diffusion in nanoporous host systems -...

CHAPTER TWO

Diffusion in Nanoporous HostSystemsRustem ValiullinFaculty of Physics and Earth Sciences, University of Leipzig, Leipzig, Germany

Contents

1.

AnnISShttp

Introduction

ual Reports on NMR Spectroscopy, Volume 79 # 2013 Elsevier Ltd.N 0066-4103 All rights reserved.://dx.doi.org/10.1016/B978-0-12-408098-0.00002-1

24
2. Phase Equilibria Under Confinements 26
2.1
Gas–liquid equilibria 26 2.2 Liquid–solid equilibria 31
3.
Diffusion During Gas–Liquid Phase Transitions 32 3.1 Surface diffusion 34 3.2 Diffusion in the gaseous phase 38 3.3 Diffusion during the formation of capillary-condensed domains 42 3.4 Diffusion under varying temperature 46 3.5 Non-equilibrium behaviour 48
4.
Diffusion During Melting/Freezing 52 5. Diffusion During Other Phase Transitions 59
5.1
Diffusion during liquid–liquid phase separation 59 5.2 Diffusion during structural transitions 61 5.3 Diffusion in supercritical phases 63
6.
Conclusions 64 Acknowledgement 65 References 65
Abstract

Diffusion is one of the key parameters controlling the progress of various processesoccurring in porous materials. For fluids confined to mesoporous solids, diffusion, inaddition to the complex structure of these materials, is found to be further diversifiedby their rich phase behaviour, giving rise to a strong coupling between the phase stateand internal dynamics. By applying NMR as a non-perturbing experimental technique,the results of systematic studies of fluid microscopic and macroscopic dynamicsconfined to mesoporous solids are reviewed. In particular, the correlations betweendiffusive dynamics and the details of gas–liquid and solid–liquid coexistences areexplored. Selected examples concerning dynamic behaviour accompanying othertypes of transitions are presented.

23

http://dx.doi.org/10.1016/B978-0-12-408098-0.00002-1

24 Rustem Valiullin

Key Words: Diffusion, Phase transitions, Nanoporous solids, Pulsed field gradientNMR

1. INTRODUCTION

Fluids under condition of spatial confinements often exhibit physical

properties different from those found in their bulk states. This is brought

about by (i) the introduction of additional interactions with the pore walls

and (ii) the confinement effects (this general term is used here to address a

variety of phenomena resulting from finite-size effects or effects of reduced

dimensionality). Thus, geometric confinements may give rise to restricted

diffusion, which is widely used as a tool for structural characterization.1

As scale goes down to the range of nanometres, the overall impact of the

inter-molecular interactions may become comparable to the fluid–pore

wall interactions. Superimposed with pure confinement effects, these com-

peting interactions may appreciably change some properties of the fluids,

including, in particular, elementary mechanisms of molecular diffusion.

Among the methods of studying molecular diffusion in porous media,

the pulsed field gradient technique of NMR spectroscopy (PFG NMR) is

of particular relevance.2,3 The technique is based on the creation of an initial

nuclear coherence and following its loss due to the molecular displacements

in an applied magnetic field gradient. Thus, due to its non-invasive nature, it

allows the observation of molecular migration without interfering with the

internal processes on a time scale from milliseconds to seconds. The exper-

iments can be designed to trace transport under both equilibrium and non-

equilibrium conditions. In the most relevant cases, however, transport in

porous materials, which are in the focus of this contribution, occurs close

to equilibrium, rendering the diffusive dynamics being the decisive mode

for molecular transport. The parameters of diffusive motion, such as self-

diffusivity, are essential ingredients for understanding and predicting the

progress of various physico-chemical processes, including those occurring

in porous solids.

PFG NMR can probe molecular displacements on a length scale from

hundreds of nanometres to hundreds of micrometres. Therefore, the infor-

mation accessible for fluids confined to porous matrices depends on

the characteristic length scale of the porous material under study. Consider-

ing typical pore sizes, porous solids may be subdivided into three major

classes, referred to as micro-, meso-, and macroporous ones. Although

25Diffusion in Nanoporous Host Systems

the distinction between them is not strictly defined, the IUPAC classifica-

tion associates these classes with porous solids with pore sizes below 2 nm,

between 2 and 50 nm, and above 50 nm, respectively.4 With this classifica-

tion, it becomes clear that direct access to structural information of porous

solids using PFG NMR is possible only for macroporous ones. Exactly for

this reason, the diffusion-based structural characterization was in the focus of

the NMR community over the past decades and substantial progress in the

understanding structure–dynamics relationships was attained.5–14 For

micro- and mesoporous adsorbents, PFG NMR can access only molecular

displacements notably exceeding their typical characteristic pore sizes. In this

way, only the so-called long-range diffusivities can be measured. Notably,

structure–dynamics correlations for these materials are more complex

and can hardly be generalized.15–17 Experimental diffusion studies in micro-

porous hosts, such as zeolites18 or activated carbons,19 were driven by

their technological value and are broadly represented in the literature.

In particular, the application of PFG NMR in zeolite sciences has been

reviewed elsewhere.20–23

A very specific class of porous materials constitute the mesoporous ones,

which only recently have attracted substantial interest. The mere fact of

mesoscalic confinement already leads to very peculiar phenomena in trans-

port of fluids confined in these materials.24–30 The most dramatic distinction

of diffusion processes in mesoporous solids, as compared to other materials,

comes from a rich phase behaviour of fluids in mesoporous solids. Thus, as

most widely known, spatial confinements lead to size-dependent alterations

of the phase equilibria characteristics, such as the equilibrium transition

temperature or pressure.31–34 Given the size-dependent character of the

phase transitions under confinement and a multitude of mesoporous solids

synthesized to date with very different geometries of the pore spaces, ranging

from perfectly ordered materials to those with completely disordered

structures,35–38 fluid phase equilibria in mesoporous hosts may show a spec-

trum of various behaviours. Exactly this family of porous solids will be in

the focus of this work, in which recent progress in experimental studies

of diffusion processes in this materials using PFG NMR will be reviewed.

The correlations between molecular transport and phase state of con-

fined fluids are, clearly, predetermined by those in bulk substances. Indeed,

molecular mobilities in gases, liquids, and solids may strongly differ.

In confined systems, different phases may coexist with each other, giving rise

to not only simple alternations of molecular trajectories in either phases

but, in some cases, also to severe modifications of the transport mechanisms.

26 Rustem Valiullin

The latter results, in particular, due to the existence of the interfaces between

the domains of different phases. This, owing to different densities or mixture

compositions in these phases, may lead to reflecting, adsorbing or partially

adsorbing boundary conditions at the interfaces, further diversifying the

transport pathways. As the resulting effect, the transport properties of the

guest ensembles may become not only simply determined by the phase com-

position but also by details of their relative spatial arrangement.

Concerning practical applications, measurements of the phase transitions

of substances confined to mesoporous solids have long been used as sensitive

tools for their structural characterization.31,33,39,40 In recent years, potential

applications exploiting changes in phase equilibria under confinement have

emerged in such areas as heterogeneous catalysis, drug delivery, gas storage,

optics, etc. For these “advanced-level” applications, in addition to the static

properties (such as phase compositions of the confined fluids), the knowl-

edge of dynamic properties of guest molecules is of crucial importance

for a purposeful functionalization of porous solids and process design.

Henceforth, establishing correlations between the pore structure, the phase

state and the microscopic dynamics of fluids in mesoporous materials turns

out to be a crucial task for both experimental and theoretical explorations.

In addition to their practical relevance, such complementary studies allow

a better understanding of the thermodynamical processes occurring in

confined spaces.41–43

In light of these strong correlations between molecular transport and

phase state, before we proceed with discussing the dynamics properties of

fluids in mesopores, Section 2 will shortly review the main relationships

between phase state and structural details of mesoporous solids. In Sections 3

and 4, diffusive dynamics during gas–liquid and solid–liquid coexistences,

being the most important ones for applied sciences, will be considered in

some detail. Finally, Section 5 will give some selected examples for the cor-

relation between translational dynamics and other transitions, occurring in

mesopore spaces.

2. PHASE EQUILIBRIA UNDER CONFINEMENTS

2.1. Gas–liquid equilibria
The strong impact of confinement upon phase equilibria and phase transi-
tions of fluids in mesoporous solids has been noticed already more than one

century ago in pioneering studies of gas adsorption by silica gel, a siliceous


material with a broad distribution of pore sizes and irregular pore shapes.44,45

This type of measurements did not require any sophisticated instrumenta-

tion and, therefore, the phenomenon has extensively been addressed

experimentally, providing thus a solid basis for further theoretical develop-

ments. The rich set of experimental data available led to a substantial progress

in the understanding of various aspects of gas sorption, including capillary

condensation and evaporation phenomena and their inter-relations with

structural properties of porous solids. As a result, the measurements of gas

adsorption have progressed to become nowadays a routinely applied

technique for the characterization of nanoporous solids.39

Mesoporous solids posses a relatively large specific area A of their inner

surface (it typically varies from tens to hundreds of square metres per gram).

Thus, due to the van der Waals interaction of guest molecules with the sur-

face of the solid framework (also referred to as the pore walls), at already low

pressure P in the surrounding gas atmosphere, a relatively large amount of

molecules may get physisorbed at the inner surface. The relationship

between the gas pressure P and the amount adsorbed a in this regime of

sub-monolayer coverage is determined by the strength of the interaction

efs between the guest molecules and the pore walls and its distribution along

the surface, either due to chemical or geometric disorder.46

With further increase of P, poly-layer adsorption occurs, that is, mole-

cules get adsorbed on top of the already adsorbed molecules. Depending on

the ratio efs/eff, where eff is the interaction energy between the fluid mole-

cules, the onset of this process may start after the full completion of the first

monolayer or may progress in parallel with it. A most simple model captur-

ing this effect is referred to as the BET (Brunauer–Emmet–Teller) model,

resulting, under certain assumptions, in the widely used BET isotherm

describing adsorption of multi-layered molecules. It is important to note that

in the regimes of multi- and sub-monolayer coverage the sorption processes

occur (in most instances) reversibly, that is, independently of the direction of

pressure variation.

Further increase of P at temperatures below the critical one, Tc, leads to

the phenomenon of capillary condensation (an analogue of the first-order

gas–liquid equilibrium transition in bulk substances) during which pore

spaces become completely filled with the liquid phase. Notably, the capillary

condensation pressure Pc is lower than the bulk equilibrium transition pres-

sure P0. Decreasing P from the state with pore space filled with the capillary-

condensed liquid results in the capillary evaporation transition (an analogue

of the liquid–gas transition), which occurs at a pressure Pe even below that of

1.00.80.60.40.20.00.0

0.2

0.4

0.6

0.8

1.0ii

ii

iii iv

Rel

ativ

e am

ount

ads

orbe

d, q

Relative pressure, P/Ps

Figure 2.1 Typical hysteretic adsorption isotherm obtained in disordered mesoporoussolids. The effect of disorder shows up as an asymmetry of the hysteresis loop, which isin contrast to parallel adsorption and desorption branches in materials with ideal porestructures. The inset shows schematically different regimes of adsorption: (i) sub-monolayer coverage, (ii) multi-layer adsorption, (iii) formation of capillary-condenseddomains, and (iv) a capillary-condensed liquid completely filling the pore space.

28 Rustem Valiullin

the capillary condensation transition Pc. Thus, gas sorption processes in

mesoporous solids usually exhibit hysteretic behaviour, as exemplified in

Fig. 2.1, signalling the non-equilibrium nature of at least one of the transi-

tions involved, capillary condensation and/or evaporation, during which the

system fails to equilibrate on a laboratory time scale.

Pioneering theories of capillary phenomena were based on classical mac-

roscopic thermodynamics and have associated shifts of the condensation and

evaporation transitions in capillaries with the menisci curvature.47 These

theories were capable to predict the occurrence of sorption hysteresis quite

in general. Indeed, the geometries of the gas–liquid interfaces during adsorp-

tion and desorption in, for example, cylindrical pores appear to be different

(concave semi-spherical upon desorption and concave cylindrical upon

adsorption), giving rise to different equilibrium transition pressures via

the Kelvin equation. Moreover, they have also highlighted the importance

of the pore geometry, the pore size distribution (ink-bottle pore configura-

tion), and the boundary conditions (open and one-end closed pores) on

phase equilibria. For example, Cohan anticipated that closing one end of

a cylindrical pore should eliminate the hysteresis by providing a semi-

spherical interface at the closed pore end.48 Being generally accepted and


verified in computer simulations, the validity of the Cohan’s conjecture has

nevertheless been a subject of recent controversial discussions in the

literature.49–51

Later on, different modifications of the density functional theory and lat-

tice gas models capturing microscopic features of the adsorption phenomena

have emerged.31 Importantly, the microscopic theories pointed out a meta-

stable character of the transitions. For example, they predict that, with

changing external gas pressure, molecular ensembles in uniform pores

remain in a gas-like or a liquid-like state beyond the point of the true ther-

modynamical transition, that is, persist in a state of a local minimum in the

free energy. This is continued until the barrier separating the local and the

global energy minima becomes sufficiently small to be surmounted via fluc-

tuations. However, in order to make it possible for the liquid-like and the

gas-like states to coexist simultaneously within the porous matrix of disor-

dered materials, these theories have to incorporate effects of the geometrical

heterogeneity. This significantly complicates the problem and makes it

dependent on the chosen framework.

The understanding of the phenomenon of sorption hysteresis highly

benefited from the applications of advanced computer-based calculations

and simulation approaches.31,34,52,53 In particular, the results obtained using

a non-local density functional theory pointed out that in sufficiently big

pores condensation occurs at the vapour-like spinodal, while desorption

takes place at the equilibrium.54 The same conclusion has been drawn using

molecular dynamics simulations of the molecular behaviour in one- and

both-ends open pores.55 The phenomenon of pore blocking, which is

widely accepted to contribute to the development of hysteresis,56,57 has

as well been addressed. Thus, using an ink-bottle pore as confining geom-

etry, it was found that liquid can evaporate from a large cavity even if the

neck of the ink bottle remains filled with the capillary-condensed phase.55

At the same time, the relevance of the pore-blocking mechanism was also

verified,58 emphasizing the importance of the details of the pore structure

and the involved interactions.

A particular advantage of the modelling approaches is the possibility to

address the problem of disorder inherent in the vast majority of porous solids

by implementing pore heterogeneity effects into the microscopic theories.

Exactly in this way one of the central questions about the inter-relation

between hysteresis and phase transition in disordered mesoporous materials

has recently been addressed using mean-field theory and Monte Carlo sim-

ulations for a lattice gas model.50,59,60 Importantly, this modelling approach

30 Rustem Valiullin

reproduced well the shape of adsorption isotherms for fluids in Vycor porous

glass (a material which is often considered as a standardmaterial for validating

new theoretical approaches). The calculations indicated that the hysteresis

can be understood in terms of the effects of the spatial disorder in the material

upon the density distribution in the system. A similar conclusion has also

been drawn from the experimental results on nitrogen adsorption in porous

silicon with tubular pore morphology with excluded network effects.61,62 In

particular, hysteresis is associated with the appearance of a very large number

of metastable states, which are minima in the local free energy corresponding

to different spatial distributions of the adsorbed fluid within the void space of

the porous material.

Recent progress achieved in the area of chemical synthesis of ordered

mesoporous materials has further advanced our understanding of confined

fluids and allowed a better description on single pore and ink-bottle pore

levels. Thus, the availability of well-defined ink-bottle systems allowed a

direct experimental verification of the cavitation phenomenon,55 earlier

predicted in computer simulation studies of confined fluids, and ramification

of the conditions of pore-blocking-controlled desorption.63 In particular, it

has experimentally been demonstrated that, for mesoporous materials con-

taining mesoporous voids connected to the external gas phase via smaller

mesopores (necks), two evaporation scenarios from the larger mesopores

are possible64,65: (i) With decreasing vapour pressure, first the necks empty

at the pore equilibrium pressure, thus actuating evaporation from enclosed

large pores. (ii) With sufficiently narrow necks (such that the first-order

liquid–gas transition in them is inhibited by the confinements), fluid in

the larger pores can spontaneously empty at the spinodal point via mass

transfer through the necks, which remain to contain a high-density fluid.

The transition between these two regimes in terms of the neck size was

found to be about 5 nm for nitrogen at 77 K (obviously, it depends on

the sorptive thermodynamic properties).

While there is a great deal of experimental data for adsorption isotherms

including hysteresis, much less attention has been given to the relaxation

dynamics for systems exhibiting hysteresis. There are only a few experimen-

tal reports in the literature indicating that equilibration kinetics may slow

down in the hysteresis region. Thus, Rajniak et al. have suggested an ana-

lytical description of the observed behaviour by modelling of the effective

diffusivities as a function of the external conditions and topology of the

porous space and adopting it to describe the obtained kinetics.66 This is a

rather difficult problem by itself and involves a priori assumptions about a


diffusion-controlled density relaxation. Data of recent experimental studies,

however, are found to be inconsistent with this assumption.42,61,62,67

Indeed, recent dynamic Monte Carlo simulations for the lattice model of

a fluid in Vycor glass60 suggested that the relaxation processes in the hyster-

esis regime for longer times are dominated by activated barrier crossings

between minima in the local free energy. This is an intrinsically slower pro-

cess than the relaxation through mass transfer to or from the external surfaces

of the material that dominates for states outside the hysteresis regime.

2.2. Liquid–solid equilibriaLiquid–solid equilibria in confined spaces are less understood, compared to

the gas–liquid one. One particular reason for this is the complexity of the

liquid–solid transitions, which, even in bulk state, may show quite different

features originating, for example, from the different mechanisms giving rise

to the phase transitions.68 Thus, solids or molecular crystals made up of the

same atoms or molecules may have quite different crystal structures, which

may further be determined by the confinements, including the geometry of

the confining spaces and the surface chemistry.69–73 In what follows, how-

ever, we will not be dealing with all these fine details such as atomic struc-

tures and will confine ourselves to considering only general relationships

between the pore size, pore morphology and macroscopic fluid properties

and the freezing and melting behaviour. In particular, only liquids wetting

the pore walls will be considered.

Under these circumstances, the problem becomes reminiscent to gas–

liquid transitions under confinement.74 This analogy stems from the fact that

the shifts of the gas–liquid transition points in small pores are, in first approx-

imation, captured by the Kelvin equation. Description of the solid–liquid

equilibria may similarly be done using the Gibbs–Thomson equation having

the same structure. They both predict pore size-dependent shifts of the tran-

sition pressure and temperature being proportional to the inverse pore size,

resulting from the competition between the bulk and interface free energies.

As well, there are many experimental evidences that, in complete analogy to

multi-layer adsorption, the phenomenon of surface pre-melting75 leads to the

existence of a liquid-like, disordered layer between the pore walls and the fro-

zen crystalline core in the pore interior, whose purpose is tominimize the total

energy due to mismatch in the crystalline structures between two solids.76–78

Thus, many features discussed in the preceding section do as well apply here

and, therefore, will only briefly be discussed.

32 Rustem Valiullin

One of the central questions for understanding solid–liquid equilibria of

fluids in porous solids on the basis of macroscopic thermodynamics is the

location of the equilibrium transition temperature. In gas sorption, for mate-

rials with ideal pore structures, it is associated with capillary evaporation.

In total analogy, one may associate the freezing transition with the equilib-

rium transition (it has to be assumed, however, that the bulk frozen phase is

supplied at the pore openings for avoiding any nucleation delay).79 Thus,

metastability in the freezing–melting hysteresis is associated with delayed

melting due to the barriers in the free energy to nucleate liquid “bridges”

in pores with a frozen fluid in their core parts. In the absence of an external

frozen phase at the pore openings, pore size-dependent metastable freezing

due to delayed homogeneous nucleation is observed for pore sizes above the

critical one.80 It should be noted, however, that this picture has not been

fully supported and the coexistence temperature has also been associated

with the melting temperature (see, e.g. Refs. 81,82).

The lattermay hold in disorderedmaterials. Here, the pore-blocking effect

has experimentally been proven to render the freezing transition metasta-

ble.83,84 On the other hand, metastability of the melting branch due to radial

melting in bigger pores can be avoided at temperatures corresponding tometa-

stable axial melting of smaller pores. Thus, in disordered materials (or even

ordered materials in the presence of structural defects), it is the melting branch

rather than the freezing one which may better describe the condition of inter-

phase coexistence. This conjecture was experimentally evidenced in a recent

work reporting freezing and scanning behaviour in Vycor porous glass with

disordered pore structure.85 As the main conclusions of this study, it was

found that (i) the freezing is controlled by strong pore blocking, leading to

the formation of percolating continuous clusters of the frozen phase spanning

over neighbouring pore sections. Thus, being initiated at the pore openings

by supplying the frozen bulk phase, macroscopically extended regions

containing frozen andmolten parts can be formed in porous monolithic parti-

cles. (ii) Melting, in contrast, is found to be a more local property, rendering

the transition to occur homogeneously over the whole pore network.

3. DIFFUSION DURING GAS–LIQUID PHASETRANSITIONS

Being of technological relevance in the areas of, for example, gas

separation or heterogeneous catalysis, most comprehensively addressed is

transport of fluids for gas–liquid coexistence in porous solids.86–89 Quite


generally, the overall molecular trajectories under these conditions may

include time intervals of surface diffusion, diffusion in multi-layers confined

between the pore walls and the gaseous phase in the pore interiors, diffusion

in the capillary-condensed domains, and Knudsen diffusion in the gaseous

phase. Before considering these different transport mechanisms separately,

it is instructive to show first the overall behaviour, which may further be

used as a guide to discuss, in more detail, all features and to trace them back

to their microdynamic origins.

As such an example, Fig. 2.2 shows the amounts adsorbed and the effec-

tive molecular diffusivities of acetone measured using PFG NMR in meso-

porous silicon90 at different gas pressures of the surrounding atmosphere.42

This type of simultaneous measurement of both quantities can easily be

realized by directly connecting a reservoir, storing the vapour of a selected

gas at a given gas pressure, to the NMR sample, positioned in an NMR

probe head in a magnet, containing the porous material under study

(see for more details, Refs. 91,92). The thus obtained diffusivities show

quite a complex behaviour, which appears to be generic for different meso-

porous solids, irrespective of their fine structure. In what follows, we pro-

ceed in our considerations from the low to the high gas pressure regions.

In particular, we discuss first surface diffusion in the regime of sub-

monolayer coverage (region i in Fig. 2.2). Thereafter, we discuss

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.2

0.4

0.6

0.8

0.0

1.0

Diff

usiv

ity (

10−9

m2 /

s)

Relative pressure, P/P0

iviii iii

Am

ount adsorbed, q

Figure 2.2 The amount adsorbed (triangles, right axis) and the effective diffusivities(squares, left axis) of acetone in mesoporous silicon with 10 nm pore sizes as a functionof the relative gas pressure P/P0 as measured using PFG NMR during adsorption. Themeasurements are performed at room temperature. For the different adsorptionregimes indicated by the Roman numerals, see Fig. 2.1.

34 Rustem Valiullin

conditions under which diffusion in the gaseous phase may affect the over-

all mass transfer. Finally, transport in the regime of coexisting multi-layers,

capillary-condensed domains, and domains with gaseous phase will be

considered.

3.1. Surface diffusionBeing one of the key points for understanding chemical reactions occurring

at surfaces or crystal growth processes, the phenomenon of surface diffusion

has been thoroughly addressed in the literature (see, e.g. reviews 93–95).

On considering the progress in the theoretical studies of surface diffusion,

one becomes aware that an accurate theoretical treatment has been per-

formed in only some limiting cases, such as for particles on homogeneous

surfaces interacting via hard core-like inter-molecular interaction. The anal-

ysis becomes much more complicated for heterogeneous surfaces. While

analytical solutions are still possible for non-interacting particles (e.g.

at very low surface coverage) on lattices with quenched disorder, particle

ensembles under such conditions can hardly be analyzed.96,97 Under

certain assumptions about the microscopic nature of the jump process,

however, some general properties can be obtained.98–100 Appreciable

progress in the understanding of surface diffusion has been attained using

computer modelling. By illuminating some general patterns in the diffus-

ion process, these studies have shown a very rich behaviour depending

on the inter-molecular and adsorbate–substrate interactions involved, the

distributions of site and/or barrier energies, the temperature and the particle

concentration.101–104

Thanks to the high-surface area in mesoporous solids, allowing to

accommodate sufficiently large molecular ensembles on their surfaces, it

has recently been shown that PFG NMR can be used to trace surface dif-

fusion in these materials.105,106 This possibility broadens substantially the

class of physical systems in which this process can be studied. Thus,

Fig. 2.3 shows diffusivities of n-heptane measured in Vycor porous glass

as measured using PFGNMR technique at 246 K. The diffusivities are plot-

ted as a function of the surface coverage c, which is defined as c¼y/ym,where ym is the amount adsorbed corresponding to a one monolayer

coverage.

An important finding revealed by these results is the observation of dif-

fusivities increasing with increasing surface coverage. This finding is in strik-

ing contrast to the data on diffusion of adatoms on flat metal surfaces, which

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.5

1.0

1.5

2.0

2.5

Diff

usiv

ity (

10−1

1 m

2 /s)

Surface coverage, c

Figure 2.3 Surface diffusivities of n-heptane inmesoporous silicon plotted as a functionof surface coverage c. The measurements were performed at 191 K, ensuring a suffi-ciently low pressure of the gas atmosphere.


typically show the exactly opposite behaviour: diffusivities in those systems

are typically found to decrease with increasing surface loading.93 Here, the

process is governed by an increasing degree of mutual restrictions for the

erratic hopping motion of guest atoms on the surfaces, with increasing sur-

face occupancy. Before associating the results of Fig. 2.3 with some specific

features of surface diffusion, any possible artefacts have to be ruled out.

In particular, the overall mass transfer probed using PFG NMR at these

low pore loadings may be affected by molecular flights through the gaseous

phase (see relevant discussion in Section 3.2). By choosing n-heptane, a liq-

uid with a relatively low pressure of its saturated vapour, as an adsorbate and

by performing all measurements at the relatively low temperature of 191 K,

it was ensured that the data of Fig. 2.3 are free of this disturbing effect of the

gaseous phase.

Surface diffusivities increasing with increasing surface coverage can be

assigned to the energetic heterogeneity of the surface.99 The latter is associ-

ated with the existence of a distribution fs(E) of site energies E due to struc-

tural defects on the surface and/or chemical disorder. With a site-energy

disorder along the surface, energetically favourable sites will preferentially

be occupied at low-surface coverages, that is, the probability to find a

particle on a site with higher adsorption energy will be higher. Due to

the activated character of the jump process, this leads to a corresponding

distribution W(E)/exp{�E/kT} of the jump rates.

36 Rustem Valiullin

With such a condition, the solution of the diffusion problem for a single

particle can be obtained within the frame of the so-called random trap model,

resulting in a diffusivity a2hW�1i�1, where the brackets denote averaging

over all surface sites and a is the inter-site distance.96,107 It is worth noting that

the solution does exist only if there exists a finite average residence time

hW�1i and if the model implies that there is no correlation between the ener-

gies of neighbouring sites, that is, if a random energy topography may

assumed. The particle ensemblesmay be treated similarly.However, the prob-

abilities of site occupancy, p(E), have to be properly accounted for

Ds ¼ðfs Eð Þp Eð ÞW�1 Eð ÞdE ð2:1Þ

Forbidding multiple site occupancies, p(E) has naturally to be chosen to

follow the Fermi–Dirac statistics, p(E)¼ (1þexp{(E�m)/kT})�1, where mis the chemical potential of the surface ensemble. Such an occupancy factor

explains increasing diffusivity with increasing surface coverage or, corre-

spondingly, chemical potential: a new particle added to the system will

occupy a site with a lower surface energy as compared to those already occu-

pied. Therefore, the overall transition rate, obtained by averaging over the

whole ensemble, will increase.

By taking account of the excluded double occupancy of the surface sites,

the average activation energy is expected to decrease with increasing surface

coverage. Figure 2.4 shows surface diffusivities for n-heptane in mesoporous

3.6 4.0 4.4 4.8 5.2

10−11

10−10

Diff

usiv

ity (

m2 /

s)

c = 1.0 c = 0.83c = 0.66c = 0.50c = 0.35

103/T (K−1)

Figure 2.4 Arrhenius plot of the surface diffusivities for n-heptane in mesoporoussilicon for different surface coverages as indicated in the figure. Lines show the bestfit of Eq. (2.2) to the experimental data.


silicon at different surface loadings in the Arrhenius co-ordinates. In the

range of temperatures studied, the experimental data of Fig. 2.4 are found

to follow the Arrhenius behaviour

Ds ¼Ds0 exp �Ea=kTf g ð2:2Þwhere Ea is the activation energy for diffusion andDs0 is the pre-exponential

factor. The activation energies Ea, which are proportional to the slopes to

the experimental data in Fig. 2.4, clearly indicate decreasing Ea with increas-

ing c. This further supports that it is the effect of surface heterogeneity upon

surface diffusion.108 The same conclusion has been drawn by performing

more elaborated experiments involving two different molecular species,

one of which has been used to intentionally block the sites with highest

energies of adsorption.105

To rationalize the dependence of the surface diffusivities on the surface

coverage c in systems with site-energy disorder, models considering distri-

butions of trapping times may be used.99,100 In the frame of these models,

the surface diffusivity Ds is shown to be

Ds¼D0

1� cð Þ2c

expm�E0

kT

� �, ð2:3Þ

where D0 is the diffusivity in the limit of c!0, m is the chemical potential,

and E0 is the reference site energy, which could be associated with, for

example, the average energy over all surface sites. Eq. (2.3) also includes

the site-blocking effect which, in the mean-field approximation, is given

by the term (1� c).

In our experiments, the chemical potential is controlled externally and is

related to the external gas pressure P. Using the ideal gas approximation for

the chemical potential, m¼m0þkT ln(P/P0), where m0 is the chemical

potential of the gas under standard conditions, Eq. (2.3) may be rewritten as

Ds ¼D�0 1� cð Þ2P

c, ð2:4Þ

where all parameters, independent of surface coverage, have been collected

into one parameter D�0. Eq. (2.4) predicts diffusivities diverging for high c,

which is not the case in the experiments. Such behaviour may be associated

with the fact that for guest–host systems with low energy of physisorption,

the onset of multi-layer adsorption may start already at c<1. This is also the

case for surfaces with strong surface roughness. In this case, the adsorbed

38 Rustem Valiullin

molecules may exchange their positions even if all neighbour sites are

occupied by hopping on top of neighbouring adsorbed molecules. This

especially can become effective with increasing surface coverage for mole-

cules occupying sites with lowest site energies. Thus, this mechanism pro-

vides finite surface self-diffusivities even at full surface coverage. To correct

Eq. (2.4) for this mechanism in the region of high c, a simple mean-field-like

approach may be used, in which c can be considered as the probability that a

neighbour site is occupied. By introducing a surface diffusivity Dsf at full

surface coverage, the effective surface diffusivity Ds,eff may, in this way,

be noted as

Ds,eff ¼Dsþ cDsf , ð2:5Þ

where Ds is given by Eq. (2.4). It has been shown that such modelling

approach nicely reproduced all features revealed by the experimental

data.106

3.2. Diffusion in the gaseous phaseTo rationalize conditions under which mass transfer in partially filled porous

media can be enhanced due to molecular flights in the gaseous phase, let us

consider molecular trajectories which are composed of time intervals in which

themolecules perform erratic hops along the inner surface of a porousmaterial

and time intervals of propagations in the gaseous phase. For relatively light

molecules and high temperatures, molecules adsorbed on the pore walls or

on top of the already formed adsorbed layers may occasionally get desorbed

and perform flights in the gaseous phase. Due to the low gas pressures, for

which the mean-free path length in the bulk gas notably exceeds the pore

dimensions in themesoporous solids, upon being desorbed, themolecules will

perform the so-called Knudsen flights, that is, they experience ballistic flights

until they hit the pore wall again and get adsorbed.

A sufficiently long trajectory, for which the distance between its initial

( r!

i) and final ( r!f ) positions notably exceeds the characteristic pore size of a

porous material, can be considered as a stochastic one. This implication is

further supported by the fact that the spin-echo diffusion attenuations, mea-

sured using PFG NMR for fluids in mesoporous solids under these condi-

tions, typically exhibit a mono-exponential form, revealing that diffusion on

the time scales studied is a Gaussian process. The statistics of such trajectories

can therefore be fully described by an effective diffusivity


Deff ¼r!f � r

!i

� �2� �6t

, ð2:6Þ

where t is the time interval over which the trajectories were sampled and

h. . .i conventionally means the ensemble average. The term r!

f � r!i

� �in Eq. (2.6) can be expanded to the sum of the individual displacements s

along the surface (referred to with the subscript s) and in the gaseous phase

(referred to with the subscript g). By collecting the displacements in the two

different phases separately, but keeping their order within each phase

unchanged, Eq. (2.6) may be rewritten as

Deff ¼ 1

6t

Xi

s!s,iþ

Xj

s!g,j

!2* +: ð2:7Þ

In this way, all sub-trajectories of diffusive motion along the surface

become combined to one long, continuous trajectory and all individual

flights as well become joined together to build a single trajectory composed

of continuous flights in the gas phase.

Opening the brackets in Eq. (2.7), one obtains

Deff ¼ 1

6t

Xi

s!s,i

!2* +þ 1

6t

Xj

s!g,j

!2* +: ð2:8Þ

Note that in Eq. (2.8) the cross-term has been omitted. Indeed, doing so,

we have assumed that distribution of the flight angles with respect to the

surface normal, following the desorption events, is distributed according

to the Lambert law. Under this condition, the directions of the flights

become uncorrelated with those during the preceding surface excursions.

Therefore, the average over the cross-term becomes zero.

The first term on the right-hand side of Eq. (2.8) does refer to the dis-

placements along the surface. Due to the stochastic nature of both the acti-

vatedmolecular hops on the surface and the diffusion process in the adsorbed

multi-layers, it is evident that there are no correlations between two subse-

quent displacements separated by a flight in the gaseous phase (there might

be exceptions for highly ordered porous materials). Therefore, one may

introduce effective surface diffusivity as

40 Rustem Valiullin

Des¼ 1

6ts

Xi

s!s,i

!2* +, ð2:9Þ

where ts is the fraction of the total time t spent by the molecule on the sur-

face. Exactly in the same way, the second term on the right-hand side of

Eq. (2.8) may generally be associated with the effective diffusivity Deg in

the gaseous phase,

Deg¼ 1

6tg

Xj

s!g,j

!2* +, ð2:10Þ

leading finally to

Deff ¼ ts

tDesþ tg

tDeg: ð2:11Þ

For systems under equilibrium, the ratios ts/t and tg/t are equal to the

fractions of the molecules in the two coexisting phases. Defining them as

ps¼ ts/t and pg¼ tg/t, Eq. (2.11) becomes

Deff ¼ psDesþ pgDeg, ð2:12Þwhich, in the context of NMR, is often referred to as the fast-exchange

equation.With this equation in hand, the contribution of gas phase diffusion

to the effective, long-range diffusivity as probed by PFGNMRmay now be

established.

To do this, let us first interrelate the fraction pg (recall that ps¼1�pg)

with the amount adsorbed y.We base our inter-relation on the two straight-

forward equations, namely NgasþNads¼N and VgasþVads¼V, where Ngas

and Nads are the number of molecules in the gaseous and liquid phases and

Vgas¼mNgas/rgas and Vads¼mNads/rads are the volumes occupied by them

(rgas and rads are the densities in the gaseous and adsorbed phases, respec-

tively, and m is the molecular mass). With y¼Vads/V and taking account

of rgas�rads, it may thus be noted91

pg¼ 1�yy

rgasrads

¼ 1�yy

PM

RT, ð2:13Þ

where the density rgas has been approached by that for ideal gases and M is

the molar mass. Thus, the effective diffusivity appears to be correlated with

the adsorption isotherm, similarly as observed already with Eqs. (2.4) and

(2.5) for surface diffusion.


To quantify the relative contribution of diffusion through the gaseous

phase to Deff, let us now estimate Deg. Most simply, this quantification is

done for pore systems of tubular geometry, like channels of mesoporous sil-

icon orMCM-41. In this case, it becomes immediately obvious that the pro-

cess associated with Eq. (2.10) resembles the classical process of Knudsen

diffusion in cylindrical channels109 with an effective tube diameter

deff ¼ dffiffiffiffiffiffiffiffiffiffi1�y

p(where d is the tube diameter and the correction takes

account of the thickness of multi-layers adsorbed on the pore walls89).

Indeed, it does refer to random flights of a point-like particle in an ideal

cylindrical tube subjected to diffusive reflections at the tube wall, that is,

exactly to the process we are considering in our case. The coefficient of dif-

fusion, referred to as the Knudsen diffusivity, is found to be109

DK ¼ 1

3d�u¼ 1

3d

ffiffiffiffiffiffiffiffiffiffi8RT

pM

r, ð2:14Þ

where �u is the average thermal velocity.

Clearly, Deg depends on the pore geometry, but it turns out that devi-

ation from a cylindrical pore morphology will result in only a differing

numerical factor in Eq. (2.14), which accounts for the chord-length

distribution in the pore space considered.110,111 Thus, considering a limiting

case of random flights in isolated spherical voids with diameters of dsp(the starting positions of two subsequent Knudsen flights are uncorrelated),

Deg results as112

DK,sp ¼ 1

8dsp�u: ð2:15Þ

Presumably, for all other geometries of pore spaces, Deg will assume a

value confined by the two limiting ones given by Eqs. (2.14) and (2.15).

Notably, the difference between them is relatively small. Thus, estimates

of Deg, based on the use of Eq. (2.14), as often done in the literature, turn

out to be well justified.

With Eq. (2.13) for pg and with Eq. (2.14), as an estimate for Deg, the

term pgDeg can now be readily calculated and compared to the experimen-

tally measured diffusivities. As an example, Fig. 2.5 shows the respective data

for acetone in mesoporous silicon with channel-like pore structure. The

comparison reveals that, for this particular fluid and for room temperature,

the gaseous phase in the pore interior notably contributes to the overall mass

transfer. This is found to be valid in the regime of sub-monolayer adsorption

as well as in the regime of multi-layer adsorption. Recalling that Deg is

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Diff

usiv

ity (

10−9

m2 /

s)


Figure 2.5 The rectangles show the data of Fig. 2.2 for the effective diffusivities of ace-tone in mesoporous silicon. The line is the contribution pgDeg of diffusion through thegaseous phase to overall diffusion calculated with Eqs. (2.13) and (2.14).

42 Rustem Valiullin

constant, it is noteworthy that the formation of the maximum in the mea-

sured diffusivities with increasing pressure is due to the maximum in the

value of pgDeg and, in particular, of pg. Approaching the adsorption isotherm

by a certain model, such as the conventional BET equation, the position of

the maximum can readily be obtained via Eq. (2.13).91,113

3.3. Diffusion during the formation of capillary-condenseddomains

It is interesting to note that the data of Fig. 2.5 in the regime of the formation

of capillary-condensed phase, which according to Fig. 2.2 is found between

0.7<P/P0<0.9, are quantitatively captured by Eq. (2.12) with psDes rep-

laced by pcDc,

Deff ¼ pcDcþpgDeg: ð2:16ÞHere, pc(�1) andDc are the relative fractions and the diffusivities of mol-

ecules in the capillary-condensed phase, respectively. At a first glance, it

seems that the line of reasoning used in the preceding section directly applies

also in this case and one may immediately proceed with Eq. (2.16),

which, with Eq. (2.13), automatically takes account of the reduced volume

available for the gaseous phase due to the formation of domains with

capillary-condensed liquid. Some differences may result from the modifica-

tions of the chord-length distributions when spatial extension of the


domains with the gaseous phase will approach the characteristic scale length

of the pore system, but they are expected to be minor.

However, it turns out that the mere fact of the existence of domains of

the gaseous phase, spatially isolated from each other, can lead to a

dramatic change in the statistics of the Knudsen flights determined by the

microscopic details of the molecular behaviour at the gas–liquid interface.

Indeed, the main assumption underlying Eqs. (2.14) or (2.15) was the sto-

chastic nature of the trajectories associated with Eq. (2.10). This assumption

may break under the condition of capillary condensation, where the thus

formed domains of the gaseous phase may form closed volumes. Thus,

the coefficient Deg in Eq. (2.16) will not have the simple meaning of a

Knudsen-like diffusivity anymore. It has rather to be reanalyzed. One

has to keep in mind that the diffusivity Dc may also change due to an

increased tortuosity of the sub-space occupied by the capillary-condensed

phase by a factor depending on the details of spatial arrangement of the

gas-filled domains.

Taking a closer look intoDeg, it appears that the key role is played by the

probability plg for a molecule to cross the liquid–gas interface during a single

collision with it.88,112 To illustrate this, let us consider a molecule which has

just entered a domain of capillary-condensed liquid after performing a

Knudsen flight. Due to the laws of diffusion, it will return back to the same

liquid–gas interface with a higher probability rather than to similar interfaces

formed by other domains of the gaseous phase. Now, if the probability plg is

relatively high, this molecule will leave the liquid domain in close proximity

of the point, where it has been adsorbed. That means that the flight direc-

tions between two subsequent flight events will be anti-correlated. In the

limiting case of the extremely high escape probability plg from the liquid into

the gaseous phase, the molecule will predominantly perform back-and-forth

flights leading to a very slow growth of the mean square displacements

acquired in the gaseous phase as compared to that in the liquid domains. This

situation is illustrated in Fig. 2.6A.

If, however, the probability plg is sufficiently low, then the molecule

will experience many crossing trials before it could leave the liquid phase.

Thus, in the time interval between adsorption and desorption events, just

by random diffusion, the molecule may be displaced over a sufficiently

long distance, such that the memory about the interface orientation at

the entering (adsorption) point may become totally forgotten. In this

way, randomization of the emission direction is provided, as exemplified

by Fig. 2.6B.

B

A

Figure 2.6 Schematic representation of trajectories in a porous medium containingdomains of capillary-condensed liquid (in gray) and spatially separated domains of gas-eous phase (in white). The solid lines show diffusive motion in the capillary-condensedliquid and the broken lines Knudsen flights in the gaseous phase. The cases (A) and (B)refer to high and low transition probabilities plg, respectively (see text).

44 Rustem Valiullin

Exactly under the latter condition, Eq. (2.15) has been derived for a

model system containing isolated spherical voids as shown in Fig. 2.6. With

increasing plg, the emerging anti-correlation in the flight directions will

more and more hinder the growth of the mean-squared displacements.

As it has been shown in Ref. 112, the long-time behaviour will still be dif-

fusive, but with the diffusivities lower than the Knudsen-like one. Figure 2.7

demonstrates this for the particular case of spherical domains of the gaseous

phase.

To make a connection to experimental situations, that is, to establish the

range of plg typically found in mesoporous solids at partial pore fillings, sim-

ple gas-kinetic analysis can be used. An important fact to appreciate, in this

respect, is that at a given pressure P the liquid and gas in the gas-filled

domains must be in equilibrium with each other as well as with the external

gas. This can be assured by equating the molecular fluxes onto and out of all

interfaces involved. In this way, plg has been shown to be112

plg¼2

9

P�ukT

d40D0

, ð2:17Þ

where D0 is the liquid diffusivity and d0 is the molecular diameter. With

typical parameters for most low-molecular organic liquids and for typical

pressures in the adsorption hysteresis region, corresponding to coexisting

10−3 10−2

10−2

10−1

100

10−1 100

Nor

mal

ized

diff

usiv

ity

Transition probability, plg

Figure 2.7 The ratio of long-time diffusivities Deg resulting from Monte Carlo simula-tions for the case of isolated spherical domains of gaseous phase to the diffusivitypredicted for a purely stochastic trajectory as given by Eq. (2.15) (broken line) as a func-tion of the transition probability plg of molecules through the liquid–gas interface.112


gas-filled regions with domains of capillary liquid, plg is estimated to be of the

order of 10�3–10�2 at room temperatures.

Inspecting Fig. 2.7, it appears that, with such low transition probabilities

plg, the trajectories composed of Knudsen flights in spherical voids are almost

random, that is, no appreciable effects of anti-correlations can be expected.

For materials with random pore structures, in which the irregular gas-filled

domain surfaces introduce additional randomization, this is fulfilled even

better. Thus, we may generally conclude that for all mesoporous materials

irrespective of the details of density distribution along the pore matrices,

Eq. (2.16) is generally valid with Deg being approached by its limiting con-

stant value corresponding to purely diffusive Knudsen flights.

With the progress in the area of chemical synthesis of orderedmesoporous

materials, the validity of Eq. (2.16) can directly be tested. Thus, Fig. 2.8

shows the diffusivity data for PIB-IL material at different gas pressures as

measured using PFG NMR. Ordered silica PIB-IL material is produced

by using aqueous surfactant mixtures and liquid crystal templating.114

A block copolymer containing a polyisobutylene and a polyethylene oxide

blocks is used to generate well-defined spherical mesopores of ca. 20 nm

in diameter. Small worm-like mesopores with a diameter of about 2–3 nm

are introduced by using a small ionic surfactant. This special combination

of templates results in a hexagonally arranged set of spherical mesopores

inter-connected via worm-like mesopores. In the region of gas pressures

between 0.3<P/P0<0.8, attained on adsorption, a capillary-condensed

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Diff

usiv

ity (

10−1

0 m

2 s−1

)


0.0

0.5

1.0

Ads

orpt

ion,

q

Figure 2.8 Effective diffusivities (filled symbols, left axis) and amount adsorbed(open symbols, right axis) for cyclohexane in PIB-IL measured using PFG NMR on theadsorption (circles) and desorption (triangles) branches at T¼297 K and the diffusivitiescalculated for the adsorption branch (rectangles) using Eq. (2.16) with Deg given byEq. (2.15).112

46 Rustem Valiullin

liquid in the worm-like mesopores coexists with a gaseous phase in the

spherical mesopores. The system does thus provide a nice experimental

model, which resembles the main features of spatially coexisting domains

of the adsorbed and gaseous phases in mesopores. Importantly, because all

structural parameters in this case are well known, the direct quantification

of the diffusivities can readily be done. The rectangles in Fig. 2.8 show

the effective diffusivities calculated via Eq. (2.16) with Deg approached by

Eq. (2.15). The almost perfect coincidence between the measured and cal-

culated data proves that the Knudsen flights in spatially isolated domains of

the gaseous phase are, in fact, of the required stochastic nature. This excellent

agreement found further justifies the use of the concepts of the fast-exchange

approach and of Knudsen diffusivity as applied to mesoporous materials at

partial pore fillings, including those having multiple porosities.115

3.4. Diffusion under varying temperatureAt this point, it is worth discussing what are the consequences of the

established strong correlations between the microscopic dynamics and phase

state in mesopores if, instead of pressure variation, other thermodynamical

parameters are varied. In first instance, it does refer to experiments under

varying temperature. Most easily, such measurements can be performed

by first equilibrating a mesoporous solid with a gas at a given pressure


and temperature, and, thereafter, closing the volume and measuring diffu-

sivities as a function of temperature. First, we are going to consider purely

mesoporous materials, in which we shall demonstrate that, with varying

temperature, the diffusivities may happen to deviate from a simple Arrhenius

behaviour.

As such an example, Fig. 2.9 shows diffusivities for n-pentane in Vycor

porous glass as a function of temperature.116 Two different situations have

been considered. In the first case, the Vycor glass monolith was largely over-

saturated by the liquid. In the temperature range studied, thus, the intra-pore

space was always filled by the capillary-condensed phase. The figure reports

only the diffusivity data for the intra-pore liquid, which are found to nicely

follow the Arrhenius behaviour.

For the second sample in which, at the lowest temperatures considered,

only the mesopores were filled with the liquid, upon increasing temperature

a strong deviation from the Arrhenius behaviour has been found at high

temperatures. This finding has been associated with the fact that, with

increasing temperature, evaporation (either due to gas invasion or due to

cavitation) leads to the formation of domains of gaseous phase in the

1/T (103 K−1)

Diff

usiv

ity, D

eff (

10−9

m2 s

−1)

2.80.4

0.6

0.81

2

4

6

3.2 3.6 4.0 4.4

Figure 2.9 Arrhenius plots of the effective diffusivities of n-pentane in Vycor porousglass measured in closed volume samples with increasing temperature. At the lowesttemperature, one sample contained excess liquid surrounding the porous particle(circles), while in the second one only the mesopore space was filled with the capillarycondensate (triangles). The solid line shows a fit of the Arrhenius equation to the datarepresented by circles. The dashed line shows the theoretical prediction according toEq. (2.16).

48 Rustem Valiullin

porous solid. Thus, not only increasing kinetic energy of the molecules

contributes to increasing molecular mobilities. A strong increase of pg with

increasing temperature in the region of the coexisting capillary-condensed

and gaseous phases does as well contribute to an increase in the

measured diffusivities so that the activation energy exceeds that of the pure

liquid. The theoretical modelling presented in the preceding sections

may easily be adopted to capture also the conditions of this type of

experiment. The thus obtained predictions are found to be in good

agreement with the experimental results as demonstrated, for example, by

the dashed line in Fig. 2.9 which perfectly coincide with the data

measured. Thus, the increase of the activation energy for diffusion as

compared to that in the liquid phase has to be associated with the increase

of pg controlled by the heat of desorption which, in this case, is the heat of

evaporation.

3.5. Non-equilibrium behaviourWith Eq. (2.16), the inter-relation between the phase composition and the

effective diffusivity is established. Here, by phase composition we refer to

the amount adsorbed y, which determines the fractions pc and pg. This fact

solely explains already the formation of the diffusion hysteresis when the dif-

fusivities are plotted versus the gas pressure, as shown, for example, in

Fig. 2.8. Indeed, this appears to be a trivial reflection of the underlying

adsorption hysteresis. One may thus expect that the diffusivities at identical

pore loading obtained upon adsorption and desorption should as well be

identical. This, however, is generally found to be not the case for disordered

mesoporous materials.

Thus, Fig. 2.10 shows the diffusivity of cyclohexane in Vycor porous

glass, a material with random pore network and a typical pore size of about

6 nm, plotted versus the amount adsorbed.117 One may clearly identify that,

in the adsorption hysteresis range, the diffusivities for the adsorption and for

the desorption branches differ notably. That means that, irrespective of fast

molecular motion, states with the same average density have different diffu-

sivities. In addition to simple transitions from the empty state by adsorption

and from the fully loaded state by desorption also the experimental data fol-

lowing desorption from a partially loaded state are shown. Remarkably,

depending on the history of the system preparation, again the state of iden-

tical pore loading results in different diffusivities.

0.0 0.2 0.4

Amount adsorbed, q

0.6 0.8 1.0

2.5

3.0

3.5

4.0

4.5

5.0

0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0

Am

ount

ads

orbe

d, q

Relative pressure, P/PS

Diff

usiv

ity (

10−1

0 m

2 /s)

Figure 2.10 Effective diffusivities of cyclohexane in Vycor porous glass measured usingPFG NMR at 297 K as a function of the amount adsorbed. Different symbols refer to dif-ferent pathways of the system preparation via adsorption (open circles), desorptionfrom completely filled state (filled circles), and desorption from the partially filled state(triangles) as indicated in the inset.


Understanding this behaviour requires an assessment of how the fluid

density distributions on adsorption and desorption (or in adsorption or

desorption scans) may be different for the same overall average fluid density.

One of the key points is that, on desorption, the liquid-like regions may

correspond to somewhat expanded (or stretched) liquid states with density

as much as 10% lower than the average density when the porous solid is

saturated with liquid at the pressure P0. Such a behaviour can be easily

revealed, for example, by inspecting the adsorption data in Fig. 2.8 obtained

on desorption. Stretching does not occur, however, on adsorption. The

stretching effect means that, for one and the same overall concentration,

the liquid phase occupies a notably larger part of the pore space during desorp-

tion than during adsorption. Though the diffusivity in the stretched liquid is

somewhat higher than in the dense liquid (see Fig. 2.8), it is still very much

smaller than in the gaseous phase. As a consequence, the larger volumes occu-

pied by the gaseous phase during adsorption lead, for a given total amount

adsorbed, to notably larger overall diffusivities than during desorption.

Besides this general tendency in the “history dependence” of the diffu-

sivities for a given y, one has clearly to be aware of further influences related

50 Rustem Valiullin

to the differences in the density distribution within the porous space. Dif-

ferent geometric configurations of the domains containing capillary-

condensed and gaseous phases may yield slight but still notable differences

in the respective diffusivities due to different chord-length distributions in

the gaseous domains or due to restricted diffusion effects owing to low tran-

sition probabilities plg. Cumulatively, they may further contribute to the

existence of the diffusivity spectrum. We note that—as unveiled by the

constancy of the measured diffusivities over time periods from hours to

days—these different (arrested) density configurations attained via different

histories of the system preparation are preserved over essentially unlimited

intervals of time. This, however, does not mean that there is no hidden

relaxation towards equilibrium. Rather, it is so slow that it can hardly be

noted within the accuracy of the diffusivity measurements.

The overall diffusivity has thus been identified as a sensitive probe of the

given state of a confined liquid. Different states emerge from the different

history of the system and are associated with different out-of-equilibrium

distributions of the fluid within the pore network. Being separated by large

barriers in the system-free energy, these states are found to remain stable over

very long intervals of time. Intriguingly, PFGNMRmay help addressing the

question whether these arrested states are completely “frozen” or slowly

relax towards the state of global minimum in the free energy.

This can be done by following the adsorption or desorption kinetics fol-

lowing stepwise changes of the pressure in the surrounding gas atmosphere.

The availability of mesoporous solids as macroscopically big monolithic par-

ticles with well-defined size and shape allows quantitative analysis of the

uptake kinetics. In particular, for the rod-like geometry of the Vycor porous

glass monolithic particles, the solution of Fick’s diffusion equation results in

the following equation for diffusion-limited uptake kinetics118

y tð Þ¼ y0þ yeq�y0

1� 4

a2

Xn¼1

1

a2nexp �a2nDeff t� � !

ð2:18Þ

where an are the positive roots of the equation J0(aa)¼0, J0 is the Bessel

function of the first kind, and a is the rod radius. Figure 2.11A shows

the experimental data for the adsorption kinetics of cyclohexane into

Vycor porous glass in the out-of-hysteresis region (steps from 0.33 to 0.38

relative pressures, see inset in Fig. 2.10).42 It shows, as well, the prediction

of Eq. (2.18) with the independently measured Deff¼5.6�10�10 m2/s and

the known value a¼3 mm for the rod radius. It is important to note that no

fitting parameters have been used and that, in Fig. 2.11A, the result of

0 2000 4000 6000 80000.20

0.21

0.22

0.23

4000 5000 6000 7000 80000.228

0.229

0.230

0.231

0.232

0.233

0 2000 4000 6000 8000 10000

0.46

0.47

0.48

0.49

0.50

0.51

0.52

0.53

6000 7000 8000 9000 10000

0.522

0.523

0.524

0.525

0.526

Ads

orpt

ion,

q

Time (s)

Ads

orpt

ion,

q

Time (s)

B

A

Figure 2.11 Uptake kinetics of cyclohexane in Vycor porous glass following pressuresteps from 40 to 45 mbar (A) and from 70 to 75 mbar (B).42 The insets display thezoomed long-time behaviour of the same data. The solid lines show Eq. (2.18) withthe independentlymeasuredDeff of Fig. 2.9. y0 and yeq are chosen to approach the valueof y attained initially and at long times, respectively. The dashed line in (B) shows thebest fit of Eq. (2.18) to the short-time data by allowing yeq to vary.


Eq. (2.18) with only independently determined parameters is shown. The

perfect coincidence between the calculation and the experimental data proves

that indeed, in the out-of-hysteresis region of the isotherm, the macroscopic

dynamics is determined by solely diffusive fluxes of the molecules.

52 Rustem Valiullin

The situation changes completely upon moving into the hysteresis regime

of the isotherm, as shown by Fig. 2.11B. In the region where the formation of

capillary-condensed liquid is evidencedby the increased slopeof the isotherm, a

formal applicationof themodel predictsmuch faster equilibration (dotted line).

This discrepancy between the observed time dependence of uptake and the

prediction on the basis of the measured diffusivities unequivocally points out

that only the early-stage uptake is controlled by diffusion. Thus, yeq in

Eq. (2.18) has to be taken such that the short-time part of the curve is

reproduced by Eq. (2.18). Therefore, yeq, which can better be defined as

yeq,diff, becomes a fitting parameter. The result of this fit is shown in the figure

by the dashed line, revealing that in this particular case, the diffusion-controlled

uptake is responsible for about 80% of the density relaxation.

At this stage, the chemical potentials of the external gas and the intra-pore

fluid equilibrate via mass transfer, which contributes primarily to the density

increase of the capillary-condensed liquid and to multi-layer adsorption.

A (quasi-equilibrium) distribution of the fluid density within the porous

material, which has been attained during the equilibration time before the

pressure quench and which does correspond to a local minimum in the free

energy, remains intact. An important consequence of the first, diffusion-

controlled uptake stage is, however, lowering of the heights of the barriers

in free energy separating the localminima.Thus, the systemmay lower its free

energy (relaxation towards the global minimum in the free energy) by reduc-

ing the gas–liquid interface area (merging of two domains, moving a domain

to a positionwith lower energy) or bynucleating newdomains (if the creation

of the interface is compensated by lowering of the chemical potential).

All these events require thermal activations. It is thus found that system

equilibration is governed by extremely slow relaxation. Any single event,

contributing to this relaxation, is accompanied by a local density distribu-

tion, which is equilibrated by diffusive mass transfer from the outer phase,

as described by Eq. (2.18), similarly to the out-of-hysteresis regime. The

characteristic time scale of the activated processes by far exceeds the diffusion

time scale and is, therefore, controlling the overall long-time dynamics.

Establishing the details of this dynamics is far from being trivial and is

challenging problem for future research.

4. DIFFUSION DURING MELTING/FREEZING

The way of analysis used to address diffusive dynamics for the

gas–liquid equilibria in porous solids may be generalized to other types of


phase equilibria. Thus, for coexisting liquid and frozen phases, the effective

diffusivities are obtained by taking the limit plg!0, that is, by making the

domains of frozen phase inaccessible for the fluid molecules. Clearly, in this

case, long-time statistics of the molecular trajectories always remains

diffusive (excluding the cases when liquid volumes form closed spaces)

and only the changing tortuosity of the space available for diffusion may

affect the limiting long-range values of the effective diffusivities.

In complete analogy with the case of gas–liquid equilibria in mesoporous

solids, reported in the preceding Section 3.3, Fig. 2.12 shows the fraction of

the liquid nitrobenzene and its respective effective diffusivities in Vycor

Temperature, T (K)

A

B

Liqu

id fr

actio

n, f

2100.0

0.2

0.4

0.6

0.8

1.0

1.2

220 230 240 250 260

Temperature, T (K)

Diff

usiv

ity, D

eff (

10−1

1 m

2 /s)

210

0.1

1

220 230 240 250 260

Figure 2.12 The fraction of liquid nitrobenzene (A) and the effective diffusivities ofnitrobenzene in the liquid phase (B) in Vycor porous glass as a function of temperaturemeasured during cooling (circles) and heating (triangles). The stars show the dataobtained on the cooling branch from a partially molten state. The lines are shown toguide the eye.

54 Rustem Valiullin

porous glass as a function of temperaturemeasured on the cooling (from a state

with the complete pore interior filled with the fluid in the liquid state in con-

tact with an external “bath” of frozen nitrobenzene at the pore openings) and

heating (from a state with completely frozen nitrobenzene in the pore interior)

branches. All measurements have been performed with an excess liquid phase

surrounding the Vycor porous glass particles. Upon cooling from room tem-

peratures, first the excess phase has been frozen around the equilibrium

solid–liquid transition temperature of nitrobenzene (T0¼278.7 K). Upon

further cooling, the phase transition observed at T�215 K indicates the

strong suppression of the freezing temperature. Heating from the completely

frozen (excepting for non-frozen surface layers) state reveals a melting transi-

tion atT�250 K. Thus, like capillary condensation and evaporation, freezing

and melting again observed to be subject to a hysteresis.

The effective diffusivities in Fig. 2.12B as well show pronounced

differences on cooling and heating.119 To get deeper insight into the

latter phenomenon, it is useful to re-plot the diffusivity data of Fig. 2.12B

versus the fraction of liquid phase. Such plot is shown in Fig. 2.13. Addition-

ally, because the temperature ranges of the liquid–solid equilibria on the

cooling and heating branches differ notably, the measured diffusivities

were normalized to the temperature-dependent diffusivities of supercooled

Liquid fraction, f

Nor

mal

ized

diff

usiv

ity, D

eff,n

Freezing

Scanning freezing

Melting

0.01

0.1

1

0.1 1

Figure 2.13 Normalized diffusivities of liquid nitrobenzene in Vycor porous glass as afunction of the liquid fraction attained duringmelting and freezing (transformed data ofFig. 2.11). The solid and dashed lines show proportionality with f1/3 and f3/2, respectively.


liquid nitrobenzene. In this way, it is possible to detect—if there is any—the

sole effect of thedifferent geometric arrangements of the two typesof domains

(liquid and frozen) on the overall diffusivity. The data of Fig. 2.13 indeed

show a well-notable hysteresis between the normalized diffusivities Deff,n

obtained during cooling and heating, starting from the completely molten

and completely frozen states, respectively.

A hint for rationalizing the origin of the melting–freezing diffusion hys-

teresis is provided by the data obtained upon cooling, showing almost no

dependency of Deff,n on the liquid fraction f down to about f¼0.3. This

finding may most straightforwardly be explained by the invasion-percola-

tion120 character of the freezing process in Vycor porous glass.85 Indeed,

in such small pores, with diameters of less than 10 nm, freezing via homo-

geneous nucleation is strongly suppressed. On the other hand, the experi-

ments were performed in a way that, at the pore openings, there was a

direct contact between the intra-pore liquid and the bulk phase of frozen

nitrobenzene surrounding the Vycor glass particle. This facilitates freezing

via solid front penetration from the particle boundaries into the porous solid

(see the schematic representation of this process in Fig. 2.14A). Due to dis-

order, freezing turns out to be strongly metastable, as a consequence of pore

blocking at narrow necks. Recalling the macroscopic dimension of the

Vycor porous glass particle used (a few millimetre), it becomes clear that

the liquid phase in the particle core forms a continuous domain with a size

notably exceeding the molecular displacements of up to 1 mm as registered in

the PFGNMR experiments. Therefore, on the length scales probed by PFG

NMR, the diffusion process appears to be not affected, that is, not restricted

by the frozen domains, down to f¼0.3.

A B

Figure 2.14 Schematic representation of the liquid–solid configurations at identical liq-uid–solid composition obtained on freezing (A) and melting (B) for a lattice modelresembling the pore structure of a disordered mesoporous materials. The frozen phaseis shown in black and the liquid phase in white.

56 Rustem Valiullin

In contrast to freezing, it is believed that melting occurs homogeneously

over the entire volume of the particle (see Fig. 2.14B). Thus, upon forma-

tion of the very first liquid domains in the pore sections with the smallest

dimensions, further melting occurs predominantly by growing of these

domains accompanied by the formation of new domains. Let us recall here

that non-frozen liquid layers between the frozen solid core in the pore inte-

riors and the pore walls are always present in the system. Thus, the space

occupied by the liquid phase in the capillary-condensed domains is always

inter-connected via the “bridges” formed by the non-frozen surface layers.

Diffusion theory yields, for such situations, lower diffusivities as compared to

the diffusivity in the capillary-condensed liquid (this can be understood in

terms of increasing tortuosity upon adding impermeable regions).14,121

How the effective diffusivity will change with increasing liquid fraction will

depend on both how the volumes of the liquid-filled domains and how the

thicknesses of the non-frozen layers will vary with temperature.121–123

Further support for this scenario of liquid–solid phase transitions in dis-

ordered spaces is supplied by the scanning experiments. They do, however,

also deliver differences in comparison with the corresponding results shown

in Fig. 2.10 for gas–liquid equilibria. Thus, Figs. 2.12B and 2.13 show also

the diffusivities measured on the freezing scanning branch. Here, the data

have been measured during cooling, starting from a state with only half

of the mesopores containing the liquid phase. This was attained upon

heating from the completely frozen state. In contrast to the diffusivities

obtained during the “main” freezing branch, these data do coincide, within

the experimental error, with those obtained on melting. This finding reveals

that, upon removing kinetic limitations for the nucleation processes, that is,

by providing the seeds of a new phase throughout the pore network, the

phase composition in random materials plays the decisive role in determin-

ingDeff. For Vycor porous glass, in this regime the normalized effective dif-

fusivities Deff,n are found to vary proportional to f1/3. The origin of this

dependency has still to be established.

Finally, it is worth shortly addressing the functional dependence of the

normalized effective diffusivitiesDeff,n on the liquid fraction f at low f. Here,

mass transfer solely occurs along non-frozen surface monolayers.77,124 In

Fig. 2.13, these are the data points for f<0.05. This conclusion is made

solely based on the fact that, for f<0.05, Deff,n behaves differently than

for higher liquid fractions. It has to be noted, however, that the normaliza-

tion has been done by taking account of the temperature dependence of the

diffusion behaviour in the supercooled liquid. The respective behaviour in


the non-frozen layers may be different and, therefore, the data for the region

f<0.05 may have an only qualitative meaning. In addition, in this region, it

is not trivial to take properly account of nuclear magnetic relaxation effects,

which shift the values of f from the real ones.

Notably, the behaviour obtained and the discussion performed for Vycor

porous glass with random structure of the pore network appears to be com-

mon for all disordered mesoporous materials as supported by our experi-

ments performed with mesoporous silicon with intentionally created

disorder.125 In more detail, we have prepared a sample with tubular pore

geometry with stepwise varied pore diameters along the pore axes as shown

in Fig. 2.15. With a total pore length of 50 mm, each channel-like pore was

found to consist of 100 sections with different pore diameters selected ran-

domly from the five values between 5 and 10 nm. In this way, by knowing

the exact pore structure and by knowing the fluid behaviour in each such

obtained pore section, the origin of the melting/freezing diffusion hysteresis

can be traced back to the peculiarities of the liquid–solid transition processes

under confinement.

A

*

B

1 cm

10 nm

500

nm

50µm

6 nm

Figure 2.15 Schematic view of the tailor-made mesoporous silicon. By the filledregions, two possible configurations of the frozen domains on freezing (A) and melting(B) are shown. The asterisk denotes the pore segment which, upon temperaturedecrease, is frozen first.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Freezing branchMelting branchN

orm

aliz

ed d

iffus

ivity

, Def

f,n

Liquid fraction, f

Figure 2.16 Normalized diffusivities of liquid nitrobenzene in mesoporous silicon withtubular pore morphology and with intentionally varied pore diameter along the poreaxis as a function of liquid fraction attained during melting and freezing.

58 Rustem Valiullin

In analogy to the data of Fig. 2.13, Fig. 2.16 shows the normalized

diffusivities for nitrobenzene in the thus obtained mesoporous silicon with

the modulated pore structure. In this case, however, it is known in advance

that, owing to the random distribution of channel diameters along the pore

axis, one and the same fraction f of the liquid phase may correspond to quite

different configurations of the frozen sections during freezing and melting.

This is exemplified in Fig. 2.15, where two different configurations of the

frozen domains yielding the same fraction f of the liquid phase are shown.

In the first case (A), the frozen region is formed upon cooling by first

freezing, via the homogenous nucleation mechanism, the section with

the largest pore size (denoted in the figure by an asterisk) and further

progressive freezing of the neighbouring sections with sufficiently large

pores. In the second case (B), the melting occurs progressively in the sections

with sufficiently small pore sizes, determined by the Gibbs–Thompson

equation. Exactly, this difference is reflected in the hysteresis in Fig. 2.16.

As a consequence of the differences in the respective transition mechanisms,

the freezing process tends to lead to the formation of more extended regions

of the pore volume filled with either liquids or solids. This means that,

for identical fractions f of the fluid phase, the molecules within the fluid

phase are generally able to freely diffuse over notably larger distances during

freezing, than during melting.

0.0 5.0 � 103 1.0 � 104 1.5 � 104 2.0 � 104 2.5 � 104 3.0 � 104

6

8

10

12

14

T = −43 °C -> −44 °C

T = −42 °C -> −43 °C

NM

R s

pin-

echo

sig

nal i

nten

sity

(ar

b. u

.)

Time (s)

T = −41 �C -> −42 °C

Figure 2.17 Time dependence of the amount of liquid phase indicated by the NMRspin-echo signal intensity after temperature decrease as shown in the figure for nitro-benzene in mesoporous silicon with 6-nm pore diameter.


Finally, it is worth noting that, in full agreement with the non-equilibrium

behaviour reported for the gas–liquid coexistence in disordered mesoporous

solids, the same phenomena do also accompany the liquid–solid transitions in

these materials. Thus, the data of Fig. 2.17 demonstrate the freezing kinetics,

namely, the evolutions of the liquid fraction in the pore space of mesoporous

silicon with 6 nm pore diameter upon stepwise temperature decrease,

measured using the NMR spin-echo pulse sequence with a time delay of a

few milliseconds to suppress the NMR signal from the frozen phase.40,126

The thus measured kinetics are found to be extremely slow, exhibiting

a power-law-like behaviour with failure to equilibrate on the time scale

of hours.125 This phenomenon can be associated with the internal disorder

of the channels in mesoporous silicon.50,62

5. DIFFUSION DURING OTHER PHASE TRANSITIONS

5.1. Diffusion during liquid–liquid phase separation
Upon temperature quench to the two-phase region of the phase diagram,
critical phase separating liquids (we confine ourselves to considering only

binary mixtures) build domains of different molecular compositions.

If confined to random porous solids, the macroscopic phase separation is

prohibited and arrested configurations, namely, domains of the minority

60 Rustem Valiullin

component-rich phase isolated from each other by a continuous network of

the majority component-rich phase, are formed. One may note that, in this

case, the majority component resembles gas–liquid systems. Indeed, the

domains of the majority component-rich phase can be considered as a

(stretched) capillary-condensed phase and traces of the majority component

in the domains of the minority component-rich phase as a gas. The same

is valid for the second component but now in the complementary space.

Considering diffusion behaviour in these systems, distinct differences to the

gas–liquid ones may result from (i) different values of the transition proba-

bilities plg at the domain interfaces and (ii) the intrinsically diffusive character

of the molecular translational motion for the “gas-like” traces.

As revealed by the experiments, however, no appreciable differences in

the diffusivities are measured upon crossing the critical temperature for

binary liquids in mesopores, which is also in accord with theoretical predic-

tions (see, e.g. Ref. 127). It may be anticipated that, in part, this is related to

relatively high concentrations of the molecules in either phases, that is, to the

relatively high solubilities even far deep in the two-phase region of the phase

diagram. Therefore, the transition probabilities (probabilities to cross the

domain interfaces, analogue of plg) are sufficiently high to lead to appreciable

restricted diffusion effects. Further, one may note that because both the

liquid–liquid separation and the tracer or self-diffusion are controlled by

the inter-molecular interactions, the effective average interaction seen by

the molecules before and upon phase separation remains essentially identical.

The effect of phase transition upon binary separated mixtures may still be

noted at low temperatures. Let us consider, for example, the classical binary

critical mixture n-hexane-nitrobenzene of volume composition 0.64–0.36.

Upon phase separation, nitrobenzene, which is the minority component,

forms isolated domains. Due to different 1H chemical shifts, the diffusivities

of both molecules can be measured separately using Fourier transform PFG

NMR. Figure 2.18 shows the diffusivities of n-hexane and nitrobenzene in

the nitrobenzene-n-hexane mixture of critical composition at temperatures

well below the critical temperature 293.1 K, that is, deep in the two-phase

region of the phase diagram.128,129

The results obtained reveal that diffusivities of both components strik-

ingly increase upon cooling at temperatures around 270 K. For bulk mix-

tures this, however, can be easily rationalized by the fact that, around this

temperature, the macroscopically separated nitrobenzene-rich phase freezes

out. Hence, one measures (i) the diffusivity of nitrobenzene in the hexane-

rich phase, which is higher due to the lower dynamic viscosity of n-hexane

103/T (K−1)

D (

10−9

m2 /

s)

D (

10−9

m2 /

s)

3.6

0.4

0.8

1.2

1.62

2.42.8

1.2

1.6

2.4

2

2.8

A B

3.7 3.8 3.9

103/T (K−1)3.6 3.7 3.8 3.9

Figure 2.18 Arrhenius plots of the diffusivities of n-hexane (A) and nitrobenzene (B) inan n-hexane-nitrobenzene mixture of critical composition in bulk (stars) and controlledporous glass with 70-nm pore diameter (circles).


and (ii) the diffusivity of n-hexane in the n-hexane-rich phase, which is again

higher because now there is no contribution of the slower n-hexane mol-

ecules in the nitrobenzene-rich phase.

The same general trends are also observed for the confined critical mix-

ture. Here one may note, however, some differences. First of all, there is a

discernible confinement effect upon the phase transition point. Therefore,

all phenomena discussed for liquid–solid equilibria in mesopores also apply

for the mixture components. Second, the relative increase is not as big as in

the bulk mixture, which one may anticipate to result from tortuosity effects.

Finally, one may note that, quite strikingly, the diffusivities of nitrobenzene

at temperatures high above the melting point are identical for the bulk

mixture and the mixture in porous glass. That means that the confinement

effect, leading to the reduction of the diffusivity as compared to bulk liquids,

is somehow compensated. The origin of this phenomenon is still to be

completely clarified.

5.2. Diffusion during structural transitionsThe experimental data reported in the previous sections reveal that molec-

ular diffusion is a sensitive parameter to changes in the local environment of

diffusing species. It is expected to hold not only for phase transition involv-

ing changes in the physical state but also for structural transitions. In this sec-

tion, without going into much detail, we are going to demonstrate this by

considering short, chain-like molecules confined to channels of mesoporous

silicon.130

62 Rustem Valiullin

It is long known that molecules may orient at interfaces. For example,

there is experimental evidence that n-alkane molecules tend to orient at

liquid–vapour as well as at liquid–solid interfaces.131,132 Depending on tem-

perature, chain length and surface chemistry, both alignment along and per-

pendicular to the interface can be observed. The effect of nanoscalic

confinement upon molecular orientations between two parallel surfaces

has also been shown both experimentally71 and in simulations.133

Due to its strong anisotropy, mesoporous silicon with tubular pore mor-

phology provides good options to transfer this anisotropy to confined mol-

ecules. The degree of the thus created anisotropy may be followed by NMR

spectroscopy by assessing residual dipolar couplings of the confined mole-

cules.134 In bulk liquids, the nuclear dipolar interactions are averaged out

due to fast rotational motion. Under confinement in mesoscalic channels,

however, not all spatial conformations and orientations are equally probable,

giving rise to non-zero dipolar coupling, being, therefore, directly propor-

tional to the orientational order parameter.

Figure 2.19 shows the dipolar coupling constants and diffusivities of

n-eicosane (C20H42) in mesoporous silicon channels of different pore tube

diameters, varying from 5 to 10 nm. The non-zero dipolar coupling mea-

sured reveals a partial ordering of the n-eicosane molecules in the channels.

The sudden drop of the dipolar coupling constant observed in small pores is,

presumably, a manifestation of a change of the orientation director. Indeed,

the local ordering is governed by the inter-play of the surface and the

inter-molecular interactions as well as by the effect of the confinement.

Consequently, with changing pore dimension, the distribution function

Pore diameter, d (nm) Pore diameter, d (nm)

Dip

olar

cou

plin

g (H

z)

Diff

usiv

ity, D

(10

−10 m

2 /s)

5

200

250

300

350

6 7 8 9 10 5

0.2

0.4

0.6

0.8

1.0

1.2

BA

6 7 8 9 10

Figure 2.19 Dipolar couplings (A) and diffusivities for n-eicosane measured along thepore axes in mesoporous silicon channels as a function of the pore channel diameter d.


of molecular orientation will be modified correspondingly to yield a lower

free energy. Based on the molecular geometry of n-alkanes, namely, the dis-

tances between the different protons and their orientation with respect to

the molecular axis, it can be shown that alignment of the latter along the

pore axis yields higher values of the dipolar coupling. Thus, our experimen-

tal findings may be discussed in the framework of a partial orientation of the

molecules along the pore axis in bigger pores and perpendicular to the sur-

face in smaller pores. This picture is in full agreement with the data on

molecular diffusivities. As expected, stretching of the molecules along the

pore axis yields higher diffusivities in this direction.

5.3. Diffusion in supercritical phasesIn view of the multitude of issues on molecular dynamics in confined spaces

under various external conditions discussed, one may have the impression

that their main conceptual features are already highlighted and that there

are only some ramifications left still to be done by, for example, considering

complexly organized pore architectures. It turns out, however, that at least

one important phenomenon remained not considered, namely, the forma-

tion of supercritical phases in confined spaces, a phenomenon which is still

far from being understood. Interestingly, here diffusion studies may substan-

tially contribute to a better understanding of the thermodynamics of critical

phases.135–138

To demonstrate this, let us consider Vycor porous glass contained in a

closed vessel and over-saturated by liquid n-pentane. As it has been shown

in Fig. 2.9, by heating the system up to 350 K one expectedly obtains an

Arrhenius-type increase of the intra-pore fluid diffusivity. The activation

energy for diffusion almost coincides with that of the bulk liquid. This is

directly seen in Fig. 2.20, where both intra-pore and bulk diffusivities are

shown. However, something spectacular is observed at T¼438 K around

which the diffusivity of the intra-pore liquid experiences a jump and there-

after does not change appreciably, forming a kind of plateau.135 Notably, the

diffusivity of the bulk liquid does not show, in this temperature range, any

visible deviation from the Arrhenius behaviour. Only close to T¼470 K,

which is the critical temperature for n-pentane, the diffusivity of the bulk

liquid shows sharp enhancement. We associate these two facts with the

for mation of a supercritical phase in the mesopores at temperatures substan-

tially below the bulk critical temperature. In the supercritical

state, the diffusivity remains essentially constant, being determined by the

mean-free path in the pore space according to the Knudsen limit of

2.0 2.5 3.0 3.5

10−6

10−7

10−8

10−9

Pore diffusivity (Dp)

Bulk diffusivity (Db)

Diff

usiv

ity (

m2s−1

)

103/T (K−1)

Figure 2.20 Arrhenius plot of the bulk and pore fluid diffusivities for n-pentane in Vycorporous glass as a function of temperature. The vertical dashed lines show the positionsof the bulk (solid line) and pore (dashed line) critical points.

64 Rustem Valiullin

diffusion. In this regime, D/ ffiffiffiffiT

p(see Eq. 2.14), which is a weak function

and appeared, therefore, as a plateau in the narrow interval of temperatures

studied.

This study did not only represent the first concomitant evidence of the

shift of the pore critical temperature by directly measuring the fluid transport

(diffusion) properties in the pores. It also did provide directly obtained abso-

lute numbers for the diffusivities of a fluid in the supercritical state confined

in a nanoporous solid. This helps, in particular, to rationalize that around the

bulk critical temperature, that is, in the range of a dramatic increase in the

bulk diffusivities, pore diffusion can already proceed in the supercritical state.

In turn, this gives a tool to vary the diffusivity by a tiny change of the tem-

perature by choosing an appropriate porous material. In this way, this phe-

nomenon can be used as a dynamic tool for an in situ manipulation of

chemical reactions occurring in pore spaces of mesoporous solids.113 It is also

noteworthy that such type of measurements may directly provide the pore

critical temperature Tcp, which may be different from the hysteresis critical

temperature obtained from adsorption isotherms.31

6. CONCLUSIONS

With the progress in the area of chemical synthesis of mesoporous

solids and the option to intentionally design the morphology of their pore


spaces, further advance in the exploration of fluid behaviour in confined

spaces has become possible. In particular, it does refer to a better description

of phase transitions, as an important tool for structural characterization of

nanoporous solids, and of translational dynamics, as the most important

physical parameter in the majority of technological applications involving

nanoporous materials. It turns out, however, that these two phenomena

are strongly coupled and that their simultaneous exploration is of critical

importance for their deeper understanding.

As shown in this contribution, NMR spectroscopy in general, and its

technique of pulsed field gradient NMR, in particular, proves to be a pow-

erful tool for a comprehensive exploration of these two phenomena. In this

way, it has become possible to trace the microscopic dynamics for fluids con-

fined to mesoporous materials as a function of a multitude of their internal

states, attained upon variation of the external conditions, such as pressure or

temperature. Thus, not only correlations between the phase composition, as

an indicator of a particular point on the phase diagram, and diffusion was

obtained, but also correlations between the history of the system preparation

in a state with a given phase composition and the internal dynamics has been

revealed.

All the results obtained have been discussed in the frame of a theoretical

model, which has been developed to resemble closely the experimental pro-

cedures of NMR. In particular, the consideration was performed by analyz-

ing the trajectory time series, similar to the process of tracing the nuclear

spins displacements in PFGNMR experiments. Having, in addition, a good

control over the structural details of porous materials used, it was possible to

perform the analysis on a quantitative level with a nice agreement between

theory and experiment. In this way, we lay down a basis for the quantitative

prediction of transport properties of a new family of novel hierarchical

materials,139–142 which are still waiting for a systematic exploration of their

dynamic properties.112,143–145

ACKNOWLEDGEMENTThe author wishes to thank the German Science Foundation (DFG) for the financial and

organizational support in the frame of the Heisenberg Fellowship.

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