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Freezing and Melting Transitions under Mesoscalic Confinement:
Application of the Kossel−Stranski Crystal-Growth Model
D. Kondrashova and R. Valiullin*
Institute for Experimental Physics I, University of Leipzig, Leipzig, Germany
ABSTRACT: The Kossel−Stranski model of crystal growth has been adopted to
study freezing and melting of fluids in mesoporous materials. The model is found to
exhibit the key features observed in the experiments, including shifted solid−liquid
and liquid−solid transition temperatures, irreversibility between freezing and
melting, and strong impact of the pore geometry. By first analyzing fluids confined
to cylindrical pores, we obtain several important insights into the transition
mechanisms. In particular, we establish the conditions for the occurrences of
equilibrium and metastable transitions and derive exact analytical equations
interrelating the transition temperatures, confinement size, and the interaction
parameters of the Kossel crystal. Variation of the channel diameter along the channel
axis, mimicking disorder in real porous materials, is shown to strongly affect both
freezing and melting. The model predicts that the freezing transition in disordered
materials is governed by the pore blocking mechanism. The melting transition is
found to result as an interplay of two different transition mechanisms.
■
INTRODUCTION
It is well documented that freezing and melting transitions for
fluids confined to porous solids are found to occur at
temperatures deviating from the bulk, equilibrium transition
temperatures T0.1,2 Using a broad spectrum of experimental
techniques, including differential scanning calorimetry,3,4
nuclear magnetic resonance cryoporometry5−8 and relaxometry,9−11 and X-ray12−14 and neutron15−17 scattering techniques, various aspects of these phenomena have thoroughly been
addressed. In particular, these studies focused on establishing
correlations between the confinement size, pore geometry,
surface wetting, and transition temperatures and on understanding irreversibility between freezing and melting. In earlier
works, the experiments were mostly performed using
disordered pore networks, such as porous sol−gel glasses18,19
or controlled pore glasses.20−22 The complex pore structure of
these materials made, however, direct comparison of the
experimental results to theoretical predictions questionable.
The recent advent of mesoporous materials with well-ordered
pores has notably contributed to advancements in the
fundamental knowledge about confined fluids. They provided
important options to either completely decouple pure confinement effects from those arising from complexities of the pore
geometries13,23−31 or to control the degree of their coupling
using materials with ink-bottle pore geometries.32,33 Notably,
the deeper understanding of fluid behavior on a single-pore
level allowed more elaborate analyses of the experimental data
obtained for disordered materials. Despite this progress brought
about by the availability of the ordered materials, some aspects
of thermodynamics of fluids confined in disordered porous
materials have still remained scarcely studied.34−39
The shifts of the liquid−solid and solid−liquid transition
temperatures to lower values observed in porous materials as
© 2015 American Chemical Society
compared to that in the bulk state are typically associated with
the excess surface free energy of the confined crystals. Due to
the common phenomenon of “interfacial melting”, i.e., the
formation of a thin, premolten layer between two solids causing
a partial reduction of the excess interfacial energy,40,41 the
crystals formed in the pore interiors are found to be surrounded
by one-to-two monolayer thick liquid-like layers adjacent to the
pore walls. Thermodynamics of the freezing and melting
processes of such systems can be explored by analyzing their
total free energy as a function of the thickness of the liquid
layer.35,42−44 Under certain assumptions made about the
interactions and the pore geometry and by considering only
the leading terms in the free energy functional, the transition
temperatures can be shown to conform to a modified Gibbs−
Thomson law
T0 − T = K /(d − τ )
(1)
where d is the pore size, τ is the thickness of the nonfrozen
surface layer, and the proportionality factor K depends on the
thermodynamical properties of the confined substances, the
geometry of the pore space, and the phase transition
mechanisms. Corrections to this equation can further be
obtained by taking into account additional mechanisms
contributing to the phase equilibria.
As mentioned above, for predicting the freezing or melting
transition temperature by this approach, i.e., for quantifying K,
one has to assume a particular phase transition pathway. Thus,
for freezing to occur in ideal cylindrical pores (the most simple
Received: October 17, 2014
Revised: January 5, 2015
Published: January 27, 2015
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interactions of the confined substances with the pore walls,
which may notably complicate the freezing and melting
phenomena.36,38,50
In what follows, we first elaborate on the lattice model used
and on its implementation for confined fluids. Thereafter, we
apply it to examine different phenomena of phase transitions in
pores with cylindrical pore geometry. The results obtained in
this section will then be used to discuss fluid behavior in pore
spaces with structural disorder.
pore geometry frequently used in the literature), a crystal
nucleus has to be formed first. The highest temperature Tn, at
which a spherically shaped crystal seed can be nucleated, can be
estimated from the condition that, at this temperature, the
channel is able to accommodate a nucleus of the critical size.1
The nucleation process is effective at sufficiently low temperatures only, i.e., it requires a significant degree of undercooling
of the confined fluid. Hence, as soon as a stable crystal seed of
the critical size is nucleated, it will immediately initiate freezing
in the whole channel. The nucleation-related supercooling can
be, however, avoided by providing seeds of the frozen phase at
the pore openings. The latter corresponds to experimental
situations in which porous materials containing capillarycondensed liquids are brought into contact with the bulk
frozen liquid. Under this condition, freezing occurs by invasion
of the frozen phase from the outer boundaries of the porous
monoliths. The onset of this process is, obviously, found to
occur at temperatures higher than Tn. Because this temperature
depends on the confinement size, geometric disorder of the
pore space can lead to quite complex percolation patterns, thus
giving rise, once again, to the formation of metastable liquid
phases in the pores.
Analogous phenomena may be noted for the melting
transition. Similarly to the previously considered case of the
liquid−solid transition, melting is triggered by nucleation of
seeds of the liquid phase. In contrast, however, the existence of
liquid-like layers at the pore walls may facilitate the nucleation
process. Because nucleation requires the overcoming of barriers
in the free energy (crystals in the pore interior are, at these
temperatures, in a metastable state), the formation of any liquid
phase domain over the whole channel cross-section will cause
melting of the frozen phase in the whole pore. Structural
disorder may as well alter the melting process. It may, in
particular, lead to situations where different phenomena may
act in a concerted way and may, thus, result in complex patterns
of phase coexistence.
Already this compendious overview illustrated how the
disorder-induced interplay between different transition mechanisms, possibly further complicated by metastability effects,
may give rise to different transition pathways. The direct
application of theoretical predictions based on ideal pore
systems to disordered materials may, therefore, lead to faulty
conclusions. This may have important implications for methods
of structural characterization, based on the application of eq 1
for the determination of pore sizes45 or for the prediction of the
solid−liquid equilibria in industrially or environmentally
relevant processes. In order to gain deeper insights into
freezing and melting in disordered porous materials, the
application of microscopic approaches, free of the a priori
assumptions inherent to macroscopic approaches, becomes
therefore crucial. In the most simple way, this can be done
using lattice models, in which the molecules are represented by
the lattice sites and the interaction energies between the nearest
neighbors are considered to depend on whether a particular site
belongs to crystal or liquid. Despite their simplicity, these
approaches have proven to be extremely useful.46−48 In
particular, they contributed significantly to an understanding
of crystal growth processes in bulk systems. Among them, the
Kossel−Stranski model with the simplest cubic symmetry was
frequently used.49 In this work, we employ it for an in-depth
study of the generic properties of the freezing and melting
transitions of fluids confined to pores with different pore
geometries.37 In addition, we also address the effect of the
■
LATTICE MODEL
Let us consider an arbitrary cubic lattice configuration Ψj, in
which different lattice sites can be in either solid (pore walls),
fluid (liquid), or crystal (frozen liquid) states. The solid sites
are used to model the pore geometry and remain in this fixed
state. The probability P for the occurrence of a certain
configuration is
⎧
⎡
⎪
P(Ψj) = Q−1 exp⎨−⎢∑ β( −Niiϕii − Nisϕis + NF
i i)
⎪ ⎢
⎩ ⎣ i
+
⎤⎫
⎪
∑ ⎛⎝− 1 βNikϕik⎞⎠⎥⎬⎪
⎜
i≠k
⎟
2
⎥⎦⎭
(2)
where β = 1/kT is the inverse temperature, the subscripts i and
k can assume two values, namely c (crystal) or f (fluid), the
subscript s stands for solid, Nik is the number of the ik bonds, Ni
is the number of sites of type i, Q is the partition function, and
−ϕik is the potential energy of the nearest neighbor interaction
between ith and kth sites. Fc and Ff in eq 2 are the internal
energies of the crystal and fluid sites, respectively.
Taking account of only the nearest neighbor interactions, the
number of sites Ni can be expressed as
3Ni = Nii +
1
1
Nik + Nis
2
2
(3)
where k ≠ i. With Nii given by eq 3, eq 2 becomes
P(Ψj) = Q−1 exp{−β ∑ Ni(Fi − 3ϕii) − ωcf Ncf − ωcsNcs
i
− ωfsNfs}
(4)
where the following definitions have been used: ωcf = β[−ϕcf +
(ϕcc + ϕff)/2], ωcs = β[−ϕcs + ϕcc/2], and ωfs = β[−ϕfs + ϕff/
2]. The chemical potential μi, associated with a site i, is μi = Fi
− 3ϕii. The total number of sites is fixed, Nc + Nf = C. Thus
P(Ψj) = Q−1exp{βNc Δμ − βCμf − ωcf Ncf − ωcsNcs
− ωfsNfs}
(5)
where Δμ = μf − μc. The probability P(Ψl) to find a
configuration Ψl which contains one extra crystal site as
compared to the configuration Ψj (with the bond numbers
adjusted correspondingly and which are marked by an
apostrophe) is
P(Ψl ) = Q−1 exp{β(Nc + 1)Δμ − βCμf − ωcf Ncf′ − ωcsNcs′
− ωfsNfs′ }
(6)
According to the principle of microscopic reversibility, the
transition probabilities p(Ψj → Ψl) and p(Ψl → Ψj) fulfill the
condition P(Ψj)p(Ψj → Ψl) = P(Ψl)p(Ψl → Ψj). This leads to
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p(Ψj → Ψl )
p(Ψl → Ψj)
= exp{β Δμ − ωcf ΔNcf − ωcsΔNcs − ωfsΔNfs}
(7)
where ΔNcf = N′cf − Ncf, ΔNcs = N′cs − Ncs, and ΔNfs = N′fs − Nfs
are the changes in the bond numbers.
In what follows, we will confine ourselves to considering only
fluids wetting the pore walls. In addition, we will assume that
the interaction energy of the fluid sites with the walls is larger
than that of the crystal sites. In this way, we model the
occurrence of the interfacial premelting as discussed in the
Introduction. Deviations from these conditions will be touched
upon in a separate subsection. Furthermore, for the sake of
simplicity we will assume that ϕff = ϕcf. Under these conditions,
ωcf = β[(ϕcc − ϕff)/2] = βL/6, where L = 3(ϕcc − ϕff) is the
heat of fusion. Finally, taking into account that ΔNfs = −ΔNcs
and defining ωc ≡ β[−ϕcs + ϕfs + (ϕcc − ϕff)/2], eq 7 for the
ratio between the “creation” (p+) and “annihilation” (p−)
probabilities of a crystal site results as
p+
1
β ΔNcf L − ωcΔNcs
− = exp β Δμ −
6
p
{
}
Figure 1. Schematics showing (a) the xy plane cross-section of a
disordered pore and (b) a three-dimensional perspective of an ideal
pore. (a) corresponds to a temperature just slightly below T0, at which
all sites, except the bulk ones, are in the fluid state. (b) shows a typical
configuration attained at sufficiently low temperatures with the crystal
sites forming a frozen core in the pore interior.
(8)
The difference in the chemical potential Δμ upon converting
a site from the fluid to the crystal state at a given temperature T
is Δμ = L(T0 − T)/T0. By noting that ΔNcf = 2(3 − Nii), where
Nii is the resulting number of neighbors of the same type i after
the site conversion ( f for the crystal−fluid and c for the
fluid−crystal conversion, respectively), the exponent on the
right-hand side of eq 8 can be defined as
⎛N
T⎞
−β ΔG = βL⎜ ii − ⎟ − ωcΔNcs
T0 ⎠
⎝ 3
The system evolution was studied using dynamic Monte
Carlo simulations.51 In our simulations, one Monte Carlo step
consisted of two subsequent runs. In the first run, a fluid site,
randomly chosen from the pool of the pore and interfacial (see
the next paragraph) sites, was attempted to be converted to the
crystal site. The conversion was accepted with the probability
p+ as given by eq 10. This procedure was repeated Nf times. In
the second run, the same procedure was applied Nc times to the
crystal sites attempting to convert them into the fluid ones with
the probability p−. The Monte Carlo steps were repeated t
times, with t denoting the Monte Carlo time of our simulations.
The results presented were typically averaged over 200 pores
(otherwise, it is explicitly indicated in the text).
Throughout the simulations, the bulk sites were kept in
either the crystal state (for temperatures below T0) or the fluid
state (for temperatures above T0). In this way, any uncontrollable homogeneous nucleation delays for freezing in the bulk
phase were intentionally removed. Notably, this closely
resembles the conditions of typical thermoporometry experiments, in which the cooling−heating cycles are performed in
the presence of the excess bulk fluid. In addition, before the
experiments, the bulk phase is first intentionally crystallized at
low temperatures and then the temperature is raised to a
temperature just slightly below than T0.7,8 Finally, all sites in the
interfacial layer between the crystallizer and the porous material
were allowed to change their states (see Figure 1b).
In what follows, the results will be presented in the form of
the relative (equilibrium) fractions f of the pore sites obtained
in the fluid state as a function of temperature. In this way, the
results reported will correspond to the raw-data measured by
NMR cryoporometry.7,8 To provide a closer parallelism to the
experimental situations, each lattice site may be associated with
a molecule, which is either part of the crystal or liquid phases.
Therefore, we fixed the site length lm to 0.5 nm, which is a
typical molecular diameter of most low-molecular-weight
(9)
+
−
Finally, for the calculations, the probabilities p and p can be
taken as
p± =
1
1
exp ∓ β ΔG
2
2
{
}
(10)
to comply with eq 8.
In the present work, we analyzed the freezing and melting
processes of the Kossel crystal confined to straight channel-like
pores. Two different types of channels were considered. In the
first case, referred to as ideal pores, the channel diameter was
kept constant along the channel axis (z axis). To obtain such
channels, all sites satisfying a2 + b2 < r2, where r is the channel
radius and a and b are the site indices in the xy plane (the pore
center is taken at a = b = 0), were assigned to the pore sites,
while all remaining ones formed the solid sites. Note that, for
relatively small pore sizes, the actual shape of the pore
boundary appears to be corrugated. In the second case, referred
to as disordered pores, the channels were composed of short
sections of a length l. Within each section the channel diameter
was held constant (as for ideal pores). The channel diameter,
however, varied from section to section. The radii of two
adjacent pore sections were taken to be completely
uncorrelated and were chosen to follow a Gaussian distribution
with a width w centered around r0. To adopt the model to real
situations, two cut-offs, rmin and rmax, were applied to the
Gaussian distribution. At the channel openings, direct contact
to the sites resembling the bulk phase, sometime referred to as
crystallizer, was provided (see Figure 1b).
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organic liquids. Accordingly, the pore diameters will be
indicated in nanometers. The bulk transition temperature is
arbitrarily chosen to be T0 = 278 K. Two representative values
for the heat of fusion, L = 10 and 20 kJ/mol, will be considered.
These two values describe entropy changes typical for organic
liquids. With these two parameters, L and T0, and for the cubic
crystal structure, the Jackson α-factor (α = Lη1/ZRT0, where R
is the universal gas constant, η1 is the number of bonds in one
layer, and Z is the crystal coordination number)49 results as α =
2.9 and 5.8 for L = 10 and 20 kJ/mol, respectively. Notably, α >
2 holds for situations in which a layer-by-layer growth of the
crystal is preferred over surface roughening. Thus, for α = 2.9
stronger fluctuations of the crystal-fluid are expected than for α
= 5.8. To model the condition of interfacial premelting, ωc in
eq 9 was set to ωc = 3βL.
■
Figure 3. Suppressions of the transition temperatures of freezing
(circles) and melting (triangles) in cylindrical pores open at both ends
as a function of the channel diameter for two values of the heat of
fusion L = 10 (full symbols) and 20 kJ/mol (open symbols).
MELTING AND FREEZING IN IDEAL PORES
Melting-Freezing Hysteresis. Figure 2 shows a typical
result obtained for the most simple case of an ideal channel
the Gibbs−Thomson law, predicting proportionality between
ΔT and the reciprocal channel diameter d−1. Before any
quantitative comparison with eq 1 will be made, we shall first
explore in more detail the mechanisms controlling freezing and
melting. Indeed, as it has been discussed in the Introduction,
the numerical constants in eq 1 are determined not only by the
geometry of the pore space but also by the assumptions made
about the transition mechanisms. It is, therefore, necessary to
establish first the thermodynamic conditions for either of the
transitions.
Equilibrium Transition. Let us first identify which of the
transitions, if there is any, is an equilibrium one and what is the
equilibrium transition temperature in ideal cylindrical channels.
In what follows, this temperature will be referred to as the pore
equilibrium temperature, T0,p. To obtain T0,p, we have
intentionally prepared an arbitrary phase configuration
(typically resembling a large, continuous domain composed
of the crystal sites surrounded by the bath of the sites in the
liquid state) in the channel and followed its evolution at a given
temperature T. At equilibrium, i.e., if T = T0,p, the thusprepared phase configuration should first relax to a
configuration providing the lowest energy state (e.g., by
forming a liquid layer between the frozen core and the pore
walls or/and rounding the crystal shape) and then, at longer
times, fluctuate around this latter configuration. If, however, T
is higher or lower than T0,p, then there will be a long-time
kinetics leading to complete pore melting or freezing,
respectively.
Figure 4 shows the equilibration kinetics obtained with a
system initially containing about 30% of the sites in the crystal
state (forming a continuous crystal domain). For providing the
initial configuration for which the lowest-energy configuration
can be attained most efficiently, it was prepared by letting the
crystal phase grow from the crystallizer at low temperatures.
The process was stopped when the crystal phase in the channel
occupied about 30% of the total channel volume. If the thus
prepared system is allowed to evolve at a certain temperature,
the thickness of the fluid layers between the wall and the frozen
core attains quickly its equilibrium value. Further evolution on a
longer time scale, which we are interested in, is determined by
the attainment of the global minimum in the free energy. As
revealed by the data of Figure 4, there is a well-defined
temperature T0,p at which the phase composition in the pore, as
Figure 2. Fractions of the lattice sites in the fluid state within the
channel with a diameter of d = 6 nm and a length of l = 10 nm as a
function of temperature on the cooling (triangles) and heating
(circles) branches. The full and open symbols refer to the channels
open at both ends and at one end only, respectively. At temperatures
below T0 = 278 K, direct contact to the frozen bulk phase at the
channel openings was provided.
open at both ends. The freezing and melting transitions are
found to be relatively sharp and irreversible. The shape of the
hysteresis loop formed between freezing and melting closely
resembles those found in experiments with nanoporous
materials having cylindrical pores, like SBA-15.13,25 At low
temperatures, below the freezing temperature, f has nonzero
values, associated with the formation of a liquid film between
the channel walls and the frozen core in the channel interior.
The thickness of this film is found to be temperaturedependent. Notably, this finding is in a good agreement with
the literature data providing convincing evidence for the
existence of such layers possessing in high-mobility, liquid-like
states and with thicknesses varying with temperature.11,41,52−54
As revealed by the data of Figure 2, both the freezing and
melting transition temperatures can be identified with high
accuracy. Their suppression with respect to the bulk, ΔT = T0
− T, is shown in Figure 3 as a function of the channel diameter.
The suppression is found to be in qualitative agreement with
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Figure 4. Freezing and melting kinetics in cylindrical pores with d = 6 nm obtained by preparing an initial configuration with 30% of the sites in the
crystal (one domain) and 70% in the liquid states and by tracing the system evolutions at temperatures indicated in the figure (L = 10 kJ/mol in (a)
and L = 20 kJ/mol in (b)). At the equilibrium transition temperature, the magnitudes of f fluctuate around their mean values as shown by the solid
lines.
quantified by f, fluctuates around some mean value, while for T
> T0,p and T < T0,p occurrence of, respectively, complete
melting and complete freezing is observed. Most notably, the
thus-obtained temperatures T0,p coincide, within the limit of the
computational accuracy, with the freezing transition temperatures shown in Figure 3. This finding allows us to conclude
that, if the nucleation barrier is removed, freezing in ideal
cylindrical pores occurs as the equilibrium transition. The
mechanism of the transition is the axial advancement of the
solid−fluid interface and is reminiscent of the desorption
transition from cylindrical channels.
In the same line it may be argued that, whenever there will be
a mechanism for the formation of a solid−fluid interface across
the channel, melting as well will occur at T0,p. The latter
situation can be encountered, e.g., in channels closed at one
end. Because a liquid-like layer is formed at the closed end, it
will facilitate the melting process for temperatures just above
T0,p. As exemplified by the data of Figure 2, for such systems
indeed no hysteresis is observed (note that a small hysteresis in
Figure 2 has purely kinetic origin associated with slow kinetics
of freezing, which will be discussed later, and can be eliminated
by providing sufficiently long simulation times) and the
reversible transition occurs at T0,p. Once again, this is in the
complete analogy to the reversible gas−liquid and liquid−gas
transitions in capped cylindrical channels as predicted by
Cohan.55−57
This finding, namely identifying freezing and melting in ideal
pores as the equilibrium and metastable transitions, respectively, is in line with a part of the theoretical studies made
earlier.42−44 There were many subsequent works devoted to
validations of these predictions using porous materials with
well-defined, cylindrical pore geometries (like MCM-41 or
SBA-15). Some results obtained experimentally have shown
deviations from these theoretical predictions (see, e.g, related
discussions in ref 31). In particular, it was found that the
melting transition for water in MCM-41 and SBA-15 materials
with different pore diameters was reasonably captured by eq 1.
At the same time, freezing exhibited much more scattering,
questioning the applicability of eq 1 to the freezing
transition.13,24 Most importantly, the fit of eq 1 to the melting
data (taking account of one monolayer of the nonfrozen water
and using independently obtained thermodynamical parameters
for water) revealed equilibrium character of this transition. We
would like to note, however, that, in this particular case, one has
to be aware of the fact that water is a substance having a
relatively low Jackson factor α ≈ 2.0 (ref 49, page 302). This
value is very close to the critical one, when the layer-by-layer
crystal growth is not preferred over dendritic one, i.e., the
solid−fluid interface is subject to strong fluctuations. This
scenario is further supported by the finding that, upon freezing
in small pores, ice exhibits a very defective structure deviating
from bulk hexagonal ice.38 As it will be shown in the following
sections, such interface fluctuations tend to shift the melting
temperatures toward the equilibrium ones. Thus, in sufficiently
small pores melting may occur close to the equilibrium
transition temperature and the coincidence of the melting and
equilibrium transition temperatures in refs 13 and 24 may be
due to thermodynamic fluctuations. Although the effect of these
fluctuations is barely understood and is rarely discussed in the
literature, such lowering of the melting temperature, which may
ultimately reach T0,p, has already been pointed out by Petrov
and Furó.8
Freezing Temperature. By resolving the freezing mechanism in ideal pores, the freezing temperature Tf of the Kossel
crystal in cylindrical pores can readily be established. Using eq
9, the condition of equilibrium, ΔG = 0, is given by
⎛ Nii
T⎞ ⎛ω ⎞
− ⎟ − ⎜ c ⎟ΔNcs = 0.
⎜
T0 ⎠ ⎝ βL ⎠
⎝ 3
(11)
Here, the bars denote the ensemble average. For the bulk
Kossel crystal, ΔNcs = 0, hence Nii = 3. Thus, changing the state
of a site having exactly three neighbors in the crystal state does
not change the overall energy of the whole system. In the
literature, this is known as the concept of a repeatable or kink
site. In turn, if Nii is known (for bulk systems Nii = Z/2, where
Z is the total number of bonds), the equilibrium transition
temperature can be found using eq 11.
Nii for the confined Kossel crystals can be evaluated as
follows. If the crystallizer is supplied at the pore openings, the
crystal will grow in the axial direction of the channel at
temperatures just slightly below T0,p. In the radial direction, the
crystal is confined by the channel walls. Because the liquid is
considered to wet the pore walls, a nonfrozen layer will be
found between the wall and the crystal core in the channels.
Hence, one may distinguish between two different types of
repeatable sites facilitating the axial growth of the crystal. For
any site which has no direct contact with the sites belonging to
the nonfrozen layer, Nii,m = 3 like for the bulk system. The
sites in contact with the nonfrozen layer may, however, form on
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Figure 5. Freezing temperature suppression ΔT normalized by the equilibrium transition temperature T0 as a function of the inverse (a) pore and
(b) crystal diameters for L = 10 kJ/mol (full symbols) and L = 20 kJ/mol (opens symbols). The solid lines show ΔTf/T0 resulting from eq 13 with lm
= 0.5 nm.
Figure 6. Melting temperature suppression ΔT normalized by the equilibrium transition temperature T0 as a function of the inverse (a) pore and (b)
crystal diameters for L = 10 kJ/mol (full symbols) and L = 20 kJ/mol (opens symbols). The dashed and solid lines show ΔTm/T0 resulting from eqs
13 and 14 with lm = 0.5 nm, respectively.
average only Z = 5 bonds with sites in the crystal state. Hence,
for these sites Nii,p = 5/2. The mean Nii can, thus, be found as
Nii = pm Nii ,m + pp Nii ,p
crystal−liquid interface allowing axial shrinking of the crystal at
temperatures above T0,p. Hence, melting starts only at
temperatures at which the formation of the liquid domains
forced by fluctuations becomes possible. Fluctuations play an
essential role in this process and become increasingly important
with decreasing pore diameters. This is, in particular, revealed
by the data for the melting temperature suppressions obtained
in simulations which are shown in Figure 6. It is not
straightforward, however, to describe the effect of fluctuations
quantitatively. For the purpose of this work, we estimate
therefore only the highest melting temperature at which there
will be no barriers in the free energy for the process of radial
shrinking. Formally, this process may be considered to occur at
equilibrium for the outermost layers of the crystal. In contrast
to the flat interfaces analyzed in the preceding section, here we
deal, however, with convex cylindrical interfaces.
With the last remark, the melting temperature, which in what
follows will be denoted as T0,m, can be established using eq 11.
Nii can most easily be calculated by considering wide channels
allowing to neglect the finite size of the crystal lattice. Let us
consider the outermost layer of the frozen core in a channel. In
one cross-section, it contains approximately N = πdc/lm sites.
The layer adjacent to it contains approximately N′ = π(dc −
2lm)/lm sites. Hence, N − N′ sites in the outermost layer will
form only 5/2 bonds, while the rest will form 3 bonds. Using eq
12 with all relevant quantities, Nii results as (5 + N′/N)/2.
Finally, the melting temperature suppression is readily found to
be two times smaller than ΔTf:
(12)
where pm and pp are the relative fractions of the two types of
the repeatable sites. For cylindrical symmetry, pm = 1−4/Nd
and pp = 4/Nd, where Nd is the number of sites along the
diametric line of the crystal. Finally, Nii results as Nii = 3−2/Nd.
By noting that Nd = dc/lm, where dc is the crystal diameter and
lm is the site dimension, eq 11 with ΔNcs = 0 (because of a very
low probability to have the crystal-solid bonds) yields
ΔTf = T0 − T0, p =
2 lm
T0
3 dc
(13)
Figure 5 shows the data of Figure 3 plotted versus the inverse
pore and crystal diameters. In the latter case, the mean crystal
diameters for temperatures in the vicinity of Tf were obtained
by direct analysis of the simulation data. The data of Figure 5b
are found to be in perfect agreement with the predictions of eq
13. This confirms that the classical Gibbs−Thomson law
corrected for the thickness of the nonfrozen layers, as given by
eq 1, can accurately describe the freezing transition in ideal
pores. Notably, this type of correction is a common practice for
the determination of actual pore sizes.13,24,27,54
Melting Temperature. As shown in Figure 2, for ideal
channels with two open ends, at which the crystallizer is
supplied, there is a significant difference between the freezing
and melting temperatures. Melting in this case occurs at
substantially higher temperatures due to the lack of the (radial)
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ΔTm = T0 − T0,m =
1 lm
T0
3 dc
having a crystal site in contact with a solid one. Because ωc
uniquely combines all interactions parameters, this allows to
compile a global phase diagram as a function of ωc. This
diagram may further be used to analyze the effect of the
variation of different interaction terms. It is, however, more
convenient to introduce a new parameter b = (ϕcs − ϕfs)/(ϕcc
− ϕfc), so that ωc = ((1/2) − b)((βL)/3). In particular, all
previously considered situations (ωc = 3βL) corresponded to b
= −17/2. To build the phase diagram, we have performed
dynamic Monte Carlo simulations for different values of ωc in
eq 9 and studied how the freezing and melting temperatures are
altered by its variation. A typical result is shown in Figure 7. It
demonstrates the shifts ΔTf and ΔTm of the transition
temperatures as a function of b as obtained for the channels
with d = 6 nm.
(14)
By comparing the prediction of eq 14 with the simulation
data in Figure 6b, one finds reasonable agreement for large
pores, but notable deviations for smaller ones. In particular, the
most dramatic deviation is found for narrow channels and the
crystal with the lower heat of fusion, L = 10 kJ/mol.
Presumably, this is caused by the increasing role of fluctuations
with decreasing crystal diameter and decreasing L. Indeed,
fluctuations of the crystal-fluid interface with amplitudes
comparable to the pore diameter lead to the formation of
liquid bridges triggering melting by axial shrinking of the crystal
(recall that the equilibrium transition temperature T0,p is
notably lower than T0,m). Therefore, for one and the same
temperature, fluctuations will be more effective for smaller
pores. At the same time, according to eq 9, the energy penalty
for fluctuations leading to increase of the crystal−liquid
interfacial area is directly proportional to L. Thus, for crystals
with lower bond strength stronger deviations from eq 14 can be
expected.
At this point, it is important to make a remark about water,
which is often used as a test liquid for freezing and melting
experiments in porous materials. This choice is partially
determined by the fact that all thermodynamical parameters
for bulk water are known independently and with high
accuracies. However, as it has already been mentioned before,
water has a sufficiently low Jackson factor α ≈ 2.0. For the
Kossel crystal with T0 = 278.0 K, as considered in the preceding
sections, this corresponds to the heat of fusion L = 7 kJ/mol.
This reveals that the effect of thermodynamic fluctuations upon
melting will be even stronger than for the case of L = 10 kJ/mol
as exemplified in Figure 6b.
Transition Kinetics. It is interesting to note that Figure 4
indicates that, for small, identical deviations ± (T − T0,p), the
kinetics of (equilibrium) freezing transitions are slower than the
kinetics of (equilibrium) melting transitions. Remarkably, this
effect becomes stronger with decreasing L, although the
location of the equilibrium transition temperature T0,p is not
affected by L. This observation may be explained by referring to
the fact that, for the values of L and T0 used in our simulations
resulting in Jackson α-factors larger than 2, the addition/
removal of one layer to/from the crystal surface requires
overcoming of nucleation barriers. The latter are higher for
systems with larger bond energies. In full agreement, both the
freezing and melting rates are found to be notably slower in
Figure 4b for L = 20 kJ/mol than in Figure 4a for L = 10 kJ/
mol. To explain the difference between the freezing and melting
kinetics for identical L, one has to take into account the finite
length of the systems considered and the occurrence of a liquid
layer in contact with the crystal. Thus, the formation of a new
layer (in the cross-sectional plane of the channel) will require
nucleation of a critical cluster on the crystal facet, while the
removal can be facilitated by the already existing liquid sites. In
the case of weak bonds (low L), a rougher surface of the crystal
will help to overcome the nucleation barriers, making the
difference between the freezing and melting rates more modest.
This line of reasoning is also supported by the simulation data.
Effect of the Pore Wall Interaction. So far, only liquids
wetting the pore walls and having much better affinity to the
pore walls as compared to their crystalline phases have been
considered. This was intentionally provided by setting ωc in eq
9 to 3βL, i.e., by setting a sufficiently high energy penalty for
Figure 7. Shift of the freezing (filled squares) and melting (filled
circles) temperatures T0 − T as a function of the ratio b = (ϕcs − ϕfs)/
(ϕcc − ϕfc) in the channels with d = 6 nm. The simulations were
performed with L = 20 kJ/mol. For T < T0 (upper half of the figure)
the bulk phase at the pore openings was kept in the frozen state, for T
> T0 (lower half of the figure) in the liquid state.
For b < 0 (ϕcs < ϕfs), i.e., for liquids having better affinities to
the pore walls as compared to their solid phases (the
occurrence of the interfacial premelting), ΔTf and ΔTm are
found to be independent of b. This behavior emerges due to
the short-range character of the interactions in our model,
where only the first-neighbor interactions between adjacent
sites are considered. For these interaction strengths and for
temperatures not significantly lower than the transition
temperature, one layer of the sites located between the pore
walls and the frozen core always remains in the liquid state. The
transition temperatures are, thus, determined by eqs 13 and 14
with the confinement size corrected for the thickness of the
liquid layer.
For 0 < b < 1, i.e., for interaction strengths ϕfs < ϕcs < ϕcc
resembling better affinity of the crystal phase to the pore walls
than the fluid one, ΔTf is found to decrease with increasing b.
In this regime, the last term on the right hand side of eq 9 (or
eq 11) cannot be neglected (ΔNcs ≠ 0). Hence, ΔTf is altered
due to lowering of the excess surface energy of the confined
crystal. In particular, for b = 1 the excess surface energy
becomes zero and, therefore, the transition occurs at T0 (ΔTf =
0). The melting transition is found to be dramatically affected
by the occurrence of the favorable crystal-solid bonds, i.e., by
the diminishment of the liquid layers. Indeed, as it has been
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Figure 8. (a) Freezing and melting transitions in disordered channels with total lengths of 100 nm (open circles) and 750 nm (filled triangles) and
with a mean channel diameter of 8 nm. The simulations were performed with L = 10 kJ/mol. For comparison, the filled squares show the results
obtained with L = 20 kJ/mol and for a channel length of 750 nm. In both cases, the formation of liquid layers was ensured (b = −17/2). The results
shown for the short pores were averaged over 70 different channel configurations, for the long pores over 10 configurations. The dashed line
approaches the change of the volume fraction occupied by the nonfrozen surface layer. (b) Freezing and melting data obtained with L = 20 kJ/mol
and for a channel length of 750 nm replotted from the figure (a). The solid and dashed vertical lines show, respectively, the melting and freezing
transition temperatures obtained in an ideal channel with the pore diameter d = 8 nm and with L = 20 kJ/mol.
nm were used. The length l of the individual sections was 2.5
nm (we have proven that choosing longer section lengths
resulted in similar results, but required sufficiently longer
simulation times). Notably, any particular detail of the disorder
(as modeled here by assuming missing correlation between the
diameters of neighboring pores and by using relatively short
section lengths), have only quantitative, but not qualitative,
consequences.
Figure 8 shows the results obtained for the channels with
disorder. First of all, we note the occurrence of relatively wide
hysteresis loops formed by the freezing and melting branches.
The freezing transition is found to strongly depend on the
channel length, while the melting one shows no length
dependence. Finally, the transition enthalpy has a minor, but
still observable, effect on both freezing and melting.
As revealed by the data of Figure 8b, the hysteresis widths
obtained for disordered channels is found to notably exceed
that obtained in ideal pores with similar pore diameters. This
observation may be rationalized by recalling the equilibrium
character of the freezing transition occurring via the invasion of
the crystal from the pore openings. Because Tf is determined by
the channel diameter, the growth of the invading crystal phase
will be controlled by the channel sections with the smallest
diameters (the pore-blocking effect). Notably, this scenario also
naturally emerges in the macroscopic theories assuming that
freezing occurs at equilibrium.42−44 Due to the channel-like
pore geometry, the narrowest channel sections located
(statistically) close to the pore openings may effectively
postpone freezing in the whole channel to substantially low
temperatures. Indeed, the onset of freezing is found at
temperatures corresponding to freezing in ideal channels with
d = 4 nm. The longer the channels are, the higher fractions of
the sites will remain in the liquid state until these temperatures.
For the shorter pores with the total lengths comparable to an
average separation between the narrowest pores, pore blocking
will not be the dominant mechanism and the temperature
dependence of the freezing transition will be determined by the
pore size distribution. The difference between these two
situations is nicely illustrated in Figure 8a.
In contrast to freezing, melting in disordered channels may
involve two distinct mechanisms. If a crystal in a pore section of
diameter d will melt via nucleation of a liquid bridge at the
temperature T ≈ T0,m(d) as given by eq 14, it may trigger axial
discussed earlier, these layers facilitate nucleation of the liquid
bridges. Only for relatively small values of b, the composition
fluctuations in the layers adjacent to the pore walls may trigger
the melting process at temperatures T < T0. For larger b,
nucleation of the liquid bridges is strongly suppressed and
melting is, therefore, postponed to the bulk transition
temperature T0. In this case, the liquid phase supplied at the
pore openings invades into the channels.
The situations with b > 1 (ϕcs > ϕcc), i.e., when the pore
walls may facilitate heterogeneous nucleation of the crystal
phase, can be discussed in the same way by noting the
symmetry between freezing and melting. They are found to
coincide upon two subsequent reflections over the lines b = 1
and ΔT = 0. In this case, the above discussion is identically
applicable upon interchanging the crystal and fluid sites. It is
important to mention that, in our model, the crystalline
structures of the pore walls and of the crystal growing in the
pore space are identical because they both are determined by
the lattice. The mismatch between the crystalline structures in
real system may lead to more complex behavior, which is not
directly captured by our model.58
■
MELTING AND FREEZING TRANSITIONS IN
DISORDERED PORES
One-Dimensional Channels with Disorder. Disorder in
the pore structure may strongly affect the freezing and melting
processes. This is caused by the temperature-dependent
character of these transitions in confined spaces and by the
interplay of two different transition mechanisms contributing to
the melting transition. To address the effect of geometric
disorder, we made use of a most simple model of disordered
porous materials, considering it to be composed of channel-like
pores with the variation of the channel diameters along their
axes. It turns out that already this model shows all essential
features for rationalizing the complex transition phenomena in
materials with three-dimensional pore spaces.
In more detail, we composed long channels by joining short
sections, represented by ideal channels of length l. Diameters of
neighboring channel sections were uncorrelated, following a
discretized Gaussian distribution. In what follows, we present
the results for the channels with an average pore diameter d = 8
nm and a distribution width σ = 2 nm. Cut-offs at d = 4 and 14
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melting in the neighboring sections for which T0,p ≤ T0,m(d).
This phenomenon, referred to as advanced melting, has already
been discussed qualitatively in the literature and has been
shown to impact the accuracy of the pore size determination.59
Within our model, this effect may be considered on a
quantitative basis. With eqs 13 and 14, the condition of the
advanced melting means that melting in any channel section of
diameter d = dc + τ (where τ is the thickness of the liquid layer)
will be accompanied by successive melting in the neighboring
sections with diameters up to 2dc + τ. This condition is
visualized in Figure 9. Thus, at relatively low temperatures,
Notably, eqs 13 and 14 predict that, in ideal pores, the
transition temperatures should not depend on L. These
equations, however, do not include the effect of the liquidcrystal interface fluctuations, which increase with decreasing
L.49 This has already earlier been demonstrated for the melting
transition in ideal pores. The same phenomenon can also
contribute to a more effective freezing in disordered channels.
Indeed, fluctuations may facilitate the propagation of the solid−
liquid interface through a narrow channel section. This may
clearly occur if the section lengths are sufficiently short. Thus,
for fluids with higher L both transition temperatures in
disordered channels may be expected to be shifted toward
higher temperatures. This prediction is found to be in nice
agreement with the simulation results shown in Figure 3.
Structural Information Accessible. Measurements of
solid−liquid transitions are often used as a means for structural
characterization of nanoporous solids. It is therefore instructive
to discuss in more detail which type of structural information
can be obtained for disordered porous materials based on the
transition mechanisms established. Let us first consider the
melting transition, which is most frequently used to assess the
pore size distributions. With the relative fraction f of the liquid
phase in the sample measured upon heating, the crystal size
distribution (relative number of the channel sections Nc with
crystal diameter x) can be found as
dNc
1 d(f − fm )
1 d(f − fm ) dT
∝ 2
= 2
dx
dx
dT
dx
x
x
(15)
where f m is the relative volume fraction of the liquid layer. The
thus obtained distribution can be considered to approach the
pore size distribution with d = x + τ, where τ is the thickness of
the liquid layers. The derivative dT/dx in eq 15 is readily
obtained from eqs 13 or 14, resulting in
Figure 9. Diagram showing the interplay between two different
melting mechanisms in disordered pores. The lines are the equilibrium
and metastable melting transition temperatures versus inverse crystal
size dc. Melting of the frozen liquid in the channels with the diameters
up to d = dc + τ by nucleating liquid bridges will give rise to the
formation of continuous domains with the molten liquid due to axial
melting of the adjacent sections with the channel diameters up to d =
2dc + τ.
dNc
1 d(f − fm )
∝ 4
dx
dT
x
(16)
At relatively low temperatures the melting process is
controlled by radial melting of the narrowest channels.
Equation 16 combined with eq 14 can therefore be used to
assess the pore sizes corresponding to smallest ones in the real
pore size distribution. Although eq 16 can correctly predict the
range of the pore sizes, as it is demonstrated in Figure 10a, the
information about their relative fractions will be corrupted by
the simultaneous occurrence of axial melting in the neighboring
pores.
melting will predominantly occur as a concerted action of these
two mechanisms. At higher temperatures, equilibrium melting
of large pores will dominate the overall process due to the
occurrence of a large number of the liquid domains. It is
evident that, in this scenario, the channel length does not play
any decisive role.
Finally, Figure 8 demonstrates the effect of the transition
enthalpy L on both melting and freezing in disordered pores.
Figure 10. Pore size distributions derived from the melting (a) and the freezing (b) transitions shown in Figure 8 for L = 10 kJ/mol using eq 16. The
solid line is the real pore size distribution used in the simulations. The filled circles and triangles in (b) represent the data obtained for the long and
short channels, respectively. The fractions f m in eq 16 were approached by the dashed line shown in Figure 8
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As one of the significant results of this work, we found that
the Jackson α-factor, a parameter quantifying the entropy
change during the transition, plays an important role for the
melting and freezing transitions in confined spaces. Thus, for
materials with large entropy changes, the Gibbs−Thomson law,
corrected for the occurrence of nonfrozen layers adjacent to the
pore walls, is found to be in a satisfactory agreement with the
simulation data. For materials with low α-factors, however, the
fluctuations are shown to lead to notable deviations from this
law. In particular, it is found that for the melting transition
these deviations are notably stronger than for freezing. This
finding suggests that the data analysis of the freezing and
melting experiments in porous solids should be performed
taking account of the Jackson factor. This might especially be
important for water having a sufficiently small α-factor of about
2.49
By varying the binding energies, we explored how the
interaction between the crystal, the liquid, and the pore wall
affects the transition temperature. For pore walls favoring liquid
over crystal sites, the model exhibited the emergence of a liquid
layer minimizing the overall energy of the system, resembling
the phenomenon of interfacial premelting.11,41,52−54 Its
occurrence is shown to lead to the correction of the Gibbs−
Thomson equation for the thickness of this layer. Notably, this
correction is widely used for the pore size determination.13,24,27,54 Under these conditions, we found that the
transition temperatures do not depend on the interaction
strength. The same is found to be valid in the opposite case,
when the crystal-solid bonds lower the total energy, giving rise
to an epitaxial growth of the crystal on the pore walls. In this
case, both melting and freezing appeared to occur at
temperatures higher than the bulk transition temperature, but
irrespective of the substrate-crystal binding energies. For
intermediate interaction energies, we find a transition between
these two limits.
Transition mechanisms established using ordered pores
allowed for a more robust analysis of the phase transformations
occurring in disordered materials. We showed that structural
disorder renders freezing to occur under strong pore blocking
control; that is, crystal growth is restrained by the smallest
pores in the porous material. This finding is as well found to be
in agreement with theoretical predictions considering freezing
as equilibrium transitions. Exactly these smallest pores were
found to determine the onset of the melting process because of
smaller nucleation barriers in these pores. Importantly, the thus
formed domains of the liquid phase permitted advanced
melting in the neighboring pores, giving rise to a concerted
action of two different melting mechanisms. With the
increasing number of liquid domains, melting was found to
be controlled by the largest pores, postponing melting to
temperatures of their equilibrium transitions. Thus, depending
on the particular geometry of the pore structure and the pore
sizes in disordered materials or ordered materials with defects,
the melting transition may exhibit features of metastable, of
equilibrium transition, or of their mixture. To assess the
underlying mechanisms in these situations, more elaborate
experiments probing scanning freezing and freezing behavior
are required.37,39 What type of information is delivered by this
type of experiments may be efficiently educed by using our
model. Such studies are currently under progress.
In the same spirit, the range of the largest pore sizes can be
obtained by the combined use of eqs 16 and 13. This procedure
applied to the melting data of Figure 8 as well yields the correct
range of the largest pore sizes (see Figure 10a). Notably, this
latter procedure is only valid for materials with relatively broad
PSDs, i.e., having heavy tails with d > 2dmin, where dmin
represents the typical diameter of the smallest pore size.
Otherwise, the frozen liquid in materials with narrow pore size
distributions can melt in one step, simultaneously with the
onset of radial melting.
The freezing transition also contains important structural
information. Because freezing under the condition of crystal
phase growth starting from the pore openings is controlled by
the pore blocking mechanism, eqs 16 and 13 can be applied to
the freezing data to obtain the neck size distribution. This is
especially valid for long channels, resembling monolithic porous
materials. With decreasing the channel length, more larger
pores become accessible by this procedure. These features are
nicely illustrated in Figure 10b.
■
CONCLUSIONS
In the present work, we have studied the freezing and melting
phenomena in porous materials using the Kossel−Stranski
crystal growth model. According to this model, molecules in
bulk systems arrive at the crystal surface at a certain, specified
rate and leave it at a rate depending on the number of adjacent
crystal sites. We have modified the model by including
additional sites representing the porous material as a solid
matrix. The binding energies between two crystal sites and
between the crystal and sold sites have been considered to be
different. Thus, the crystal growth could occur only in the pore
spaces and the detachment rate from the crystal surface was
determined by not only the adjacent crystal sites but also the
solid ones. With only this modification, a number of
phenomena inherent to solid−liquid equilibria and phase
transitions in confined spaces have naturally emerged. The
flexibility of the model in terms of the possibility to vary
different interactions and the pore space geometry permitted us
to establish some generic features in the freezing and melting
behavior. Their deeper understanding, obtained on this
microscopic route, may serve for a more proficient analysis of
the experimental results obtained for porous materials with
complex pore structures. This may not always be accessible in
the frame of macroscopic thermodynamics requiring certain a
priori assumptions.
By addressing foremost ideal, channel-like pores, we
examined the status of the Gibbs−Thomson law for the Kossel
crystal by deriving its analogues for two different transitions.
The first one was the freezing transition for which the crystal
nucleation barriers were intentionally removed by providing the
crystalline phase at the pore openings, as often done in
thermoporometry experiments. In this case, we show
unequivocally that the freezing transition occurs at equilibrium,
i.e., without change in the overall free energy. This finding is in
agreement with some theoretical models put forward earlier.42−44 The second transition concerned melting occurring by
nucleation of liquid bridges within the channels, which is
intrinsically metastable. These two transitions were considered
because they are most frequently encountered in the experiments. By comparing the analytical results with the data
obtained using Monte Carlo simulations, we identified
conditions for the applicability of the Gibbs−Thomson law.
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■
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
The authors thank DFG (the German Science Foundation), in
particular in the frame of the research project FOR-877 “From
local constraints to macroscopic motion”, for financial support.
■
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