Surface and capillary transitions in an associating binary mixture model

12 Pages·0.71 MB·English

by  J. M. Romero-Enrique

Surface and capillary transitions in an associating binary mixture model.
J. M. Romero-Enrique,1,2,∗ L. F. Rull,2 and U. Marini Bettolo Marconi3
1Department of Mathematics, Imperial College of Science,
Technology and Medicine, 180 Queen’s Gate, London SW7 2BZ, United Kingdom
3 2Departamento de F´ısica At´omica, Molecular y Nuclear, Area de F´ısica Te´orica,
0 Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain
0 3Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia,
2 via Madonna delle Carceri, Universit`a di Camerino, 62032 Camerino, Italy

n We investigate the phase diagram of a two-component associating fluid mixture in the presence
a of selectively adsorbing substrates. The mixture is characterized by a bulk phase diagram which
J
displayspeculiar features such as closed loops of immiscibility. The presenceof thesubstrates may

0 interferethephysicalmechanism involvedintheappearanceof thesephasediagrams, leadingtoan
3 enhanced tendency to phase separate below the lower critical solution point. Three different cases
areconsidered: aplanarsolidsurfaceincontactwithabulkfluid,whiletheothertworepresenttwo
]
modelsofporoussystems,namelyaslitandanarrayoninfinitelylongparallelcylinders. Weconfirm
h
thatsurface transitions, as well as capillary transitions for a large area/volume ratio, arestabilized
c
e in the one-phase region. Applicability of our results to experiments reported in the literature is
m discussed.
-
PACSnumbers: 64.75.+g,05.70.Np,68.08.Bc,68.35.Rh
t
a
t
s
. I. INTRODUCTION the mixture is completely miscible at low temperatures,
t
a up to a temperature where it separates into two liquid
m phases. Asthetemperatureisfurtherincreasedthecom-
- In the past few years there has been a considerable plete miscibility reappears. The nicotine-water mixture
d progress in the understanding of the physics of fluids is a paradigm for such a behavior [9]. The temperature-
n
in restricted geometries [1]. The properties of liquids compositionphasediagramshowsclosed-loopsofimmis-
o
or gases confined in narrow pores appear different from cibility, characterized by the existence of a lower and an
c
[ those in the bulk. A common trend of these systems is upper critical solution point (LCSP and UCSP, respec-
the fact that condensation occurs at pressures different tively). Hirschfelder et al [10] proposed a mechanism
2
from the saturation bulk pressure and the critical tem- to explain the reentrant miscibility. The essential ingre-
v
perature for phase separationis generallylowerthan the dient is the existence of highly directional strong attrac-
8
6 corresponding bulk temperature. According to Nakan- tion(asthehydrogen-bond)betweentheunlikeparticles.
0 ishiandFisher[2,3,4,5]anincompressiblefluidmixture The dispersive (isotropic) forces between these particles
2 (with isotropic interactions) under the effect of confine- are assumed to be so weak as to favor segregationof the
1 ment displays a reduced tendency toward phase separa- species. Atlowtemperatures,theenergyfavorscomplete
2 tion at a fixed temperature, because the effective attrac- mixing. However, as the temperature increases, the en-
0
tive forces resultweakerwithin a pore. The confinement ergydecrementduetotheformationofdirectionalbonds
/
t also determines an effective dimensional reduction thus competes with the decrease of orientational entropy. As
a
m enhancing fluctuation effects. In addition, the presence aconsequence,thesystemphaseseparateswhenthetem-
of adsorbing walls may induce a surface transition in a perature raises. This mechanism has been confirmed by
-
d sub-critical fluid, even when this is in a single phase re- theoretical studies on lattice [11, 12, 13] and continu-
n gion in the bulk. Such a behavior, which is now well ousmodels[14], aswellasinrecentcomputersimulation
o understood by means of a mean field approach [6, 7, 8] studies [15, 16].
c is due essentially to the reduction of the contribution to
:
v the free energy from the the cohesive fluid-fluid forces How the presence of substrates or the confinement in
i caused by the presence of the confining walls. There apore affectthe behaviorofthese mixtures(if ina man-
X
exists, however, a class of non-simple fluid mixtures for ner similar to that of simple fluids, or if on the contrary
r which the situation is different: these are the strongly these mixtures are peculiar) is an issue of great practi-
a
associatingmixtures. Strong directionalattractiveinter- cal and theoretical interest. The presence of selectively
actionssuchashydrogenbondingorchargetransfercom- adsorbing substrates can interfere with the formation of
plexing between molecules affect dramatically the prop- bulk hydrogen-bonding, leading to a stabilization of the
erties of the fluid or solid. An interesting phenomenon two-phase region with respect to the completely misci-
thatoccursinsuchamixtureisthereentrantmiscibility: ble phase. Computer simulation studies on confined lat-
tice models [17] show the appearance of surface transi-
tions in a range of temperatures below the LCSP. The
samemechanismhasbeeninvokedtoexplainsomeexper-
∗Correspondingauthor: [email protected] iments [18, 19] that seemed to contradict the Nakanishi-
2
Fisher picture. ε =0
BB ε ε
In the present paper we shall study a lattice model
BB AB
of associating binary mixtures, introduced few years ago
by Lin and Taylor [20, 21] and reconsidered recently by
B
the present authors [22, 23, 24, 25]. In spite of its ide- ε
AA
alized nature, the model has the merit of allowing for
A
an explicit analysis of its non trivial phase diagram. As
an instance, the study of its bulk properties predicts the
ε =0
existence of closed-loops of immiscibility under constant
AB
pressure conditions. This analysis will be extended to
thebehaviorofthemixtureinpresenceofsubstratesand
V =0
under confinement. B V
A
The structure of the paper is organized as follows. In V
B
section II we briefly present the model and write explic-
itly the equations for the order parameter profiles. Sec-
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
tionIIIisdevotedtoanalyzeindetailthewettingbehav- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
iorofthemixtureinthepresenceofadsorbingsubstrates.
InsectionIVwestudythecapillarybehaviorofthesame
FIG. 1: Schematic representation of the A-B LT model in
systemwhenconfinedbetweentwoparallelplates(aslit)
the two-dimensional square lattice (the underlying lattice is
and in section V we consider the same system confined notshownforclarity). Relevantfluid-fluidandfluid-substrate
in a network of very narrow cylinders. Capillary transi- interactions are also highlighted (see text for explanation).
tions (and surface transitions for the slab geometry)will
be analyzed in terms of the physical parameters of the
system. Finally,insectionVIwepresentourconclusions.
II. THE MODEL
Thepresenceofsubstratesintroducesnewinteractions
Some years ago, Lin and Taylor proposed a lattice (see Fig. 1). We shall assume that the substrates do
modelto describe a binarymixture in whichthe twoun- notaffect the fluid-fluid interactions,i.e. there is no sur-
like species can form highly directional bonds. In more face enhancement. The interaction strength between an
detail, the Lin-Taylor (LT) model [20, 21] is an A-B A particle and the substrate is VA, if the particle is in
binary mixture lattice model, defined on a generic D- a cell in contact with the substrate, and zero otherwise.
dimensional lattice of coordination number ν. The A TheinteractionbetweentheBparticlesandthesubstrate
particles are allowed to singly occupy any of the lattice is orientationally dependent, i.e. its strength VB is non
cells. Every cell contains ν equal sub-cells, one for each vanishing only if the face of the B particle touches the
face of the mother cell. Again, each sub-cell can host substrate. The form of the substrate-fluid interactions
at most one B particle, provided the mother cell is not are consistent with the bulk interactions. Hereafter, the
occupied by an A particle. In addition to the repulsive substrate-fluidinteractionswillbeassumedasattractive,
interactions (which determine the previous occupation i.e. VA ≤0 and VB ≤0. For absolute values of VB large
rules), one includes short-ranged interactions (see Fig. enough, the substrate-B interactions compete with the
1). The interaction strength between a pair of nearest- bulk A-B interactions, so that one expect this type of
neighbor A particles is ǫ , ǫ between unlike species substrate to enhance the demixing tendency for temper-
AA AB
and ǫ between B-B particles. All these interactions atures lower than the corresponding LCSP.
BB
are non-vanishing only if the particles share a face.
The bulk properties, as well as the global phase dia-
gram, have been exhaustively analyzed in previous work To study the LT model in presence of substrates, we
[22, 23, 24, 25]. One of the main conclusions of such considerthegrand-canonicalensemble,wherethesystem
studies is that the appearance of closed loops of immis- isinthermalandchemicalequilibriumwithareservoirat
cibility occurs when the interaction between A-A and fixedtemperatureT andchemicalpotentialsµ andµ .
A B
A-B pairs are attractive (ǫ < 0, ǫ < 0), and The activities z and z are defined via z ≡exp(βµ )
AA AB A B A A
ǫ . (ǫ + min(ǫ ,0))/2. The latter condition is and z ≡ exp(βµ ), where β = 1/k T. After trac-
AB AA BB B B B
in complete agreement with the mechanism proposed by ing over all configurations of the B particles, the system
Hirschfelder et al [10] for the reentrant miscibility. In results to be isomorphic to a mono-component lattice
this work, we assume the bulk fluid-fluid parameters to gas, with renormalized interactions and chemical poten-
have values: ǫ = −1 (it defines the energy scale), tial. The details on this procedure can be found in Refs.
AA
ǫ = −0.65 and ǫ =0, so that we are in the regime [21, 23, 25]. The grand-canonicalpartition function Ξ of
AB BB
where closed-loops of immiscibility appear. the LT model can be written in terms of an equivalent
3
Ising model as: and n (i,s), the B particle density in the sub-cell s of
B
the cell i is
νN νK
Ξ = 1+2zB+zB2e−βǫBB 2 exp N H − 2 n (i,s)= 1−mi zBe−βVB (8)
(cid:0) (cid:1) ν⊥N(cid:18)b (cid:18) (cid:19)(cid:19) B 2v0 1+zBe−βVB
1+z e−βVB
B
× 1+2zB+zB2e−βǫBB! exp"Nb ∆H1 infth(ie,ssu)bis-cdeellfiinsebdyatsh:esubstrate(seeFig. 1). Otherwise,
B
νpK
+ ⊥2 !#ZDIs,iνng(K,H,∆H1) (1) n (i,s)= 1 zBe−βǫAB (1−m + m −hs s i)
B 4v0"1+zBe−βǫAB i j i j
whereN isthetotalnumberoflatticecells,N the num-
b z +z2e−βǫBB
nbeearroefstc-enlelsigahdbjoarcsenint ttohethdeirseucbtisotnratpee,rνp⊥entdhiceunlaurmtboerthoef +1+B2zB+BzB2e−βǫBB(1−mi−mj + hsisji)# (9)
substrate;ν isdefinedasthenumberofnearestneighbor
k
cells in directions parallel to the substrate. Obviously, where hsisji is the 2-spin correlation function between
ν = ν +2ν . Finally, ZIsing(K,H,∆H ) is the canon- the magnetization on the cell i and its nearest-neighbor
k ⊥ D,ν 1 j that it is associated to the sub-cell s. It is easy to see
ical partition function of the Ising model on the same
that in the homogeneous case these equations reduce to
lattice. Let us name K the effective coupling constant,
the expressions given in Refs. [23, 25].
H the effective bulk magnetic field, and ∆H , the effec-
1
The equivalence between the LT model and the Ising
tive surface magnetic field acting only on the boundary
model, allows one to obtain the surface behavior of the
sites:
mixturefromtheknowledgeofthebehaviorofthelatter.
βǫ 1 1+2z +z2e−βǫBB However, such a mapping, is not at all trivial and may
K = − AA + ln B B (2) lead in some cases to unexpected results.
4 2 p 1+zBe−βǫAB !
1 νβǫ ν
H = lnz − AA − ln(1+2z +z2e−βǫBB) III. WETTING TRANSITIONS IN THE LT
2 A 4 4 B B
MODEL
(3)
∆H = ν 1ln 1+2zB+zB2e−βǫBB Letus consideranA-Bbinarymixture inthe presence
1 ⊥"2 p 1+zBe−βVB ! of a planar, infinite and structureless substrate. Such a
substrateconfinesthefluidtothehalf-spacez ≥0. When
β 2VA the fluid (e.g., the A-particle rich phase, α) is at bulk
+ ǫ − (4)
4 AA ν coexistence, the substrate can promote the formation of
(cid:18) ⊥ (cid:19)#
drops of the opposite coexisting phase (the B-particle
rich phase β) on it, characterized by a contact angle θ.
All equilibrium properties can be obtained from the
When θ = 0, the drops spread over the substrate, i.e.
knowledge of the partition function Eq. (1) by using
thephaseβ wetscompletelytheα-substrateinterface. A
standard thermodynamical relationships. In particular,
finitevalueofθ,instead,correspondstoapartialwetting
the bulk pressure is obtained as:
situation. The crossoverbetween a partialto a complete
wetting regime is called the wetting transition [26]. The
lnΞ k T ν
p = lim k T = B ln 1+2z +z2e−βǫBB surface tensions involved in the problem (σ , σ and
V→∞ B V v0 "2 B B σ , corresponding to the α-substrate, β-suαbsstraβtse and
αβ
(cid:0) (cid:1)
α-β interfaces, respectively) are related to the contact
νK 1
− +H + lim lnZIsing (5) angle via the Young equation [27]:
2 N→∞N D,ν #
σ =σ +σ cosθ (10)
αs βs αβ
where v is the volume of lattice cell, and N ≡ V/v .
0 0
The mole fraction X (i) profile is obtained as: Thermodynamically the surface tensions correspond to
A
the excess grand-canonical free energy per interfacial
nA(i) unit area (respect to the bulk) [27]. Since both σαβ
X (i)≡ (6)
A nA(i)+ νs=1nB(i,s) and σβs have an analytical behavior at this thermody-
namic state, the wetting transition is a (surface) ther-
wheren (i),theAparticledenPsityinthecelli,isdefined modynamic phase transition, as it can be seen from Eq.
A
in terms of the local magnetization m as: (10) and the dependence of θ on the temperature. As
i
usual, appropiateorderparameterscorrespondingto the
1+m wetting transition can be defined as first derivatives of
i
n (i)= (7)
A 2v the surface tension respect to thermodynamic fields. In
0
4
p V =0 p ν ε /2 < V <0 0.3
A ΑΑ A
0.25
s| β | α s| β | α 0.2
p*
0.15
T Partial wetting T
V=0
0.1 B
p ν εΑΑ<VA< ν εΑΑ/2 p VA< ν εΑΑ 0.05 VVBB==0ε.Α5Α εΑΑ
0
s| α | β 0 0.1 0.2 0.3 0.4 0.5 0.6
s| α | β T*
Partial wetting T T FIG. 3: Wetting phase diagram for the square lattice LT
model. ThebulkparametersarefixedtoǫAB =0.65ǫAA <0,
ǫBB = 0, and VA = 0.25ǫAA. The thick solid line corre-
FIG.2: QualitativewettingphasediagramsfortheLTmodel sponds to the bulk critical line, and the thick dashed line to
for VB = 0. Thick solid lines correspond to critical lines, the liquid-vaportransition line of the A pure fluid. The thin
dashed lines to the A pure gas-liquid coexistence lines and solidlinescorrespondtothewettingtransitionlinesfordiffer-
thin solid lines to the wetting transition lines. The symbol ent values of VB. Hereafter thereduced units are defined via
s|α|β (s|β|α) means the bulk conditions where there exist T∗ ≡kBT/|ǫAA| and p∗ ≡pv0/|ǫAA|, where v0 is the volume
wetting of the β-substrate (α-substrate) interface by the α of a lattice cell. See explanation in text.
(β)phase, respectively(seetext). Thebulkconditionsunder
which thereis partial wetting are also shown.
tween the substrate and the phase α. We shall focus on
the coexistence side H = 0+ (H = 0−) for ∆H < 0
1
simple fluids, the excess number of particles per inter- (∆H >0), respectively (otherwise, only partial wetting
1
facial unit area or adsorption Γ (that corresponds to can occur). If the absolute value of the parameter is
−(∂σαs/∂µ)T,beingµthechemicalpotential)isthenat- larger than one (|∆H1/ν⊥K| > 1), then all coexistence
ural order parameter choice. In mixtures, other choices surface corresponds to a complete wetting situation. If
areavailable,e.g. thevolumeintegratedexcessmolefrac- such a condition is not fulfilled, one will observe a par-
tion per interfacial unit area. Microscopically, the wet- tial wetting situation up to a value K (dependent on
w
ting transition manifests itself as the growthof a macro- |∆H /ν K|)andbeyonditacompletewettingsituation
1 ⊥
scopicallythickβ layerintrudingintheα-substrateinter- up to K =K .
c
facecompositionprofile. Threedifferenttypesofwetting Let us consider first the case V = 0, when ∆H de-
B 1
transition may arise. If the layer width jumps suddenly pends only on T, regardless the value of z . Moreover,
B
from a microscopically finite value, the wetting transi- the sign of ∆H is uniquely determined by the differ-
1
tion is first order, and an off-coexistence surface transi- ence between V and ν ǫ /2 over the whole temper-
A ⊥ AA
tion (prewetting) is associated to it. If the layer width ature range. One can envisage various cases (see Fig.
diverges continuously, the wetting transition is named 2): if V = 0, the coexistence corresponds to a sit-
A
critical. Finally, the wetting transition can occur as the uation of complete wetting of the interface substrate-
accumulation point of an infinite sequence of first-order α phase by phase β, denoted by the symbol s|β|α. If
transitionsinthe layeringwettingcase(thesetransitions ν ǫ /2<V <0,thereexistsaregionofcompletewet-
⊥ AA A
extend to the off-coexistence region). The conditions for ting s|β|α anda lowerpressureregionofpartialwetting.
such a transition depend subtly on the fluid-fluid and As V decreases, the region of complete wetting s|β|α
A
substrate-fluid interactions, and interfacial fluctuations shrinks and eventually disappears for V = ν ǫ /2,
A ⊥ AA
can play an important role [28]. because ∆H ≡ 0. For V < ν ǫ /2, the sign of ∆H
1 A ⊥ AA 1
In the LT model two-phase coexistence occurs when changes and a region of complete wetting of the inter-
H = 0 and K > K , where K is the lattice dependent face substrate-phase β by phase α appears (s|α|β). As
c c
Ising critical coupling. Moreover, the wetting behavior VA increases, the region of partial wetting reduces and
is controlled by the ratio ∆H1/ν⊥K, that is the ratio disappears for VA ≥ν⊥ǫAA.
between the surface energy and the loss of bulk energy When V 6= 0, it is possible that in correspondence
B
due to the presence of the surface. If such a param- of different state points the substrate favors either the β
eter is positive the substrate favors the α-phase to in- or the α phase. To study such an effect it is necessary
trude between the substrate and the β-phase. On the to locate the coexistence states where ∆H = 0. Since
1
other hand, if ∆H /ν K < 0 the phase β intrudes be- V and ǫ are negative, for a temperature T all the
1 ⊥ B AB
5
0.3 0.3
0.25 0.25
p*0.2 VB=0.7εΑΑ VB=εΑΑ p*0.2 VB=εΑΑ VB=1.25εΑΑ
0.15 VB=0.5εΑΑ 0.15
VB=0.35 εΑΑ VB=2εΑΑ
0.1 0.1 VB=0.75 εΑΑ
V=0
B
0.05 0.05
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6
T* T*
FIG.4: ThesameasFig. 3butforVA=0.75ǫAA. Thedotted FIG. 5: The same as Fig. 4 but for VA = ǫAA. See text for
linescorrespondtothecoexistencepointsatwhich∆H1 =0. explanation.
See text for explanation.
substrate-A vapor. For values of V ≃ 0, ∆H > 0 for
coexistence states with pressure p larger than p (T)≡ B 1
nw all conditions of coexistence, and the configurations are
p(T,H = 0,∆H = 0) correspond to ∆H > 0, whereas
1 1 s|α|β. ForV <(V /ν )+ǫ −ǫ thereappearstates
states with lowerpressurescorrespondto ∆H <0. It is B A ⊥ AB AA
1 characterizedby∆H <0. Asaconsequence,withinthe
1
easytoprovefromEq. (4)thatanecessaryconditionfor
coexistence regionat the left of the line p (T) the sub-
theexistenceofthislineisthatV +ǫ /2<(V /ν )≤ nw
B AA A ⊥ strateadsorbspreferentiallyparticlesB, andthe wetting
ǫ /2. However, ∆H changes sign for a temperature
AA 1 is of type s|β|α. On the other hand, at the right of the
T only if the value of the pressure which corresponds
curve p (T) the wetting is of type s|α|β. The line of
to ∆H = 0 is less than the critical pressure at such nw
1 wetting transition touches tangentially the critical curve
temperature.
at the point where the curve in turn p (T) crosses the
In order to illustrate such a phenomenology, we shall nw
critical line. Such a point moves monotonically toward
study the 2D square lattice case (ν =4, ν =1). In this
⊥ higher temperatures as V decreases,driving with it the
case, only critical wetting can occur (fluctuation effects B
wetting transition line and the p (T) curve. As also
preclude any other possibility) and the value of K at nw
w happened for the ν ǫ /2 < V < 0 case, the wetting
whichthe transitionofcriticalwettingoccursisgivenby ⊥ AA A
transitionline froma s|β|α complete situationto partial
solution of the equation [29]
wetting moves to lower pressures, convergingtowardthe
A-pureliquid-vaporcoexistencelineforV →−∞(how-
exp(2K )[cosh(2K )−cosh(2∆H )]=sinh(2K ) (11) B
w w 1 w ever, along this line there is complete wetting by liquid
of the vapor-substrate interface over all the temperature
Let us first consider the case ν ǫ /2 ≤ V < 0:
⊥ AA A
range).
∆H < 0, for arbitrary values of T, z y V . More-
1 B B
over, ∆H1 becomes more negative as VB decreases, at When VA ≤ ν⊥ǫAA (see Fig. 5), for values VB near
fixed T and z (or pressure p). The typical diagrams zero, the coexistence corresponds to complete wetting
B
of complete wetting are displayed in Fig. 3. The region s|α|β. Only when VB < (VA/ν⊥) + ǫAB − ǫAA states
between the wetting transition line and the critical line of partial wetting reappear, near the line pnw(T). The
corresponds to a situation of wetting s|β|α, and the re- behaviorofthelatteraswellasthewettingtransitionline
maining coexistence region corresponds to a situation of is similar to the one observed in the previous case, with
partial wetting. The complete wetting zone increases as the difference that the right branch of the wetting tran-
V becomes more negative. sition line (that corresponds to a transition from partial
B
When (V /ν ) < ǫ /2, the substrate also favors to s|α|β complete wetting) begins at zero temperature.
A ⊥ AA
the adsorption of particles A. This determines a com- Oneexpectsthatthebehaviorofthewettingtransition
petition between A and B species. In general, when for the three-dimensional case will be qualitatively simi-
(V /ν )−ǫ /2 < 0.1ǫ β-phase preferential adsorp- lar to the two-dimensional case. However, the complete
A ⊥ AA AA
tion can only occur if V −(V /ν ) < ǫ −ǫ . Fig- wetting phase diagram for the three-dimensional Ising
B A ⊥ AB AA
ures 4 and 5 display the wetting diagrams for ǫ < model is not totally understood yet. Theoretical [30]
AA
(V /ν ) < ǫ /2 and (V /ν ) ≤ ǫ , respectively. In and simulation studies [31] show, in the case of strictly
A ⊥ AA A ⊥ AA
the first case, the complete wetting transition line orig- short-ranged forces and in the absence of an enhanced
inates at T = 0 and terminates at the temperature of surface spin-spin coupling, that the wetting transition is
complete wetting for the A-pure liquid in the interface continuous only if K is smaller than K , where K is
w R R
6
2
the value of the coupling constant corresponding to the
roughening transition (K ≈ 1.85K for the simple cu-
R c
bic lattice). However, if such condition is not fulfilled, s|β|α
the wetting transition will occur through a sequence of 1.5
first order layering transitions. Theoretical studies have
claimed that the wetting transition can be weakly first-
order instead of second order [32]. In the present work p* 1
we will consider the Ising model in the frameworkof the Partial wetting
mean-field Bragg-Williams approximation:
s|α|β
1 0.5
lim lnZ (K,H,∆H )=∆H m
Ising 1 1 1
N⊥→∞N⊥
∞ ν
+ kKm2+ν Km m +Hm 0
2 j ⊥ j j+1 j 0 0.5 1 1.5
"
Xj=1 T*
1+m 1+m
j j
− ln
2 2 FIG. 6: Wetting phase diagram for the simple cubic (sc) LT
(cid:18) (cid:19) (cid:18) (cid:19)
model in the mean field approximation. The substrate-fluid
1−m 1−m
− j ln j (12) interactions are VA = ǫAA and VB = 1.25ǫAA. The critical
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)# line(thicksolid),A-purevapor-liquidtransitionline(dashed),
and pnw(T)line (dotted)are plotted. The wettingtransition
where N is the number of cells in a layer, an {m }, points (diamonds) are also represented. The line that joins
⊥ j
j = 1,2,...∞ is the equilibrium magnetization profile them is only for eye-guide.
obtained by solving the Euler-Lagrangeequations:
m =tanh(ν Km +ν K(m +m )+H) (13)
j k j ⊥ j−1 j+1 (∆H < 0), respectively, if j ≤ j , and m = m for
1 0 j b
j > j ) as initial magnetization profiles. Starting the
where m ≡ ∆H /ν K. This set of equations must 0
0 1 ⊥
iteration with different initial profiles, in general, the al-
be solved in conjunction with the asymptotic condition
gorithm converges toward different local minima of the
m →m for j →∞, where m is the spontaneous mag-
j b b
freeenergy. Asalreadymentioned,theglobalfreeenergy
netizationpersiteinthebulkcasewithcouplingconstant
minimum gives the true equilibrium magnetization pro-
K andmagneticfieldH. Ingeneral,differentsolutionsof
file. Finally,wehaveconsidereddifferentvaluesofN ,
Eq. (13) are found. The equilibrium profile is the solu- max
since the equilibrium profile in a complete wetting situ-
tion that maximizes the functional (12). This approach
ation has a strong size dependence. On the contrary, no
neglectscapillaryfluctuations,thatitisequivalenttoas-
appreciable finite width effects occur in a partial wet-
sume K ≡ K and the wetting transition occurs as a
R c
ting situation for N large enough. In order to locate
sequence of layering transitions [33, 34]. Once the equi- max
thewettingtransition,westudiedtheequilibriumprofiles
librium magnetization profile is computed, the surface
at bulk coexistence (H = 0) and constant temperature,
tension σ is obtained via:
varying the pressure.
σ =(Ω+pV)/A (14) The wetting phase diagram for V = ǫ and V =
A AA B
1.25ǫ is plotted in Fig. 6. As one can see, the quali-
AA
where Ω = −k T lnΞ, with Ξ defined by the Eqs. (1)
B tative behaviorofthe wetting transitionis similarto the
and (12), and the pressurep by Eq. (5). The adsorption
one observed in the two dimensional case. However, the
Γ, that it is the relevant order parameter in the wetting
wettingtransitionisassociatedtoseriesoffirstorderlay-
transition, is defined as:
ering transitions between surface phases with the same
surface tension, but different adsorptions. The behav-
∞
Γ= (X (j)−Xb) (15) ior of the layeringtransitions at low temperatures shows
A A
surprising features. Fig. 7 shows the first four layering
j=1
X transitions for T∗ ≡ k T/|ǫ | = 0.25. It is observed
B AA
where X (j) is the A molar fraction profile Eq. (6) by that the first layering transitiondoes occur for pressures
A
using the approximation hs s i ≈ m m . Finally, Xb is higher than the bulk critical pressure. Consequently, for
i j i j A
its bulk value. afixedpressurebetweenthefirstlayeringtransitioncriti-
We have studied the simple cubic lattice, for which calpointandthebulkcriticalpointcorrespondingtothis
ν = 6 and ν = 1. The magnetization profiles have temperature, the layering transitionwill extend to lower
⊥
been obtained by an iterative method of the Eq. (13) temperatures than the critical temperature correspond-
for j = 1,...,N , imposing the condition m = m ing to the LCSP. This behavior is found only for s|β|α
max j b
for j > N . In order to sample the solution space, completewettinginthelowtemperaturezone. Thepres-
max
we use sharp kink profiles (m = +1 (−1) for ∆H > 0 sureisanincreasingfunctiononH forfixedtemperature
j 1
7
tuations preclude the existence of capillary transitions
0.8 if the interactions are short-ranged. Consequently, only
the 3D case will be studied. We shall consider a sim-
ple cubic lattice within the mean field approximation.
0.6 The substrate-fluidinteractions are set to V =ǫ and
A AA
V = 1.25ǫ , that stabilize the first (surface) layering
B AA
p* 0.8 transition at temperatures lower than the lower critical
0.4 p* temperature. We also assume that the separation be-
c
p*0.6 tween plates is N lattice units. The free energy of the
D
confinedmixtureisgivenbyEq. (1),withthemeanfield
0.4
0.2 partition function Z (K,H,∆H ) given by
Ising 1
0.2
-4 -2 0
Γ 1
lim lnZ (K,H,∆H )=∆H (m +m )
00 0.2 0.4 0.6 0.8 1 N⊥→∞N⊥ Ising 1 1 1 ND
X
A +ND νkKm2+ν Km m +Hm
2 j ⊥ j j+1 j
FIG. 7: Layering transitions for T∗ = 0.25 for the sc LT Xj=1"
model in the mean-field approach. The interaction couplings 1+m 1+m
j j
− ln
are thesame asin Fig. 6. The dashed line correspond to the
2 2
bulk coexistence, and the solid lines correspond to the bulk (cid:18) (cid:19) (cid:18) (cid:19)
conditions corresponding to the layering transitions. In the 1−m 1−m
j j
− ln (16)
inset, the coexistence adsorptions for the layering transition 2 2
(onlythefirstfourtransitions areplotted). Thearrowshows (cid:18) (cid:19) (cid:18) (cid:19)#
thebulk critical pressure.
The equation for the equilibrium profile is given by
Eq. (13) with the boundary condition N = N /2,
max D
for N even, or (N +1)/2, for N odd. The values
T andz (sobothK and∆H arefixed). Consequently, D D D
B 1 of m for j ≥ N are obtained from the symmetry
as K > K for the layering transition critical point, the j max
c of the problem: i.e. m = m . In addition to
corresponding pressure can only be bigger than the bulk j ND−j+1
the layering transitions (related to the phenomenology
critical pressure if the transition occurs for H >0. This
observed on the semi-infinite case), the capillary transi-
impliesthat∆H mustbenegative,andthusthewetting
1 tions results from the shift ofthe bulk coexistence under
is of type s|β|α. The physical interpretation of this phe-
confinement. Near the center of the pore both phases
nomenonisthatthesurfaceadsorbspreferentiallyBpar-
havedifferentmagnetizations,incontrastwiththelayer-
ticles. SuchamechanismcompeteswiththeA-Bbonding
ingandprewettingtransitionswheretheprofilesdifferin
(atleastclosetothe substrate). Therefore,the tendency
a localized surface region and they have the same values
to phase separate in a fluid layer close to the substrate
farenoughfromthatregion. IntheIsingmodel,theshift
isenhanced(becausethe A-Bbondingfavorsmixing)for
is always toward higher values of K (i.e. lower effective
temperatureslowerthanthe(lower)criticaltemperature.
temperature) and H negative (H positive) when ∆H
This fact was also observed in computer simulations of 1
positive (∆H negative), respectively.
a different model of associating binary mixture in Ref. 1
The cases N = 1 and N = 2 can be studied ana-
[17]. However, we must stress that this phenomenon is D D
lytically since m =m for all j. The bulk conditions for
notrelatedtoanenhancedcoupling(i.e. theinteractions j
which the capillary transition takes place are shown in
between particles in the first layer close to the substrate
Fig. 8. In spite of the fact that the temperature range
are increased respect the bulk values), that it is known
for which the capillary transition occurs is reduced, at
to stabilize surface transitions to temperatures higher to
lowtemperatures the pressurerangeis largelyincreased.
the bulk criticalone in the Ising model[2]. Onthe other
Hence,atagivenpressure,theLCSPshiftstolowertem-
hand,thecompetitionbetweensubstratepreferentialad-
peratures and can even disappear. This means that for
sorptionandtheA-Bbondingshouldbealsorelevantfor
certain values of the pressure range there are no immis-
prewetting transitions.
cibility islands in the confinedsystem, but the usual bell
shaped coexistence line terminating in a UCSP. This is
a direct consequence of the directional character of the
IV. CONFINEMENT BETWEEN SYMMETRIC substrate-particle B interaction. The kink that it is ob-
WALLS served in the capillary critical line corresponds to the
crossing to the p (T) line.
nw
We turn to consider the behavior of the Lin-Taylor For larger values of N , the behavior of the capillary
D
model when confined between two identical parallel sub- transitions are studied numerically. As N → ∞ the
D
strates. Confinement in the two-dimensional case corre- capillary critical line must move towardthe bulk critical
sponds to a quasi-one-dimensional situation. Thus fluc- curve. We shall consider the isotherm T∗ = 0.25, as a
8
2 1.5
1.5
1
p* 1 p*
0.5
0.5 0.7
p*
0.6
00 0.5 1 1.5 00 0 0.2 0.2 0.40.4XA 0.6 0.8 1
T* X
A
FIG. 8: Capillary transitions of the mean field sc LT model FIG. 9: X¯A−p phase diagrams for T∗ = 0.25 and different
(VA = ǫAA, VB = 1.25ǫAA) for ND = 1 (thin solid line) and
values of ND: ND = 1 (dashed-double dotted line), ND = 2
ND = 2 (dot-dashed line). In both cases, upper line is the
(dashed line), ND = 5 (dotted line) and ND = 15 (thick
capillary critical line, and lower line is the A-pure capillary
solid line), underthe same conditions as in Fig. 8. The bulk
gas-liquid transition. The bulk conditions under which there
transition(thinsolidline)isalsoplottedforcomparison. The
is α-β capillary coexistence are the regions enclosed by their
inset corresponds to an enlargement of the zone around the
respective lines, and the T = 0 axis. By comparison, the
bulk critical point.
bulk critical line (thick solid line) and the A pure gas-liquid
transition line (thickdashed line).
The values of the capillary critical pressure for small
N are larger than the corresponding bulk critical val-
representative case of the low temperature regime. The D
ues. By increasing N , the capillary critical pressure
excess free energy ω reads: D
decreases up to a minimum, corresponding to a pressure
less than the bulk critical pressure. From such value it
ω =(Ω+pV)/A (17)
converges monotonically to the bulk critical value (see
figure 10). This fact means that along an isobaric the
and converges to 2σ when N → ∞, with σ defined by
D
LCSP shifts to lower temperatures only for very narrow
Eq. (14). The relevant order parameter for the capillary
transitions is the average molar fraction X¯ , defined as pores. Consequently,thestabilizingmechanismoftheco-
A
existence region below the bulk LCSP seems to be only
effective for small values of N . The deviations of the
1 ND D
X¯ ≡ X (j) (18) capillary critical pressure with respect to the bulk value
A A
ND j=1 intherange5<ND ≤15followapowerlaw∆p∼ND−x,
X
with x = 1.04±0.09. This behavior is consistent with
thatconvergestothe bulk coexistencemolarfractionsas the findings for the confined lattice gas for intermediate
ND →∞. values of ND reported in Ref. [35]. The scaling behav-
We have obtained the constant temperature phase di- ior ∆p ∼ N−1/ν (N−2 in the mean-field approach) pre-
D D
agrams for T∗ = 0.25 (see Fig. 9). Various transitions dicted by the Nakanishi-Fisher theory should be obeyed
are observed for N ≥ 3. One of these transitions con- for larger values of N . However, our results do not
D D
vergestowardthe bulk phase transitionfor largesepara- extend to that region due to the numerical error in es-
tions and is identified with the true capillary transition. timating the capillary critical pressure close to the bulk
The remaining transitions are characterized by having critical point. Nevertheless, we do not expect a change
similar average molar fractions and correspond to the inthetendencyforlargevaluesofN andweexpectour
D
layering transitions observed in the presence of a single resultsto be qualitativelycorrectforallvaluesofN . It
D
substrate. The coverages (as defined by Eq. (15)) cor- is interesting to note that the slab approximation [35],
responding to the coexisting phases in these transitions which corresponds to set m =m for all j, predicts that
j
convergeveryfasttothesemi-infinitevalues,asexpected for sufficiently small T∗ the capillary critical pressure is
from their surface nature. However,the number of these larger than the bulk value for all the values of N . The
D
transitions depends strongly on N , since the capillary failure of this approach is not surprising since it gives
D
transition,whichdependssensiblyonN ,competeswith poor results for the confined lattice gas close to the cap-
D
them [35, 36]. For instance, in the case of 3 ≤ N < 10 illary critical point as it overestimatesboth K and H at
D
only the first layering transition appears, while the sec- the capillary critical point [35]. This fact alters the bal-
ond appears for N ≥10. ance between the different mechanisms involved in the
D
9
1.4
0.75
1.2
p* 0.7
p* 1 0.65 A
00 55 1100 1155
N
D
0.8
B
0.6
00 55 1100 1155
N
D
∗
FIG. 10: Capillary reduced critical pressures p versus ND
∗
for T = 0.25 (same conditions as Fig. 8). The dotted line
corresponds to the bulk critical pressure. Inset: enlargement
of the minimum zone.
shift of the critical point, leading to the wrong result FIG.11: Schematicrepresentation oftheLTmodeladsorbed
inanetworkofnarrowinfinitelylongcylinders. TheAparti-
mentioned above.
clesarerepresentedassphericalparticles,andtheBparticles
assphericalparticleswithaninteractionside(shadowcircle).
In this case, the coordination numberis ν⊥ =6.
V. THE CYLINDRICAL PORE NETWORK volume reads:
Ω k T ν
= − B ⊥ ln 1+2z˜ +z˜2e−βǫ˜BB
Finally, we consider the confinement of the LT mix- V v 2 B B
0 "
ture in a network of parallel infinitely long, cylindrical (cid:0) (cid:1)
ν K
pores. This model canbe understoodas a crudeapprox- + ln(1+2z +z2e−βǫBB)+H − ⊥ ⊥
B B 2
imation for adsorption in zeolites. It represents another
instance in which the substrate can prevent the forma- 1
− K + lnZ (K ,K ,H) (19)
tion of hydrogen bonds: the geometrical hindrance due k N Ising ⊥ k
#
to the confinement in very narrow nanopores. We shall
assume that the pore diameter is so small that the A- where z˜ ≡ z e−βVB is the effective activity of the B
B B
B bonds can form only along the direction z parallel to particlesorientedintheplaneperpendiculartothecylin-
the cylinder axis (see figure 11). The cylinder diameter der axis, and Z is canonicalpartition function of an
Ising
is taken to be less than twice the A-B collision diame- anisotropicIsingmodelcharacterizedbycouplingsK in
k
ter. In addition, the pores have diameters less than two the paralleldirectionsto thecylinderaxisandK inthe
⊥
A particle diameters and they form a two dimensional perpendicular directions, and an effective magnetic field
network with coordination number ν⊥. We allow atten- H. Kk has the same expression as the bulk coupling
uated interactions between particles in nearest neighbor constant Eq. (2), while K and H are defined as:
⊥
pores, in order to allow a capillary transition. A similar
zeolite model has been employed to study the capillary βǫ˜ 1 1+2z˜ +z˜2e−βǫ˜BB
K = − AA + ln B B (20)
transition of methane adsorbed in ALPO4-5 [37]. ⊥ 4 2 p 1+z˜Be−βǫ˜AB !
Thein-porefluid-fluidinteractionshavethesameform
as in bulk, with coupling constants ǫAA, ǫAB and ǫBB. H˜ = H +ν β ǫ −ǫ˜ − 2VA
However, the interactions between particles in nearest- ⊥ 4 AA AA ν
" (cid:18) ⊥ (cid:19)
neighbor pores are attenuated to ǫ˜ , ǫ˜ and ǫ˜
AA AB BB
(|ǫ˜ij| < |ǫij|), although the A-B and B-B interactions + 1ln 1+2zB+zB2e−βǫBB (21)
directional character remains the same as before. This 4 (cid:18)1+2z˜B+z˜B2e−βǫ˜BB(cid:19)#
system behaves as a bulk lattice model with anisotropic
interactions. The grand-canonical free energy per unit with H given by Eq. (3). One can see that confinement
10
shifts the effective magnetic field H with respect to its 0.6
0.6
bulk value with a contribution analogous to that found
0.5
for planar substrates (compare Eqs. (4) and (21)). In 0.5 T*0.4
addition, the confinement determines an anisotropy in 0.3
the couplings. The A particle molar fraction X reads: 0.4 0.2
A
n p* 0.1
A
XA = n +n1 +n2 (22) 0.3 00 0.2 0.4 0.6 0.8 1
A B B
X
A
wheren =(1+m)/2v istheAparticledensity,andn1
A 0 B 0.2
(n2)isthedensityofBparticlesorientedinaperpendic-
B
ular(parallel)directiontothecylinderaxis,respectively:
0.1
ν z˜ +z˜2e−βǫ˜BB z˜ e−βǫ˜AB
n1 = ⊥ B B + B
B 4v0 (cid:18)1+2z˜B+z˜B2e−βǫ˜BB 1+z˜Be−βǫ˜AB(cid:19) 00 0.1 0.2 0.3 0.4 0.5 0.6
ν z˜ +z˜2e−βǫ˜BB z˜ e−βǫ˜AB T*
− ⊥ B B − B
4v0 (cid:18)1+2z˜B+z˜B2e−βǫ˜BB 1+z˜Be−βǫ˜AB(cid:19) FIG. 12: Capillary transitions in the network of cylindrical
∂lnZ /N
Ising pores. Thereservoirconditionscorrespondingtothecapillary
×
∂K critical lines are represented for different values of α: α =
(cid:18) ⊥ (cid:19)
0.2 (dashed line), α = 0.4 (dotted-dashed line) and α = 0.6
ν z˜ +z˜2e−βǫ˜BB
− ⊥ B B m (23) (dotted line). The bulk critical line (solid line) and the A
2v01+2z˜B+z˜B2e−βǫ˜BB pure vapor-liquid line (long dashed line) are also plotted. In
∗
the inset, the phase diagrams corresponding to p = 0.07
1 z +z2e−βǫBB z e−βǫAB
n2 = B B + B for the same values of α (the symbol meaning is thesame as
B 2v0 (cid:18)1+2zB+zB2e−βǫBB 1+zBe−βǫAB(cid:19) previously)andtheircomparisonwiththebulkphasediagram
1 z +z2e−βǫBB z e−βǫAB (solid line).
− B B − B
2v0 (cid:18)1+2zB+zB2e−βǫBB 1+zBe−βǫAB(cid:19)
∂lnZ /N
Ising
×
∂K
(cid:18) k (cid:19)
1 z +z2e−βǫBB
− B B m (24)
v01+2zB+zB2e−βǫBB bulk value) less than the isotropic bulk value, the sys-
temcanundergoaphasetransitionforK largeenough.
⊥
Phases characterized by different molar fractions X
A Note that since K = −βαǫ /4, this condition is ful-
may coexist when the spontaneous (i.e. H˜ = 0) magne- ⊥ AA
filled in the low temperature region. This fact is also
tization m=±|m|=6 0. We consider the case ǫ =−1,
AA observedin the mean-fieldapproach,for whichthe coex-
ǫ =−0.6andǫ =0,thatcorrespondstoacasethat
AB BB istence condition is ν K +2K ≥1.
⊥ ⊥ k
presentsclosed-loopsofimmiscibilityinthebulk. Onthe
other hand, as the hydrogen bond interactions are very
short-ranged, we will take ǫ˜AB = ǫ˜BB = 0 as physically Figure 12 shows the reservoir conditions under which
sensible values. Moreover, ǫ˜AA is set to be αǫAA, where the capillary phase transition occurs for different val-
α is a positive number less than one. ues of α. There also exist immiscibility islands for the
Inordertodiscriminatebetweendifferentmechanisms, capillary transition within the porous medium (see in-
we chose the substrate-fluid interactions in such a way set in Fig. 12), but the corresponding LCSP occurs
that H˜ = H, i.e. VA = ν⊥(1−α)ǫAA/2 and VB = 0. at temperatures lower than those of the bulk LCSP at
Hence, the preferential substrate adsorption effect (very the same pressure. As α, increases the temperature and
similar to the one studied in the slab geometry) is sup- pressure range for which one observes capillary transi-
pressed and we can focus on the coupling anisotropy. tion increases. Hence, the immiscibility islands become
We have studied the 2D square lattice case, for largerwithα,butmainlyinthehightemperatureregion.
which analytical expressions for ZD,ν are known in the In fact, the temperature corresponding to the LCSP is
anisotropic case [38]. However, we expect that the fea- rather insensitive to the value of α, and it is always be-
tures obtained for this case will be qualitatively correct low the temperature corresponding to the bulk LCSP.
also for the 3D case. From the exact solution, the val- Our findings can be understood by a simple physical ar-
uesof(K⊥,Kk)thatcorrespondtothephasecoexistence gument. Thegeometryforbidstheformationofhydrogen
satisfy the following relationship: bondings in the directions perpendicular to the cylinder
axis. So, the confinement effectively reduces the A-B
sinh(2K )sinh(2K )≥1 (25)
k ⊥ bonds respect to the bulk case, and consequently the
whereequalityholdsonlyatthe criticalpoint. Itisclear phase separation is enhanced in the lower temperature
from such an expression that, even for K (equal to the regime, in complete agreement with our results.
k