamic stability of the adsorbed film.

Recently, Woo et al. applied Monte Carlo simulations on model systems at

the molecular level ([32]). The authors showed that, for a disordered pore network,

attractive interactions between pore walls and the fluid can suppress the macroscopic

phase separation. This makes the density relaxation rate, i.e. the redistribution of

the fluid in the pores, increasingly slow. Most importantly, it has been shown that

sorption processes in such disordered systems are controlled by the presence of an

abundance of free energy minima in a very rough energetic landscape ([33]). These

local minima in the free energy are separated by barriers which can be overcome by

the thermally induced fluctuations of the fluid. In the temperature range below the

5


critical temperature, where the adsorption hysteresis is observed, these finite barriers

dominate the static and dynamic behavior of fluids in the pores ([33, 31, 32]).

The Mean Field Kinetic Theory (MFKT) approach has been applied by Monson

and coauthors to study phase transitions of fluids under confinement ([25, 34]). This

method allows, by numerically solving a set of relevant equations, the calculation of

the ensemble average of many dynamical trajectories of the system evolution. For

sufficiently long times, MFKT yields the thermodynamic behavior identical to that

of Mean Field Theory (MFT), predicting the metastable states characterized by the

local minima of the free energy.

Despite the achieved progress in the understanding of the adsorption hysteresis

phenomenon, there are still open issues. They are primarily related to the existence

of different kinds of disorder in real porous materials, such as chemical and geo-

metrical disorder. Another important issue is the facilitation of well defined porous

materials with well ordered pore structure at the macroscopic length scale, as ad-

dressed in [35]. Every material exhibits plenty of specific properties like pore wall

heterogeneity, pore interconnections or solid-fluid interactions with specific probe

molecule. One of the perhaps most important questions which only recently has

become addressed by the scientific community concerns the internal dynamics ac-

companying hysteresis phenomenon ([36, 37, 32, 38]).

It is worth mentioning that beside the importance of the fundamental under-

standing of the phase transitions at the mesoscopic length scale, the practical issues

are important as well. The interpretation of the adsorption experiments is still one

of the most widely used tools for the characterisation of porous materials. Only

detailed knowledge of the processes in the pores allows the development of corre-

sponding methods which can be utilised to access structural properties correctly.

1.2 Aims of this Work

While a wealth of studies is devoted to the nature and the thermodynamic equilib-

rium of phases within mesoporous solids, much less investigations were carried out

to describe and understand the transport properties of mesopore-confined phases.

It is obvious, of course, that the presence of different phase states or distributions

will strongly affect the corresponding transport properties within the mesopores.

In recent years, NMR progressed to a level that provides a number of approaches

6

1.2. AIMS OF THIS WORK

to analyse different aspects of molecular dynamics in porous materials with inhomo-

geneities of the porous structure on very different length scales ([39]), including the

possibility to quantify molecular diffusivities in mesopores under different external

conditions. The simultaneously measured NMR signal intensity provides the option

to correlate the transport properties with the phase state in the pores. Moreover, by

stepwise changing the external conditions, e.g., the vapour pressure, one may create

a gradient of the chemical potential between the gas phase and the confined fluid

allowing to follow its equilibration by means of NMR. In this way, the results of

macroscopic and microscopic techniques may be compared to reveal information on

the fluid behavior which, so far, was inaccessible ([40, 41, 42, 43, 44]). Altogether,

a set of NMR approaches allow to address various aspects of molecular dynamics in

mesoporous adsorbents of different pore architecture and macro-organisation.

The main goals of the present work, thus, may be summarised as follows:

• To correlate molecular self-diffusivities of fluids confined in mesoporous ma-

trices with different pore structures as provided by pulsed field gradient NMR

with their phase state as controlled by the chemical potential of the surround-

ing gas phase. In particular, one of the main questions to address will be

to probe whether different fluid configurations in mesopores, as revealed by

adsorption isotherms and scanning curves, are characterised by different ef-

fective diffusivities and what kind of novel information the latter may yield

about fluid behavior in pores.

• To compare the results on molecular transport properties as revealed by mi-

croscopic (PFG NMR) and macroscopic (sorption kinetics) methods and mea-

sured at identical conditions. In this way, by having simultaneously access to

two different transport properties which may be controlled by different inter-

nal mechanisms, we may highlight these mechanisms and may, in more detail,

address dynamical aspects accompanying the adsorption hysteresis.

• To address the effect of disorder on fluid behavior during sorption experiments,

by means of the Mean Field Theory of a lattice gas. By the use of linear pores,

excluding network effects, disorder effects by intentionally created geometrical

and chemical heterogeneities, can be studied in a most efficient way and com-

pared to the results of our experimental studies of relevant phenomena using

mesoporous silicon with linear pores.

7

Chapter 2

Basics

In this section we outline some basic concepts which are essential for the under-

standing of the subsequent experimental results and will be used throughout this

work.

2.1 Liquid-Gas Phase Transition

One of the most fundamental properties of liquid-gas interfaces is the surface energy,

also referred to as surface tension, γ, of the liquid surface. It is defined as the

proportionality constant between the work necessary to increase the liquid surface

area, ∆W , and the change of the surface area, ∆A

∆W = γ ·∆A (2.1)

In general, surface tension depends on the composition of the liquid and vapour,

temperature, and pressure, but it is independent of the area. Detailed discussion of

the surface tension can be found in [45].

The surface tension tends to minimise the surface area. If the pressure on one

side of the liquid-gas interface is larger then on the other side, the surface may

become curved, like a rubber membrane. The curvature of the surface is related to

the pressure difference, ∆P , via the Young-Laplace equation

∆P = γ

(1

R1

+1

R2

)(2.2)

with R1 and R2 tow principal radii of curvature. ∆P is also called the Laplace

9

CHAPTER 2. BASICS

pressure ([45]). For a spherical droplet with a radius R we have R1 = R2 and

the curvature in Eq. (2.2) becomes 2/R. In the case of a cylinder of radius r, the

convenient choice is R1 = r and R2 =∞, so that curvature is 1/r.

The vapour pressure over a liquid surface, which is in the thermodynamic equi-

librium with the liquid is called saturated vapour pressure. As a consequence of the

Young-Laplace Eq. (2.2), the saturated vapour pressure over a planar liquid surface

(R1 = R2 = ∞) is larger than for the case of a curved liquid surface. The depen-

dence of the saturated vapour pressure on the curvature of the liquid is given by the

Kelvin equation

RT lnP

P0

= γVm

(1

R1

+1

R2

), (2.3)

where P is the vapour pressure above the curved surface, P0 is that above the

flat surface. Vm denotes the molar volume of the liquid. This equation is only valid

in thermodynamic equilibrium, which is not always the case as we will see later. For

a spherical surface with radius r, Eq. (2.3) becomes

RT lnP

P0

=2γVmr

. (2.4)

2.2 Capillary Condensation

An important application of the Kelvin equation is the description of the capillary

condensation. It is the process of condensation in small capillaries and pores at

vapour pressures below the saturated vapour pressure P0. The Kelvin equation

introduced above is valid for a droplet surrounded by the vapour phase. For a bubble

in the liquid, as applies to capillary condensation, the radius of the curvature, r, is

negative and Eq. (2.4) becomes

RT lnP

P0

= −2γVmr

. (2.5)

The Kelvin equation (2.5) does not take account of any fluid-wall interaction.

The consequence of the latter is that there exists an adsorbed liquid-like layer on the

surface of the pore walls which has to be taken into consideration for the description

of the experimental data. This can be done by modifying the Kelvin equation ([10]).

For example, for a cylindrical pore it then reads

10

2.3. ADSORPTION MECHANISMS

RT lnP0

P0

= − 2γ cos θ

∆ρ(r − tc). (2.6)

In this equation, the contact angle, θ, of the liquid meniscus against the pore

wall and the thickness of the adsorbed layer on the pore wall, tc, take account of the

presence of the adsorbed layer. The contact angle can be considered as a measure

of the fluid-wall interaction strength. In the case of complete wetting, θ = 0, which

will be considered to be valid for our experiments, ∆ρ = ρl − ρg is the difference

between the bulk liquid density and the gas density ([5]).

2.3 Adsorption Mechanisms

In mesoporous materials with pore sizes ranging from 2 to 50 nm (IUPAC 1985

classification [1]), the capillary condensation is a prominent process having a strong

influence on the molecular transport through the pores. Some examples of such

materials include porous glasses ([46]), MCM-41 ([47, 48]), SBA-15 ([15, 43]), elec-

trochemically etched silicon ([49, 50]), or anodic aluminium oxide ([51]).

In mesoporous materials, the sorption behavior depends not only on the fluid-

wall interaction strength, but also on the attractive interactions between the fluid

molecules. This leads to the occurrence of multilayer adsorption and capillary con-

densation in the pore. As predicted by the Kelvin equation (2.5), the pore conden-

sation happens at a gas pressure P lower than the bulk saturated vapour pressure

P0. Keeping the temperature constant and varying the external gas pressure, simul-

taneously recording the amount adsorbed at each pressure, one can obtained the

adsorption isotherm. The adsorption isotherms can be used to analyse the pore size

distribution, surface area, pore volume, fluid-wall interaction strength, and other

properties.

In Figure 2.1(a), the IUPAC classification of the sorption isotherms are presented.

The detailed discussion of the shape of adsorption and desorption isotherms can

be found in ([1, 4, 5]). As one can see in Figure 2.1(a), type IV and V exhibit

a hysteresis loop, i.e. the adsorption and desorption isotherms do not coincide

over a certain region of external pressures. The type IV isotherm is typical for

mesoporous adsorbents. At low pressures, first an adsorbate monolayer is formed

on the pore surface, which is followed by the multilayer formation. The point B in

Figure 2.1(a) is often taken to indicate the stage at which the monolayer coverage

11

CHAPTER 2. BASICS

(a) Sorption isotherms (b) Hysteresis loops

Figure 2.1: Types of sorption isotherms and hysteresis loops (IUPAC 1985, [1]).

is complete. One should keep in mind, that the concept of monolayer adsorption

works only on the perfect planar surface. A real surface possesses some degree of

roughness ([52, 38, 53, 54]), which makes adsorption to progress not homogeneously.

The amount of molecules adsorbed on the external sample surface is negligible in

comparison to that on the pore wall ([3]), since nanoporous materials typically

possess a very large internal surface area (e.g. 250 m2 per gram of Vycor 7930 [55]).

The onset of the hysteresis loop usually marks the beginning of the capillary

condensation in the pores. At the external pressure corresponding to the upper

closure point of the hysteresis loop, the pores are completely filled with liquid.

Type V hysteresis loop is a typical sign of a weak fluid-wall interaction. It is less

common, but observed with certain porous adsorbents ([1]).

2.3.1 Adsorption Hysteresis

All mechanisms leading to and having impact on the adsorption hysteresis are still

not completely understood. In [5], three models generally used for the explanation

of the hysteresis phenomenon are presented:

• Independent Pores. The hysteresis is assumed to be an intrinsic property of

12


a single pore, reflecting the existence of metastable fluid states. That means

that during adsorption, fluid inside the pore remains in the gaseous state,

although the liquid-filled pore would be thermodynamically more preferable.

The metastable adsorption branch terminates at a vapour-like spinodal, where

the limit of stability for the metastable states is attained and the fluid sponta-

neously condenses. Here one assumes that the desorption isotherm corresponds

to the equilibrium transition and might be taken, therefore, for the pore size

analysis.

Cohan explained in [10] this behavior macroscopically in the following way:

The shape of the meniscus during the condensation is cylindrical and spheri-

cal during the evaporation, which leads to different pressures P of the phase

transition according to the Kelvin equation (Eq. (2.5)). The hysteresis loop

expected for this case is of type H1 (see Figure 2.1(b)). Typical materials with

such a hysteresis shape are MCM-41 ([56]) or SBA-15 ([57]).

• Pore Network. The H2 type adsorption hysteresis is explained as a consequence

of the interconnectivity of pores ([21, 22]). In such systems, the distribution

of pore sizes and the pore shape is not well-defined or irregular. A sharp step

on the desorption isotherm is usually understood as a sign of interconnection

of the pores. If a pore connected to the external vapour phase via a smaller

pore, in many cases the smaller pore acts as a neck (often referred to as an

”ink-bottle” pore [8]).

In the cases when adsorption is expected to happen homogeneously over the

entire volume of a porous material, desorption may happen by different mech-

anisms: percolation, i.e. the pore space is emptied progressively when the

condition of emptying of the smallest pores, blocking the excess to the ex-

ternal gas phase, is fulfilled; cavitation, i.e. formation of gas bubbles in the

pore interior. The latter corresponds to the condition of spinodal evaporation,

when the limit of stability of the liquid is reached.

Typical representative of disordered porous materials are porous glasses like

Vycor ([46]), or disordered sol-gel glasses.

• Disordered Pores. The most realistic feature of nanoporous materials is the

existence of a structural disorder. The disordered pores may be considered

as a pore network, however, with a rather undefined structure. Thus, for

13

CHAPTER 2. BASICS

understanding of the adsorption experiments, more realistic models need to

be applied ([58, 32]).

The reconstruction method applied by Woo and Monson ([32]) based on the

spinodal decomposition developed by Cahn ([59]) allows to find the mate-

rial structure which matches the available experimental material data. This

method provides a good agreement with the results of the adsorption experi-

ments in disordered porous glass ([60]). More general is the so-called mimetic

simulation, which mimics the development of the pore structure during the

materials facilitation ([61]).

One of the widely discussed concepts assumes that the adsorption hysteresis orig-

inates from the metastable states of the fluid inside the porous matrix ([10, 2, 62, 16,

63, 32, 41, 44]). This metastability may lead to a very slow density relaxation be-

havior in the hysteresis region in disordered materials ([32]). Kierlik et al. ([16, 33])

have shown that the main features of the capillary condensation in disordered solids

result from the appearance of a complex free energy landscape. As a consequence

of this fact that the global minimum of the free energy cannot be attained on the

laboratory time scale, for one and the same system different techniques observe the

same isotherms. With increasing temperature, the equilibration controlled by the

fluctuations and redistribution of the fluid in the pores should become faster and

finally, at some critical temperature, Th, which should be smaller than the bulk crit-

ical temperature, Tc, the hysteresis loop should disappear. Indeed, such a behavior

is observed in experiments ([62, 64, 65, 4]).

2.3.2 Sorption Scanning Curves

One of the most important proofs of the applicability of a given model for the adsorp-

tion may be related to the shape of the so-called scanning curves ([2]). Where the

major adsorption and desorption isotherms are obtained by a step-by-step change of

the external pressure from zero to the saturated vapour pressure and by the recording

amount adsorbed, the scanning isotherms are obtained by an incomplete filling and

draining of the pores. In Fig. 2.2, schematic representations of the scanning curves

for independent channels (a) and interconnected channels (b-d) are presented, fol-

lowing the discussion in [12, 62, 25, 66].

Scanning curves for independent pores attain the adsorption and desorption

14


Figure 2.2: Schematics of the scanning curve behaviour. Dashed lines representthe major isotherms, solid lines the scanning curves. (a): Independent pores,adsorption (1-2)/desorption (2-1) scanning curves and the scanning loop (3-4-3).(b,c,d): Interconnected pores with desorption scanning curve (5-6), adsorption scan-ning curve (7-8) and two scanning loops starting on the adsorption isotherm (9-10-9)and on the desorption isotherm (11-12-11). Arrows show the direction of the pressurevariation and the corresponding amount adsorbed, respectively.

branches at a pressure different from that of the closure points of the major hys-

teresis loop (Fig. 2.2(a) 1-2 and 2-1). In contrast, scanning curves for dependent

systems (Fig. 2.2(b,c) 5-6 and 7-8) attain the major hysteresis loop at its closure

points ([66]). For an assembly of independent domains, scanning curves are typically

reversible so that it is not possible to observe subloops (Fig. 2.2(a) 3-4-3) within

the main hysteresis loop. On the contrary, non-congruent subloops, i.e. loops of

different shape, may be observed for a collection of interacting domains (Fig. 2.2(d)

9-10-9 and 11-12-11); the non-congruence is due to the dependence of the sorption

processes in the interconnected pores. Additionally, there is an effect of variation in

the adsorbed film at the pore surface.

The scanning curve experiments yield isotherms which are located inside the

15

CHAPTER 2. BASICS

boundary adsorption and desorption isotherms. It is possible to draw an essentially

infinite number of different scanning curves within the major hysteresis loop.

2.4 Diffusion

Molecular diffusion is one of the most fundamental processes in the nature ([67]).

Because molecules do possess thermal energy, they are in continuous movement. Due

to intermolecular collisions, this movement starting from some characteristic time

may become totally uncorrelated. This microscopic, irregular, so-called Brownian

motion in the absence of any gradients (temperature, concentration, etc.) is called

self-diffusion. On the other hand, any gradient in (quite generally) the chemical

potential lead to¿ molecular fluxes which can be observed macroscopically. The

rate of transfer of a diffusing substance through a unit area is proportional to the

concentration gradient measured normal to this area. This proportionality is known

as Fick’s first law of diffusion ([68, 69])

~J(~r, t) = −D∇c(~r, t), (2.7)

where ~J is the flux and c the concentration at position ~r at time t. The propor-

tionality constant D is generally referred to as the transport diffusion coefficient.

The minus sign indicates the direction of flow: from larger to smaller concentrations.

The conservation of mass yields

∂c(~r, t)

∂t= −∇ · ~J(~r, t). (2.8)

Combining Eqs. (2.7) and (2.8) one obtains Fick’s second law of diffusion:

∂c(~r, t)

∂t= D∇2c(~r, t). (2.9)

Notably, this diffusion equation (2.9) remains valid also under equilibrium condi-

tions. In this case, however, one has to replace the concentration by the probability

P (~r0, ~r1, t), to find a particle which has started at position ~r0 after time t at position

~r1. The proportionality factor in Eq. (2.9) is then referred to as the coefficient of

self-diffusion D0.

With the initial condition P (~r0, ~r1, t) = δ(~r1 − ~r0) and the boundary condition

P → 0 as ~r1 → ∞ one obtains the solution of Eq. (2.9), the so-called diffusion

16

2.4. DIFFUSION

propagator, given by the relation

P (~r0, ~r1, t) = (4πD0t)− 3

2 exp

(−(~r1 − ~r0)2

4D0t

)(2.10)

As we can see from Eq. (2.10), the radial distribution function of the molecules

in an infinitely large system is Gaussian. The width of this probability function

increases with time and the function is completely characterized by the diffusion

parameter D0. The mean-squared displacement of free diffusion can be calculated

from Eq. (2.10) and is given by

〈(~r1 − ~r0)2〉 = dD0t (2.11)

where d=2, 4, or 6, for one-, two-, or three-dimensional motion, respectively.

Eq. (2.11) is known as the Einstein’s relation. It provides a direct correlation be-

tween the diffusivity D0, as defined by Fick’s second law, and the time dependence

of the mean-squared displacement which is a most easily observable quantitative

property of Brownian motion.

2.4.1 Diffusion in Pores

Variation of molecular concentration in the pores, by which from now on we will

understand pore filling, may change the character of the diffusion process. Capillary

condensation, the different types of adsorption and molecular exchange between the

coexisting phases may be of crucial influence for the transport phenomena ([70, 40]).

At the beginning of the adsorption process, i.e. for low concentrations corresponding

to coverages of one surface monolayer or less, molecular diffusion can proceed via two

mechanisms. The first one is the diffusion in the vapour phase which proceeds as a

sequence of collisions either between the molecules or with the pore walls. The latter

is known as Knudsen diffusion ([71]). Knudsen diffusion occurs when the number of

molecule-wall collisions is dominant, which is the case for sufficiently diluted gases.

For an infinitely-long cylindrical pore of a diameter d, the Knudsen self-diffusion

coefficient is given by ([72])

DK =d

3

√8RT

πM, (2.12)

where M denotes the molar mass, R is the universal gas constant.

17

CHAPTER 2. BASICS

With increasing gas density, the amount of the molecules adsorbed on the surface

also increases. As discussed in [70], surface diffusion and diffusion of the multilayered

molecules depend in a complex way on the concentration and the pore parameters.

The mechanism of surface diffusion at an early stage of adsorption is most likely

a thermally induced hopping of the molecules between the adsorption sites on the

microscopically rough surface. A detailed overview of the various surface models

can be found in [70].

At higher concentrations, when capillary condensation occurs, the effective dif-

fusivity becomes equal to the adsorbed liquid-like phase diffusivity in a pore, Da.

Under equilibrium conditions, Da is a function of the fluid-wall interaction as well as

of pore geometry. The self-diffusion coefficient under confinement is, thus, usually

smaller than the bulk self-diffusivity, D0.

18

Chapter 3

Materials and Methods

3.1 Materials

For the study of hysteresis phenomena, two different types of porous systems have

been used, namely porous glasses and etched porous silicon. The porous glasses

represent a group of materials with a highly interconnected random pore network,

including the option of a hierarchical pore architecture. Electrochemically etched

porous silicon films represent a material with isolated, i.e. non-interconnected, par-

allel channels.

3.1.1 Porous Glasses

Vycor porous glass. One of the very widely studied model mesoporous systems

with interconnected pores is Vycor 7930 porous glass, which has become one of

the widely accepted standards for the verification of the existing models for the

description of adsorption phenomena. It is an open-cell porous glass with highly

interconnected random pores with a relatively narrow pore size distribution (PSD)

around an average pore diameter of about 6 nm. The pores allocate 28% of the

material volume and possess an internal area of about 250 m2 per gram ([55]). The

monolithic Vycor samples used in this work have the shape of a rod with a diameter

ranging from 3 to 6 mm and with length of 12 mm. The material was purchased from

Advanced Glass and Ceramics (Holden MA, USA [73]). Fig. 3.1 shows the sorption

isotherms of nitrogen in Vycor 7930, obtained by BelSorp Mini II apparatus (see

Sec. 3.3.2). The asymmetric shape of the hysteresis loop of type H2 is typical for

such highly interconnected materials.

19

CHAPTER 3. MATERIALS AND METHODS

Figure 3.1: Adsorption (open squares) and desorption (black squares) isotherms ofN2 in Vycor 7930 at 77 K. Lines are guide to the eye.

CPG. The second porous glass used is the so-called controlled porous glass (CPG)

FD121, purchased from European Reference Materials (Berlin, Germany). It has

narrow PSD with the mean pore diameter of about 15 nm, the internal surface area

is of about 160 m2 per gram ([74]). The spherical particles with an internal meso-

porous structure have diameters of about 100 micrometres. The sorption isotherms

of nitrogen in FD121 at 77 K obtained by means of BelSorp Mini II are presented

in Fig. 3.2. The late and steep adsorption step shows the relatively large pore size

and narrow PSD.

PID-IL. The hierarchically structured porous silica glass material PID-IL, con-

sisting of spherical cavities with 20 nm diameter, connected via channels of 3 nm

diameter, have been kindly provided by the Smarsly group (Institute of Physical

Chemistry, Giessen University, Giessen, Germany). The bulky particles are of about

1 mm size. In Fig. 3.3 the adsorption isotherms of nitrogen in PIB-IL materials are

presented. The wide hysteresis loop is typical for materials where the desorption

from bigger cavities is prevented by the narrow necks. Half of the pore volume

persists in the small channels connecting the spheres, as can be seen from the low

20

3.1. MATERIALS

Figure 3.2: Adsorption (open squares) and desorption (black squares) isotherms ofN2 in FD121 CPG at 77 K. Lines are guide to the eye.

pressure range of the adsorption isotherm.

Before all experiments, these materials were kept in a strong oxidiser (35% hy-

drogen peroxide) at 380 K to remove organic contaminants followed by a cleaning

at 500 K under vacuum.

3.1.2 Porous Silicon

Mesoporous silicon (PSi) ([75, 49, 76]) is a representative of porous materials the

mesostructure of which can be intentionally made quite anisotropic. Due to its very

attractive structural properties it has attracted a lot of scientific interest ([77, 78, 38,

79, 50, 80, 81, 54]). Especially, by a proper tuning of the fabrication conditions it can

be prepared to consist of macroscopically long, linear pores. The electrochemically

etching procedure also allows control of the pore shape by varying the pore diameter

along the channel direction, making PSi an attractive material to verify theoretical

predictions.

PSi used in our work was prepared as a porous film consisting of non-interconnected

21


Figure 3.3: Adsorption (open squares) and desorption (black squares) isotherms ofN2 in PIB-IL material at 77 K. Lines are guide to the eye.

parallel channels with a typical channel length of a few tens of micrometers and a

mean pore diameter of about 6 nm. The material was fabricated in our department

by Dipl.-Ing. Alexey Khokhlov. The samples have been prepared by electrochemical

etching of single-crystalline (100)-oriented p-type Si wavers with a resistivity of 2-

5 mΩcm−2. The electrolyte contained HF acid (48%) and ethanol in a ratio 1:1. To

produce PSi samples with both pore ends open, PSi films have been removed from

the substrate by an electropolishing step with a current density of 700 mAcm−2 ap-

plied for 2-3 seconds. To obtain a material with long channels closed at one end, the

substrate has not been removed. The adsorption/desorption isotherms of nitrogen

at 77 K obtained with BelSorp Mini II (see Section 3.3.2) are shown in Fig.3.4.

3.2 Pulsed Field Gradient NMR

The Pulsed Field Gradient NMR (PFG NMR) technique is an established method

for the measurements of molecular self-diffusivities. The application of a specially

designed sequence of radio frequency (RF) pulses and magnetic field gradient pulses

22

3.2. PULSED FIELD GRADIENT NMR

Figure 3.4: Adsorption (open squares) and desorption (black squares) isotherms ofN2 in PSi material at 77 K. Lines are guide to the eye.

leads to the formation of a nuclear spin echo, the intensity of which depends on

the sequence parameters, the nuclei under study and the molecular self-diffusivity

of the species carrying the nuclei. Acting only upon the nuclear magnetic moment,

the (PFG) NMR technique does not perturb the system under study, and is thus

of non-invasive nature ([82]). A comprehensive introduction into the PFG NMR

measurement technique can be found in [82, 83, 84] and here we only briefly mention

some basic points.

Two important types of information are accessible by NMR: The amount of

molecules, as derived from the signal intensity of the free induction decay (FID), and

the molecular self-diffusion coefficient obtained by means of the PFG NMR method.

The former can be measured as a function of time by recording the FID signal

intensity during the uptake/release process of the adsorbate molecules into/from the

pores. Thus, we can follow the sorption dynamics in a very direct way. During the

adsorption of the molecules in the pores, the longitudinal nuclear magnetic relaxation

time, T1, does not change significantly. However, the transverse relaxation times T2

can change considerably. Thus, the latter has to be analysed and the data should

23


be corrected accordingly. In Fig. 3.5, the adsorption and desorption isotherms of

cyclohexane in Vycor porous glass measured at 297 K are presented. The isotherms

obtained by NMR (open stars, black stars) and the volumetric (open squares, black

squares) adsorption measurement (see Sec. 3.3.1) show the same qualitative and

quantitative behavior, supporting the validity and correctness of the adsorption

measurement by means of the NMR FID signal.

Figure 3.5: Adsorption (open squares) and desorption (black squares) isotherms forcyclohexane in Vycor 7930 at 297 K obtained by volumetric measurement and thoseobtained by means of NMR (adsorption: open stars, desorption: black stars).

The self-diffusivities presented in this work, are obtained from the spin echo

attenuation measured using the stimulated echo and 13-interval pulse sequences

([84]). For the general case of anisotropic diffusion, i.e. if there is an orientational

dependence of molecular mobility, the spin echo attenuation is given by

Ψ(q,∆) = exp(−D · q2∆) (3.1)

where ∆ is the observation time, q = γgδ with γ - the gyromagnetic ratio,

and g and δ the gradient pulse amplitude and duration, and where tensor D =

Dxx cos2 αx + Dyy cos2 αy + Dzz cos2 αz stands for the diffusivity in the direction of

24

3.2. PULSED FIELD GRADIENT NMR

the applied magnetic gradient. αi denote the angles between the field gradient and

the directions of the principle tensor axes ([82]).

In the case of MCM-41 ([40]) and SBA-15 ([43]), where the diffusion may be

assumed to occur predominantly in channel direction, one has to integrate over all

directions, yielding ([43])

Ψ(q,∆) =1

2

∫ π

0

exp(−q2∆(Dpar cos2 θ +Dperp sin2 θ)

)× sin θdθ, (3.2)

with Dpar and Dperp being the self-diffusivities parallel and perpendicular to the

channel direction. Performing this integration, one obtains

Ψ(q,∆) =

√π

2exp

(−q2δDperp

) erf(√

q2∆(Dpar −Dperp))

√q2∆(Dpar −Dperp)

. (3.3)

In the case of isotropic molecular motion, as can be found for sufficiently long

observation times in random porous glasses, Dxx = Dyy = Dzz, and Eq. (3.1) can

be simply used for the gradient applied along the z axis

Ψ(q,∆) = exp(−Dzq2∆) (3.4)

In opposite to the signal intensity, which is essentially unaffected by the gas

phase, the contribution of the gas phase to molecular transport may be very signif-

icant. The self-diffusivity of saturated cyclohexane vapour at room temperature is

of order of magnitude of 10−6 m2s−1 ([85]), while D0 of the liquid cyclohexane is of

about 10−9 m2s−1. As discussed in [86, 40], the effective self-diffusivity,De, in porous

solids obtained from the spin echo attenuation for sufficiently long observation time

is

De = pgDg + paDa

with Da denoting the self-diffusion coefficient in the adsorbed phase, Dg the

diffusivity in the gaseous phase in the pore interior (coefficient of Knudsen diffusion),

pa and pg are the relative fractions of the molecules in these phases. In more detail

this will be discussed in section 4.1.

25


3.3 Adsorption Measurement

3.3.1 Adsorption from Vapour Phase

To perform adsorption experiments, a computer-controlled adsorption setup (in

what follows referred to as VaporControl) was built. It allows to prepare a vapour

of a liquid at a desired pressure in a reservoir, which, thereafter, can be brought

into contact with the porous substance under study. In Fig. 3.6 the schematics of

the adsorption setup is presented. Opening the valves v1 and v2, one can increase

the vapour pressure in the gas reservoir (res). A sufficiently large reservoir (3 litres)

is taken to maintain a desired pressure constant during the adsorption/desorption

experiments. By using a turbomolecular pump (tmp) it is possible to decrease the

pressure in the gas reservoir, but also to prepare the samples for the measurements.

In other words, this latter option allows an in situ activation of sample materials

by keeping them in an oven at high temperature and then simultaneously evacuate.

The pump used here is the diaphragm vacuum pump with a turbomolecular pump

(Pfeiffer-Vacuum Pumping Station TSH 071 E). The pressure is controlled by means

of two capacitance sensors (p, with measuring ranges from 0.0001 to 10 mbars and

from 10 to 1000 mbars, connected to the digital dual gauge unit (dg).

To keep the system temperature constant, a thermostat unit has been designed.

The whole vapour handling system is put into a plexiglass box (box) with a vol-

ume of 90 litres. The box is tempered by an Omron E5CK temperature controller.

The temperature sensor (Pt100) is placed on the gas reservoir. The remotely con-

trolled laboratory power supply (PS3000B by EA-Elektro-Automatik, Germany) is

connected to three heating mats, each of 20 Watts power. As heating agent, the

ambient room air is used. These heating elements are controlled by the Omron

E5CK. Being a PID regulator, it thus allows to maintain a very high temperature

stability.

To control the adsorption measurements remotely, stepping motors (Sanyo Step-

Syn 103G7702517) have been used as actuators for the valves. The stepping motors

are connected to RN-Motor (rn0,rn1) driver units ([87]). The stepping motors allow

a precise positioning (200 steps/360 degree) and are thus well applicable as valve

actuators.

Temperature controller, pressure gauge and the valve controlling hardware are

connected to a PC and can be remotely controlled by software. For this purpose,

26

3.3. ADSORPTION MEASUREMENT

θ

rn0

rn1

liq

res

tmps

pdg

tc

fan

h

box

v0usb

v1v2

v3

electric linedata line

Figure 3.6: Schematic of VaporControl adsorption setup: liq - flask with adsorbateliquid; res - big reservoir for gas phase preparation; tmp - turbo molecular pump;s - adsorbent sample inside the spectrometer; box - thermostat box; p - pressuresensors; dg - digital pressure display; tc - digital temperature controller and powersupply; h - heating mats; v0..v3 - valves with motor actuators; rn0, rn1 - valve motordriver; usb - RS-232 to USB converter unit.

a graphical user interface (see Fig. 3.7) has been written in Delphi for Windows

operating systems. This software provides the control of the valve state as well as

the controlling of the temperature and pressure values. The built-in OLE (Object

Linking and Embedding) server provides the access to the VaporControl functions

from custom software for tailored user applications. Such an application in our

case is the measurement of the adsorption/desorption isotherms or other loading-

dependent properties of porous materials by using an additional measurement device

(such as PFG NMR).

For the volumetric measurements, the sorption experiment control logic has been

implemented in a VBA script (Visual Basic for Applications). This makes the con-

trolling and presentation of the adsorption experiment via Microsoft Excel possible.

27


Figure 3.7: VaporControl GUI

3.3.2 BelSorp Mini II

BelSorp Mini II is a computer controlled gas handling system that is equipped with

diaphragm pressure gauges with a pressure range up to 133 kPa. The accuracy of

the pressure sensors is 0.25%. For the adsorption measurement the sample cell is

immersed in liquid nitrogen to keep the experiment temperature at 77 K. For each

data point the sample is exposed to the vapour pressure for 900 seconds. Pres-

sure equilibration outside of the hysteresis loop is typically completed after about 1

minute. Inside the hysteresis loop, times of about 900 seconds were needed so that

no observable change of the pressure. The room temperature around the apparatus

has been kept constant by the air conditioning.

28

3.4. MEAN FIELD THEORY APPROACH

3.4 Mean Field Theory Approach

With the rapid development of computer technology, the application of compu-

tational techniques has become an essential part of almost any branch of science.

Because the molecular systems generally consist of a large number of interacting par-

ticles it is sometimes difficult to describe their certain properties. The introduction of

computer simulation techniques such as Monte Carlo and Molecular Dynamics([88])

allowed to gain an insight into the microscopic world of single molecules and molec-

ular ensembles. For the description of the confined fluids, lattice gas models have

attracted a lot of attention ([89, 24, 23, 90, 16, 25]). Manor artefacts caused by the

coarse description of the system by a lattice gas are compensated by the simplicity

and the computational efficiency of the model. The application of the mean field

theory (MFT) to the lattice model is especially reasonable in elucidating the nature

of the adsorption hysteresis for fluids confined in mesoporous matrices ([16, 32, 34]).

This approach allows a very efficient calculation of the fluid states depending on

the external driving forces such as chemical potential or temperature, making the

static MFT an appropriate tool to study quasi-equilibrium configurations and phase

transitions. However, neither fluctuations nor a time scale are incorporated in this

approach, so that the dynamics of density relaxation cannot be investigated. In

[34], the application of mean field kinetic theory (MFKT) to the lattice gas model

is described for confined fluids. This method is based on the calculation of flux

at any sites by means of hopping probabilities between the neighboring sites, de-

termined within the mean field approximation. For long times, MFKT yields the

thermodynamic behavior identical to that calculated by MFT.

In this work, MFT is utilised to study adsorption hysteresis in electrochemically

etched porous silicon. In the following, a short overview of the method presented by

Monson in [34] is given. The Hamiltonian of a lattice gas system with only nearest

neighbor interaction considered is given by

H = − ε2

∑i

∑a

nini+a +∑i

niφi, (3.5)

where ε denotes the nearest neighbor interaction and ni is the occupancy at

site i. The external field φi at site i is calculated from the solid-fluid to fluid-fluid

interaction ratio y = wsf/wff via

29


φi = −∑a

(1− ti+a)y, (3.6)

with ti being 1 for lattice sites accessible to fluid and 0 else. i denotes lattice

coordinates and a the vector to the nearest neighbor sites for site i. If we express

the occupation of a lattice site by the fluid density, 0 ≤ ρi ≤ 1, the Helmholtz free

energy, F , for such a system becomes ([34])

F = kT∑i

[ρi ln ρi + (1− ρi) ln(1− ρi)]−ε

2

∑i

∑a

ρiρa +∑i

ρiφi. (3.7)

The fluid density, ρi, is related to the total number of molecules, N , via∑

i ρi =

N . Thus, at equilibrium, the essential condition

∂F

∂ρi− µ = 0 ∀i (3.8)

should be fulfilled for fixed N, V, T , where µ is the chemical potential, being

uniform everywhere in the system at equilibrium.

Using (3.7) and (3.8), one obtains

kT ln

[ρi

1− ρi

]− ε∑a

ρi+a + φi − µ = 0 ∀i. (3.9)

For the fluid density at site i, we can rewrite (3.9) as

ρi =1

1 + exp(−H∗i ), (3.10)

with H∗i = 1/T ∗∑a

ρi+a − φi/ε+ µ/ε and the temperature T ∗ = kT/ε.

The equations (3.10) are solved simultaneously by an iterative method to yield

the equilibrium density distribution.

30

Chapter 4

Random Pore Network

In this chapter, the experimental study of the adsorption dynamics in disordered

porous materials is presented. As model materials, porous glasses were used. Cre-

ated by phase separation in an alkali borosilicate glass at high temperatures, followed

by leaching of the phase soluble to acids, Vycor glass represents an ideal random

porous matrix with 3-dimensional pore structure ([46]). The adsorption experiments

reveal a narrow pore size distribution, though the pore size is barely defined in such

a disordered structure and the pore size distribution (PSD) is rather an estimate of

the length scale of the confinement which influences the capillary condensation.

Measurement of adsorption isotherms is the classical characterisation technique

of supreme importance ([61, 6, 5]). The shape of the adsorption and desorption

isotherms can be analysed to obtain the properties of the porous material and the

effects of the confinement on the adsorbate molecules. The methods of analysis are

usually based on the Kelvin equation (see Chapter 2.2) which is only valid for the

thermodynamic equilibrium. Another approach is the construction of the sorption

isotherms in model pores, e.g. by the Density Functional Theory (DFT) and Grand

Canonical Monte Carlo (GCMC) simulation ([91, 92, 93]). The density distribution

of the adsorbed fluid in pores is calculated by minimizing the corresponding grand

potential and the isotherms obtained in such a way can be fitted to the experimen-

tally obtained sorption isotherms to derive the PSD. Again, this method assumes

the system to be in the thermodynamic equilibrium.

The phenomenon of adsorption hysteresis itself is already a sign of the departure

from thermodynamic equilibrium ([4, 16, 32]). This fact raises the question which

isotherm should be used for pore structure analysis. In [94], Neimark at al. present

31

CHAPTER 4. RANDOM PORE NETWORK

a combined nonlocal DFT and MC study of the adsorption hysteresis in MCM-41-

like material. The authors argue that the desorption branch follows the theoretical

line of equilibrium transitions while the adsorption branch is close to the theoretical

vapour-like spinodal. In a subsequent publication ([63]), the same authors have

observed the existence of multiple internal states of equal density, revealed by the

DFT.

Recent mean-field density theory studies of a disordered lattice-gas model sug-

gest the presence of a rugged free energy landscape in the hysteresis region ([16, 32]).

With the onset of capillary condensation, the system exhibits a very large number

of spatial arrangements of the fluid in a disordered structure which have the same

average density. These states are metastable and represent the local minima of free

energy. The evolution of the system towards the equilibrium, i.e. to the global

free energy minimum, involves transitions between these states via activated bar-

rier crossing. This process is intrinsically slow and exceeds the experimental time

scale. In [32], this is shown by MC simulations for the lattice models of a fluid in

Vycor porous glass. It is suggested, that the hysteresis can occur even without an

underlying phase transition.

4.1 Adsorption and Diffusion Hysteresis

Additionally to the direct measurement of the amount adsorbed by NMR (see

Chap. 3), pulsed field gradient NMR gives us a unique possibility to access self-

diffusivities of a fluid at different pore loadings. Thus, microscopic information

reflecting the internal density states contained in the diffusivities can be correlated

with the amount adsorbed.

In Fig. 4.1 (bottom), the adsorption and desorption isotherms for cyclohexane

in Vycor at 297 K measured by NMR are presented. One may recognise a well-

pronounced hysteresis loop of type H2. The amount adsorbed, θ, is given in nor-

malised units by dividing the actually measured FID signal intensity by that ob-

tained at full pore loading. The latter is achieved at vapour pressures only slightly

below the saturated vapour pressure, P0. Corresponding effective self-diffusivities

(see Sec. 2.4.1) obtained by means of PFG NMR are shown in the top of Fig. 4.1

as a function of the relative pressure, P/P0. The diffusivities have been measured

applying the stimulated echo sequence with the observation time δ = 10 ms after

32

4.1. ADSORPTION AND DIFFUSION HYSTERESIS

sufficiently long equilibration times following a pressure step, so that no measurable

change in the amount adsorbed was observed. Thus, the measured diffusivities may

thought as those obtained under (quasi)equilibrium conditions.

Figure 4.1: Top: Effective self-diffusivities of cyclohexane in Vycor 7930 at 297 Kmeasured upon increasing (adsorption, open circles) and decreasing (desorption,black circles) the vapour pressure, obtained by PFG NMR. Bottom: Correspondingadsorption (open squares) and desorption (black squares) isotherms. Lines are guideto eye.

One of the most important observations is that, in line with the adsorption

hysteresis, also a hysteresis loop of the self-diffusivities can clearly be observed.

Such behavior of the self-diffusivities of organic molecules in Vycor, porous silicon

and MCM-41 has already been investigated by means of PFG NMR by Valiullin

et al. in [40, 56]. Certainly, in the representation of Fig. 4.1, one may recognise

that there is a correlation between the hysteresis loops in the diffusivities and the

amount adsorbed. In [40], the analytical model for the complex dependence of the

effective self-diffusivities on the amount adsorbed has been presented. This model

assumes that under experimental conditions, as given by PFG NMR, the gaseous

and the adsorbed phases inside the pores are subjected to fast exchange on the

experimental time scale. Thus, the effective self-diffusivity, De, obtained by the

PFG NMR experiment can be estimated by

33


De = pgDg + paDa, (4.1)

where pg and pa = 1−pg refer to the relative fractions of molecules in the gaseous

and adsorbed phases. The diffusivity Dg can be approximated by the Knudsen

diffusion coefficient(see Sec. 2.4.1) for an effective pore diameter de = d ·√

1− θ([40]). The relative fraction of the gas pg can be estimated from the adsorption

isotherm via

pg =ρgρa

1− θθ

, (4.2)

with θ being the relative amount adsorbed, which can be related to the external

gas pressure via the adsorption isotherm. ρg/ρa is the ratio of the densities in the

gaseous and adsorbed phases.

Figure 4.2: Effective self-diffusivities of cyclohexane in Vycor 7930 at 297 K duringadsorption (open circles) and desorption (black circles) obtained by PFG NMR. Bythe lines, De calculated for the adsorption (dashed line) and the desorption (solidline) branches are shown.

In Fig. 4.2, the calculated effective self-diffusivities qualitatively mimic the de-

34


pendence of the measured self-diffusion coefficient on the external pressure. This

reveals that the hysteretic behavior of the self-diffusivities is primarily determined

by the specific contribution of the gaseous phase, pg, with changing gas pressure.

The PFG NMR allows us to explore the total probability distribution of molecular

displacements (see Secs. 2.4 and 3.2) via the spin-echo diffusion attenuation func-

tion. We have found a mono-exponential dependence of the echo intensity on the

square of the applied gradient strength (Fig. 4.3). During the observation time, the

average molecular displacement is of the order of several microns which is by three

orders of magnitude larger than the structure size of the Vycor porous glass. This

reveals the clear evidence of the Gaussian propagation and implies the fast exchange

between different regimes of the molecular mobilities, i.e. between the adsorbed, the

capillary condensed, and the gaseous phases, respectively ([82, 40]).

Figure 4.3: Spin-echo diffusion attenuation function for cyclohexane in Vycor atP/P0 = 0.52 on the adsorption branch measured at 297 K (black squares) by PFGNMR. The solid line represents the best fit using Eq. (3.4).

Thus, a straightforward explanation of the general trends of the diffusivities may

be drawn. The main contributions to the average self-diffusivity are as follows:

• The fluid adsorbed on the pore walls dominates molecular transport at low

35


pressures. The diffusion coefficients are expected to be very slow. The low

density of the gaseous phase at low pressures minimises the contribution of

the molecular transport through the vapour phase in the pore interior.

• The gaseous phase in the pore interior provides the largest fractional con-

tribution to overall transport in the intermediate pressure region, where the

gas density is high enough, so that the fast transport in the gaseous phase

significantly contributes to the effective diffusivity.

• The capillary-condensed phase in the pores restricts the transport at higher

pressures due to the slower diffusivities as compared to the gaseous phase.

The ”competition” between the increasing transport through the gaseous phase

due to increasing gas density and the decrease of the space available for the gaseous

phase results in the maximum of the effective self-diffusion coefficients as a function

of pore loading, and thus, of the external gas pressure. In nice agreement with

this expectation, the diffusivities on adsorption notably exceed those measured on

desorption.

A more detailed description of the transport properties of fluids in the adsorbed

phase, in particular of the diffusivities Da, which is certainly not a straightforward

task for such an inhomogeneous medium ([70]), will improve the quantitative agree-

ment between the model and the experiment. Another question to be further ex-

plored concerns the validity of the Knudsen model for random pores in the presence

of the adsorbed phase. The Eq. (4.1) has used been only as a first approximation,

its applicability is constrained by certain assumptions made during the derivation

([40]). How good it captures the regime of capillary condensation is still has to be

studied in more detail.

One of the most remarkable features of the results presented in Fig. 4.1 emerges

when the diffusivities are presented as function of the relative concentration, i.e. of

the amount adsorbed θ. This dependence is shown in Fig. 4.4 for the adsorption

and the desorption branches. Importantly, these isotherms do not coincide, revealing

different internal density distributions of the same number of the adsorbed molecules

inside the disordered porous matrix. The different values of De at other equal

conditions may be considered as a manifestation of the history-dependent adsorbate

distribution!

Considering the importance of the history how a state has been achieved for

the adsorbate configuration, such a behavior should be expected to be even more

36


Figure 4.4: The diffusivities, De, plotted versus the amount adsorbed, θ, obtainedfrom the sorption isotherms. open circlesrepresent the adsorption isotherms, blackcirclesrepresent desorption. Lines are guide to eye.

pronounced in scanning experiments ([2, 16, 25]), i.e. by performing incomplete

filling/draining cycles (see Sec. 2.3.2). In Fig. 4.5(a), the data obtained by the des-

orption scanning experiments are presented. The adsorption has been performed

until the external gas pressure has attained 0.65 P0 (black diamonds) or 0.68 P0

(black triangles), followed by desorption upon which all relevant measurements

have been performed. The respective effective self-diffusion coefficients are shown

in Fig. 4.5(b).

As it has been shown earlier ([12, 33]), there should be a whole hierarchy of

subloops inside the major adsorption (open squares) and desorption (black squares)

loop. These subcycles are obtained, when, e.g., the desorption scanning curve is not

continued to pressures below the hysteresis range, but reverses back to the adsorption

one. As shown in Fig. 4.6, the desorption scanning curves and the corresponding dif-

fusivities from 0.65 P0 (black stars) have been reversed into the adsorption scanning

curve at 0.44 P0 (open stars). Similarly, one obtains internal loops by incomplete

adsorption scanning curves starting on the desorption branch at 0.43 P0 and increas-

37


ing the pressure to 0.59 P0 (open circles) followed by a desorption scanning sequence

(black circles). The amount adsorbed versus the relative pressure is found to yield

dependencies consistent with measurements presented by Everett in [17].

Recent theoretical work using mean field theory ([16, 25]) and MC simulations

([32]) provide an explanation of the observed sorption behavior in terms of the

multiplicity of metastable states associated with different distributions of the same

amount of molecules in the pore network. Within the main hysteresis region for a

given chemical potential or pressure, there is an infinite number of metastable states

with different densities characterized by the local minima of the free energy ([32]).

These differences in the density distributions are reflected by the behavior of the

scanning curves of the corresponding self-diffusivities at the given external pressure

as shown in Figs. 4.5 and 4.6.

The self-diffusivity scanning curves in Fig. 4.6 exhibit two further important

features:

• Return point memory : After an incomplete sorption cycle, the system returns

to its initial state. This is a feature of many systems exhibiting hysteresis

including magnets ([37, 95]), suggesting that the main driving force of the

evolution in such systems are the external conditions, and further thermal

equilibration is prohibited by the high energy barriers between the local free

energy minima.

• Lack of congruence, i.e. two different subloops are in general not parallel to

each other: This is a signature of the networked pores, since the independent-

pore model predicts exact congruence ([16, 66]).

Combining the self-diffusion experiments (Figs. 4.5(b) and 4.6(b)) with sorption

experiments (Figs. 4.5(a) and 4.6(a)) one obtains a whole map of the diffusivities as

function of the pore loading, θ, as presented in Fig. 4.7. Remarkably, the resulting

representation reveals states with the same average density but with different diffu-

sivities. The understanding this behavior requires an assessment of the differences

in the fluid density distributions with the same average density but attained via

different sorption ”histories”.

Following mechanisms leading to the behavior as observed in Fig. 4.7 may be

anticipated:

• During desorption, the liquid-like phase may be stretched (or expanded), i.e.

38


Figure 4.5: The relative amount of cyclohexane adsorbed (a) in Vycor and thecorresponding self-diffusivities (b) at 297 K as a function of relative pressure. Thedesorption scanning curves start from 0.68 P0 (black triangles) and 0.65 P0 (black di-amonds) obtained after incomplete filling. The boundary adsorption (open squares)and desorption (black squares) isotherms envelop the scanning curves. The lines areguide to eye.

39


Figure 4.6: The relative amount of cyclohexane adsorbed (a)in Vycor and the cor-responding self-diffusivities (b) at 297 K as a function of the relative pressure. Thedesorption scanning isotherms begin on the boundary adsorption isotherm at 0.65 P0

(black stars) and is reversed at 0.44 P0 (open stars). The adsorption scanning curvefrom 0.43 P0 to 0.59 P0 (open circles) is reversed to 0.43 P0 (black circles). The linesare guide to eye.

40


Figure 4.7: The diffusivities, De, in Figs. 4.5(b) and 4.6(b) plotted versus amountadsorbed, θ, from Figs. 4.5(a) and 4.6(a).

the density of the liquid can be lower than that of the fluid density at the sat-

urated vapour pressure, P0. Notably, such stretching due to a strong surface

field may also occur on adsorption but in much lesser extent. These stretched

states on desorption are primarily caused by the pore-blocking ([22]) effects

in a disordered pore network. This so-called ”ink-bottle” geometry has been

the subject of several recent simulations and theoretical studies ([60, 26]). Re-

gardless whether the desorption occurs via cavitation or via pore blocking,

the liquid in the wider pore region is in a stretched state. This density dif-

ference means, that during desorption, the same number of molecules in the

capillary-condensed phase occupy notably larger part of the pore space than

during adsorption. Additionally, the diffusivity in the stretched phase is some-

what higher than in the dense liquid, but still significantly smaller than in the

gaseous phase. This effect is stronger in materials with large cavities and small

necks. The availability of such materials nowadays allow that this point may

be confirmed experimentally. Thus, Fig. 4.8 shows the sorption isotherms as

well as the corresponding self-diffusivities for cyclohexane the PIB-IL porous

41


silica ([96]) measured at 297 K. This hierarchical pure SiO2 porous material

consists of large spherical cavities of 20 nm diameter, connected by the chan-

nels of 3 nm diameter (see Sec. 3.1). The decrease of the amount adsorbed

during desorption, before the steep knee at 0.35 P0 may be associated with

desorption from the exterior pores. The increase of the corresponding self-

diffusivities on the other hand, reveals a decrease of the liquid density, i.e.

stretching of the liquid.

• Different distributions of the adsorbed fluid within the sample may lead to the

different diffusivities during adsorption and desorption. In [97], the spatial

correlations in the pores of Vycor on filling and draining of n-hexane were

studied. Ultrasonic attenuation and light scattering studies have shown that,

during adsorption, the pore filling proceeds uniformly over the sample. During

capillary condensation vapour bubbles persist in the pores, until the pores are

completely filled. No long-range correlations between the bubbles have been

observed, i.e. the pores fill independently. By contrast, the desorption process

is accompanied by long-range correlations in the liquid distribution, which

can be modelled by invasion-percolation ([98]). However, our experiments

indicate homogeneity of the fluid distribution in the entire sample, since the

diffusion propagator (see Chap. 3) results in an ideal Gaussian. This means

that, for distances of several micrometres, as traced by PFG NMR, we have

identical filling properties. Otherwise we would have observed a distribution

of the self-diffusion coefficients. This reveals that the pores get empty via

the gas invasion, i.e. the liquid-gas interface percolates from the boundary

into the sample interior. However, during desorption the interplay between

the pore-blocking and cavitation may result in more extended regions of the

liquid and gaseous phases than during adsorption. We anticipate that the

latter mechanism can give rise to different diffusivities due to different effective

”tortuosities” (here we understood tortuosity in a more general sense rather

than as a mere geometrical parameter of the pore space), i.e. due to differently

weighted molecular propagation paths in the system. However, one should also

be aware of further effects related to differences in the fluid density within the

pore space.

42

4.2. SORPTION KINETICS: STRONG SURFACE FIELD

Figure 4.8: Top: Effective self-diffusivities of cyclohexane at 297 K as a functionof the external pressure in PIB-IL during adsorption (open circles) and the desorp-tion (black circles). Bottom: Corresponding adsorption (open squares) and desorp-tion (black squares) isotherms. The lines are guide to eye.

4.2 Sorption Kinetics: Strong Surface Field

One of the most straightforward methods to illuminate the mechanisms of the ad-

sorption is the analysis of the transient sorption behavior. Fig. 4.9 shows results

of a transient sorption experiment correlated with the information from the diffu-

sion studies. The uptake kinetics in the pressure range outside the hysteresis loop

measured after a stepwise change of the pressure from 0.16 to 0.24 P0 is shown in

Fig. 4.9(a), while Fig. 4.9(b) shows the adsorption kinetics measured after a step

form 0.48 to 0.56 P0, i.e., in the hysteresis region. The change of the pore loading

from θ0 at the starting pressure to θeq at the target pressure at quasi-equilibrium

is given in relative units (note that although the true equilibrium in Fig. 4.9(b)

is not attained, the change of θ at long times is sufficiently small so that such a

representation is reasonable). The figure presents typical examples of the uptake

kinetics following a relatively small stepwise change of the external pressure. Note

also that the gas reservoir of the adsorption setup was designed to be large enough

so that there was essentially no change of the external pressure during the uptake

43


experiment.

With the independently determined diffusivity and assuming that diffusion is

the rate-limiting process one may calculate the expected uptake. This function

can be obtained by solving the diffusion equation (2.9) with appropriate initial

and boundary conditions for an infinitely long cylindrical region with a radius r,

mimicking the shape of our Vycor material. The corresponding solution is ([99]):

θ(t)

θ∞= 1−

∞∑n=1

4

r2α2n

exp(−Deα2nt), (4.3)

where αn are the roots of J0(rαn) = 0, J0(x) is the Bessel function of the first

kind of zero order.

In Fig. 4.9, the diffusion-controlled uptake curves, as given by Eq. (4.3), are

plotted by the dotted lines. Eq. (4.3) does not contain any fitting parameters, while

the effective self-diffusivities De have been measured independently as shown in

Fig. 4.7.

Importantly, the diffusion model reproduces the experimental data for the re-

gion out of hysteresis (Fig. 4.9(a)), but fails in the hysteresis region (Fig. 4.9(b)).

Slower equilibration in the hysteresis region has been noted before ([100, 38, 101]).

In [100], based on comparing experimental uptake curves with micro-kinetic models,

it has been assigned to a decrease of the effective diffusivities in the region of cap-

illary condensation and related to percolating properties of the system. The direct

measurement of the diffusivities (Fig. 4.1) reveals, however, that the slowing down

of the uptake process cannot be described by a decrease of the diffusivities. We

explain this observation by the fundamental difference in the nature of the density

relaxation dynamics for the states within the hysteresis region compared to those

out of this region. As one may see from Fig. 4.9(b), even after several hours for

the pressure step inside the hysteresis loop the equilibrium is still not achieved. On

the opposite, the diffusion-limited uptake outside the hysteresis region attains the

equilibrium after less than one hour.

Following the arguments that Woo et al. present in their study of the adsorption

dynamics of a lattice gas model of confined fluid by means of the MC simulations

([32]), we may identify a two-stage mechanism relevant for the transient uptake in

the hysteresis region:

• Adsorption at low pressures is limited by the diffusion of the guest molecules

44


Figure 4.9: Transient sorption of cyclohexane in Vycor porous glass cylinder (diam-eter 3 mm, length 12 mm) at 297 K measured by NMR. Typical kinetic data (blacksquares) obtained upon stepwise change of the external gas pressure from 0.16 to0.24 P0 (a) and 0.48 to 0.56 P0 (b). The inset of (b) shows the long-time part of thedata (b), axis quantities and units are the same as in main figure. The dotted linesrepresent the kinetics calculated via the diffusion equation (4.3). The solid line in(b) show the results from Eq. (4.5) with parameters τ0 = 600 s, τa = 5182 s.

45


into the pore space, including the formation of an adsorbed layer on the pore

wall. Since, at this stage, the whole pore space is accessible to mass transfer

from the surrounding atmosphere, the dynamics is purely diffusive. The global

equilibrium, with the chemical potential is uniform over the whole system, may

be attained very fast on the experimental time scale.

• With increasing density, capillary condensation occurs, followed by a growth

of the domains with the capillary-condensed liquid inside the pore structure.

The formation process of the liquid bridges is essentially an activated one,

i.e. driven by thermally induced fluid fluctuations. In parallel, the system

may further evolve, i.e. move to the global minimum in the free energy by

redistribution of such domains. This, however, is an activated process requir-

ing the transition of the free energy barriers between the local minima of the

free energy. The jumps of the system from one local minimum to another

due to macroscopic fluctuations creates a local density perturbation. This

perturbation is quickly equilibrated (to local equilibrium) via the diffusion of

the molecules from the surrounding pores and, therefore, from the external

gaseous phase which leads to further uptake. Because the redistribution pro-

cess may by far exceed the experimental time scale, the global equilibrium is

made essentially unattainable.

Let us reconsider the observation of slow kinetics in the adsorption hysteresis

region within the frame of existing models for such processes. We have already

mentioned that the density redistribution and the subsequent growth of the liquid-

phase are prerequisites of uptake. These processes are essentially activated in nature,

like the density fluctuations in random-field Ising systems ([102, 36]). In the frame

of this model, the system evolution requires crossing of barriers of height bξψ > kT ,

where ξ is the characteristic size of a droplet, ψ > 1, kT is the Boltzmann factor,

and b is a constant determined by the fluid properties. Thus, thermally activated

crossing of free energy barriers results in relaxation times exponentially diverging

with ξ. In [36], it was shown that the overall relaxation is described by the sum of

two components corresponding to diffusive and activated dynamics. The dynamic

correlation function corresponding to the latter part is found to generally follow the

form ([36, 103])

S(t) ∝ exp (− [ln(t)/ ln(t0)]p) (4.4)

46


where t0 is a typical microscopic time and the exponent p takes account of a

distribution of the barrier heights. Eq. (4.4) may be adopted to describe the adsorp-

tion kinetics in the hysteresis region at late stages of the uptake. For this purpose,

uptake in thus stage may be rewritten as

θ(t) ∝ 1− exp (− [ln(t/τ0)/ ln(τ0/τa)]p) , (4.5)

where τa is the characteristic relaxation time for activated dynamic. It is related

to the average rate of the fluid density fluctuations initiating further uptake. τ0 is the

characteristic microscopic time, necessary for relaxation of the density perturbations

via molecular diffusion. It depends on the sample geometry and, for an infinite

cylinder with the radius r one has τ0 = r2/15De. With p = 3 and a calculated value

of τ0 = 600 s (with the known values of r and D0) one obtains an excellent fit to the

experimental data in the late stage of uptake as presented in the inset of Fig. 4.9(b).

As shown in [41], the dynamic behavior obtained by Monte Carlo simulations ([32])

is in a very good agreement with Eq. (4.5).

Diffusion control of molecular uptake outside the hysteresis loop is further sup-

ported by uptake experiments using Vycor glass particles with different sizes. In

Fig. 4.10, the relative amount adsorbed by increase of pressure from 0.16 to 0.24 P0

is presented as a function of the re-scaled time, namely of tDe/r2, for three different

cylindrical Vycor samples with the same height (12 mm), but different diameters (3,

4, and 6 mm). As it is demonstrated, all curves collapse into one, thus confirming

the validity of diffusion scaling. Re-scaling of uptake kinetics in the same way in-

side the hysteresis loop yields notable difference between the different sample sizes,

confirming the absence of diffusion control under these conditions.

The extremely slow process of activated density relaxation observed in the range

of the hysteresis loop explains why the hysteresis, although representing a depar-

ture from equilibrium, is experimentally reproducible by various methods under

the same conditions. With increasing temperature, the Boltzmann factor kT may

become comparable to bξψ, providing the global density equilibration on the exper-

imental time scale. This leads, as reported in [65, 4], to the shrinking or even to the

disappearance of the hysteresis loop at the so-called hysteresis critical temperature.

47


Figure 4.10: Transient sorption of cyclohexane in Vycor porous glass cylinder withdiameters of 3 (solid line), 4 (crosses), and 6 (open circles) mm and height of 12 mm,measured at 297 K via the intensity of the NMR FID signal. The kinetics for theadsorption step 0.16 to 0.24 P0 are presented as function of dimension-less timetDer

−2.

4.3 Sorption Kinetics: Weak Surface Field

A strong argument supporting the activated nature of density relaxation in the hys-

teresis region is provided by uptake kinetics. In the case of Vycor porous glass with a

sufficiently low porosity and small pore sizes, leading to a strong surface field acting

upon confined fluid, we argue that the limiting mechanism is the fluctuation-driven

process of the fluid redistribution within the porous matrix. With increasing poros-

ity and, possibly, pore size, where the material may be considered to give rise to a

weak surface field, one may expect that nucleation of the very first nucleus, namely

in small regions containing a capillary-condensed liquid, can limit the adsorption

process. One of such materials is the controlled porous glass (CPG) FD121 (see

Sec. 3.1). Produced by a similar procedure as Vycor porous glass, CPG has a ran-

dom pore structure with a narrow pore size distribution around 15 nm. These pores

are significantly larger than in Vycor, which is reflected by the late condensation

48

4.3. SORPTION KINETICS: WEAK SURFACE FIELD

step as presented in Fig. 4.11. Moreover, the relatively parallel condensation and

evaporation transitions in the isotherms reveal the very open, networked pore struc-

ture. The adsorption and desorption isotherms of cyclohexane in FD121 at 300 K

as well as the corresponding effective self-diffusivities were measured by PFG NMR.

The particles are approximately spherical, with diameters ranging from 100 to 200

microns. The material gives rise to a prominent hysteresis loop as can be seen in

Fig. 4.11.

Figure 4.11: Top: Effective self-diffusivities of cyclohexane in ERM FD121 at 300 Kduring adsorption (open circles) and desorption (black circles) obtained by PFGNMR. Bottom: Corresponding adsorption (open squares) and desorption (blacksquares) isotherms. Lines are guide to the eye.

The uptake process outside the hysteresis loop is purely diffusion-limited and will

not be discussed here in detail. More important is the transient uptake behavior

inside the hysteresis region. In Fig. 4.12 the uptake curve is shown as recorded

upon a stepwise pressure change from 0.85 to 0.89 P0, well within the hysteresis

range. After a very fast diffusion controlled uptake at the short times, the long-time

behavior can be explained by the ”ageing” model proposed in [104]. It is based on

the hypothesis, that capillary condensation arises through an activated process, i.e.

by crossing of energy barriers for droplet nucleation.

49


Figure 4.12: Transient sorption of cyclohexane in FD121 spherical porous glassparticles at 300 K measured by NMR. The kinetics for the adsorption step 0.85to 0.89 P0 is presented as a function of time. The linear long-time uptake at thelogarithmic time scale reveals the nucleation-limited processes.

A similar dependence is observed for the PIB-IL hierarchical porous silica ma-

terial (isotherms given in Fig. 4.8). After the smaller necks are filled, the uptake

kinetics during the pressure step 0.77 to 0.88 P0 is limited by the activated nucleation

of liquid phase in the big spherical cavities. This may be easily seen in Fig. 4.13,

where the long-time uptake follows the logarithmic time dependence, similar to the

FD121 sample. During the diffusion-limited density relaxation, the global equilib-

rium in these approximately spherical particles with 1 mm diameter is found to be

established after circa 600 seconds (see Eq. (2.11)). The uptake curve presented in

Fig. 4.13 however, is not in equilibrium after 12 000 seconds!

4.4 Summary

With the present investigations we provide an extensive experimental study corre-

lating phase behavior and transport of confined fluids in mesoporous materials by

50

4.4. SUMMARY

Figure 4.13: Transient sorption of cyclohexane in PIB-IL at 297 K measured byNMR. The kinetics for the adsorption step 0.77 to 0.88 P0 is presented as a func-tion of time. The linear long-time uptake at the logarithmic time scale reveals thenucleation-limited processes.

means of NMR. Using materials with different pore structures and by comparing

the results of microscopic and macroscopic measurements of transport properties

we elucidate very general mechanisms which may account for the development of

adsorption and diffusion hysteresis in the systems under study.

As we have shown, adsorption in the mesoporous materials in the range of the

adsorption hysteresis dramatically slows down as compared to that out of the hys-

teresis region. In the latter case, we have been able to demonstrate that the uptake

is solely controlled by the equilibration of the created gradient in the chemical poten-

tials between the external gas phase and the confined fluid via molecular diffusion.

In the former case, however, slowing down of the uptake process cannot be explained

by a decrease of the diffusivities in pores - we prove experimentally that, with the

independently measured diffusivities, the relevant analytical models overpredict the

rate of equilibration. This unequivocally means that, in the hysteresis regime, there

exist two time scales where different mechanisms dominate the process of mass trans-

fer process. While on the short-time scale the equilibration is of diffusive character,

51


the long-time dynamics is controlled by thermally activated crossings of barriers in

the free energy of the whole system.

Being almost pure silica glasses, the three materials used in this work have almost

the same chemical compositions. In all presented studies, the same probe molecule,

namely cyclohexane, was used and some of the experiments were performed at the

same conditions (e.g., temperature). The obtained results reveal that the different

pore structure of the Vycor porous glass, controlled porous glass FD121, and PIB-IL

material may lead to different mechanisms of the slowing down of kinetics in the

hysteresis region.

In Vycor, with comparably small pore diameters of about 6 nm and a low porosity

of 28%, the pore surface plays the major role, determining the behavior of the fluid.

After the fast diffusive transport with capillary condensation of liquid bridges at the

early stage of uptake, the further adsorption is controlled by an activated growth

and the redistribution of the adsorbed phase. On the opposite, adsorption in the

similar random structure of FD121, but with bigger pore diameters (of about 15 nm)

is rather limited by the delayed nucleation of the regions with capillary-condensed

fluid. Certainly, the activated redistribution of the adsorbed phase can occur in this

”big” pores too. It becomes apparent that in this case the respective, characteristic

time scale of this latter activated process can be much longer than for Vycor porous

glass, since the energetic barriers to be overcome increase with increasing pore sizes.

However, we restrict ourselves from definitive conclusions on this issue, which would

certainly require further experimental work and theoretical analysis.

To confirm that nucleation-limited uptake can also occur in sufficiently big pores

and can slow down the kinetics, adsorption kinetics has also been probed in a mate-

rial with hierarchical pore structure, where the big spherical cavities (with diameters

of 20 nm) are connected via small necks (diameter 3 nm). Such a structural organi-

zation can substantially suppress the redistribution of the capillary-condensed phase

between the cavities. Indeed, the slowing down of kinetics in the hysteresis region

proves the relevance of the discussed mechanism, which also may be applied for the

analysis of the mass transfer processes in random porous glass with high porosity.

The measurement of adsorption is an established tool for the characterisation

of mesoporous materials. Assuming thermodynamic equilibrium in the system, the

pore size distribution can be calculated utilising the respective theories based on

the macroscopic Kelvin equation. However, the metastable nature of the adsorption

hysteresis raises the practical question about the most appropriate procedure for the

52

4.4. SUMMARY

estimate of structural properties. Our experiments confirm the theoretically found

non-equilibrium nature of adsorption hysteresis ([16, 63, 32]) and do provide novel

information for theoretical analysis.

To our knowledge, we presented for the first time the experimental proof for

the decoupling between the fast (diffusive) and slow (activated density distribution)

modes, which are responsible for the occurrence of adsorption hysteresis in meso-

porous materials. This work provides a natural explanation of this phenomenon

based on the specific dynamical features of the process: After a stepwise pressure

change, diffusion-controlled uptake brings the system into a quasi-equilibrium regime

and the further evolution follows the thermally activated fluctuations of the fluid

([32]).

The multiplicity of the internal density states inside the hysteresis loop is clearly

reflected by the behavior of the self-diffusivities, where the same number of molecules

in random mesopores are found to exhibit different transport properties, depend-

ing on the history how a particular state has been attained. Further theoretical

exploration of this phenomenon may lead to an approach providing a novel type

of information on micro-mesostructural details of fluid distribution in mesoporous

matrices by analysing the measured transport characteristics of the confined fluids.

53

Chapter 5

One-Dimensional Channels

It has already long ago been noted by Everett ([2]) that for an array of independent

pores with different pore sizes both desorption and adsorption branches should be

affected in the same way, i.e. these two branches should be parallel to each other.

Experiments, however, often reveal asymmetric hysteresis loops (H2-type). This

is typically the case for materials with highly networked pore structures, such as

random porous glasses as presented in Chapter 4. Therefore, the asymmetry of the

hysteresis is generally considered as a consequence of interconnectivity of the pores

([22, 4]).

Despite many experimental studies devoted to the understanding of the relation-

ship between pore geometry and sorption behavior, limited possibilities for a control

over the pore structure precluded definitive answers. The advent of template-based

mesoporous materials was expected to substantially contribute to the verification

of the existing theoretical predictions. However, it has become evident that the

experimental results obtained using these materials still may suffer from some ”non-

ideality” effects. These include, first of all, the occurrence of some defects in their

structure, such as the existence of interconnections between individual channels

(known, e.g., for SBA-15 material [43]). Another type of complications may arise

from finite-size effects.

Recently, a new type of materials obtained using electrochemical etching of sin-

gle crystals, namely porous silicon (PSi), have emerged as a promising, potential

candidate for studying the effects of pore structure on phase equilibria in pores.

It has been shown that by proper tuning of the fabrication conditions, PSi with

independent, linear pores of microscopic extensions (up to a few hundreds of mi-

55

CHAPTER 5. ONE-DIMENSIONAL CHANNELS

Figure 5.1: Nitrogen sorption isotherms in PSi at 77 K with one end (adsorption opencircles, desorption black circles) and two ends (adsorption open stars, desorptionblack stars) open. The isotherms are obtained by BelSorp Mini II. Inset: schematicsof the two systems. Lines are guide to the eye.

crometres long) can be obtained ([49]). A number of different experimental methods

have been used to prove the absence of intersections between individual channels

([78, 54]). Importantly, the fabrication procedure also allows to control the shape of

the pores ([105, 38, 50]). Providing such attractive options for a structure control,

PSi has been extensively used for experimental studies ([77, 78, 38, 106, 80, 81]).

However, the experiments revealed some unexpected, apparently counterintuitive

results.

In the process of electrochemical fabrication of PSi, a porous film is grown on a

silicon substrate. This provides a simple means to prepare channel-like pores open

at both end (upon detaching the porous film from the substrate by the use of an

electro-polishing current pulse) or only at one end (leaving the porous film on the

substrate) end. These two materials allow the verification of an important issue

in the sorption behavior, namely the identification of the equilibrium transition.

56

Figure 5.2: Nitrogen desorption scanning curves measured in PSi at 77 K. Afterincomplete adsorption up to 0.82 P0 (black circles) and 0.80 P0 (black triangles),the desorption has been measured. The desorption scanning curves are enveloped bythe boundary adsorption (open squares) and desorption (black squares) isotherms.The isotherms are obtained by BelSorp Mini II. Lines are guide to the eye.

Following the classical work by Cohan [10], the adsorption hysteresis in a cylindrical

pore, open at both ends, is due to a delayed menisci formation upon adsorption.

Thus, closing one end should remove the hysteresis. The experiments with PSi,

however, have shown identical adsorption isotherms irrelevant of whether the porous

film is removed from the substrate or not ([107, 54]), as can be seen in Fig. 5.1. This

finding has questioned the applicability of Cohan’s model to PSi.

The second interesting observation was that PSi exhibit H2-type hysteresis, al-

though the individual channels are isolated from each other. Additionally, the scan-

ning behavior observed in the sorption experiments is very similar to those for the

networked materials (Vycor, CPG), as presented in Figure 5.2. The dependencies

of the self-diffusivities of cyclohexane in PSi on the external gas pressure measured

using PFG NMR at 297 K as well show the behavior similar to disordered porous

glasses (Fig. 5.3). In Vycor, we can explain the increasing self-diffusivities on the ad-

57


Figure 5.3: Top: Effective self-diffusivities of cyclohexane in PSi at 297 K ob-tained by PFG NMR along the adsorption (open circles) and desorption (black cir-cles) branches. Bottom: Adsorption (open squares) and desorption (black squares)isotherms. Lines are guide to the eye.

sorption branch by the contribution of the gaseous phase to the overall mass transfer

in the lower pressure region. With starting capillary condensation, the influence of

the homogeneously distributed liquid-like phase leads to a decrease of the effective

diffusivities. On the desorption branch, the liquid is kept in the pore in a stretched

state which results in the slight increase of the diffusivities, until the steep evapo-

ration from the pores occurs. Thus, electrochemically etched porous silica material

exhibit a sorption behavior similar to the systems with quenched disorder, although

it is fabricated in a way that it possess independent parallel channels (see Sec. 3.1).

Generally, hysteresis loops of type H2 are believed to result from network ef-

fects, where both, pore blocking and percolation phenomena may contribute to the

observed asymmetry of the hysteresis loop ([22]). In addition to such an asym-

metry, the network effects result in specific types of the desorption and adsorption

scanning curves ([11, 12, 2]). Exactly such a behavior of scanning sorption curves

typical of interconnected structures was found for PSi too ([80]). Keeping in mind

the tubular pore geometry in PSi, without intersections between individual chan-

nels, explanations of all these experimental results often include a hypothesis about

58

5.1. MODEL

the occurrence of some inter-pore interaction leading to a cooperative evaporation

process from the pores. As one of the possible mechanisms of such an interaction,

the existence of a liquid film on the external surface of PSi has been suggested ([80]).

Wallacher et al. have performed adsorption experiments with PSi with an ink-

bottle morphology of the pores ([38]). Interestingly, they found an apparently iden-

tical hysteresis behaviour irrespective of whether the bottle-part of the PSi channels

had direct contact to the gas phase or only through the narrow neck. That means

that, in the latter case, the larger pores empty even if the necks remained filled

with liquid. Although this behavior, i.e., evaporation via the cavitation process, is

known to occur under certain conditions ([26, 30, 108]), the difference ∆d of only

about 1 nm in the diameters of the bottle and the neck parts in [38] was too small to

support this scenario. In addition, the authors found a very slow density relaxation

in the hysteresis region, which followed a stretched-exponential form with a stretch-

ing exponent less than one. This has been attributed to the effect of a quenched

disorder of the order ∆d, which, subsequently, has been anticipated to account for

the identical hysteresis behaviour for two different pore geometries.

In the light of such challenging experimental results, we have recently used Mean

Field Theory (MFT) of a lattice gas in order to explore how disorder in linear

pores may affect sorption behavior ([54]). It was found that all the experimental

findings described above can be comprehensively explained, taking account of a

strong mesoscalic disorder of the pore diameter. The main goal of this work is

to provide more detailed information about the influence of the different types of

disorder (geometrical and chemical) by means of MFT calculations.

5.1 Model

In the used Mean Field Theory approach (see Sec. 3.4), we consider a pore com-

posed of a random set of slit pore segments arranged along the x axis and infinitely

extended in y direction (Fig. 5.4). This is the simplest model of isolated pores with

disorder. The density is independent of y direction ([34]). It has been shown that

this geometry is qualitatively similar to the cylindrical pore (as can be seen by com-

paring the results in [109] for cylindrical pores with those in [60] for slit pores), but

reduces the problem to two dimensions, which helps to avoid additional confinement

effects of the lattice gas model and considerably saves computer time. For each seg-

59


ment, the height of the pore Hi is varied randomly in such a way that the overall

pore size distribution (PSD) has a Gaussian shape (Fig. 5.5). Periodic boundary

conditions are applied in the x and z directions. For the lattice sites occupied by

the pore walls, which thus are not accessible to fluid, the boundary conditions for z

direction have no influence. The total length of the pore, L, is the sum of the single

segment lengths Li. The first 10 lattice columns and the last 10 lattice columns

represent the external bulk gas, which is kept at the desired chemical potential, µ.

Figure 5.4: Schematic representation of a pore consisting of slit segments with vary-ing height Hi along the x axis and constant length Li. The pore is infinitely extendedin the y direction.

In the present work, we have considered tree types of disorder.

(i) Mesoscalic disorder, which is modelled by the variation of the segment size along

the channel direction. We fixed the segment length, Li, to 10 lattice sites

(variation of the segment length does not affect the qualitative behavior). The

visualisations of such a disorder are presented for several adsorption states in

Figs. 5.7 and 5.10 for a solid-fluid interaction ratio of y = 2 (see, for more

details on the model, Chap. 3).

(ii) Geometrical roughness of the pore wall. It is modelled by randomly adding

up to 10 single solid sites on the surface of a segment. In order to keep the

pore volume constant, simultaneously one wall site has been removed from the

surface. This roughness slightly changes the PSD making it slightly wider, but

the mean pore size remains constant (Fig. 5.8).

(iii) Chemical heterogeneity of the surface, which can be modelled by the variation

of the solid-fluid to fluid-fluid interaction ratio, y. Fig. 5.9 shows the fluid

states in a pore with surface field variation as it would be produced by the

pore wall roughness shown in Fig. 5.8. At the early stage of the isotherms

60

5.2. EFFECT OF MESOSCOPIC DISORDER ANDSURFACE ROUGHNESS

one can see the difference between the attraction strength of the wall sites

characterized by the different fluid density at the same chemical potential.

The fluid density profiles are presented by gray, and the color is scaled in such

a way that white corresponds to zero density and black to the liquid density. All

calculations have been performed at T ∗ = 1 which is 2/3 of the bulk critical temper-

ature for the simple cubic lattice gas in MFT. Note for the comparison that nitrogen

at 77 K is at about 61% of its bulk critical temperature.

In our model, adsorption of the fluid on the external surface was not allowed.

This simplification has no significant influence on the adsorption behavior and the

isotherms, since the internal wall area is much larger than the external surface. The

adsorption isotherms presented in our work are calculated for a pore which consists

of 500 segments. It has been tested that averaging over many random realisations

(i.e. a random array of segments with the same PSD), with the PSD kept constant,

does not affect the qualitative picture of the isotherms. The fluid density in the

pores is calculated for a sequence of external chemical potentials by fixing the value

in the bulk regions. The relative fluid density is plotted versus the relative activity

z = λ/λ0 = P/P0, where λ = exp(µ/kT ) is proportional to the pressure.

To study the influence of the pore size and its inhomogeneity, four different

realisations with a Gaussian PSD have been considered: (A) - with a segment size

from 4 to 8 lattice units: (B) - 6 to 10 lattice units; (C) - 8 to 12 lattice units;

(D) - 4 to 12 lattice units. The PSD for (A), (B) and (C) have the same shape

(Fig.5.5(a)), but are shifted towards a higher mean value. The PSD for (B) and (D)

have different widths, but the same mean value of 8 lattice units (Fig. 5.5(b)).

5.2 Effect of Mesoscopic Disorder and

Surface Roughness

First we are going to address the influence of the mesoscalic roughness on the ad-

sorption/desorption behavior in a single channel. This mesoscalic roughness is char-

acterized by a segment size variation along the pore. In Figure 5.6, we demonstrate

the adsorption and desorption isotherms for linear channels with the four different

pore size distributions shown in Fig. 5.5. In the case of a flat homogeneous sur-

face (solid line), for the (A) type channel we observe a sharp step in the amount

adsorbed at z = λ/λ0 ≈ 0.35 for y = 2. This step reflects the formation of a liq-

61


Figure 5.5: Studied pore size distributions. (a) Three PSD with the same dispersionbut different mean values: (A) 4 to 8 lattice units, (B) 6 to 10, (C) 8 to 12. (b) TwoPSD with the same mean value but different dispersions: (B) 6 to 10 lattice unitsand (D) 4 to 12. Lines emphasise the Gaussian shape of the PSD.

uid layer on the pore wall (as visualised in Fig. 5.7 for z = 0.40). Further uptake

in the (A)-type channel is controlled by the growth of the condensed liquid and

by capillary condensation in the segments in the order of increasing segment size

Hi. This can be recognised from the visualisations of the fluid density profiles in

Fig. 5.7, z ≥ 0.76, and is reflected by the steep jumps in the adsorption isotherm

(Fig. 5.6(A)). After the pores are filled completely, desorption first occurs by a de-

crease of the liquid density in the pore, i.e. by a stretching of the liquid. In Fig. 5.7

at z = 0.65, one may see an exemplification of the stretched liquid characterised

by a lower density. The fluid is kept in the pores by the pore blocking due to the

existence of pore segments with a sufficiently small size (4 lattice units in our case,

statistically distributed), representing necks. When the evaporation condition for

these necks is attained (z ' 0.64), the capillary-condensed phase evaporates from

all pores, creating a knee-like behavior in the desorption isotherm.

With increasing mean pore size, as in the case of the (B)-type channel, the ad-

sorption behavior changes. Although the layering transition on the flat homogeneous

surface occurs in the same manner as in the former case (Fig. 5.6(B)), the capillary

condensation occurs in one step upon surface covering by the fluid film. Thus, the

formation of the liquid film on the pore walls may be considered as a process making

the effective segment size almost uniform along the entire channel. In Fig. 5.10, one

can see the adsorption step from z = 0.93, where the mesoscopically-rough surface

62


Figure 5.6: Adsorption and desorption isotherms calculated for the slits with PSD(A), (B), (C), and (D). Solid lines: Isotherms for flat surface; black squares: Roughsurface; crosses: Chemical heterogeneity corresponding to geometrical roughness.The isotherms are obtained at T ∗ = 1.

is covered by a film of different thickness, to z = 0.94 where condensation occurs all

over the channel. This process creates a steep jump in the amount adsorbed in the

isotherm (Fig.5.6(B)). During the desorption, the pore blocking effect dominates the

emptying of the pores (z = 0.80 to z = 0.79), similar to the (A)-type channel.

The influence of the liquid layer, which covers the surface, becomes more sig-

nificant with increasing mean pore size. For the (C) channel with a mean segment

size of 10 lattice units, formation of a polylayer on the surface is observed. This

is reflected by the stepping in the adsorption isotherm in Fig. 5.6(C) at z = 0.92.

Desorption is again governed by pore blocking effect, as observed in cases (A) and

(B).

Widening of PSD (case (D)) around the same mean value of 8 lattice units (case

(B)), allows cavitation in bigger segments (e.g., with a size of 12 lattice units), before

63


0.200.400.750.760.820.840.870.950.650.64

Figure 5.7: Visualization of fluid density states in the channel (A) without geomet-rical roughness. Adsorption and desorption from top to bottom. The z values aregiven on the right of the pictures.

0.200.400.600.750.840.950.640.630.590.570.38

Figure 5.8: Visualisation of fluid density states in the channel (A) with geometricalroughness. Adsorption and desorption from top to bottom. The z values are givenon the right of the pictures.

the evaporation condition for the necks (4 lattice units) is fulfilled. Since the volume

of the neck segments is very small compared to the volume of the bubble created

by the cavitation, one observes a very steep decrease in the amount adsorbed as

indicated in Fig. 5.6(D) at z ≈ 0.65. Cavitation cannot be distinguished therefore

from pore blocking by the adsorption isotherm, but can be recognised in the density

profiles (density profiles are not presented here).

Introducing microscopic roughness of the surface does not change the qualitative

picture. As shown in Fig. 5.6 (black squares), for all the channels with different

PSD as studied in this work, the surface roughness smoothes the layering transition

due to the inhomogeneity of the surface field created by this pore wall roughness.

64


0.200.400.760.820.840.950.650.64

Figure 5.9: Visualization of fluid density states in the channel (A) with chemicalheterogeneity corresponding to geometrical roughness. Adsorption and desorptionfrom top to bottom. The z values are given on the right of the pictures.

0.200.400.600.930.950.800.79

Figure 5.10: Visualization of fluid density states in the (B) pore with homogeneousflat surface. Adsorption and desorption from top to bottom. The z values are givenon the right of the pictures.

Such a roughness does not change the mean pore size, but varies the segment size

locally, creating in some cases smaller and in some cases bigger pores. In Fig. 5.8,

some selected fluid density profiles for the (A)-type channel during adsorption and

desorption are presented. The wider PSD yields a more gradual adsorption isotherm

(Fig.5.6 (A)) because adsorption is now governed by the capillary condensation and

by the growth of the condensed bridges of capillary-condensed phase (z = 0.75). In

contrast to the flat homogeneous surface, desorption is controlled by a first stretching

of the liquid (z = 0.64) which is followed by cavitation (z = 0.63) and by the

evaporation of the remaining liquid phase (z = 0.57). The roughness can slightly

narrow the pore segments, creating very narrow necks. On the desorption, the limit

of the thermodynamic stability of the liquid in the larger pore segments may thus

be attained earlier than the condition of the evaporation from the necks. This

effect is even stronger pronounced in the case of the (D)-type channel, where the

necks connect significantly bigger cavities. In Fig. 5.12 one may recognise such

a desorption, controlled by the cavitation in bigger segments (step z = 0.66 to

65


0.200.400.600.820.830.850.890.950.680.67

Figure 5.11: Visualization of fluid density states in the (B) pore with geometricalroughness. Adsorption and desorption from top to bottom. The z values are givenon the right of the pictures.

0.200.400.600.820.890.950.660.650.64

Figure 5.12: Visualization of fluid density states in the channel (D) with geometricalroughness. Adsorption and desorption from top to bottom. The z values are givenon the right of the pictures.

z = 0.65), in parallel to the pore blocking.

We have found that the wall roughness has a significant influence on the adsorp-

tion behavior, when the characteristic length scale of the roughness is comparable to

the pore size. This is further supported by the visualisation of the sorption process

in the (B)-type channel shown in Fig. 5.11. With wider necks, the pore blocking

dominates and no cavitation process in the pores is observed. After a slight decrease

in the amount adsorbed, due to the density loss of the liquid (z = 0.68), the pore

get empty at z = 0.67, corresponding to the condition when evaporation from the

necks becomes thermodynamically favourable.

66

5.3. CHEMICAL HETEROGENEITY

5.3 Chemical Heterogeneity

To study the relation between the surface roughness and the chemical heterogeneity

of the surface, we have considered a distribution of the surface field on the flat surface

of the channels. Fig. 5.6 shows the isotherms obtained with a disordered surface field

(crosses). The chemical heterogeneity of the pore wall makes the adsorption behavior

in the early stage very similar to that obtained with the geometrical disorder. After

the surface layer is formed, the impact of the chemical heterogeneity on the next

layer formation or the capillary condensation reduces. In Fig. 5.9 one may note

continuous adsorption on the surface, relevant for heterogeneous surfaces, leading

to the continuous increase of the amount adsorbed (z = 0.20). When the surface is

covered by the liquid layer (z = 0.40), no or a very slight difference in the isotherms

is observed, as compared to the flat surface in Fig. 5.7.

The shape of the hysteresis loops is found to be very similar to that for the

homogeneous surface with y = 2. In Fig. 5.13, the effect of different distributions

of the y parameter is presented for the case of the (A)-type channel. As one may

expect, the isotherms for the continuous random variation of y in the range of 1 to

3, 2 to 4, 3 to 5, and 2 to 6 show a higher amount adsorbed at the same activity

with increasing solid-fluid attraction strength and a distinct variation of the shape

of the isotherms at early adsorption stage with varying y.

As mentioned above, the impact of the chemical heterogeneity is similar to that

of the surface roughness. If we assign the surface field distribution as provided by the

geometrically rough surface to a flat surface, the isotherms do coincide on the early

stage of adsorption, where the surface heterogeneity plays the dominant role. This

can be observed by comparing the isotherm for the rough surface (black squares) to

those for the chemically heterogeneous surface (crosses) in Fig. 5.6.

5.4 Role of External Surface

Another experimental observation we are going to address with help of MFT con-

cerns the identical adsorption/desorption behavior in the PSi with pores which are

open on both ends or only on one end. In [78], the authors have taken this finding

as an indicate of the importance of a liquid layer covering the whole surface of the

porous material. A similar assumption has been made in [80] to describe the shape

of the adsorption hysteresis in PSi. Since the external area of such a mesoporous

67


Figure 5.13: Adsorption/desorption isotherms for the (A) pore with heterogeneoussurface field. 1 ≤ y ≤ 3 (black stars), 2 ≤ y ≤ 4 (black squares), 3 ≤ y ≤ 5 (crosses),2 ≤ y ≤ 6 (open circles). T ∗ = 1. The lines are guide to the eye.

substrate is negligible as compared to the internal area, we do not expect this ef-

fect to play a sufficient role. In Fig. 5.14, the cumulative adsorption/desorption

isotherms are shown for two independent slit pores (height of 6 and 8 lattice units)

and two pores of the same shape connected by a liquid film over the external sur-

face. The inset of Fig. 5.14 shows the visualisations of the profile density for both

cases at full pore loading. The slight difference of the desorption isotherms are due

to small difference in the shape of the liquid-vapour menisci and disappears with

increasing pore length. Thus, no effect of the external surface is revealed by our

MFT calculations.

5.5 Effect of Pore Openings

In order to understand the coincidence of the isotherms for pores open at both ends

or only at one end as observed in the experiments ([78, 54]), we have performed

additional calculations for the channels with a rough surface (see Fig. 5.12) with one

68

5.5. EFFECT OF PORE OPENINGS

Figure 5.14: Adsorption/desorption isotherms for two independent pores (solid line)and two pores connected via the fluid film on the substrate surface (black squares).The schematics of both systems show the fluid state at completely filled pores.(y = 2, T ∗ = 1)

end closed. As we have discussed before, the desorption is controlled by cavitation

and pore blocking. If channel emptying is controlled by pore blocking effects, the

isotherms are expected to coincide. Due to the random distribution of the segment

sizes, Hi, and the sufficient length of the pore, there are always narrow necks close

to both pore ends. If the condition for evaporation from these necks is fulfilled, it

makes no difference whether the fluid evaporates from the bigger segments over a

single or over both pore openings.

Fig. 5.15 shows the adsorption/desorption isotherms for the (D)-type channel

with the pores open at both ends and at only one end. As observed before, one

finds a strong impact of cavitation on desorption. As the most important feature of

Fig. 5.15, the channels open at both ends or only on one end are found to give rise to

identical adsorption isotherms! This is in complete agreement with the experimental

results mentioned before. The inset in Fig. 5.15 shows the hysteresis loops for the

simple slit pore open at both ends and closed at one end with the pore size of 8

69


Figure 5.15: Adsorption and desorption isotherms calculated for the (D)-type poreopen at both ends (solid line) and open at only one end (black squares). Inset showsthe adsorption/desorption in a simple slit pore open at both ends (dotted line) andclosed at one end (solid line).y = 2, T ∗ = 1.

lattice units, which is the mean segment size of the (D)-type channel. As has been

already discussed elsewhere ([26]), there is little or no hysteresis in a linear pore

with one end open and a pronounced hysteresis if both ends are open, which is in

agreement with Cohan ([10]). The step in the adsorption isotherm at z ≈ 0.9 for the

pore open at both ends marks the second layering transition in this lattice model.

5.6 Discussion

We have highlighted by the application of Mean Field Theory to a lattice gas model

that three types of disorder affect the adsorption/desorption behavior at different

length scales and at different stages of the uptake. It is found that chemical het-

erogeneity, created by varying the local surface field strength in a pore with the

geometrically smooth pore wall, affects only the low-pressure regime, namely the ad-

sorption of the first monolayer on the pore walls. It generally smoothes the layering

70

5.6. DISCUSSION

transitions (two-dimensional condensation transition) at an early stage of adsorp-

tion at low external activities as can be seen in Fig. 5.6 for all channel types studied.

Comparing the isotherms for the a flat homogeneous surface with chemically hetero-

geneous surface one may clearly see the different behavior of the isotherms during

adsorption. If there occurs a sharp layering transition on the homogeneous surface

(at z ≈ 0.35 in Fig. 5.6), the chemically heterogeneous surface adsorbs different

amounts of molecules at the same z, depending on the surface attraction strength.

In Fig. 5.9, the fluid profile for z = 0.18 is presented. One may also note the vari-

ation of the fluid density adsorbed on the surface, where a higher fluid density is

observed closer to the stronger adsorbing sites.

After the entire surface is covered by the liquid, the capillary condensation or

evaporation and, thus, the shape of the hysteresis loop is not affected anymore, by

the variation of the surface field. This becomes apparent upon inspection of Fig. 5.6,

where for all channel types, the isotherms for homogeneous (solid lines) and chemi-

cally heterogeneous (crosses) surfaces coincide for z ≥ 0.40. However, for sufficiently

small pores, the effect of chemical heterogeneity can be more pronounced. Thus,

for the homogeneous surface of the-(A) type channel we observe that desorption is

controlled by the pore blocking (in Fig. 5.7 step z = 0.65 to z = 0.64). Surface field

variation for the same type of channel, i.e., creating stronger and weaker adsorption

sites, is found to result in the cavitation in bigger segments.

The uptake at low external activities is found to become steeper with increas-

ing surface field. Additionally, MFT shows that the distribution of the attraction

strengths, given by y, plays a significant role in determining the shape of the low-

pressure part of the adsorption isotherms (compare, e.g., the cases 3 ≤ y ≤ 5 and

2 ≤ y ≤ 6).

Quite similar to the chemical heterogeneity, the pore wall (microscopic) rough-

ness produces a variation of the surface field, thus having a strong impact on the

layering transitions. In Fig. 5.6 (A-D), the isotherms for the case of surface rough-

ness (black squares) and chemical heterogeneity (crosses) may be compared. The

reversible adsorption/desorption isotherms at an early stage of uptake coincide, un-

til the surface monolayer is formed (z ≈ 0.40). In contrast to the chemical het-

erogeneity, the surface roughness produces a variation of the segment sizes which

can change the thermodynamical conditions for capillary condensation. Indeed, we

find that capillary condensation occurs first in the thus created necks, followed by

the growth of the formed liquid bridges, which is reflected by the higher amount

71


adsorbed in Fig. 5.6(A). The additional necks created by the surface roughness may

also have an impact on the desorption process.

In sufficiently small pores the surface roughness may become comparable to the

pore size. This can enforce the cavitation pores in the pore segments disconnected

from the external gas phase by small necks due to strong pore blocking effect. For

the (A)- and (D)-type channels with the smallest segments with the width of 4

lattice units, the necks are significantly narrowed by the wall roughness sites. In

Fig. 5.8 and Fig. 5.12 one clearly observes the cavitation occurring in the (A)- and

(D)-type pores at the activity of z ≈ 0.65 during the desorption. Since the ratio

of the total pore volume to the volume of the liquid bridges remaining after the

cavitation is bigger for the (A)-type channel, one may see in the Fig. 5.6(A) a

slow decrease of the amount adsorbed (z ≈ 0.60) after the sharp knee (z = 0.64).

This decrease reflects the evaporation from the necks. Desorption from the (D)-type

channel shows a similar behavior. First, a slight decrease of the amount adsorbed due

to the decrease of the liquid density, i.e., stretching of the liquid (Fig. 5.12 z ≥ 0.66),

is observed. This is followed by a combination of cavitation and evaporation from

the liquid bridges (z < 0.66). It has already been observed by Molecular Dynamics

and Monte Carlo simulations, that the mass transfer can also occur through necks

filled with liquid ([9, 31, 26, 27]). In [26], the authors emphasise its dependence on

the model parameters, pore geometry, and the temperature, which is in complete

agreement with our calculations.

With increasing pore size (the cases of the (B)- and (C)-type channels) one ob-

serves that the desorption behavior is solely controlled by the pore blocking as in the

case of the flat homogeneous surface (Fig. 5.7). Our calculations show that without

geometrical roughness, the hysteresis loop is characterized by steep condensation

jumps due to capillary condensation in segments with different width (in the order

given by the Kelvin equation, see Chap. 2) and sharp desorption due to the pore

blocking. When the surface roughness may change the PSD significantly (relevant

for small pores), the hysteresis loop exhibits the typical asymmetry (H2 type [110]),

as observed in the experiments with PSi. With increasing pore size, the influence of

the wall roughness decreases, as possibly the case for the anodic aluminium oxide

([111, 51]). The hysteresis loop becomes more and more symmetric of the type H1.

For MCM-41 material, it has already been suggested in [65] that the adsorption

hysteresis (H1 type) does not originate from the pore blocking, but rather from the

metastability of the multilayer film in a single pore. For such a small pore size

72

5.7. CONCLUSIONS

(the case of MCM-41), MFT suggests that some defects on the pore walls (both of

geometrical and chemical nature) may affect the sorption properties.

It is worth noting that in our calculations the modelled surface roughness corre-

sponds to the atomistic disorder. Mesoscalic disorder requires a significant pore size

variation. As one may see in Fig. 5.6, our model calculations reveal that the main

qualitative properties of the hysteresis loop are governed by the mesoscalic disorder.

Comparing the isotherms with different type of disorder, we may recognise that the

surface roughness and the chemical heterogeneity determine only fine details.

An important observation is shown in Figure 5.14. We have observed that there

is no necessity for an interaction between the neighbouring channels which has been

suggested in [106, 80, 112] for a material to exhibit H2 hysteresis type isotherms.

We could rather - by the results shown in Fig. 5.14 - demonstrate that the observed

coincidence may be considered as a simple consequence of mesoscalic heterogeneity!

5.7 Conclusions

In this chapter, we presented the study of the influence of the geometrical and chem-

ical disorder in linear pores on the adsorption/desorption behavior by means of the

mean field theory. This structural model was used to capture the main properties of

electrochemically etched porous silicon. In contrast to the analogous template-based

materials with channel-like pores such as SBA-15, MCM-41 or anodic aluminium

oxide, mesoporous silicon has adsorption properties similar to that of disordered

materials with a network of mesopores (random porous glasses). Considerations of

the model presented here suggests that these properties (asymmetric hysteresis of

type H2, irrelevance of closing one end) can be explained using one and the same

concept assuming the existence of mesoscalic disorder, namely a distribution of a

pore dimension, exceeding disorder on the atomistic level. In this sense, linear pores

with a statistically varying pore diameter, exhibit all properties of three-dimensional

pore networks.

Visualization of the density distributions for states along the isotherms helped

us to elucidate some basic features of adsorption and desorption processes in linear

disordered pores. At low activities the isotherm is associated with the covering of

the pore walls with adsorbed layers. Importantly, the small-scale surface roughness

is only of importance in determining the proper isotherm curvature before onset of

73


hysteresis by, e.g., smearing out signatures of the 2D surface condensation transition.

At intermediate activities we see condensation of liquid bridges where the pore

widths are smallest. For the closed pore, these condensations may have already

occurred before pore condensation is underway at the closed end. This explains

why Cohan’s analysis [10], which applies to an idealised smooth-walled pore, is not

applicable here ([78]). At higher activities we have condensation of liquid bridges

in regions of higher pore diameter as well as growth of liquid bridges condensed at

lower activity. Through these processes the system progressively fills with liquid.

On desorption, the model predicts first a loss of density leading to an expanded

liquid throughout the pore. Further decrease of the gas pressure leads to a more

significant loss of density through a combination of cavitation ([30, 9]) and evapora-

tion from liquid menisci (delayed by pore blocking). Importantly, as a consequence

of strong disorder, the very first cavities may occur in the pore body far away from

the pore ends. This may help to rationalise puzzling desorption behavior from the

ink-bottle systems observed in [38]. Irrespective of whether the bottle-part has a

direct contact to the bulk phase or not, desorption is initiated by cavities formed

in the pore body. Thus, isotherms for the two ink-bottle-like configurations in [38]

become largely indistinguishable.

In summary, our experiments and theoretical calculations have identified the

effects of quenched disorder in the channel pores of PSi as the directing feature for

adsorption hysteresis. Importantly, our calculations suggest that this disorder has

to be relatively pronounced, exceeding disorder on an atomistic level. Thus, the

channel pores of PSi turn out to exhibit all effects more commonly associated with

three-dimensional disordered networks. In addition, however, their simple geometry

makes them an ideal model system for experimental observation and theoretical

analysis.

74

Chapter 6

Summary

In recent years, progress in the development of novel synthesis strategies has led to

the discovery of a large number of porous materials with controlled architectures

and pore sizes in the mesoporous range. The pore spaces in these materials are

sufficiently large that they can accommodate assemblies of molecules in condensed

(liquid-like or solid-like) states at low temperature. An important feature of these

materials is the phenomenon of hysteresis. Thus, the amount of a gas contained by

the material at a given bulk pressure is higher on desorption than on adsorption.

This indicates a failure of the system to equilibrate.

In the present work, we present an experimental study in which microscopic

and macroscopic aspects of the relaxation dynamics associated with hysteresis are

quantified by direct measurement. Using NMR techniques and porous glasses with

different properties as model systems, we have explored the relationship between

microscopic translational mobility (i.e. molecular self-diffusion) and global uptake

dynamics. For states outside the hysteresis region the relaxation process is found to

be essentially diffusive in character. Within the hysteresis region, however, the re-

laxation dynamics is dominated by activated rearrangement of the adsorbate density

within the host material, i.e. by an intrinsically slower process.

The latter leads to many interesting features of confined fluid systems, which

can be probed experimentally. One of them is a remarkably long ”memory” of the

past when the actual amount of molecules in the pores dramatically depends on the

history of how the external conditions have been changed. We demonstrate that

the intrinsic diffusivity as measured by NMR serves as an excellent probe of the

history-dependent states of the confined fluid. A remarkable feature of our results

75

CHAPTER 6. SUMMARY

are differences in diffusivity between out-of-equilibrium states with the same density

within the hysteresis loop. This reflects different spatial distributions of the confined

fluid that accompany the arrested equilibration of the system in this region.

Many features of adsorption behavior in random porous glasses are determined

by the disorder in their structural properties. The ability to exert a significant degree

of pore structure control in mesoporous silicon has made it an attractive material for

the experimental investigation of the relationship between pore structure, capillary

condensation and hysteresis phenomena. Using both experimental measurements

and a lattice gas model in mean field theory, we have investigated the role of pore

size inhomogeneities and surface roughness on capillary condensation of nitrogen at

77 K in porous silicon with linear pores. We find that this material has more in

common with disordered materials such as Vycor glass than the idealised smooth-

walled cylindrical pores discussed in the classical adsorption literature. We provide

strong evidence that this behavior comes from the complexity of the processes within

independent linear pores, arising from the pore size inhomogeneities along the pore

axis, rather than from cooperative effects between different pores.

76

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83

Acknowledgements

I wish to thank first of all the people who introduced science to me, my supervisorsDr. Rustem Valiullin and Prof. Dr. Jorg Karger. This is a great opportunity toexpress my respect to them for their patience, encouragement and competent guid-ance. Dr. Rustem Valiullin has been holding me to a high standard and enforcingstrict validations for each research result, and thus teaching me how to do research.I have been amazingly fortunate to have thus excellent working environment.

I am deeply grateful to Prof. Dr. Petrik Galvosas for the long discussions thathelped me sort out the technical details of my work. Together with Stefan Schlayer,they provided me the support for the PFG NMR and the know-how which has beenindispensable to achieve my scientific goals.

My deepest gratitude is to Prof. Dr. Peter A. Monson, who teached me the firststeps in the domain of the Mean Field Theory. Without his inspiration in this field ofscience, motivating discussion and ideas, this deep insight into the thermodynamicsof phase transitions under confinement would not have been possible. I am verythankful for the care and the pleasant atmosphere during my stay at the Universityof Massachusetts (Amherst, USA) to him and his colleagues, John Rajadayakaran,Barry Husowitz, Lin Jin and Lingling Jiaz.

I wish to thank Prof. Dr. Wolfhard Janke, Dr. PD Michael Bachmann andPD Dr. Siegfried Fritzsche who sparked my interest in computer science duringmy student time. In their lectures, for the first time I encountered the world ofMonte Carlo and Molecular Dynamics.

I owe my gratitude to all those people from Prof. Karger’s group who havemade this dissertation possible and because of whom my graduate experience hasbeen one that I will cherish forever. Especially I shall thank Dipl.-Phys. MuslimDvoyashkin for his high scientific level stimulating me to stay tuned, Dipl.-Ing.Alexey Khokhlov for his industriousness facilitating porous silicon material, Dipl.-Chem. Katrin Kunze, Dipl.-Phys. Cordula Barbel Krause and Lutz Moschkowitz formaking things simple, and Prof. Dr. Dieter Freude for sharing his worldly wisdom.

There is no doubt that the steady progress of my work would not have beenpossible without the continuous support by the staff of the mechanical workshop ofour faculty together with glassblower Peter Fatum. I have a deep respect to themfor their capabilities and industry.

The financial support of the German Research Foundation (DFG), the ResearchAcademy Leipzig (Forschungsakademie Leipzig) and Max-Buchner-Research Foun-

85

dation (Max-Buchner-Forschungsstiftung) is gratefully acknowledged.

Many friends have helped me stay sane through these years. I greatly value theirfriendship and I deeply appreciate their belief in me.

Most importantly, I wish to thank my family for everything which cannot be putinto words.

86

Appendix

VaporControl Reference

The VaporControl adsorption setup can be controlled either by the GUI or textbased scripts. Following graphical user control elements are implemented:

• RNControl: Accesses an RNMotor board ([87]), connected via RS-232 COMinterface (Tab. 6.4)

• RNMotor: Indirect accesses a stepping motor connected to an RNMotor board(Tab. 6.5)

• DGControl: Accesses an Pfeiffer Vacuum DualGauge TPx261 pressure sensorunit connected via RS-232 COM interface (Tab. 6.7)

• OmronControl: Accesses an Omron E5CK thermostat connected via RS-232COM interface (Tab. 6.6)

• COMControl: Parent element for graphical user control elements. Possesses nofunctionality besides connecting to a RS-232 COM interface of an arbitrarymodule. RNControl, DGControl, OmronControl inherit all properties fromCOMControl element (Tab. 6.3)

Table 6.2 gives the overview of the VaporControl GUI commands.A built-in OLE interface object VC3Client.Communicator allows the remote

control of the VaporControl from user made applications (Tab. 6.1).

SendCommand commandline send a command line to VaporControl GUIReturnValue the return value of last operationErrorCode the error code of last operation

Table 6.1: VC3Client.Communicator OLE object command reference

87

EXIT close VaporControl GUISTOP abort the program executionX val set left window coordinate to val

Y val set top window coordinate to val

W val set window width to val

H val set window height to val

LOG text add text to logging windowWAIT s delay the program execution by s secondsSAVE filename save current control state to a script file filename

LOAD filename load and execute script from file filename

SET key val set an internal variable key to value val

GET key returns the value of an internal variable key

ADD control name adds a control and assigns the control nameFollowing hardware controls are implemented:RN: Control for an RNMotor stepping motor driver unitDG: Control for a Pfeiffer Vacuum TP26x DualGaugepressure sensor unitOMRON: Control for an OMRON temperature controllerunit

DEL name remove a control with name name

BG filename set the layout image stored in file filename

MODE mode set the GUI operating mode to name

Following modes are available:0: Idle mode, the controls cannot be moved or resized1: Design mode, the controls can be moved and resized2: Execution mode, internal mode used for scripts exe-cution

LIST list all controls

Table 6.2: GUI command reference

88

NAME name assign the control name, only letters a-z and numbers0-9 may be used, no spaces!

X val set left control coordinate to val

Y val set top control coordinate to val

W val set control width to val

H val set control height to val

ACTIVE state set control state to state

Following states are possible:0: inactive, no refresh, no connection to hardware1: active

COM param value set a COM port parameter param to a value value

Following parameter are available:PORT port: COM port nameBAUD rate: COM baud rate (110, 300, 600, 1200, 2400,4800, 9600, 14400, 19200, 38400, 56000, 57600, 115200)PARITY parity: COM parity (None, Odd, Even, Mark,Space)DATABITS bits: COM data bits number (5, 6, 7, 8)STOPBITS bits: COM stop bits number (1, 1.5, 2)TIMEOUT READINTERVAL ms: COM read interval in ms,specifies the maximum time allowed to elapse betweenthe arrival of two characters on the communications lineTIMEOUT READCONST ms: COM read interval in ms,specifies the constant used to calculate the total time-out period for read operationsTIMEOUT READMULT ms: COM read interval in ms, spec-ifies the multiplier used to calculate the total time-outperiod for read operations.TIMEOUT WRITECONST ms: COM read interval in ms,specifies the constant used to calculate the total time-out period for write operations.TIMEOUT WRITEMULT ms: COM read interval in ms,specifies the multiplier used to calculate the total time-out period for write operationsTimeout = (MULTIPLIER * number-of-bytes) + CON-STANT

Table 6.3: COMControl command reference

89

MOTOR m ... send command with parameter to the RNMotor controlm (0 or 1)

VREF v set the board reference voltage. Caution, wrong valuemay damage the board! See RNMotor reference ([87]).

Table 6.4: RNControl command reference

NAME name assign the control name, only letters a-z and numbers0-9 may be used, no spaces!

X val set left control coordinate to val

Y val set top control coordinate to val

W val set control width to val

H val set control height to val

SPEED val set stepping motor speed to 0 ≤ val ≤ 255CURRENT i set stepping motor maximal current to 0 ≤ i ≤ 255.

Real current value in A = i / 100POS pos assign the position of the valve to pos, i.e. the valve

state. No action on the valve is performed.MAXPOS pos assign the position pos corresponding to the open valveMINPOS pos assign the position pos corresponding to the closed valveOPENDIR dir assign the rotation direction

0 : open counterclockwise1 : open clockwise

GOTO pos actuate the valve to position pos

OPEN actuate the valve to maximal positionCLOSE actuate the valve to minimal positionSTOP stops the current operation

Table 6.5: RNMotor command reference

TI t set refresh timer interval to t in msREFRESH read the temperature valueT return the temperature value from sensor

Table 6.6: OmronControl command reference

TI t set refresh timer interval to t in msREFRESH read the pressure and status information from the sen-

sorsP n return the pressure value from sensor n (0 or 1)STATUS n return the status value from sensor n (0 or 1). For status

information see the DualGauge manual.

Table 6.7: DGControl command reference

90

List of Publications

Journal Publications

• Exploration of Molecular Dynamics During Transient Sorption of Fluids inMesoporous Materials, Valiullin R., Naumov S., Galvosas P., Karger J., WooH.-J., Porcheron F., Monson P. A., Nature 443, 965 (2006)

• Diffusion Hysteresis in Nanoporous Materials, Naumov S., Valiullin R., GalvosasP., Karger J., Monson P. A., Eur. Phys. J. Special Topics 141, 107 (2007)

• Dynamical Aspects of the Adsoption Hysteresis Phenomenon, Valiullin R.,Naumov S., Galvosas P., Karger J., Monson P. A., Magn. Reson. Imaging,25, 481 (2007)

• Tracing Pore Connectivity and Architecture in Nanostructured Silica SBA-15,S. Naumov, R.Valiullin, J. Karger, R Pitchumani, M.-O. Coppens, Microp-orous and Mesoporous Materials, 110 (2008) 3740

• Charge Transport and Mass Transport in Imidazolium Based Ionic Liquids,J. Sangoro, A. Serghei, S. Naumov, P. Galvosas, J. Karger, C. Wespe, F.Bordusa, and F. Kremer, Phys. Rev. E 77, (2008), 051202

• Electrical Conductivity and Translational Diffusion in the 1-butyl-3-methylimidazolium tetra-fluoroborate Ionic Liquid, J. Sangoro, C. Iacob,A. Serghei, S. Naumov, P. Galvosas, J. Kaerger, C. Wespe, F. Bordusa, A.Stoppa, J. Hunger, R. Buchner, and F. Kremer, Journal of Chemical Physics,128 (2008), 214509,

• Probing Memory Effects in Confined Fluids via Diffusion Measurements, S.Naumov, R. Valiullin, P.A. Monson, and J. Karger, Langmuir 24 (2008), 6429-6432

• Understanding Capillary Condensation and Hysteresis in Porous Silicon: Net-work Effects within Independent Pores, S. Naumov, A. Khokhlov, R. Valiullin,and J. Karger, P.A. Monson, Physical Review E 78, Rapid Communication,060601, (2008)

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• Charge Transport and Glassy Dynamics in Imidazole-Based Liquids, C. Iacob,J. R. Sangoro, A. Serghei, S. Naumov, Y. Korth, J. Karger, C. Friedrich, andF. Kremer, The Journal of Chemical Physics 129, 234511 (2008)

• Charge Transport and Dipolar Relaxations in Hyper-Branched Polyamide Amines,J. Sangoro, G. Turky, M.A. Rehim, C. Iacob, S. Naumov, A. Ghoneim, J.Karger, F. Kremer, Macromolecules, (2009) accepted

• Pulsed Field Gradient NMR Study of Surface Diffusion in Mesoporous Ad-sorbents, M. Dvoyashkin, A. Khokhlov, S. Naumov, R. Valiullin, Microporousand Mesoporous Materials (2009) accepted

• Understanding Network Effects in Adsorption/Desorption in Mesoporous Ma-terials with Independent Channels, S. Naumov, R. Valiullin, and Jorg Karger,P.A. Monson, submitted

Oral Presentations

• Diffusion Hysteresis in Porous Materials, 3rd International Workshop on Dy-namics in Confinement, Grenoble (2006)

• Hysteresis Phenomena in Mesoporous Materials, 5th International ResearchTraining Group Diffusion in Porous Materials (2006)

• Adsorption Hysteresis in Mesoporous Materials, AMPERE NMR SummerSchool (2007), Bukowina Tatrzan’ska, Poland

• Diffusion Scanning Hysteresis Loops in Nanopores, Fundamentals Of Adsorp-tion 9, (2007), Giardini Naxos, Sicily Italy

• Adsorption Hysteresis in Mesoporous Materials, 6th International ResearchTraining Group Diffusion in Porous Materials (2007)

• Overview of the Department of Interface Physics, 1st Young Researchers Meet-ing INSIDE POReS, February 2008, Delft, The Netherlands

• Diffusion Processes in Mesoporous Adsorbents Probed by PFG NMR, 20.Deutsche Zeolith-Tagung, March 2008, Halle, Germany

• Phase Transitions under Confinement: Deeper Insight using NMR, AMPERENMR Summer School 2008, Wierzba, Poland

• Phase Behavior of Fluids in Porous Silicon Materials and their Textural Char-acterization, The 5th International Workshop on Characterization of PorousMaterials from Angstroms to Millimeters, June 2009 New Brunswick, NJ,USA

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Poster Presentations

• Study of History Dependence of Adsorption and Self-difusion Processes inPorous Media with Help of PFG NMR, 8th International Bologna Conferenceon Magnetic Resonance in Porous Media (2006), Bologna, Italien

• Dynamics and Phase Transitions Under Confinement, Fundamentals Of Ad-sorption 9, (2007), Giardini Naxos, Sicily Italy

• Adsorption Hysteresis Phenomena in Mesopores, Diffusion Fundamentals 2,(2007), L’Aquila, Italy

• Hysteresis Phenomena in Porous Materials, Meeting of the Review Panel ofthe Defence of the International Research Training Group ”Diffusion in PorousMaterials”, Leipzig (2008)

• Tracing Pore Connectivity and Architecture in Nanostructured Silica SBA-15, Meeting of the Review Panel of the Defence of the International ResearchTraining Group ”Diffusion in Porous Materials”, Leipzig (2008)

• Adsorption Hysteresis Phenomena in Mesopores, EUROMAR Magnetic Res-onance Conference, July 2009, Goteborg, Sweden (2009)

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Curriculum Vitae

Personal information

Family name NaumovFirst name SergejDate of birth 31.10.1980Place of birth Pskov (Russia)Nationality GermanE-Mail [email protected]

Education

1987 - 1990 Primary school, Pskov, USSR1990 - 1994 Secondary school, Pskov, Russia1994 Emigration to Germany1994 - 1999 Secondary school, Leipzig, Germany2000 - 2005 Physics studies, Leipzig University, Germany2005 Topic of diploma thesis: ”NMR Study of Adsorption and Desorp-

tion Phenomena in Porous Media”Since 2006 Ph.D. student in the Department of Interface Physics, Faculty for

Physics and Earth Science Leipzig UniversityAssociated member of the International Research Training Group”Diffusion in Porous Materials”

Occupational development

1999 - 2000 Military service2000 - 2005 Student assistant in the Institute of Surface Modification, Leipzig,

Germany.Scope of duties: Computer based quantum chemical calculations ofmolecular properties

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Selbstandigkeitserklarung

Hiermit erklare ich, dass ich die vorliegende Arbeit selbstandig und ohne unzulassigeHilfe oder Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Ichversichere, dass Dritte von mir weder unmittelbar noch mittelbar geldwerte Leis-tungen fr Arbeiten erhalten haben, die im Zusammenhang mit dem Inhalt der vor-liegenden Dissertation stehen, und dass die vorgelegte Arbeit weder im Inland nochim Ausland in gleicher oder ahnlicher Form einer anderen Prufungsbehorde zumZwecke einer Dissertation oder eines anderen Prufungsverfahrens vorgelegt und inihrer Gesamtheit noch nicht verffentlicht wurde. Alles aus anderen Quellen odervon anderen Personen ubernommene Material, das in der Arbeit verwendet wurdeoder auf das direkt Bezug genommen wird, wurde als solches kenntlich gemacht.Insbesondere wurden alle Personen genannt, die direkt an der Entstehung der vor-liegenden Arbeit beteiligt waren.Es haben keine erfolglosen Promotionsversuche stattgefunden.Die Promotionsordnung vom 11. Juni 2008 wird anerkannt.

Leipzig, den 04.03.2009

Sergej Naumov

Synopsis

Introduction

In recent years, considerable progress has been achieved in the development of noveltailor-made mesoporous materials with well-defined structural properties. An in-herent feature of molecular ensembles in mesopores is the interplay between thefluid-pore wall and fluid-fluid interactions. It may give rise to various specific phe-nomena of the confined fluids. A classical example of such phenomena, which stillremains a subject of controversial discussions ([35, 5, 6]), is the adsorption hystere-sis: at the same external conditions, the amount of guest molecules adsorbed by themesoporous host is higher upon decreasing external gas pressure than upon increas-ing. This indicates the failure of the system to equilibrate during the experiment([4]).

This thesis addresses the equilibrium and dynamic fluid properties under meso-porous confinement. Taking advantage of the Pulsed Field Gradient (PFG) NMRtechnique, the molecular self-diffusivities of fluids in mesopores with different porestructures are correlated with the phase state as controlled by the external gas phase.Additionally, the molecular transport properties, as revealed by microscopic (self-diffusivities) and macroscopic (transient sorption) methods, are compared. Thishelps to highlight the underlying mechanisms and address in more detail dynamicaspects accompanying the adsorption hysteresis. By means of the Mean Field The-ory (MFT) of lattice gas, the effect of disorder on fluid sorption behavior is addressed.Excluding network effects by using a linear pore, effects of internal disorder by anintentionally created geometrical and chemical heterogeneity are studied and com-pared with our experimental findings.

Materials and Methods

For the experimental study of the hysteresis phenomena, two different types ofporous systems have been used, namely ”interconnected” and ”non-interconnected”pore systems. Vycor 7930 porous glass with a mean pore size of 6 nm and a ”con-trolled porous glass” (CPG) with mean pore size of 15 nm represent highly inter-connected random pore network. Another interconnected but ordered hierarchicalsystem is the PID-IL porous silica ([113]), consisting of spherical cavities with a di-

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ameter of about 20 nm connected via small channels of 3 nm diameter. Electrochem-ically etched porous silicon films belong to the materials with non-interconnectedparallel channels, with a mean diameter of about 6 nm.

Results and Discussion

Interconnected Pores

In addition to the well known adsorption hysteresis loop, the hysteresis behaviorof the self-diffusivities was obtained by PFG NMR (Fig. 6.1(a)). Obviously, theadsorption hysteresis and the hysteresis loop of the self-diffusivities are correlated.The mono-exponential dependence of the NMR spin echo decay on the applied fieldgradient strength reveal the fast exchange of the molecules during the observationtime of the diffusion experiment. Thus, we may explain the behaviour of the effectiveself-diffusivities by the contribution of the fast transport in the gaseous and the slowone in the adsorbed or capillary-condensed phases ([40]). Calculating the effectiveself-diffusivities in this way and assuming the Knudsen regime for the diffusion ingas ([114]), we have found a good qualitative agreement.

(a) (b)

Figure 6.1: (a): Effective self-diffusivities of cyclohexane adsorbed in Vycor 7930at 297 K measured by PFG NMR upon increasing (open circles) and decreasing(black circles) gas pressure and corresponding amount adsorbed (open squares) anddesorbed (black squares) plotted as a function of the relative pressure P/P0 with P0

denoting the saturated vapour pressure. (b): Diffusivities plotted versus the amountadsorbed obtained from the sorption isotherms, adsorption path (open circles) anddesorption path (black circles). Lines are guide to eye

One of the most remarkable features Fig. 6.1(a) emerges when the diffusivities arepresented as a function of the relative amount adsorbed (Fig. 6.1(b)): one and thesame number of molecules exhibit different effective self-diffusivities on adsorption

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and desorption! Thus a novel means for reflecting different internal density distri-butions has been revealed ([44]). The so-called scanning curve experiments, whereincomplete adsorption/desorption cycles are performed (as shown in Fig. 6.2(a)),yield even a whole map of self-diffusivities inside the major loop (Fig. 6.2(b)). Thisclearly manifests a history-dependent adsorbate distribution in pores!

(a) (b)

Figure 6.2: (a): Relative amount of cyclohexane adsorbed in Vycor 7930 at 297 Kas a function of relative pressure. The desorption scanning isotherm begins on theboundary adsorption isotherm at 0.65 P0 (black stars) and is reversed at 0.44 P0

(open stars). The adsorption scanning curve from 0.43 P0 to 0.59 P0 (open circles) isreversed to 0.43 P0 (black circles). (b): Corresponding self-diffusivities as a functionof relative amount adsorbed. Lines are guide to eye

The subloops measured inside the major loop exhibit two further importantfeatures ([44]):

• Return point memory, i.e. after an incomplete sorption cycle, the systemreturns to its initial state. This feature suggests that the external conditionsare the main driving force of the evolution in such systems. Further thermalequilibration is prohibited by the high energy barriers between the minima inlocal free energy ([37, 95]).

• Lack of congruence, i.e. two different subloops are in general not parallel toeach other. This is a signature of networked pores, since the independent-poremodel would predict exact congruence ([16, 66]).

One of the most straightforward methods to illuminate the mechanisms of ad-sorption is the analysis of the transient sorption behavior. The results of the tran-sient sorption experiment outside the hysteresis region (Fig. 6.3(a)) and inside it(Fig. 6.3(b)) may be correlated with the information from the diffusion studies. Inthis way, we have been able to identify two mechanisms determining the uptakekinetics([41]):

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• Adsorption at low pressures is limited by the diffusion of the fluid moleculesinto the pore space with the formation of an adsorbed layer on the pore wall.Since, at this stage, the whole pore space is accessible to the mass transportfrom external gas phase, the dynamics is purely diffusive. The global equi-librium may be attained very fast on the experimental time scale, where thechemical potential is uniform over the whole system.

• With increasing density, capillary condensation occurs, followed by a growthof the domains with the capillary-condensed liquid inside the porous structure.In parallel, the system may further evolve, i.e. move to the global minimumin the free energy by redistribution of such domains. This, however, is anactivated process requiring crossing the barriers between the local minima ofthe free-energy. If, due to a microscopic fluctuation, the system jumps fromone local minimum to another, this creates spatially local density perturbation.The latter is quickly equilibrated (to local equilibrium) via the diffusion of themolecules from the surrounding pores and, therefore, from the external gaseousphase which leads to further uptake.

(a) (b)

Figure 6.3: Sorption kinetics data of cyclohexane in a Vycor 7930 cylinder (diameter3 mm, length 12 mm) at 297 K measured by NMR. Typical kinetic data (blacksquares) obtained upon a stepwise change of the external gas pressure outside thehysteresis loop from 0.16 to 0.24 P0 (a) and inside the hysteresis loop from 0.48to 0.56 P0 (b). The inset of (b) shows the long-time part of the data (b), axisquantities and units are the same as in main figure. The dotted lines represent thekinetics calculated via the diffusion equation. The solid line in (b) is calculated forthe activated uptake processes.

Similar behavior has been observed in materials with networked but bigger pores.In the case of Vycor porous glass with sufficiently low porosity leading to a strongsurface field acting upon confined fluids, we argue that the limiting mechanism isthe fluctuation-driven process of the fluid redistribution within the porous matrix.

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With increasing porosity (as in the case of CPG and PIB-IL) and, possibly, poresize, where the material may be considered to give rise to a weak surface field, onemay expect that nucleation of the very first nucleus, namely small regions containingcapillary-condensed liquid, may limit the adsorption process.

To our knowledge, we presented for the first time the experimental proof forthe decoupling between the fast (diffusive) and slow (activated density distribu-tion) modes, which is responsible for the occurrence of the adsorption hysteresisin mesoporous materials. This work provides a natural explanation of this phe-nomenon based on the specific dynamical features of the process: After a stepwisepressure change, diffusion-controlled uptake brings the system into a regime of quasi-equilibrium where further evolution is brought about by the thermally activatedfluctuations of the fluid ([32, 114, 41]).

One-Dimensional Channels

Here, the study of the influence of the geometrical and chemical disorder in lin-ear pores on the adsorption/desorption behavior by means of the MFT ([34]) ispresented. This structural model is used to capture the main properties of the elec-trochemically etched porous silicon (PSi). We consider a pore composed of a randomset of slit pore segments creating a linear channel. This is the simplest model of iso-lated pores with disorder. The size of the pore segments is varied randomly in sucha way that the overall pore size distribution (PSD) has a Gaussian shape. The poreopenings are in contact with the bulk gas kept at the desired chemical potential.

Different types of disorder have been studied using this model:

• Mesoscalic disorder, which is modelled by the variation of the segment sizealong the channel direction

• Geometrical roughness of the pore wall, modelled by randomly adding solidsites on the surface of a segment

• Chemical heterogeneity of the surface, which can be modelled varying the ratiobetween the solid-fluid and fluid-fluid interaction.

Opposite to the analogous template-based materials with non-interconnectedchannel-like pores such as SBA-15, MCM-41 or anodic aluminium oxide, mesoporoussilicon has adsorption properties of disordered materials with a network of mesopores(random porous glasses) as can be seen from the shape of the sorption isothermsand the desorption scanning curves (Fig. 6.4(a)). Additionally, the behavior of theself-diffusivities is very similar to that observed in Vycor (compare Figs. 6.4(b) and6.1(a)).

The theoretical analysis using this model suggests that these properties (asym-metric hysteresis of type H2, irrelevance of closing one end) can be explained my

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(a) (b)

Figure 6.4: (a) Nitrogen desorption scanning curves measured in electrochemi-cally etched porous silicon at 77 K. Here, desorption followed already after incom-plete adsorption, to 0.82 P0 (black circles) and 0.80 P0 (black triangles), respec-tively. The desorption scanning curves are enveloped by the ”boundary” adsorp-tion (open squares) and desorption (black squares) isotherms. (b) top: Effectiveself-diffusivities of cyclohexane in PSi at 297 K obtained by PFG NMR along theadsorption (open circles) and desorption (black circles) branches. Bottom: Adsorp-tion (open squares) and desorption (black squares) isotherms. Lines are guide tothe eye.

assuming the existence of mesoscalic disorder, namely a distribution of pore dimen-sions, exceeding the disorder on the atomistic level. In this sense, linear pores with astatistically varying pore diameter, exhibit all properties of three-dimensional porenetworks.

0.400.750.950.640.63

Figure 6.5: Visualisation of fluid density states in the channel with geometricalroughness. The segment size ranges from 4 to 8 lattice units. Adsorption anddesorption from top to bottom. The P/P0 values are given on the right of thepictures, first increasing for adsorption and then decreasing for desorption.

Visualization of the density distributions (for an exemplification see Fig. 6.5) forstates along the isotherms helped us to elucidate some basic features of adsorptionand desorption processes in linear disordered pores ([54]):

• At low gas pressures, the isotherm is associated with the covering of the porewalls with adsorbed layers. Importantly, the small-scale surface roughness is

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only of importance in determining the proper isotherm curvature before on-set of hysteresis by, e.g., smearing out signatures of a 2D surface condensationtransition. At intermediate pressures we see the condensation of liquid bridgeswhere the pore width is smallest. For the pores closed at one end this conden-sation may have occurred already before pore condensation is underway at theclosed end. At higher pressures we observe the condensation of liquid bridgesin regions of a higher pore diameter as well as a growth of liquid bridges con-densed at lower activity. Through these processes the system progressivelyfills with liquid

• On desorption, the model predicts first a loss of density leading to an expandedliquid throughout the pore. Further decrease of the gas pressure leads to a moresignificant loss of density through a combination of cavitation and evaporationfrom liquid menisci (delayed by pore blocking). Importantly, as a consequenceof strong disorder, the very first cavities may occur in the pore body far awayfrom the pore ends. This makes the adsorption behavior of a channel open atone end indistinguishable to that open on both ends.

Summary

In summary, our experiments and theoretical calculations have identified the effectsof quenched disorder in the channel pores of electrochemically etched porous siliconas the directing feature for adsorption hysteresis. Importantly, our calculationssuggest that this disorder has to be relatively pronounced, exceeding disorder onan atomistic level. Thus, the channel pores of PSi turn out to exhibit all effectsmore commonly associated with three-dimensional disordered networks. In addition,however, their simple geometry makes them an ideal model system for experimentalobservation and theoretical analysis.

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Hysteresis Phenomena in Me So Porous Materials

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