Hysteresis Phenomena in Mesoporous Materials

Transcription

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 Naumov
Hysteresis Phenomena in Mesoporous Materials
Universität Leipzig, Dissertation, 2009
95 pages, 112 references, 37 figures, 7 tables
This thesis deals with the recent efforts to elucidate the origin of the adsorption
hysteresis phenomenon typical for mesoporous materials. Utilizing the capabilities
of pulsed field gradient nuclear magnetic resonance, the macroscopic information,
accessible by transient sorption experiments, and the microscopic information, provided by the effective self-diffusivities, have been correlated and thus shown to yield
further insight into the adsorption dynamics and the equilibrium properties of guest
molecules in mesopores. In particular, two mechanisms of molecular transport,
namely self-diffusion and activated redistribution of the fluid in the pores, have
been elucidated.
Basing on this finding, an explanation for the slowing down of the transient
uptake with the onset of capillary condensation, observed in experiments, has been
given. The activated nature of nucleation, growth and redistribution of the fluid
phase inside the pores prevents equilibration on an experimental time scale.
Adsorption behavior in electrochemically etched porous silicon with linear pores
has been studied by means of Mean Field Theory. It has been shown that the directing feature of many puzzling observations is the existence of a mesoscalic disorder,
exceeding the disorder on an atomistic level. Thus, the linear, non-interconnected
channels in mesoporous silicon turn out to exhibit all effects commonly associated
with three-dimensional disordered networks.
i
Contents
1 Introduction
1.1 Hysteresis Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Aims of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Basics
2.1 Liquid-Gas Phase Transition . . .
2.2 Capillary Condensation . . . . . .
2.3 Adsorption Mechanisms . . . . .
2.3.1 Adsorption Hysteresis . .
2.3.2 Sorption Scanning Curves
2.4 Diffusion . . . . . . . . . . . . . .
2.4.1 Diffusion in Pores . . . . .
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1
1
6
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9
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3 Materials and Methods
3.1 Materials . . . . . . . . . . . . . . . .
3.1.1 Porous Glasses . . . . . . . . .
3.1.2 Porous Silicon . . . . . . . . . .
3.2 Pulsed Field Gradient NMR . . . . . .
3.3 Adsorption Measurement . . . . . . . .
3.3.1 Adsorption from Vapour Phase
3.3.2 BelSorp Mini II . . . . . . . . .
3.4 Mean Field Theory Approach . . . . .
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19
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4 Random Pore Network
4.1 Adsorption and Diffusion Hysteresis . .
4.2 Sorption Kinetics: Strong Surface Field
4.3 Sorption Kinetics: Weak Surface Field
4.4 Summary . . . . . . . . . . . . . . . .
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31
32
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48
50
5 One-Dimensional Channels
55
5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Effect of Mesoscopic Disorder and
Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Chemical Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 67
iii
5.4
5.5
5.6
5.7
Role of External Surface
Effect of Pore Openings
Discussion . . . . . . . .
Conclusions . . . . . . .
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67
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6 Summary
75
Bibliography
77
Acknowledgements
85
Appendix
87
VaporControl 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 external 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 prop1
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 condensation 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 interior 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 adsorption 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 fluidwall 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 typical 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 parallel 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 publications 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 au4
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 materials 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 macroscopic 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 evaporation 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 occurs 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 geometrical disorder. Another important issue is the facilitation of well defined porous
materials with well ordered pore structure at the macroscopic length scale, as addressed 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 accompanying hysteresis phenomenon ([36, 37, 32, 38]).
It is worth mentioning that beside the importance of the fundamental understanding 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 corresponding 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 equilibrium 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 inhomogeneities 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 matrices with different pore structures as provided by pulsed field gradient NMR
with their phase state as controlled by the chemical potential of the surrounding 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 effective 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 microscopic (PFG NMR) and macroscopic (sorption kinetics) methods and measured at identical conditions. In this way, by having simultaneously access to
two different transport properties which may be controlled by different internal 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 compared 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 understanding 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
1
+
R1 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 equilibrium 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 dependence of the saturated vapour pressure on the curvature of the liquid is given by the
Kelvin equation
P
RT ln
= γVm
P0
1
1
+
R1 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 ln
2.2
P
2γVm
=
.
P0
r
(2.4)
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 ln
2γVm
P
=−
.
P0
r
(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 ln
2γ cos θ
P0
=−
.
P0
∆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]), electrochemically etched silicon ([49, 50]), or anodic aluminium oxide ([51]).
In mesoporous materials, the sorption behavior depends not only on the fluidwall interaction strength, but also on the attractive interactions between the fluid
molecules. This leads to the occurrence of multilayer adsorption and capillary condensation in the pore. As predicted by the Kelvin equation (2.5), the pore condensation happens at a gas pressure P lower than the bulk saturated vapour pressure
P0 . Keeping the temperature constant and varying the external gas pressure, simultaneously 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 spontaneously 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 spherical 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 mechanisms: percolation, i.e. the pore space is emptied progressively when the
condition of emptying of the smallest pores, blocking the excess to the external 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 material structure which matches the available experimental material data. This
method provides a good agreement with the results of the adsorption experiments 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 originates 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 behavior 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 critical 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 adsorption 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, following 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 represent
the 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 scanning 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 pressure
variation and the corresponding amount adsorbed, respectively.
branches at a pressure different from that of the closure points of the major hysteresis 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])
~ r, t) = −D∇c(~r, t),
J(~
(2.7)
where J~ is the flux and c the concentration at position ~r at time t. The proportionality 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)
~ r, t).
= −∇ · J(~
∂t
(2.8)
Combining Eqs. (2.7) and (2.8) one obtains Fick’s second law of diffusion:
∂c(~r, t)
= D∇2 c(~r, t).
∂t
(2.9)
Notably, this diffusion equation (2.9) remains valid also under equilibrium conditions. 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
− 32
P (~r0 , ~r1 , t) = (4πD0 t)
(~r1 − ~r0 )2
exp −
4D0 t
(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
h(~r1 − ~r0 )2 i = dD0 t
(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 between 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])
r
d 8RT
,
DK =
3 πM
where M denotes the molar mass, R is the universal gas constant.
17
(2.12)
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 diffusivity 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, parallel 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 of
N2 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 mesoporous 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, consisting 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
N2 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% hydrogen 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
N2 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 25 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 applied 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
N2 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 for
cyclohexane in Vycor 7930 at 297 K obtained by volumetric measurement and those
obtained 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 · q 2 ∆)
(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])
1
Ψ(q, ∆) =
2
Z
π
exp −q 2 ∆(Dpar cos2 θ + Dperp sin2 θ) × sin θdθ,
(3.2)
0
with Dpar and Dperp being the self-diffusivities parallel and perpendicular to the
channel direction. Performing this integration, one obtains
p
√
2
q ∆(Dpar − Dperp )
erf
π
p
exp −q 2 δDperp
.
Ψ(q, ∆) =
2
q 2 ∆(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(−Dz q 2 ∆)
(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 significant. The self-diffusivity of saturated cyclohexane vapour at room temperature is
of order of magnitude of 10−6 m2 s−1 ([85]), while D0 of the liquid cyclohexane is of
about 10−9 m2 s−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 = pg Dg + pa Da
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
3.3.1
Adsorption Measurement
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 volume 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 controlled 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 StepSyn 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
dg
p
v3
v2
rn0
v0
v1
tc
θ
liq
res
h
box
s
usb
rn1
fan
tmp
electric line
data line
Figure 3.6: Schematic of VaporControl adsorption setup: liq - flask with adsorbate
liquid; res - big reservoir for gas phase preparation; tmp - turbo molecular pump;
s - adsorbent sample inside the spectrometer; box - thermostat box; p - pressure
sensors; dg - digital pressure display; tc - digital temperature controller and power
supply; h - heating mats; v0..v3 - valves with motor actuators; rn0, rn1 - valve motor
driver; 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 loadingdependent 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 controlling 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. Pressure 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 computational techniques has become an essential part of almost any branch of science.
Because the molecular systems generally consist of a large number of interacting particles 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 molecular 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, determined 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=−
X
XX
ni ni+a +
ni φi ,
2 i a
i
(3.5)
where 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 = −
X
(1 − ti+a )y,
(3.6)
a
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
X
[ρi ln ρi + (1 − ρi ) ln(1 − ρi )] −
i
X
XX
ρi ρa +
ρi φi .
2 i a
i
The fluid density, ρi , is related to the total number of molecules, N , via
N . Thus, at equilibrium, the essential condition
∂F
−µ=0
∂ρi
∀i
(3.7)
P
i
ρi =
(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
X
ρi
ρi+a + φi − µ = 0
kT ln
−
1 − ρi
a
∀i.
(3.9)
For the fluid density at site i, we can rewrite (3.9) as
ρi =
with Hi∗ = 1/T ∗
X
1
,
1 + exp(−Hi∗ )
(3.10)
ρi+a − φi / + µ/ and the temperature T ∗ = kT /.
a
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. Created 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 experimentally 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-41like 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 suggest 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 barrier 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 selfdiffusivities 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 wellpronounced hysteresis loop of type H2. The amount adsorbed, θ, is given in normalised units by dividing the actually measured FID signal intensity by that obtained 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 K
measured upon increasing (adsorption, open circles) and decreasing (desorption,
black circles) the vapour pressure, obtained by PFG NMR. Bottom: Corresponding
adsorption (open squares) and desorption (black squares) isotherms. Lines are guide
to 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 = pg Dg + pa Da ,
(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 =
ρg 1 − θ
,
ρa θ
(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 during
adsorption (open circles) and desorption (black circles) obtained by PFG NMR. By
the lines, De calculated for the adsorption (dashed line) and the desorption (solid
line) branches are shown.
In Fig. 4.2, the calculated effective self-diffusivities qualitatively mimic the de34
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 function. 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 at
P/P0 = 0.52 on the adsorption branch measured at 297 K (black squares) by PFG
NMR. 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 contribution 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 agreement between the model and the experiment. Another question to be further explored 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, obtained
from the sorption isotherms. open circlesrepresent the adsorption isotherms, black
circlesrepresent 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 desorption 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 diffusivities 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 increas37
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 independentpore 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 diffusivities. 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 the
corresponding self-diffusivities (b) at 297 K as a function of relative pressure. The
desorption scanning curves start from 0.68 P0 (black triangles) and 0.65 P0 (black diamonds) obtained after incomplete filling. The boundary adsorption (open squares)
and desorption (black squares) isotherms envelop the scanning curves. The lines are
guide to eye.
39
Figure 4.6: The relative amount of cyclohexane adsorbed (a)in Vycor and the corresponding self-diffusivities (b) at 297 K as a function of the relative pressure. The
desorption 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 curve
from 0.43 P0 to 0.59 P0 (open circles) is reversed to 0.43 P0 (black circles). The lines
are guide to eye.
40
Figure 4.7: The diffusivities, De , in Figs. 4.5(b) and 4.6(b) plotted versus amount
adsorbed, 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 saturated 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]). Regardless whether the desorption occurs via cavitation or via pore blocking,
the liquid in the wider pore region is in a stretched state. This density difference 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 somewhat 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 channels 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 selfdiffusivities 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 function
of the external pressure in PIB-IL during adsorption (open circles) and the desorption (black circles). Bottom: Corresponding adsorption (open squares) and desorption (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 adsorption is the analysis of the transient sorption behavior. Fig. 4.9 shows results
of a transient sorption experiment correlated with the information from the diffusion 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]):
∞
X
θ(t)
4
=1−
exp(−De αn2 t),
2
2
θ∞
r
α
n
n=1
(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 region 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 capillary 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 (diameter 3 mm, length 12 mm) at 297 K measured by NMR. Typical kinetic data (black
squares) obtained upon stepwise change of the external gas pressure from 0.16 to
0.24 P0 (a) and 0.48 to 0.56 P0 (b). The inset of (b) shows the long-time part of the
data (b), axis quantities and units are the same as in main figure. The dotted lines
represent 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 requiring 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 process 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 liquidphase 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 bξ ψ > 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 )
46
(4.4)
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 adsorption 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 supported 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 inside 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 departure from equilibrium, is experimentally reproducible by various methods under
the same conditions. With increasing temperature, the Boltzmann factor kT may
become comparable to bξ ψ , providing the global density equilibration on the experimental 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 with
diameters 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 the
adsorption step 0.16 to 0.24 P0 are presented as function of dimension-less time
tDe r−2 .
4.3
Sorption Kinetics: Weak Surface Field
A strong argument supporting the activated nature of density relaxation in the hysteresis 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 porosity 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 random 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 structure. 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 K
during adsorption (open circles) and desorption (black circles) obtained by PFG
NMR. Bottom: Corresponding adsorption (open squares) and desorption (black
squares) 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 glass
particles at 300 K measured by NMR. The kinetics for the adsorption step 0.85
to 0.89 P0 is presented as a function of time. The linear long-time uptake at the
logarithmic time scale reveals the nucleation-limited processes.
A similar dependence is observed for the PIB-IL hierarchical porous silica material (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 equilibrium 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 correlating 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 by
NMR. The kinetics for the adsorption step 0.77 to 0.88 P0 is presented as a function of time. The linear long-time uptake at the logarithmic time scale reveals the
nucleation-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 hysteresis 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 potentials 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 transfer 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 material 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 organization 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 mesoporous 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, depending 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 relationship 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 ”nonideality” 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 single 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 mi55
CHAPTER 5. ONE-DIMENSIONAL CHANNELS
Figure 5.1: Nitrogen sorption isotherms in PSi at 77 K with one end (adsorption open
circles, desorption black circles) and two ends (adsorption open stars, desorption
black stars) open. The isotherms are obtained by BelSorp Mini II. Inset: schematics
of 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. After
incomplete 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 by
the 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, although the individual channels are isolated from each other. Additionally, the scanning 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 ad57
Figure 5.3: Top: Effective self-diffusivities of cyclohexane in PSi at 297 K obtained by PFG NMR along the adsorption (open circles) and desorption (black circles) 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 evaporation 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 effects, where both, pore blocking and percolation phenomena may contribute to the
observed asymmetry of the hysteresis loop ([22]). In addition to such an asymmetry, 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 channels, 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 inkbottle morphology of the pores ([38]). Interestingly, they found an apparently identical 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 stretching 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 composed 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 comparing 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 seg59
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 varying height Hi along the x axis and constant length Li . The pore is infinitely extended
in 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 AND
SURFACE 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 temperature 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 adsorption/desorption behavior in a single channel. This mesoscalic roughness is characterized 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 surface (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 liq61
Figure 5.5: Studied pore size distributions. (a) Three PSD with the same dispersion
but different mean values: (A) 4 to 8 lattice units, (B) 6 to 10, (C) 8 to 12. (b) Two
PSD with the same mean value but different dispersions: (B) 6 to 10 lattice units
and (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 decrease 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 adsorption 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
SURFACE ROUGHNESS
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: Rough
surface; 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 significant 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
0.20
0.40
0.75
0.76
0.82
0.84
0.87
0.95
0.65
0.64
Figure 5.7: Visualization of fluid density states in the channel (A) without geometrical roughness. Adsorption and desorption from top to bottom. The z values are
given on the right of the pictures.
0.20
0.40
0.60
0.75
0.84
0.95
0.64
0.63
0.59
0.57
0.38
Figure 5.8: Visualisation of fluid density states in the channel (A) with geometrical
roughness. Adsorption and desorption from top to bottom. The z values are given
on 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
SURFACE ROUGHNESS
0.20
0.40
0.76
0.82
0.84
0.95
0.65
0.64
Figure 5.9: Visualization of fluid density states in the channel (A) with chemical
heterogeneity corresponding to geometrical roughness. Adsorption and desorption
from top to bottom. The z values are given on the right of the pictures.
0.20
0.40
0.60
0.93
0.95
0.80
0.79
Figure 5.10: Visualization of fluid density states in the (B) pore with homogeneous
flat surface. Adsorption and desorption from top to bottom. The z values are given
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
0.20
0.40
0.60
0.82
0.83
0.85
0.89
0.95
0.68
0.67
Figure 5.11: Visualization of fluid density states in the (B) pore with geometrical
0.20
0.40
0.60
0.82
0.89
0.95
0.66
0.65
0.64
Figure 5.12: Visualization of fluid density states in the channel (D) with geometrical
z = 0.65), in parallel to the pore blocking.
We have found that the wall roughness has a significant influence on the adsorption 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 concerns 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 heterogeneous
surface 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 effect 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 surface. 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 pore
open at both ends (solid line) and open at only one end (black squares). Inset shows
the adsorption/desorption in a simple slit pore open at both ends (dotted line) and
closed 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 heterogeneity, 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 adsorption 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 adsorption 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 heterogeneous 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 variation 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 chemically 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 increasing 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 lowpressure 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) roughness 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 roughness (black squares) and chemical heterogeneity (crosses) may be compared. The
reversible adsorption/desorption isotherms at an early stage of uptake coincide, until the surface monolayer is formed (z ≈ 0.40). In contrast to the chemical heterogeneity, 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 observes 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 corresponds 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 chemical 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 evaporation 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 relaxation 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 smoothwalled 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 supervisors
Dr. Rustem Valiullin and Prof. Dr. Jörg Kärger. This is a great opportunity to
express my respect to them for their patience, encouragement and competent guidance. Dr. Rustem Valiullin has been holding me to a high standard and enforcing
strict 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 that
helped 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 been
indispensable to achieve my scientific goals.
My deepest gratitude is to Prof. Dr. Peter A. Monson, who teached me the first
steps in the domain of the Mean Field Theory. Without his inspiration in this field of
science, motivating discussion and ideas, this deep insight into the thermodynamics
of phase transitions under confinement would not have been possible. I am very
thankful for the care and the pleasant atmosphere during my stay at the University
of 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 and
PD Dr. Siegfried Fritzsche who sparked my interest in computer science during
my student time. In their lectures, for the first time I encountered the world of
Monte Carlo and Molecular Dynamics.
I owe my gratitude to all those people from Prof. Kärger’s group who have
made this dissertation possible and because of whom my graduate experience has
been one that I will cherish forever. Especially I shall thank Dipl.-Phys. Muslim
Dvoyashkin 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 Bärbel Krause and Lutz Moschkowitz for
making 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 been
possible without the continuous support by the staff of the mechanical workshop of
our faculty together with glassblower Peter Fatum. I have a deep respect to them
for their capabilities and industry.
The financial support of the German Research Foundation (DFG), the Research
Academy Leipzig (Forschungsakademie Leipzig) and Max-Buchner-Research Foun85
dation (Max-Buchner-Forschungsstiftung) is gratefully acknowledged.
Many friends have helped me stay sane through these years. I greatly value their
friendship and I deeply appreciate their belief in me.
Most importantly, I wish to thank my family for everything which cannot be put
into words.
86
Appendix
VaporControl Reference
The VaporControl adsorption setup can be controlled either by the GUI or text
based scripts. Following graphical user control elements are implemented:
• RNControl: Accesses an RNMotor board ([87]), connected via RS-232 COM
interface (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 sensor
unit connected via RS-232 COM interface (Tab. 6.7)
• OmronControl: Accesses an Omron E5CK thermostat connected via RS-232
COM interface (Tab. 6.6)
• COMControl: Parent element for graphical user control elements. Possesses no
functionality besides connecting to a RS-232 COM interface of an arbitrary
module. RNControl, DGControl, OmronControl inherit all properties from
COMControl 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
ReturnValue
ErrorCode
send a command line to VaporControl GUI
the return value of last operation
the error code of last operation
Table 6.1: VC3Client.Communicator OLE object command reference
87
EXIT
STOP
X val
Y val
W val
H val
LOG text
WAIT s
SAVE filename
LOAD filename
SET key val
GET key
ADD control name
DEL name
BG filename
MODE mode
LIST
close VaporControl GUI
abort the program execution
set left window coordinate to val
set top window coordinate to val
set window width to val
set window height to val
add text to logging window
delay the program execution by s seconds
save current control state to a script file filename
load and execute script from file filename
set an internal variable key to value val
returns the value of an internal variable key
adds a control and assigns the control name
Following hardware controls are implemented:
RN: Control for an RNMotor stepping motor driver unit
DG: Control for a Pfeiffer Vacuum TP26x DualGauge
pressure sensor unit
OMRON: Control for an OMRON temperature controller
unit
remove a control with name name
set the layout image stored in file filename
set the GUI operating mode to name
Following modes are available:
0: Idle mode, the controls cannot be moved or resized
1: Design mode, the controls can be moved and resized
2: Execution mode, internal mode used for scripts execution
list all controls
Table 6.2: GUI command reference
88
assign the control name, only letters a-z and numbers
0-9 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 hardware
1: active
COM param value set a COM port parameter param to a value value
Following parameter are available:
PORT port: COM port name
BAUD 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 between
the arrival of two characters on the communications line
TIMEOUT READCONST ms: COM read interval in ms,
specifies the constant used to calculate the total timeout period for read operations
TIMEOUT READMULT ms: COM read interval in ms, specifies the multiplier used to calculate the total time-out
period for read operations.
TIMEOUT WRITECONST ms: COM read interval in ms,
specifies the constant used to calculate the total timeout period for write operations.
TIMEOUT WRITEMULT ms: COM read interval in ms,
specifies the multiplier used to calculate the total timeout period for write operations
Timeout = (MULTIPLIER * number-of-bytes) + CONSTANT
NAME name
Table 6.3: COMControl command reference
89
MOTOR m ... send command with parameter to the RNMotor control
m (0 or 1)
VREF v
set the board reference voltage. Caution, wrong value
may damage the board! See RNMotor reference ([87]).
Table 6.4: RNControl command reference
NAME name
X val
Y val
W val
H val
SPEED val
CURRENT i
POS pos
MAXPOS pos
MINPOS pos
OPENDIR dir
GOTO pos
OPEN
CLOSE
STOP
assign the control name, only letters a-z and numbers
0-9 may be used, no spaces!
set left control coordinate to val
set top control coordinate to val
set control width to val
set control height to val
set stepping motor speed to 0 ≤ val ≤ 255
set stepping motor maximal current to 0 ≤ i ≤ 255.
Real current value in A = i / 100
assign the position of the valve to pos, i.e. the valve
state. No action on the valve is performed.
assign the position pos corresponding to the open valve
assign the position pos corresponding to the closed valve
assign the rotation direction
0 : open counterclockwise
1 : open clockwise
actuate the valve to position pos
actuate the valve to maximal position
actuate the valve to minimal position
stops the current operation
Table 6.5: RNMotor command reference
TI t
set refresh timer interval to t in ms
REFRESH read the temperature value
T
return the temperature value from sensor
Table 6.6: OmronControl command reference
TI t
REFRESH
P n
STATUS n
set refresh timer interval to t in ms
read the pressure and status information from the sensors
return the pressure value from sensor n (0 or 1)
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 in
Mesoporous Materials, Valiullin R., Naumov S., Galvosas P., Kärger J., Woo
H.-J., Porcheron F., Monson P. A., Nature 443, 965 (2006)
• Diffusion Hysteresis in Nanoporous Materials, Naumov S., Valiullin R., Galvosas
P., Kärger 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., Kärger 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. Kärger, R Pitchumani, M.-O. Coppens, Microporous 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. Kärger, C. Wespe, F.
Bordusa, and F. Kremer, Phys. Rev. E 77, (2008), 051202
• Electrical Conductivity and Translational Diffusion in the 1-butyl3-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. Kärger, Langmuir 24 (2008), 64296432
• Understanding Capillary Condensation and Hysteresis in Porous Silicon: Network Effects within Independent Pores, S. Naumov, A. Khokhlov, R. Valiullin,
and J. Kärger, P.A. Monson, Physical Review E 78, Rapid Communication,
060601, (2008)
91
• Charge Transport and Glassy Dynamics in Imidazole-Based Liquids, C. Iacob,
J. R. Sangoro, A. Serghei, S. Naumov, Y. Korth, J. Kärger, C. Friedrich, and
F. 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.
Kärger, F. Kremer, Macromolecules, (2009) accepted
• Pulsed Field Gradient NMR Study of Surface Diffusion in Mesoporous Adsorbents, M. Dvoyashkin, A. Khokhlov, S. Naumov, R. Valiullin, Microporous
and Mesoporous Materials (2009) accepted
• Understanding Network Effects in Adsorption/Desorption in Mesoporous Materials with Independent Channels, S. Naumov, R. Valiullin, and Jörg Kärger,
P.A. Monson, submitted
Oral Presentations
• Diffusion Hysteresis in Porous Materials, 3rd International Workshop on Dynamics in Confinement, Grenoble (2006)
• Hysteresis Phenomena in Mesoporous Materials, 5th International Research
Training Group Diffusion in Porous Materials (2006)
• Adsorption Hysteresis in Mesoporous Materials, AMPERE NMR Summer
School (2007), Bukowina Tatrzan’ska, Poland
• Diffusion Scanning Hysteresis Loops in Nanopores, Fundamentals Of Adsorption 9, (2007), Giardini Naxos, Sicily Italy
• Adsorption Hysteresis in Mesoporous Materials, 6th International Research
Training Group Diffusion in Porous Materials (2007)
• Overview of the Department of Interface Physics, 1st Young Researchers Meeting 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, AMPERE
NMR Summer School 2008, Wierzba, Poland
• Phase Behavior of Fluids in Porous Silicon Materials and their Textural Characterization, The 5th International Workshop on Characterization of Porous
Materials from Angstroms to Millimeters, June 2009 New Brunswick, NJ,
USA
92
Poster Presentations
• Study of History Dependence of Adsorption and Self-difusion Processes in
Porous Media with Help of PFG NMR, 8th International Bologna Conference
on Magnetic Resonance in Porous Media (2006), Bologna, Italien
• Dynamics and Phase Transitions Under Confinement, Fundamentals Of Adsorption 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 of
the Defence of the International Research Training Group ”Diffusion in Porous
Materials”, Leipzig (2008)
• Tracing Pore Connectivity and Architecture in Nanostructured Silica SBA15, Meeting of the Review Panel of the Defence of the International Research
Training Group ”Diffusion in Porous Materials”, Leipzig (2008)
• Adsorption Hysteresis Phenomena in Mesopores, EUROMAR Magnetic Resonance Conference, July 2009, Göteborg, Sweden (2009)
93
Curriculum Vitae
Personal information
Family name
First name
Date of birth
Place of birth
Nationality
E-Mail
Naumov
Sergej
31.10.1980
Pskov (Russia)
German
sergej.naumov@uni-leipzig.de
Education
1987
1990
1994
1994
2000
2005
- 1990
- 1994
- 1999
- 2005
Since 2006
Primary school, Pskov, USSR
Secondary school, Pskov, Russia
Emigration to Germany
Secondary school, Leipzig, Germany
Physics studies, Leipzig University, Germany
Topic of diploma thesis: ”NMR Study of Adsorption and Desorption Phenomena in Porous Media”
Ph.D. student in the Department of Interface Physics, Faculty for
Physics and Earth Science Leipzig University
Associated member of the International Research Training Group
”Diffusion in Porous Materials”
Occupational development
1999 - 2000
2000 - 2005
Military service
Student assistant in the Institute of Surface Modification, Leipzig,
Germany.
Scope of duties: Computer based quantum chemical calculations of
molecular properties
95
Selbständigkeitserklärung
Hiermit erkläre ich, dass ich die vorliegende Arbeit selbständig und ohne unzulässige
Hilfe oder Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Ich
versichere, dass Dritte von mir weder unmittelbar noch mittelbar geldwerte Leistungen fr Arbeiten erhalten haben, die im Zusammenhang mit dem Inhalt der vorliegenden Dissertation stehen, und dass die vorgelegte Arbeit weder im Inland noch
im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde zum
Zwecke einer Dissertation oder eines anderen Prüfungsverfahrens vorgelegt und in
ihrer Gesamtheit noch nicht verffentlicht wurde. Alles aus anderen Quellen oder
von anderen Personen übernommene Material, das in der Arbeit verwendet wurde
oder auf das direkt Bezug genommen wird, wurde als solches kenntlich gemacht.
Insbesondere wurden alle Personen genannt, die direkt an der Entstehung der vorliegenden Arbeit beteiligt waren.
Es haben keine erfolglosen Promotionsversuche stattgefunden.
Die Promotionsordnung vom 11. Juni 2008 wird anerkannt.
Leipzig, den 04.03.2009
Sergej Naumov
98
Synopsis
Introduction
In recent years, considerable progress has been achieved in the development of novel
tailor-made mesoporous materials with well-defined structural properties. An inherent feature of molecular ensembles in mesopores is the interplay between the
fluid-pore wall and fluid-fluid interactions. It may give rise to various specific phenomena of the confined fluids. A classical example of such phenomena, which still
remains a subject of controversial discussions ([35, 5, 6]), is the adsorption hysteresis: at the same external conditions, the amount of guest molecules adsorbed by the
mesoporous host is higher upon decreasing external gas pressure than upon increasing. This indicates the failure of the system to equilibrate during the experiment
([4]).
This thesis addresses the equilibrium and dynamic fluid properties under mesoporous confinement. Taking advantage of the Pulsed Field Gradient (PFG) NMR
technique, the molecular self-diffusivities of fluids in mesopores with different pore
structures are correlated with the phase state as controlled by the external gas phase.
Additionally, the molecular transport properties, as revealed by microscopic (selfdiffusivities) and macroscopic (transient sorption) methods, are compared. This
helps to highlight the underlying mechanisms and address in more detail dynamic
aspects accompanying the adsorption hysteresis. By means of the Mean Field Theory (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 an
intentionally created geometrical and chemical heterogeneity are studied and compared with our experimental findings.
Materials and Methods
For the experimental study of the hysteresis phenomena, two different types of
porous 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 ”controlled porous glass” (CPG) with mean pore size of 15 nm represent highly interconnected random pore network. Another interconnected but ordered hierarchical
system is the PID-IL porous silica ([113]), consisting of spherical cavities with a di99
ameter of about 20 nm connected via small channels of 3 nm diameter. Electrochemically etched porous silicon films belong to the materials with non-interconnected
parallel 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 behavior
of the self-diffusivities was obtained by PFG NMR (Fig. 6.1(a)). Obviously, the
adsorption hysteresis and the hysteresis loop of the self-diffusivities are correlated.
The mono-exponential dependence of the NMR spin echo decay on the applied field
gradient strength reveal the fast exchange of the molecules during the observation
time of the diffusion experiment. Thus, we may explain the behaviour of the effective
self-diffusivities by the contribution of the fast transport in the gaseous and the slow
one in the adsorbed or capillary-condensed phases ([40]). Calculating the effective
self-diffusivities in this way and assuming the Knudsen regime for the diffusion in
gas ([114]), we have found a good qualitative agreement.
(a)
(b)
Figure 6.1: (a): Effective self-diffusivities of cyclohexane adsorbed in Vycor 7930
at 297 K measured by PFG NMR upon increasing (open circles) and decreasing
(black circles) gas pressure and corresponding amount adsorbed (open squares) and
desorbed (black squares) plotted as a function of the relative pressure P/P0 with P0
denoting the saturated vapour pressure. (b): Diffusivities plotted versus the amount
adsorbed obtained from the sorption isotherms, adsorption path (open circles) and
desorption path (black circles). Lines are guide to eye
One of the most remarkable features Fig. 6.1(a) emerges when the diffusivities are
presented as a function of the relative amount adsorbed (Fig. 6.1(b)): one and the
same number of molecules exhibit different effective self-diffusivities on adsorption
100
and desorption! Thus a novel means for reflecting different internal density distributions has been revealed ([44]). The so-called scanning curve experiments, where
incomplete 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)). This
clearly manifests a history-dependent adsorbate distribution in pores!
(a)
(b)
Figure 6.2: (a): Relative amount of cyclohexane adsorbed in Vycor 7930 at 297 K
as a function of relative pressure. The desorption scanning isotherm begins on the
boundary 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) is
reversed to 0.43 P0 (black circles). (b): Corresponding self-diffusivities as a function
of relative amount adsorbed. Lines are guide to eye
The subloops measured inside the major loop exhibit two further important
features ([44]):
• Return point memory, i.e. after an incomplete sorption cycle, the system
returns to its initial state. This feature suggests that the external conditions
are the main driving force of the evolution in such systems. Further thermal
equilibration is prohibited by the high energy barriers between the minima in
local free energy ([37, 95]).
• Lack of congruence, i.e. two different subloops are in general not parallel to
each other. This is a signature of networked pores, since the independent-pore
model would predict exact congruence ([16, 66]).
One of the most straightforward methods to illuminate the mechanisms of adsorption is the analysis of the transient sorption behavior. The results of the transient 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. In
this way, we have been able to identify two mechanisms determining the uptake
kinetics([41]):
101
• Adsorption at low pressures is limited by the diffusion of the fluid molecules
into 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 transport
from external gas phase, the dynamics is purely diffusive. The global equilibrium may be attained very fast on the experimental time scale, where the
chemical potential is uniform over the whole system.
• With increasing density, capillary condensation occurs, followed by a growth
of the domains with the capillary-condensed liquid inside the porous structure.
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 requiring crossing the barriers between the local minima of
the free-energy. If, due to a microscopic fluctuation, the system jumps from
one local minimum to another, this creates spatially local density perturbation.
The latter 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.
(a)
(b)
Figure 6.3: Sorption kinetics data of cyclohexane in a Vycor 7930 cylinder (diameter
3 mm, length 12 mm) at 297 K measured by NMR. Typical kinetic data (black
squares) obtained upon a stepwise change of the external gas pressure outside the
hysteresis loop from 0.16 to 0.24 P0 (a) and inside the hysteresis loop from 0.48
to 0.56 P0 (b). The inset of (b) shows the long-time part of the data (b), axis
quantities and units are the same as in main figure. The dotted lines represent the
kinetics calculated via the diffusion equation. The solid line in (b) is calculated for
the 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 strong
surface field acting upon confined fluids, we argue that the limiting mechanism is
the fluctuation-driven process of the fluid redistribution within the porous matrix.
102
With increasing porosity (as in the case of CPG and PIB-IL) 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 small regions containing
capillary-condensed liquid, may limit the adsorption process.
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 is responsible for the occurrence of the adsorption hysteresis
in mesoporous 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 regime of quasiequilibrium where further evolution is brought about by the thermally activated
fluctuations of the fluid ([32, 114, 41]).
One-Dimensional Channels
Here, the study of the influence of the geometrical and chemical disorder in linear pores on the adsorption/desorption behavior by means of the MFT ([34]) is
presented. This structural model is used to capture the main properties of the electrochemically etched porous silicon (PSi). We consider a pore composed of a random
set of slit pore segments creating a linear channel. This is the simplest model of isolated pores with disorder. The size of the pore segments is varied randomly in such
a way that the overall pore size distribution (PSD) has a Gaussian shape. The pore
openings 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 size
along the channel direction
• Geometrical roughness of the pore wall, modelled by randomly adding solid
sites on the surface of a segment
• Chemical heterogeneity of the surface, which can be modelled varying the ratio
between the solid-fluid and fluid-fluid interaction.
Opposite to the analogous template-based materials with non-interconnected
channel-like pores such as SBA-15, MCM-41 or anodic aluminium oxide, mesoporous
silicon has adsorption properties of disordered materials with a network of mesopores
(random porous glasses) as can be seen from the shape of the sorption isotherms
and the desorption scanning curves (Fig. 6.4(a)). Additionally, the behavior of the
self-diffusivities is very similar to that observed in Vycor (compare Figs. 6.4(b) and
6.1(a)).
The theoretical analysis using this model suggests that these properties (asymmetric hysteresis of type H2, irrelevance of closing one end) can be explained my
103
(a)
(b)
Figure 6.4: (a) Nitrogen desorption scanning curves measured in electrochemically etched porous silicon at 77 K. Here, desorption followed already after incomplete adsorption, to 0.82 P0 (black circles) and 0.80 P0 (black triangles), respectively. The desorption scanning curves are enveloped by the ”boundary” adsorption (open squares) and desorption (black squares) isotherms. (b) top: Effective
self-diffusivities of cyclohexane in PSi at 297 K obtained by PFG NMR along the
adsorption (open circles) and desorption (black circles) branches. Bottom: Adsorption (open squares) and desorption (black squares) isotherms. Lines are guide to
the eye.
assuming the existence of mesoscalic disorder, namely a distribution of pore dimensions, exceeding the disorder on the atomistic level. In this sense, linear pores with a
statistically varying pore diameter, exhibit all properties of three-dimensional pore
networks.
0.40
0.75
0.95
0.64
0.63
Figure 6.5: Visualisation of fluid density states in the channel with geometrical
roughness. The segment size ranges from 4 to 8 lattice units. Adsorption and
desorption from top to bottom. The P/P0 values are given on the right of the
pictures, first increasing for adsorption and then decreasing for desorption.
Visualization of the density distributions (for an exemplification see Fig. 6.5) for
states along the isotherms helped us to elucidate some basic features of adsorption
and desorption processes in linear disordered pores ([54]):
• At low gas pressures, the isotherm is associated with the covering of the pore
walls with adsorbed layers. Importantly, the small-scale surface roughness is
104
only of importance in determining the proper isotherm curvature before onset of hysteresis by, e.g., smearing out signatures of a 2D surface condensation
transition. At intermediate pressures we see the condensation of liquid bridges
where the pore width is smallest. For the pores closed at one end this condensation may have occurred already before pore condensation is underway at the
closed end. At higher pressures we observe the condensation of liquid bridges
in regions of a higher pore diameter as well as a 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 and evaporation
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 makes the adsorption behavior of a channel open at
one end indistinguishable to that open on both ends.
Summary
In summary, our experiments and theoretical calculations have identified the effects
of quenched disorder in the channel pores of electrochemically etched porous silicon
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.
105

Hysteresis Phenomena in Mesoporous Materials

Transcription

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