Phase transitions of fluids in heterogeneous pores
We study phase behaviour of a model fluid confined between two unlike parallel walls in the presence of long range (dispersion) forces. Predictions obtained from macroscopic (geometric) and mesoscopic arguments are compared with numerical solutions of a non-local density functional theory. Two capil...
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| Cite this: | Phase transitions of fluids in heterogeneous pores / A. Malijevský // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13604: 1–18. — Бібліогр.: 27 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1557962025-02-23T18:58:01Z Phase transitions of fluids in heterogeneous pores Фазовi переходи плинiв в гетерогенних порах Malijevský, A. We study phase behaviour of a model fluid confined between two unlike parallel walls in the presence of long range (dispersion) forces. Predictions obtained from macroscopic (geometric) and mesoscopic arguments are compared with numerical solutions of a non-local density functional theory. Two capillary models are considered. For a capillary comprising two (differently) adsorbing walls we show that simple geometric arguments lead to the generalized Kelvin equation locating very accurately capillary condensation, provided both walls are only partially wet. If at least one of the walls is in complete wetting regime, the Kelvin equation should be modified by capturing the effect of thick wetting films by including Derjaguin's correction. Within the second model, we consider a capillary formed of two competing walls, so that one tends to be wet and the other dry. In this case, an interface localized-delocalized transition occurs at bulk two-phase coexistence and a temperature T*(L) depending on the pore width L. A mean-field analysis shows that for walls exhibiting first-order wetting transition at a temperature Tw, Ts > T*(L) > Tw, where the spinodal temperature Ts can be associated with the prewetting critical temperature, which also determines a critical pore width below which the interface localized-delocalized transition does not occur. If the walls exhibit critical wetting, the transition is shifted below Tw and for a model with the binding potential W(l)=A(T)l⁻²+B(T)l⁻³+..., where l is the location of the liquid-gas interface, the transition can be characterized by a dimensionless parameter κ=B/(AL), so that the fluid configuration with delocalized interface is stable in the interval between κ=-2/3 and κ ~ -0.23. Ми вивчаємо фазову поведiнку модельного плину, обмеженого двома рiзними паралельними стiнками у присутностi далекосяжних (дисперсiйних) сил. Передбачення, отриманi на основi макроскопiчних (геометричних) i мезоскопiчних аргументiв, порiвнюються з числовими розв’язками теорiї функцiоналу нелокальної густини. Розглянуто двi капiлярнi моделi. У випадку капiлярної моделi, що має двi (по-рiзному) адсорбуючi поверхнi, простi геометричнi аргументи приводять до узагальненого рiвняння Кельвiна, яке дуже точно локалiзує капiлярну конденсацiю, за умови, якщо обидвi стiнки лише частково змочуються. Якщо принаймнi одна зi стiнок знаходиться у режимi повного змочування, рiвняння Кельвiна слiд видозмiнити за рахунок ефекту товщини змочуючих плiвок, включивши поправку Дєрягiна. У другiй моделi розглянуто капiляр, утворений з двох конкуруючих стiнок: одна з них схильна до змочування, а друга — не змочується. У цьому випадку на границi роздiлу вiдбувається локалiзований-делокалiзований перехiд при двофазному спiвiснуваннi в об’ємi i при температурi T ∗(L), яка залежить вiд ширини пори L. Аналiз, виконаний за допомогою теорiї середнього поля, показує, що для стiнок, якi проявляють перехiд змочування першого роду при температурi Tw, Ts > T ∗(L) > Tw, де температура спiнодалi Ts може бути зв’язана з критичною температурою попереднього змочування, яка також визначає критичну ширину пори, нижче якої локалiзований-делокалiзований перехiд на границi роздiлу не вiдбувається. Якщо ж стiнки проявляють критичне змочування, тодi перехiд змiщується нижче за Tw, а для моделi з потенцiалом W (`) = A(T )` ⁻² + B(T )` ⁻³ + ···, де ` — мiсцезнаходження границi роздiлу мiж рiдиною i газом, перехiд можна характеризувати за допомогою безрозмiрного параметра κ = B/(AL), в результатi чого конфiгурацiя плину з делокалiзованою границею роздiлу є стабiльною в iнтервалi мiж κ = −2/3 i κ ≈ −0.23. The financial support from the Czech Science Foundation, project number 13-09914S is acknowledged. 2016 Article Phase transitions of fluids in heterogeneous pores / A. Malijevský // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13604: 1–18. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 68.08.Bc, 05.70.Np, 05.70.Fh DOI:10.5488/CMP.19.13604 arXiv:1512.08957 https://nasplib.isofts.kiev.ua/handle/123456789/155796 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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English |
| description |
We study phase behaviour of a model fluid confined between two unlike parallel walls in the presence of long range (dispersion) forces. Predictions obtained from macroscopic (geometric) and mesoscopic arguments are compared with numerical solutions of a non-local density functional theory. Two capillary models are considered. For a capillary comprising two (differently) adsorbing walls we show that simple geometric arguments lead to the generalized Kelvin equation locating very accurately capillary condensation, provided both walls are only partially wet. If at least one of the walls is in complete wetting regime, the Kelvin equation should be modified by capturing the effect of thick wetting films by including Derjaguin's correction. Within the second model, we consider a capillary formed of two competing walls, so that one tends to be wet and the other dry. In this case, an interface localized-delocalized transition occurs at bulk two-phase coexistence and a temperature T*(L) depending on the pore width L. A mean-field analysis shows that for walls exhibiting first-order wetting transition at a temperature Tw, Ts > T*(L) > Tw, where the spinodal temperature Ts can be associated with the prewetting critical temperature, which also determines a critical pore width below which the interface localized-delocalized transition does not occur. If the walls exhibit critical wetting, the transition is shifted below Tw and for a model with the binding potential W(l)=A(T)l⁻²+B(T)l⁻³+..., where l is the location of the liquid-gas interface, the transition can be characterized by a dimensionless parameter κ=B/(AL), so that the fluid configuration with delocalized interface is stable in the interval between κ=-2/3 and κ ~ -0.23. |
| format |
Article |
| author |
Malijevský, A. |
| spellingShingle |
Malijevský, A. Phase transitions of fluids in heterogeneous pores Condensed Matter Physics |
| author_facet |
Malijevský, A. |
| author_sort |
Malijevský, A. |
| title |
Phase transitions of fluids in heterogeneous pores |
| title_short |
Phase transitions of fluids in heterogeneous pores |
| title_full |
Phase transitions of fluids in heterogeneous pores |
| title_fullStr |
Phase transitions of fluids in heterogeneous pores |
| title_full_unstemmed |
Phase transitions of fluids in heterogeneous pores |
| title_sort |
phase transitions of fluids in heterogeneous pores |
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Інститут фізики конденсованих систем НАН України |
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2016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/155796 |
| citation_txt |
Phase transitions of fluids in heterogeneous pores / A. Malijevský // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13604: 1–18. — Бібліогр.: 27 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT malijevskya phasetransitionsoffluidsinheterogeneouspores AT malijevskya fazoviperehodiplinivvgeterogennihporah |
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2025-11-24T12:55:36Z |
| last_indexed |
2025-11-24T12:55:36Z |
| _version_ |
1849676462879145984 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 1, 13604: 1–18
DOI: 10.5488/CMP.19.13604
http://www.icmp.lviv.ua/journal
Phase transitions of fluids in heterogeneous pores
A. Malijevský1,2
1 Department of Physical Chemistry, University of Chemistry and Technology Prague,
166 28 Praha 6, Czech Republic
2 Laboratory of Aerosols Chemistry and Physics, Institute of Chemical Process Fundamentals,
Academy of Sciences, 16502 Prague 6, Czech Republic
Received November 13, 2015, in final form December 22, 2015
We study phase behaviour of a model fluid confined between two unlike parallel walls in the presence of long
range (dispersion) forces. Predictions obtained from macroscopic (geometric) and mesoscopic arguments are
compared with numerical solutions of a non-local density functional theory. Two capillary models are consid-
ered. For a capillary comprising of two (differently) adsorbing walls we show that simple geometric arguments
lead to the generalized Kelvin equation locating capillary condensation very accurately, provided both walls
are only partially wet. If at least one of the walls is in complete wetting regime, the Kelvin equation should be
modified by capturing the effect of thick wetting films by including Derjaguin’s correction. Within the second
model, we consider a capillary formed of two competing walls, so that one tends to be wet and the other dry. In
this case, an interface localized-delocalized transition occurs at bulk two-phase coexistence and a temperature
T∗(L) depending on the pore width L. A mean-field analysis shows that for walls exhibiting first-order wetting
transition at a temperature Tw, Ts > T∗(L) > Tw, where the spinodal temperature Ts can be associated withthe prewetting critical point, which also determines a critical pore width below which the interface localized-
delocalized transition does not occur. If the walls exhibit critical wetting, the transition is shifted below Tw andfor a model with the binding potentialW (`) = A(T )`−2 +B(T )`−3 +·· · , where ` is the location of the liquid-
gas interface, the transition can be characterized by a dimensionless parameter κ = B/(AL), so that the fluid
configuration with delocalized interface is stable in the interval between κ=−2/3 and κ≈−0.23.
Key words: capillary condensation, wetting, Kelvin equation, adsorption, density functional theory,
fundamental measure theory
PACS: 68.08.Bc, 05.70.Np, 05.70.Fh
1. Introduction
It is very well known that structure and phase behaviour of a confined fluid is quite distinct from that
of its bulk counterpart. A familiar example of a confining geometry is a slit pore formed by two parallel,
identical and infinite plates, a distance L apart. The combination of finite-size effects and fluid adsorption
at the walls leads to a shift in the liquid-vapour phase boundary and in the critical point compared to a
bulk fluid [1–5]. The location of this capillary condensation (or evaporation) transition is macroscopically
described by Kelvin’s equation which predicts that the chemical potential at which the vapour in the slit
condenses into the liquid-like phase is shifted from its saturation value µsat by an amount
δµ≡µsat−µcc = 2γcosθ
(ρl−ρv)L
, (1.1)
where ρl and ρg are the coexisting bulk liquid and gas densities, respectively, γ is the liquid-gas surface
tension and θ is the contact angle of a macrosocpic liquid droplet sitting on isolated wall. The role of
wetting layers adsorbed at the walls when θ = 0, i.e., for T > Tw, where Tw is the wetting temperature
corresponding to a semi-infinite system L →∞, has also been appreciated but this effect is mostly quan-
titative, such that the presence of the layers effectively reduces the pore width [6]. If the isolated walls
© A. Malijevský, 2016 13604-1
http://dx.doi.org/10.5488/CMP.19.13604
http://www.icmp.lviv.ua/journal
A. Malijevský
exhibit first-order wetting transition at T = Tw (and µ=µsat), a prewetting transition corresponding to a
finite jump in the wetting layers thickness can also occur, although the transition is typically metastable
with respect to the capillary condensation unless the pore width is fairly large [7].
The scenario of a liquid-vapour coexistence can, however, be very different in pores made of unlike
walls. This was demonstrated by Parry and Evans [8, 9] who proposed a model, treated within the Lan-
dau theory in the language of magnets, with perfectly antisymmetric surface fields of the walls. They
have shown that, for sufficiently large pores, the capillary condensation transition is replaced by the
interface localization-delocalization phase transition which occurs at a two-phase bulk coexistence at a
temperature T ∗(L) near the wetting temperature Tw of the wall with the affinity to “+” phase (which,
owing to the symmetry of the system, is identical to the wetting temperature of the opposing wall which
tends to be wetted by “−” phase). The transition separates a regime, present for T < T ∗(L), when the
equilibrium density profile corresponds to a very thick “+” or to a very thick “−” phase with a “+−” inter-
face pinned to either of the walls, from the high temperature regime at which the interface is unbounded
from either of the walls. In the latter case, sometimes referred to as soft-mode phase, the interface finds a
compromise between the antagonistic wetting preferences of the walls, such that it develops around the
midpoint of the pore being a subject of large fluctuations causing the interface to wander along the pore.
Compared with the former model of pores with identical walls, the case of antisymmetric walls makes
much closer link with wetting properties of the walls. In the theory of wetting phenomena, the concept
of a binding potential proved to be very useful (at least on a mean-field level) and can be used to describe
the phase behaviour of the fluid confined between antisymmetric walls. If the walls, when isolated, ex-
hibit critical wetting as assumed in references [8, 9], the binding potential at each wall acquires a single
minimum whose location shifts continuously from the wall to infinity as the temperature increases to-
wards Tw along the phase coexistence line [µ=µsat(T )]. It means that, when the pore is sufficiently wide,
the binding potential has two minima that are of the same depth and are located symmetrically around
the midpoint of the pore. There are thus two equally stable solutions for the density profile correspond-
ing to large and small adsorptions. The adsorption difference, or equivalently the distance between the
two minima, decreases with an increasing temperature and ultimately disappears at T ∗(L), which is a
finite-size shift of Tw. Above T ∗(L), the binding potential landscape adopts a U -shape with a very shal-
low single minimumwhich enables the interface to drift around the centre with a very small free energy
cost.
In contrast to pores with identical walls, the nature of wetting transition at the (isolated) walls be-
comes much more important when the walls are antisymmetric. If they exhibit first-order wetting, the
characteristic feature of the binding potential is a competition between a minimum at a finite distance
from the wall with the unbounded state corresponding to a minimum at infinity. Assuming the pore is
sufficiently wide, the binding potential now possesses three local minima, such that two minima near the
walls are the global minima for temperatures below the finite-size shifted wetting temperature T ∗(L),
whilst the middle minimum becomes a global minimum for T > T ∗(L). The nature of the transition (at
fixed L and T varying) thus reflects the nature of wetting at the walls: while the transition is continuous
for a critical wetting, in which case two potential minima continuously merge in the middle of the pore, it
becomes discontinuous for first-order wetting due to a jump in the location of the global minimum. Con-
sequently, T ∗(L) is a critical point for critical wetting, whereas it is a triple point for first-order wetting.
However, all these conclusions are only valid for sufficiently large pores. By decreasing the pore width,
the space to accommodate all three minima of the binding potential is reduced and when the middle
minimum disappears, the order of the transition becomes second order [10, 11].
These predictions have been verified by extensive Monte Carlo simulations by Binder et al. for Ising-
like models [12–17]. More recently, the properties of the interface localization-delocalization transition
were also studied for fluid models of soft matter systems [18–20]. Compared to magnets, however, the sit-
uation with fluids is a bit more intricate. Firstly, owing to unequal entropy of coexisting phases, they lack
the perfect symmetry of Ising-like models which makes the concept of antisymmetric walls less clear.
Secondly, ubiquity of dispersion forces in fluid systems is an important extra ingredient to be consid-
ered which, in fact, prevents complete drying. The latter problem can be avoided by considering binary
(colloid-polymer) mixtures such as in references [18–20]. The long-range dispersion interaction was in-
cluded in reference [21] by Stewart and Evans for a simple, one-component fluid; in this study, extensive
density functional theory (DFT) calculations have been made to confirm scaling predictions for several
13604-2
Phase transitions of fluids in heterogeneous pores
thermodynamic quantities. The wall parameters have been set such that they ensure a complete wet-
ting on one wall and a complete drying on the opposite wall and that, for the given fixed subcritical
temperature, the corresponding Hamaker constants, i.e., the coefficients of the lowest order term in the
respective binding potentials, are identical. However, even if neglecting the higher order contributions
in the binding potential, the system does not exhibit the same “antisymmetry” as in the case of mag-
nets. This is because the potential exerted by both walls must contain a repulsive part for the walls to
be impenetrable, but only one of them (the “solvophilic” wall) also contains an attractive portion. This
produces a binding potential well so that there exists a wetting temperature Tw below which the wall is
only partially wet. By contrast, the other, “solvophobic” wall is purely repulsive and thus dried (i.e., wet
by gas) at all temperatures. Consequently, the binding potential landscape for a fluid confined between
these competing walls is clearly different from that described above for magnetic systems. Furthermore,
since the Hamaker constants depend on a temperature (via coexisting densities), any change in the tem-
perature would necessarily break the “antisymmetry” of the system, which is thus not a property of the
system itself but also depends on thermodynamic parameters.
In this paper, we study the phase behaviour of a fluid confined between asymmetric walls that inter-
act with the fluid via dispersion forces. There are two models that we use to represent such a heteroge-
neous pore.Within the first model, bothwalls exhibit thewetting transition but at different temperatures.
We present simple geometric arguments that lead to the extended Kelvin equation which predicts the lo-
cation of capillary condensation in the pore. For the case, when the temperature of the system is greater
than the wetting temperature of at least one of the walls, we modify Kelvin’s equation by incorporating
the effect of the presence of the wetting layer(s). Both of these predictions are tested against numerical
results obtained from a non-local density functional theory (DFT). The second model represents the case
of antisymmetric walls, so that one of the walls is wetted by liquid whereas the other wall by gas. We
present a mean-field analysis for the location of the interface localization-delocalization transition. The
comparison with DFT shows that the analytic predictions are surprisingly accurate down to very narrow
pores at least for the case, when one wall exhibits first-order wetting transition (as opposed to critical
wetting) and the other wall is completely dried (wetted by gas) at all temperatures.
The remainder of the paper is organized as follows. In section 2 we present geometric arguments
to determine the location of capillary condensation for pores with two unlike walls and also show how
this extended Kelvin’s equation can be modified to embrace Derjaguin’s correction due to the presence
of wetting layers. In this section we also consider the pore model consisting of antisymmetric walls for
which we present a mean-field analysis to locate the interface localization-delocalization transition. In
section 3 we set the molecular model and show the DFT results. Although the planar symmetry permits
to treat the system as a one-dimensional problem, we employ a two-dimensional DFT to test the plausibil-
ity of geometric arguments. We then examine the properties of the interface localization-delocalization
transition for antisymmetric walls by determining the binding potential and study its behaviour as the
temperature and pore width vary. We summarise and discuss our results in the concluding section 4.
2. Heuristic arguments
We start by recalling macroscopic arguments leading to a condition for a liquid-vapour equilibrium
in a homogeneous pore, i.e., a pore made of identical walls that exhibit a wetting transition (by liquid) at
a temperature Tw. Within the purely macroscopic treatment, the distance between the walls L is taken
to be large and it is assumed that a first-order transition between a state corresponding to a gas-like
and a liquid-like phase occurs for any temperature T below the bulk critical temperature Tc. We then
expect that the pressure p (or the chemical potential µ) at which the transition occurs in the pore is
shifted below the saturation value psat (or µsat). Based on the surface thermodynamics, this value can
be determined by a simple free energy balance for a low- and a high-density state which leads to the
well known Kelvin’s equation (1.1). Apart from this thermodynamic picture, Kelvin’s equation has also a
geometric interpretation as it is illustrated in figure 1. Based on this approach, one considers a single pore
in which both phases coexist. Since the equilibrium occurs off bulk two-phase coexistence, the interface
between the gas and vapour phases in the pore must be curved in the direction perpendicular to the
walls with a Laplace radius R = γ/δp , where δp = psat − p. Assuming that δp is small, we can write
13604-3
A. Malijevský
Figure 1. (Color online) A schematic picture illustrating a coexistence of liquid-like and gas-like phases in
a homogeneous pore of a macroscopic width L. In the left panel, a contact angle at the walls is assumed
to be θ > 0 meaning that the temperature of the system is below the wetting temperature. The radius
R of the cylindrical meniscus separating the phases is given by the Laplace pressure R = γ/δµ(ρl−ρv).
In the left panel, the contact angle is zero, so that the meniscus meets tangentially the wetting layers of
thickness `π.
δp = δµ(ρl −ρv) according to the Gibbs-Duhem relation, where ρl and ρv are the particle densities of
coexisting liquid and vapour phases, respectively. Substituting from R = L/(2cosθ) yields equation (1.1).
Kelvin’s equation has proven to be fairly accurate down to surprisingly small values of L for temper-
atures T < Tw. Above the wetting temperature, when liquid layers adsorb at the walls, the appropriate
geometric picture of the phase coexistence in the pore is shown in the right-hand panel of figure 1. In this
case, the cylindrically-shaped meniscus tangentially connects the wetting films that are of thickness `π.
For sufficiently wide pores, `π can be considered as a thickness of a complete wetting layer adsorbed on
an isolated wall. This geometric interpretation suggests that the pore width L appearing in the denomi-
nator of equation (1.1) should be replaced by its effective value Leff = L−2`π. Explicit calculations based
on a free energy balance including the effective interaction between interfaces show that Leff depends on
the range of the molecular forces and that Leff = L −3`π when long range nonretarded dispersion forces
are involved.
We now turn our attention to a heterogeneous pore, where, within the macroscopic treatment, the
walls are characterised by contact angles θ1 and θ2. For the rest of the paper we shall assume without
any loss of generality that the contact angle at the left-hand wall θ1 is not larger than θ2 and that θ1 Éπ/2
[if θ1 > π/2, the problem is either reversed spatially (if θ2 < π/2) or thermodynamically (if θ2 > π/2) in
which case the gas and liquid phases interchange their roles]. We can again construct a geometric picture
of the liquid-vapour coexistence in the pore, as is shown in figure 2. Now, the center of the meniscus is no
more in the centre of the pore but is shifted towards the right-hand wall or even beyond in case of θ2 >
π/2. A simple geometry then leads to the following generalization of Kelvin’s equation for heterogeneous
pores [8]:
δµ≡µsat−µcc = γ(cosθ1 +cosθ2)
(ρl−ρv)L
. (2.1)
Usually, Kelvin’s equation is interpreted such that it tells us what is the chemical potential at the
capillary condensation for a given pore width L. This view can also be reversed and we can ask what is
the equilibrium distance L when we fix δµ and T . From the geometric construction shown in figure 2 it
is clear that the “more heterogenous” the pore is, the shorter the distance L must be for thermodynamic
criteria to be met.
Next, we wish to adopt a more microscopic view for the case when either θ1 = 0 or θ2 = 0 (or both).
We then expect that the wetting layer of a thickness `(i )
π which is formed at the wall i will somewhat mod-
ify the purely macroscopic prediction of equation (2.1). When a wetting layer of thickness `(i )
π intrudes
between a single wall i and the bulk gas, the corresponding surface free energy of the wall-gas interface
13604-4
Phase transitions of fluids in heterogeneous pores
Figure 2. (Color obline) Sketch of a vapour-liquid coexistence in a heterogeneous pore of a width L. In
this picture, the left-hand wall is partially wet and the right-hand wall is either partially wet but with
a larger contact angle (left-hand panel) or partially dried (θ2 > π/2, as shown in the right-hand panel).
Accordingly, the centre of the meniscus of a radius R = γ/δµ(ρl−ρv) is now closer to the right-hand wall.
is as follows:
γwi g = γwi l+γ+Wi
(
`(i )
π
)
, (2.2)
where γwi l is the surface tension between the wall i and the liquid, and
Wi (`) = δp`+ Ai
`2 + Bi
`3 +·· · (2.3)
is the effective potential between the wall-liquid and liquid-gas interfaces due to the wetting layer of
a thickness `. The coefficient Ai is called the Hamaker constant which must be positive for T > Tw.
This form of the binding potential assumes that the wall-fluid or fluid-fluid interactions are dominated
by nonretarded dispersion forces at large distances. The minimum of W determines the equilibrium
thickness of the wetting layer
`(i )
π ≈
(
2Ai
δp
)1/3
, T Ê T (i )
w . (2.4)
If only a microscopic wetting layer forms at the wall, i.e., if T < T (i )
w , then the global minimum of
Wi (`) is negative and the liquid-vapour interface is pinned to the wall to a microscopic distance `(i )
π even
for δp = 0, in which case the comparison of (2.2) with Young’s equation reveals that
Wi (`(i )
π ) = γ(cosθi −1) , T < T (i )
w . (2.5)
Now, in the pore of a large width L, we assume that the thickness of the wetting layer adsorbed at
either wall is the same as the one on a single wall. The free-energy difference per unit area between
a low-density state (with the grand-potential per unit area ωg = −pL +γw1g +γw2g) and a high-density
state (with the grand-potential per unit area ωg =−p+
l
L+γw1l+γw2l, where p+
l
denotes a pressure of the
metastable liquid at a given µ<µsat and T ) is then given by
∆ω=ωg−ωl = (p+
l
−p)L+2γ+W1(`(1)
π )+W2(`(2)
π ) , (2.6)
where we have neglected effective interactions other than between the liquid-gas interface and the near-
est wall.
In particular, when θ1 = 0 and θ2 > 0, a substitution of equations (2.3) and (2.5) into (2.6) gives
∆ω= (p+
l
−p)L+δp`(1)
π + A1(T )(
`(1)
π
)2 +γ(1+cosθ2) . (2.7)
13604-5
A. Malijevský
Identifying δp ≈ p −p+
l
≈ δµ(ρl−ρg) to first order in δµ and using equation (2.4), we obtain
δµ= γ(1+cosθ2)(
L− 3
2`
(1)
π
)
(ρl−ρg)
. (2.8)
This result can be generalised as follows:
δµ= γ(cosθ1 +cosθ2)[
L− 3
2
(
`(1)
π +`(2)
π
)]
(ρl−ρg)
, (2.9)
where `(i )
π = 0 if cosθi > 0.
The case of “antisymmetric” walls when cosθ1 = −cosθ2 deserves special attention. The modified
Kelvin equation implies that the only phase transition can occur at µ = µsat. In this case, apart from the
gas-like and liquid-like states, a configuration consisting of a thick film of liquid of width ` and a film of
gas of thickness L −` with the liquid-gas delocalized interface parallel to the walls should be also taken
into account. The appropriate grand potential per unit area of this configuration is:
ωdeloc(`) = γw1l+γ+γw2g+
A1
`2 + A2
(L−`)2 + B1
`3 + B2
(L−`)3 , (2.10)
where the higher-order terms were neglected.
Assuming that the walls are perfectly opposite, A1 = A2 ≡ A and B1 = B2 ≡ B we separately consider
the cases T < Tw1 = Tw2 ≡ Tw and T > Tw. The respective binding potentials are shown in figure 3 for
the walls that undergo critical (upper panels) and first-order (lower panels) wetting transitions, when L
is macroscopic. From the inspection of figure 3 we can conclude that for L macroscopic Tw represents
a triple point for first-order wetting in which case three minima of W (`) corresponding to low-density,
high-density and delocalized states are of equal depths. For critical wetting, there are only two minima
in the binding potential below Tw. Upon increasing the temperature, the two minima are getting closer
to each other continuously and finally merge at the midpoint of the pore at Tw which thus represents a
critical temperature in this case.
0
0
0
T<TW
0
T>TW
T<TW
T>TW
W
W
W
W
Figure 3. Schematic behaviour of a binding potential in a macroscopically wide pore with perfectly anti-
symmetric walls. Below the wetting temperature Tw, the binding potential has two minima located near
the walls that correspond to a high-density and a low-density state in which case the pore is predomi-
nantly filled by one phase with only a microscopic layer of the other fluid phase adsorbed on one of the
walls. For T > Tw, there is a single minimum of the biding potential in the middle of the pore, so that
thick liquid-like and gas-like films form at respective walls. The upper panel describes the case when the
walls exhibit critical wetting and the lower panel is appropriate for first-order wetting.
13604-6
Phase transitions of fluids in heterogeneous pores
We now wish to include the finite-size effects due to finite values of L, still considering the binding
potential of the form (2.3). The grand potential of the delocalized state with the liquid-gas interface in the
middle of the pore is as follows:
ωdeloc = γw1l+γ+γw2g+
8A
L2 + 16B
L3 , (2.11)
whereas the grand potentials of the low-density configuration
ωg = γw1g+γw2g (2.12)
and the high-density configuration
ωl = γw1l+γw2l (2.13)
are identical by symmetry. For T < Tw, the comparison between (2.12) (say) and (2.11), yields, upon using
Young’s equation,
γcosθ1 = γ+ 8A
L2 , (2.14)
where we have neglected the contributions of O (L−3). For T . Tw it follows that
−θ2
1 = 8A
γL2 , (2.15)
which can only be satisfied when the Hamaker constant is negative, i.e., only when the walls exhibit
critical wetting (in which case the extreme of the binding potential at the centre of the pore is a local
maximum). In this case, the critical temperature T ∗
at which the system breaks the symmetry is given
implicitly by
θ1(T ∗) = 1
L
√
8|A(T ∗)|
γ(T ∗)
. (2.16)
Using an abbreviation t ≡ (Tw−T ) and noting that θ(T ) ∼ t 3/2
and A ∼ t for critical wetting and small t ,
we obtain an asymptotic behaviour of T ∗(L):
T ∗ = Tw− f (T ∗)
L
, L →∞, (2.17)
where f (T ∗) > 0, in agreement with the result obtained on the basis of finite-size scaling arguments [8]
T ∗ = Tw−L−1/βs , where βs is the surface critical exponent for critical wetting, for algebraically decaying
binding potential (2.3) in three bulk dimensions.
In a more microscopic manner, the phase behaviour of a fluid confined between antisymmetric walls
exhibiting critical wetting can be analyzed for T < Tw by comparing the grand potential of the delocalized
state (2.11) with the localized (to wall 1) one. The latter is obtained by a substitution of `=−3B/2A (recall
that A < 0 and B > 0 for critical wetting below Tw), as given by a minimization of (2.3), back to (2.3) with
δp = 0:
ωloc = γw1l+γ+γw2g+
4
27
A3
B 2 + A(
L+ 3
2
B
A
)2 + B(
L+ 3
2
B
A
)3 . (2.18)
Balancing ωloc and ωdeloc leads to the equation
4
27
1
κ2 + 1(
1+ 3
2κ
)2 + κ(
1+ 3
2κ
)3 −16κ= 8, (2.19)
where we have introduced a dimensionless parameter κ ≡ B/AL. Equation (2.19) can be solved graph-
ically as is shown in figure 4 revealing four solutions with three of them, corresponding to negative κ,
being relevant. This analysis suggests that within the interval between κ=−2/3, where the LHS of equa-
tion (2.19) has a pole and κ ≈ −0.23, the delocalized state is the stable solution, while the localized state
is the more stable solution otherwise. There is another point κ=−1/3 at which the localized and delocal-
ized states are equally stable; however, this solution seems to be just an artifact of the current analysis
13604-7
A. Malijevský
Figure 4. A graphical solution of equation (2.19). The curve (blue line) corresponds to the left-hand side
of equation (2.19) as a function of κ = B/AL, the horizontal line (red) belongs to the right-hand side of
equation (2.19). The plot in the right-hand panelmagnifies the graph in the interval κ=−0.37 to κ=−0.22
which presumably disappears when higher order terms of O (L−4) in the binding potential are taken into
account. We note that this result is not inconsistent with the asymptotic result given by equation (2.17),
since in the latter case L is large and A is small, so that |κ| is not necessarily small in this limit.
For T > Tw, there is only a delocalized state present for the walls undergoing critical wetting. For
the walls exhibiting first-order wetting transition, the localized states become metastable with respect
to the delocalized one in the macroscopic limit of the separation of the walls. For L finite, the binding
potential is non-vanishing in the middle of the pore which makes the delocalized state less stable. The
grand potential per unit area for the localized state is given by
ωloc = γw1l+γ+γw2g+W1(`loc)+W2(L−`loc), (2.20)
where `loc is the minimum of the binding potential near the (left-hand) wall. In order to describe the
binding potential for the case of first-order wetting (see figure 3), we should consider one more term
in (2.3)
Wi (`) = Ai
`2 + Bi
`3 + Ci
`4 +·· · , (2.21)
where Ai , Ci > 0 and Bi < 0 and where we set δp = 0. Minimization of Wi (`) gives `loc = −3B/4A −p
9B 2/A2 −32C /A/4, where we again assume that the walls are perfectly antisymmetric (A1 = A2, B1 =
B2 and C1 =C2 and thusW1 = W2 ≡ W ). The grand potential per unit area for the delocalized state is as
follows:
ωdeloc = γw1l+γ+γw2g+2W (L/2) . (2.22)
The energy barrier ofW (`) disappears when its local minimum and maximum coincide which leads
to the condition for the spinodal temperature Ts:
B 2
AC
= 32
9
, T = Ts . (2.23)
Above Ts, only the delocalized state is present. For Tw < T < Ts, there exists a wall separation L∗(T ) for
which ωloc =ωdeloc, which determines the location of the interface delocalization transition. At this wall
separation, the presence of the walls rises (relative to the case of L macroscopic) the central minimum of
the binding potential (corresponding to the delocalized state) up to the level of the minima near the walls
(corresponding to the localized states). For T ≈ Ts we approximately obtain
Lc ≡ L∗(Ts) ≈ 3p
2
|B |
A
, (2.24)
13604-8
Phase transitions of fluids in heterogeneous pores
which is the critical value of L below which the interface localized-delocalized transition never occurs.
In the opposite limit, L →∞, the transition occurs right at Tw in line with the macroscopic arguments.
Below Tw, only the states with the interface pinned to either walls are stable for any value of L.
Finally, we note that in a more general case of competing walls when A1 , A2, the position of the
delocalized interface is shifted towards the wall with a weaker surface field, such that
`eq ≈ L(
1+ 3pA2/A1
) . (2.25)
3. Density functional theory
In order o describe microscopic properties of a simple fluid in heterogeneous pores we adopt a clas-
sical density function theory [22]. Within DFT, the equilibrium density profile is found by minimizing the
grand potential functional
Ω[ρ] =F [ρ]+
∫
dr
[
V (r)−µ]
ρ(r) , (3.1)
where V (r) is the external field due to the walls and µ is the chemical potential. The intrinsic free energy
functionalF can be split into the contribution from the ideal gas and the remaining excess part
F [ρ] =Fid[ρ]+Fex[ρ] , (3.2)
whereFid[ρ] = kBT
∫
drρ(r)
[
ln(Λ3ρ(r))−1
]
withΛ being the thermal de Broglie wavelength that can be
set to unity.
The fluid model is characterised by a pair potential consisting of a hard-sphere repulsion of the range
of σ, and a Lennard-Jones-like attractive portion given by
u(r ) =
0, r <σ,
−4ε
(
σ
r
)6 , σ< r < rc,
0 , r Ê rc,
(3.3)
where the cut-off is set to rc = 2.5σ. The repulsive contribution to the excess free energy functional
is approximated using Rosenfeld’s fundamental measure theory [23] and the attractive part is treated
within the mean-field approximation:
Fex[ρ] = kBT
∫
drΦ({nα})+ 1
2
∫
drρ(r)
∫
dr′ρ(r′)u(|r− r′|) , (3.4)
The free energy density Φ is a function of a set of three independent weighted densities {nα} for which
we use the original Rosenfeld prescription.
The external field due to the parallel impenetrable walls a distance L apart is given by V (r) =V (z) =
V1(z)+V2(L− z), where
Vi (z) =
{ ∞ , z <σ ,
− 2
3
πεwi ρwi σ
6
z3 , z Êσ .
(3.5)
The parameter ρwi is a density of uniformly distributed atoms forming the wall i , such that each atom
exerts a potential u(r ) according to expression (3.3) with a parameter εwi replacing ε and without a cut-
off, i.e., for rc =∞. The potential Vi of a single wall i is given by integrating all contributions of atoms
over the entire wall. The diameter σ of the wall atoms has been chosen equal to that of fluid atoms. Since
the parameter ρwi is always associated with εwi , we can characterise the wall strength by tuning just a
single parameter εwi by setting ρw1 = ρw2 =σ−3
.
In order to make a link with the more phenomenological model introduced in the previous section,
the Hamaker constant defined by equation (2.3) can be expressed in terms of the microscopic param-
eters using the sharp-kink approximation [24]. For the interaction between wall-liquid and liquid-gas
interfaces we have
Ai = 1
3
πεwi ρwi (ρl−ρg)σ6 , (3.6)
Bi =−2
3
πεwi ρwi (ρl−ρg)σ7 , (3.7)
13604-9
A. Malijevský
where it should be noted that Ai is always positive in ourmodel owing to the fact that the fluid-fluid inter-
action (3.3) is only short ranged and thus does not contribute to Ai . For the interaction between wall-gas
and gas-liquid interfaces, the expressions in equations (3.6) and (3.7) have opposite signs. However, for
“antisymmetric” walls, εw1 = −εw2 , in which case A1 = A2 and B1 = B2. This “antisymmetry” is decep-
tive, however, even if εw1 = −εw2 one does not expect B1 and B2 to be identical beyond the sharp-kink
approximation, since these second-order contributions are due to excluded volume effects at the walls
and it is reasonable to expect that those of the wall-liquid interface are much stronger than those of the
wall-gas interface. Indeed, one does not expect the existence of a drying temperature at all for the purely
repulsive wall, so that a link between amicroscopic and amesoscopic model at the level of the sharp-kink
approximation is fully justified only up to the leading order with a coefficient Ai .
The minimization of (3.1) is carried out numerically on a 2D grid with spacing dx = dz = 0.05σ. Al-
though the symmetry of the external field permits to treat the problem as one-dimensional ρ(r) = ρ(z), we
also wish to test the geometric arguments for microscopic values of L by constructing equilibrium density
profiles of a single pore filled with liquid- and gas-like coexisting phases, in which case ρ(r) = ρ(x, z). For
a given set of model and thermodynamic parameters, the 1D DFT results are used as boundary condi-
tions for the 2D DFT, such that we set ρ2D(x, zmax) = ρlow
1D (x) and ρ2D(x, zmin) = ρ
high
1D (x). Here, zmax and
zmin are respectively the maximal and minimal values of the vertical coordinate z which are considered
within the 2D calculations and are set to zmin = 0 and zmax = 40σ. The functions ρlow
1D (x) and ρhigh
1D (x) are
1D density profiles corresponding to the low and high density states, respectively. The low density state
is a configuration in which the system is filled primarily with a gas (the liquid-gas interface is pinned to
the left-hand wall), whereas the high density state is a configuration in which the system is either filled
with liquid (the interface is pinned to the right-hand wall) or a delocalized state in which case the liquid-
gas interface is around the centre of the pore. Further details and particularly the implementation of
Rosenfeld’s functional within the 2D-DFT treatment can be found in reference [25].
We start the discussion of our numerical DFT results by examining the wetting properties of a single
wall. In figure 5 we display the temperature dependence of the contact angle for two walls with surface
fields εw1 = 1.2ε (wall 1) and εw2 = ε (wall 2). The walls exhibit first-order wetting transition at tempera-
tures Tw1 = 0.83Tc and Tw2 = 0.93Tc, with kBTc/ε= 1.414 corresponding to the bulk critical temperature
[26]. We now consider a pore formed by a wall 1 and a wall 2 at a temperature T = 0.81Tc (kBT /ε= 1.15)
at which both walls, when separated, are only partially wet. In figure 6 we display the equilibrium 2D
density profiles corresponding to the liquid-gas coexistence in pores of widths L = 20σ, L = 10σ, and
L = 5σ. In this case, the walls of the pore in the low-density phase are only microscopically wet and the
liquid-gas meniscus meets both walls at angles that appear in a good agreement with the predicted val-
ues of the respective contact angles (cf. figure 5), including, somewhat surprisingly, even the narrow pore
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
(d
eg
)
k
B
T/
w
=1.2
w
=
Figure 5. (Color online) Variation of the contact angle with temperature for walls with strengths εw1 =
1.2ε and εw2 = ε.
13604-10
Phase transitions of fluids in heterogeneous pores
Figure 6. (Color online) Two-dimensional equilibrium density profiles of a fluid confined between walls
with surface fields εw1 = 1.2ε and εw2 = ε at a temperature T = 0.81Tc (kBT /ε = 1.15) which is below
both Tw1 and Tw2 . The contact angles at the single walls are θ1 = 29° and θ2 = 68°, as given by the results
displayed in figure 5. The width of the pores accessible for fluid particles is (from left to right) L = 20σ,
L = 10σ, and L = 5σ. All configurations correspond to the liquid-gas coexistence in the respective pores,
i.e., the capillary condensation which occurs at the chemical potentials that are shifted below µsat =
−4.0309ε by the values δµ(L = 20σ) = 0.0275ε, δµ(L = 10σ) = 0.0547ε and δµ(L = 5σ) = 0.1934ε.
with L = 5σ. We have further determined the location of capillary condensation for each of the pores,
and the results are in perfect agreement with the modified Kelvin’s equation (2.1), cf. table 1.
We next consider a temperature T = 0.92Tc (kBT /ε = 1.3), in which case the wall 2 is still partially
wet but the wall 1, when separated and at bulk two-phase coexistence, would be completely wet. The
equilibrium density profiles at two-phase coexistence are shown in figure 7. Now, the comparison of the
DFT results with themacroscopic Kelvin equation (2.1) is less satisfactory than in the previous case, where
both walls are non-wet, especially for narrow pores, see table 1. Nevertheless, if we employ the modified
Kelvin equation as given by (2.9), we obtain a prediction in a very good agreement with DFT, provided the
capillaries are sufficiently wide. For instance, for L = 50σ, the relative difference in δµ between DFT and
the modified Kelvin equation is only 4% (compared to 18% for the macroscopic Kelvin equation). Only
for those large capillaries, the assumptions leading to equation (2.9) are justified. For the pores widths
considered here, one can still use equation (2.9) but with `π replaced by the real thickness of the wetting
layer. This can be read off from the 1D density profiles by determining e.g., the Gibbs dividing surface of
the liquid-gas interface. The inclusion of this correction substantially improves the agreement with the
Table 1. The predicted values of a location of capillary condensation, as given by Kelvin’s equation (2.1)
and its modified version taking into account the effect of wetting layers [equation (2.9)], are compared
with DFT results for three pore widths and temperatures T < Tw1 < Tw2 , Tw1 < T < Tw2 and Tw1 < Tw2 <
T . The comparison is made for the heterogeneous pore with εw1 = 1.2ε and εw2 = ε.
Temperature Pore width DFT equation (2.1) equation (2.9)
L = 20σ δµ= 0.027502ε δµ= 0.026998ε –
kBT /ε= 1.15 L = 10σ δµ= 0.054670ε δµ= 0.053996ε –
L = 5σ δµ= 0.101590ε δµ= 0.107991ε –
L = 20σ δµ= 0.027415ε δµ= 0.020330ε δµ= 0.026229ε
kBT /ε= 1.3 L = 10σ δµ= 0.059021ε δµ= 0.040635ε δµ= 0.058079ε
L = 5σ δµ= 0.121618ε δµ= 0.081302ε δµ= 0.116158ε
kBT /ε= 1.35 L = 20σ δµ= 0.0234047ε δµ= 0.014348ε δµ= 0.026087ε
13604-11
A. Malijevský
Figure 7. Two-dimensional equilibrium density profiles of a fluid confined between walls with surface
fields εw1 = 1.2ε and εw2 = ε at a temperature T = 0.92Tc (kBT /ε = 1.3) which is below Tw2 but above
Tw1 . The contact angle of the partially wet wall is θ1 = 22°, as given by the results displayed in figure 5.
The width of the pores accessible for fluid particles is (from left to right) L = 20σ, L = 10σ, and L = 5σ.
All configurations correspond to the liquid-gas coexistence in the respective pores, i.e., the capillary con-
densation which occurs at the chemical potentials that are shifted below µsat = −3.9651ε by the values
δµ(L = 20σ) = 0.0275ε, δµ(L = 10σ) = 0.0590ε and δµ(L = 5σ) = 0.1216ε.
DFT results, as is shown in table 1.
Finally, if we increase the temperature above Tw2 , then both walls are covered with wetting films
but of different thickness, as can be seen from the equilibrium density profile for L = 20σ and µ=µcc in
figure 8. As expected, the presence of the wetting layers deteriorates the quality of the prediction given by
the macroscopic Kelvin equation [equation (2.1)] but the modified Kelvin equation [equation (2.9)] is still
fairly accurate, when again both `(i )
π are replaced by the respective film thicknesses, see table 1. At this
temperature (T = 0.95Tc), there is no capillary condensation for the intermediate (L = 10σ) and narrow
(L = 5σ) pores. While the µ−Ω dependence (not shown here) is smooth for the narrowest pore. It exhibits
a kink for the intermediate pore suggesting that the critical pore width at this temperature is Lc ≈ 10σ.
Figure 8. Two-dimensional equilibrium density profile of a fluid confined between walls with surface
fields εw1 = 1.2ε and εw2 = ε at a temperature T = 0.95Tc (kBT /ε = 1.35) which is above both Tw1 and
Tw2 . The width of the pores accessible for fluid particles is L = 20σ and the configuration corresponds to
the liquid-gas coexistence and occurs at the chemical potentials that is shifted below µsat =−3.9569ε by
the value δµ(L = 20σ) = 0.0234ε.
13604-12
Phase transitions of fluids in heterogeneous pores
At last, we turn our attention to the case of asymmetric walls, such that one wall tends to be wet and
the other dry. To this end, we consider a capillary in which the wall 1 has the same strength as before, i.e.,
εw1 = 1.2ε but the opposite wall is purely repulsive. The repulsive wall exerts the potential according to
(3.5) with a negative surface field εw2 =−1.2ε. Although the Hamaker constants and thus the asymptotic
behaviour of binding potentials for both walls are assumed to be the same, it is only the attractive wall
(wall 1) which exhibits (first order) wetting transition at temperature Tw. The purely repulsive wall (wall
2), does not induce a drying transition, i.e., the wall is always completely wet by gas. Consequently, the
binding potential of wall 2 is purely repulsive (monotonously decaying) in contrast with the binding
potential of wall 1 which, at least for sufficiently low temperatures, has two competing minima that are
of equal depth at T = Tw. In figure 9 we display numerically constructed binding potentials for this model
pore using a constrained minimization of (3.1) at bulk two-phase coexistence for several representative
temperatures and a relatively wide pore (L = 50σ). For T < Tw, the global minimum in the total binding
potential lies near wall 1, which means that the liquid-gas interface is pinned to the adsorbing wall and
thus the system is in a low-density state for µ É µsat. On the other hand, if the two-phase coexistence is
approached from the liquid state, the pore exhibits capillary emptying for some µ>µsat, according to the
Kelvin equation (2.1). For T ≈ Tw, the binding potential has two equally deep minima: one near the wall
1 and the other at the middle of the pore. For a perfectly antisymmetric system (such as for magnets),
this temperature would be identified with a triple point, where the states with a liquid-gas interface near
both walls and the state with the liquid-gas interface in the centre of the pore are all equally stable. In
this model, however, the configuration corresponding to the pore filled with liquid is missing (at µ=µsat),
so that the transition at a temperature T ∗ ≈ Tw can be interpreted as a thin-to-thick transition in some
analogy with prewetting transition on a single wall. However, at a single wall, the prewetting transition
is induced by the appearance of a non-zero term δp` in the binding potential (2.3), i.e., because the
system is away from bulk coexistence, which shifts a local minimum of W (`) at ` = ∞ (at saturation,
δp = 0) to a finite distance from the wall (for δp > 0). For our system, the shift of the minimum is due to
the presence of the second wall. For a temperature T > Tw, the minimum ofW (`) at `= L/2 becomes a
global minimum corresponding to a delocalized interface. Above the spinodal temperature Ts, the energy
barrier disappears and the configuration with the bounded interface ceases to be even metastable.
0 10 20 30 40 50
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
W
2 /
l/
k
B
T/ =1.15
k
B
T/ =1.18
k
B
T/ =1.25
k
B
T/ =1.3
Figure 9. Binding potentials in a capillary with competing walls (εw1 = −εw2 = 1.2ε) and a width L =
50σ. For all temperatures below the spinodal temperature Ts the binding potential exhibits two minima,
corresponding to the liquid-gas interface being pinned at wall 1 and a delocalized interface with a mean
location in the pore midpoint. The competition between the two minima determines the equilibrium
configuration of the fluid. For this wide pore, the crossover temperature T∗
between the states, at which
the interface is bounded to the wall (T < T∗
) and delocalized (T > T∗
) is T∗ ≈ Tw (kBTw/ε = 1.18 for
wall 1). The highest temperature (kBT = 1.3ε) is already above Ts and thus the binding potential has only
one minimum in the middle of the pore. Also note a very small energetic barrier for kBT = 1.25ε.
13604-13
A. Malijevský
0 5 10 15 20
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
W
2 /
l/
k
B
T/ =1.18
k
B
T/ =1.2
k
B
T/ =1.25
Figure 10. Binding potentials in a capillary with competing walls (εw1 = −εw2 = 1.2ε) and a width L =
20σ. In this case, in contrast to the case of a wide pore shown if figure 9, the temperature T∗
at which
two different fluid configurations coexist is above Tw and corresponds to kBT ≈ 1.2ε.
Thus far, the DFT results are in line with the predictions obtained from macroscopic considerations.
However, if the pore width is reduced, the midpoint minimum of the binding potential becomes more
affected by the (repulsive) interaction between the liquid-gas interface with both walls. Consequently,
the midpoint minimum is pushed upwards which brings about a two-phase coexistence at T ∗ > Tw, as
illustrated in figure 10 for the pore width of L = 20σ. In figure 11 we also show a density profile for a tem-
perature kBT = 1.2ε at which the states corresponding to the localized and delocalized interface coexist.
As we reduce the pore width even more, T ∗(L) still increases and terminates at Ts which determines the
critical pore width Lc = L(T ∗ = Ts ) below which no phase transition occurs at µ = µsat. In figure 12 we
display binding potentials for a near critical pore width (L = 10σ); note that the two-phase coexistence
occurs now at a temperature kBT /ε = 1.25 for which the binding potential for a single wall (or a wide
pore) has only a very weak minimum near the wall (cf. figure 9). Also shown here is the binding potential
for T > Ts which does not permit any phase transition at µ=µsat for any pore width.
Figure 11. Two-dimensional density profile of a fluid confined between parallel walls with strengths εw =
1.2ε (left-hand wall) and εw =−1.2ε (right-hand wall), separated by a distance L = 20σ. The temperature
of the system kBT = 1.2ε corresponds to the interface localized-delocalized transition (µ=µsat).
13604-14
Phase transitions of fluids in heterogeneous pores
0 2 4 6 8 10
0.00
0.02
0.04
0.06
0.08
W
2 /
l/
k
B
T/ =1.2
k
B
T/ =1.25
k
B
T/ =1.3
Figure 12. Binding potentials in a capillary with competing walls (εw1 = −εw2 = 1.2ε) and a width L =
10σ. The temperature T∗
at which two different fluid configurations coexist is now kBT ≈ 1.25ε, which
is only slightly below the spinodal temperature (cf. figure 9). For T > Ts , the binding potential has only
one minimum in the pore midpoint.
4. Summary and concluding remarks
In this work we have studied the phase behaviour of a simple fluid confined between two parallel
walls with different surface fields in the presence of long-range dispersion forces. We have shown that
simple geometric arguments lead to the generalized macroscopic Kelvin equation which, when at least
one of the walls is in complete wetting regime, should be modified by capturing the effect of thick wetting
films by including Derjaguin’s correction. We have also shown that simple mean-field arguments can be
invoked to describe the fluid phase behaviour in a capillary at which one wall tends to be wet while
the other tends to be dry and which exhibits interface localized-delocalized transition. Some of these
predictions have been tested using a microscopic non-local density functional theory. The main results of
this work can be summarised as follows:
1. Simple geometric arguments lead to a macroscopic prediction for a location (a shift in the chemical
potential relative to that of bulk saturated fluid) of the capillary condensation/evaporation in a
pore made of different walls. This leads to the (generalized) Kelvin equation (2.1) which has been
tested against DFT calculations and the comparison revealed a very good agreement between these
two approaches even for very narrow pores provided both walls are in a partial wetting/drying
regime. The geometric interpretation of Kelvin’s equation was also supported by constructing 2D
density profiles. However, if the contact angle of at least one wall is zero (or π), the predictions
of the macroscopic Kelvin equation worsen significantly, especially as the pore width decreases.
Nevertheless, a substantial improvement of the Kelvin equation can be achieved by taking into
account the effect due to the presence of wetting (or drying) films. The film thicknesses that appear
in this corrected Kelvin equation (2.9) can be determined from equation (2.4) provided the pore is
wide enough (L & 50σ, with σ being the molecular diameter).
2. We have then revisited a model of a capillary slit with competing walls. Here, the Kelvin equation
as well as symmetry considerations dictate that any phase coexistence must occur at µ = µsat. In
this case, the liquid-vapour interface is parallel to the walls and can be pinned to either of the walls
(in which case the system is in a condensed or evaporated state) or can be unbounded from both
walls around a halfway between the walls; the latter state is referred to as a soft mode phase or a
delocalized interface. Nature and the location of a transition when the interface unbinds (interface
localized-delocalized transition) are determined by an interplay between the wetting properties of
the walls and the finite-size effects due to finite separation L of the walls:
13604-15
A. Malijevský
(a) According to macroscopic arguments (L À σ), the transition occurs at T ∗ = Tw = Td, assum-
ing the walls are perfectly antisymmetric. If the walls exhibit continuous wetting/drying tran-
sition at Tw, then T ∗
is a critical temperature, below which either of the localized state is
equally stable. As T → T −
w
, the liquid-gas unbinds continuously from a given wall and moves
towards the pore midpoint. Above Tw only a single state corresponding to a delocalized inter-
face is present. If the walls undergo first-order wetting transition, then T ∗ = Tw (and µ=µsat)
is a triple point at which all three configurations coexist. These macroscopic predictions were
verified by DFT calculations for pores of widths L & 50σ.
(b) As the pore width is reduced, T ∗
does not coincide with Tw anymore but becomes a function
of L. The behaviour of T ∗(L) then strongly depends on the nature of the wetting transition of
the walls and can be described by mean-field mesoscopic arguments that capture the effect
of the effective interaction between the liquid-gas interface and the attractive tails of the wall
potentials. For thewalls exhibiting first-order wetting transition, the analysis shows that there
are three temperature regimes. For T < Tw, the only stable solutions correspond to a bounded
interface for any values of L. For Tw < T < Ts, there is an interface localized-delocalized tran-
sition for the wall separation L∗(T ) which decreases with an increasing T . Above Ts, there is
no transition and there is a single configuration corresponding to the delocalized interface.
The spinodal temperature Ts is characterised by a disappearance of the energy barrier in the
binding potential and can be associated with the surface critical point of prewetting transi-
tion. The spinodal temperature determines the critical width Lc = L∗(Ts), which is a minimal
value of L allowing the interface localization-delocalization transition. Our DFT results sug-
gest that Lc is slightly below 10σ, in a reasonable agreement with the mesoscopic prediction
as given by equation (2.24) with the parameters A and B determined from equations (3.6) and
(3.7), which yields Lc ≈ 6σ. Conversely, these results can also be interpreted such that for each
Lc < L <∞ there exists a temperature T ∗
below which the fluid undergoes capillary filling
(as µ approaches µsat from below) or emptying (as µ approaches µsat from above) and above
which no transition is present and the interface is delocalized.
(c) If the walls exhibit critical wetting at Tw then, in contrast to the case of first-order wetting, the
finite-size effects favour interface delocalization, so that the interface localized-delocalized
transition is shifted below Tw. For very large L, this shift is proportional to L−1
with a (non-
universal) temperature dependent amplitude that can be in principle determined from a
given molecular model and this result is consistent with the one obtained on the basis of
finite-size scaling [8]. For moderately large values of L and the model with the binding poten-
tialW (`) = A(T )`−2+B(T )`−3+·· · , themesoscopic analysis suggests that the phase behaviour
can be characterized by a dimensionless parameter κ= B/AL, such that the localized-deloca-
lized transition occurs for κ=−2/3 and κ≈−0.24with the delocalized interface being a stable
solution within this interval.
3. Within the microscopic DFT model that has been considered in this work, the walls exert long
range (decaying as z−3
for large z) potential while the fluid-fluid interaction was taken to be only
short-ranged. This particular choice of the interaction model requires some comments:
(a) In terms of the study of interface localized-delocalized transition, this model has one advan-
tage and one disadvantage. The great advantage is that the competing walls can be made an-
tisymmetric by simply choosing εw1 = −εw2 and this antisymmetry is maintained regardless
of the temperature. This is because the only temperature dependent factor in both Hamaker
constants (corresponding to the situation when the adsorbed film at the wall is a liquid or a
gas phase) is (ρl−ρg). Therefore, the Hamaker constants certainly vary with temperature but
in identical manner for both walls which allowed us to study the phase behaviour of a fluid
between antisymmetric walls for various temperatures. This is in contrast with the model
used in reference [21], where the fluid-fluid interaction is also long-ranged which produces
an extra temperature dependent factor in the respective Hamaker constants Aw1∝ (ρl−ρw)
and Aw2∝ (ρg−ρw) that are identical at only one particular temperature. On the other hand,
13604-16
Phase transitions of fluids in heterogeneous pores
a disadvantage of our model is that since the Hamaker constant is always positive, a possi-
bility that the walls undergo critical wetting is excluded, hence we could only test the cases
when the walls exhibit first-order wetting.
(b) Setting εw1 = −εw2 makes the wall 2 purely repulsive, so that even though A1 = A2 the walls
cannot be viewed as perfectly antisymmetric in the sense Tw = Td (Td is the drying temper-
ature of the wall 2), because the wall 2 is completely dried at all temperatures for which
vapour and liquid may coexist. This broken (anti)symmetry is reflected in the binding poten-
tial for the capillary with competing walls which exhibits at most two minima, missing a local
minimum near wall 2. In contrast with the fully antisymmetric model (considered within the
macro- and mesoscopic pictures), the temperature T ∗
for first-order wetting is not a triple
temperature but a temperature appropriate to an ordinary first-order transition at which the
liquid-gas interface jumps from the proximity of wall 1 to the centre of the pore in some anal-
ogy to prewetting transition (in contrast to the latter, the transition takes place at µ=µsat and
the jump is determined by L). Also note that below Tw, cosθ1 + cosθ2 appearing in Kelvin’s
equation becomes negative, meaning that for T < Tw a two phase coexistence (capillary evap-
oration) occurs at µ> µsat. Finally, if the competing walls have different Hamaker constants,
the interface localized-delocalized transition is still possible but with the position of the delo-
calized interface near `= L/[1+ (A2/A1)1/3] rather than `= L/2.
Acknowledgements
Thefinancial support from the Czech Science Foundation, project number 13-09914S is acknowledged.
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http://dx.doi.org/10.1063/1.445087
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http://dx.doi.org/10.1088/0953-8984/2/46/001
http://dx.doi.org/10.1088/0034-4885/62/12/201
http://dx.doi.org/10.1016/0009-2614(85)85111-3
http://dx.doi.org/10.1103/PhysRevA.32.3817
http://dx.doi.org/10.1103/PhysRevLett.64.439
http://dx.doi.org/10.1016/0378-4371(92)90089-9
http://dx.doi.org/10.1209/0295-5075/14/5/015
http://dx.doi.org/10.1103/PhysRevE.58.3353
http://dx.doi.org/10.1103/PhysRevLett.74.298
http://dx.doi.org/10.1103/PhysRevE.51.2823
http://dx.doi.org/10.1103/PhysRevE.53.5023
http://dx.doi.org/10.1023/A:1022173600263
http://dx.doi.org/10.1103/PhysRevE.71.046705
http://dx.doi.org/10.1007/s10955-009-9710-8
http://dx.doi.org/10.1209/0295-5075/77/60002
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http://dx.doi.org/10.1103/PhysRevE.86.031601
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Фазовi переходи плинiв в гетерогенних порах
A.Малiєвский1,2
1 Вiддiл фiзичної хiмiї, Празький унiверситет хiмiчної технологiї, 166 28 Прага 6, Чеська Республiка
2 Лабораторiя аерозольної хiмiї i фiзики, Iнститут основ хiмiчних процесiв, Академiя наук,
16502 Прага 6, Чеська Республiка
Ми вивчаємо фазову поведiнку модельного плину, обмеженого двома рiзними паралельними стiнками
у присутностi далекосяжних (дисперсiйних) сил. Передбачення, отриманi на основi макроскопiчних (гео-
метричних) i мезоскопiчних аргументiв, порiвнюються з числовими розв’язками теорiї функцiоналу не-
локальної густини. Розглянуто двi капiлярнi моделi. У випадку капiлярної моделi, що має двi (по-рiзному)
адсорбуючi поверхнi, простi геометричнi аргументи приводять до узагальненого рiвняння Кельвiна, яке
дуже точно локалiзує капiлярну конденсацiю, за умови, якщо обидвi стiнки лише частково змочуються.
Якщо принаймнi одна зi стiнок знаходиться у режимi повного змочування, рiвняння Кельвiна слiд видо-
змiнити за рахунок ефекту товщини змочуючих плiвок, включивши поправку Дєрягiна. У другiй моделi
розглянуто капiляр, утворений з двох конкуруючих стiнок: одна з них схильна до змочування, а друга—
не змочується. У цьому випадку на границi роздiлу вiдбувається локалiзований-делокалiзований пере-
хiд при двофазному спiвiснуваннi в об’ємi i при температурi T∗(L), яка залежить вiд ширини пори L.
Аналiз, виконаний за допомогою теорiї середнього поля, показує, що для стiнок, якi проявляють пере-
хiд змочування першого роду при температурi Tw, Ts > T∗(L) > Tw, де температура спiнодалi Ts мо-же бути зв’язана з критичною температурою попереднього змочування, яка також визначає критичну
ширину пори, нижче якої локалiзований-делокалiзований перехiд на границi роздiлу не вiдбувається.
Якщо ж стiнки проявляють критичне змочування, тодi перехiд змiщується нижче за Tw, а для моделi зпотенцiалом W (`) = A(T )`−2 +B(T )`−3 + ·· · , де ` — мiсцезнаходження границi роздiлу мiж рiдиною i
газом, перехiд можна характеризувати за допомогою безрозмiрного параметра κ= B/(AL), в результатi
чого конфiгурацiя плину з делокалiзованою границею роздiлу є стабiльною в iнтервалi мiж κ = −2/3 i
κ≈−0.23.
Ключовi слова: капiлярна конденсацiя, змочування, рiвняння Кельвiна, адсорбцiя, теорiя функцiоналу
густини, теорiя фундаментальної мiри
13604-18
http://dx.doi.org/10.1039/f29868201763
Introduction
Heuristic arguments
Density functional theory
Summary and concluding remarks
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