Phase behavior of symmetric binary mixtures with partially miscible components in spherical pores. Density functional approach

Journal of Molecular Liquids 112 (2004) 81–89

Phase behavior of symmetric binary mixtures with partially miscible components in spherical pores. Density functional approach Alexandr Malijevskya, Orest Piziob,*, Andrzej Patrykiejewc, Stefan Sokolowskic a Prague Institute of Chemical Technology, Prague, Czech Republic Instituto de Quimica, UNAM, Circuito Exterior, Coyoacan 04510, Mexico, DF, Mexico c Department for the Modelling of Physico-Chemical Processes, Maria Curie-Sklodowska University, 20031 Lublin, Poland b

Abstract We investigate adsorption in spherical pores of model symmetric binary mixtures exhibiting demixing in bulk phase, by using a density functional approach. Our focus is on the evaluation of the upper part of the coexistence envelope for the first-order phase transitions in adsorbed fluids, and the lines separating mixed and demixed phases. We show that the scenario for phase transitions is sensitive to the pore radius. This parameter can change the type of the phase diagram for the confined fluid comparing to its bulk counterpart. In particular, for not very big spherical pores and for strong wall–fluid interactions, the capillary condensation between mixed phases is strongly depressed, such that only condensation from mixed gas to demixed liquid phase terminating at tricritical point is observed. 䊚 2003 Elsevier B.V. All rights reserved. PACS: 68.43.De; 68.43.Fg; 64.70.-p; 64.70.Fx Keywords: Adsorption of mixtures; Spherical cavities; Phase transitions; Demixing

1. Introduction It is our honor and pleasure to contribute to this special issue dedicated to the 60th birthday of Myroslav Holovko. We appreciate his contributions into several topics of research in statistical mechanics of fluids, in particular, into the research in non-uniform fluids. Single-component fluids confined to micropores can exhibit several types of phase behavior, due to the competition between fluid–solid and fluid–fluid interactions, see e.g. w1x. In the majority of technologically important processes, we usually deal with the adsorption of mixtures rather than of one-component fluids. This is one of the reasons to investigate adsorption of fluid mixtures in pores. The number of theoretical and simulational works dedicated to that problem grows permanently, cf. Refs. w2–16x. Description of phase behavior of bulk fluid mixtures w17–22x, in spite of certain advances, remains incomplete in several aspects. Although several possible phase diagram topologies have been placed into a number of *Corresponding author. Fax: q52-5-616-2217. E-mail address: [email protected] (O. Pizio).

categories w22x, the knowledge which microscopic features are responsible for yielding a given category is still not satisfactory. The confinement of a fluid mixture leads to further complications. Consequently, theoretical description of phase behavior of confined binary mixtures remains at its beginning stage. Recently, in series of publications w23–25x we have undertaken the study of that problem within the framework of a density functional approach. However, our attention has been restricted to the slit-like geometry of confinement. The aim of the present study is to extend such a description for binary mixtures in spherical pores, i.e. in cavities. The model used by us can be, though as a very simple model of adsorption, the adsorbents of a ‘zeolitetype’, i.e. in an ensemble of cavities, connected together by narrow channels. The adsorbed fluid is assumed to be in equilibrium with a bulk fluid. The mechanism for achieving this equilibrium, as well as the mechanism for the mass interchange between cavities is not specified in our formalism. We assume that such mechanism does not affect spherical symmetry of the pore, and also that the exsistence of the connections between cavities can be neglected, so we can consider only a single

0167-7322/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-7322(03)00278-2

82

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

cavity. Therefore the basic thermodynamic condition controlling equilibrium at fixed temperature is the equality of chemical potentials of all components in confined and in bulk systems. To perform our study, we use a simple model of a binary mixture. This model is known in the literature as the symmetric binary mixture model w14,20,21x. This model includes two species, is1, 2, of equal diameters, s1ss2ss thus, it ignores the influence of the size ratio on the phase behavior). Additivity of diameters is assumed, s12s0.5(s1qs2)ss. The interactions between like particles are chosen equal, with identical functional form (e.g. Lennard–Jones potential), and are characterized by the same energy parameters, ´11s ´22s´. The ‘cross’ interaction potential between unlike particles has also the same functional form, and is characterized by the energy parameter ´12-´. The reduction of the number of potential parameters permits for transparent interpretation of the results, and allows for establishing clear links between microscopic quantities and the resulting phase behavior. We also assume that spherical pore walls are energetically uniform, i.e. the pore walls–fluid particle potentials depend only on the distance normal to the wall. For simplicity and in order to reduce the number of the parameters to a minimum, the fluid–solid potentials are chosen independent of adsorbate species. In spite of apparent simplicity, a homogeneous binary symmetric mixtures exhibit quite interesting thermodynamic behavior. Of particular importance is the modification of the coexistence curve for the first-order gas– liquid phase transition by the occurrence of the second-order demixing transition. If the value of the ratio ´12 y ´ changes, then a non-trivial interplay between these two transitions can take place. Wilding et al. w21x have proposed a classification scheme, which distinguishes three types of topology of the phase diagram for such simple binary mixture. It is not known at present how and to what extend each of the types of the bulk phase diagrams is affected by the confinement in spherical pores. The principal objective of our work is to consider such a problem. We shall investigate the phase behavior of a set of model fluids, exhibiting different demixing in a bulk system. The results obtained here permit us to discuss general trends and to classify topology of the phase diagrams for inhomogeneous binary symmetric mixtures. They can serve as an useful benchmark for further investigations of more sophisticated models. 2. The models and theory As we have already emphasized, the system in question is a two-component fluid of spherical particles of species 1 and 2 interacting via the Lennard–Jones (LJ) potential, truncated for technical reasons,

S LJ r-rcut Tuij Žr., uijŽr.sU T V0, rGrcut

(1)

wB s E12 B s E6z F yC F |, uLJ ij Žr.s4´ijxC D rG ~ yD r G

(2)

where rcut is the cut-off distance. We would like to introduce the reduced units from the very beginning. Throughout this work, the parameters s and ´ are chosen as units of length and energy, respectively. Then, the reduced temperature is defined as common, T*s kTy ´. In all our calculations the cut-off distance was the same for all the components, rcut y ss2.5. The choice of the value of the energy parameter for the Lennard–Jones potential between unlike particles, ´12 y ´, is crucial. In this work, we study three fluids with ´12 y ´s0.75, 0.65 and 0.55. These models are abbreviated as M75, M65 and M55, respectively, for the sake of convenience. In the bulk, these models yield three different types of the phase diagrams described below. The fluid is confined in a spherical pore of the radius Rc. The adsorbing potential V(r) is identical for both species. To calculate this potential for the spherical pore, we assume that the surface of the cavity consists of a set of ‘smeared out’ Lennard–Jones centers w26x. Therefore the total cavity surface—a test particle potential— is calculated by integration over the whole set of attractive elements to yield,

|

12

6

VŽr.s4´gs dr9 wysgsysŽr9.z~ ywysgsysŽr9.z~ ,

µ

x

|

x

|

∂

(3)

where r is the distance from the center of the cavity, s(r9) is the distance from the point (0, 0, r) to a point on the surface of the cavity of the radius Rc. Without loss of generality, we have chosen sgs y ss0.5 for both components. The parameter ´gs, in what follows, is measured in ´ units. Note that in the case of a planar surface Eq. (3) reduces to the usual Lennard–Jones (10,4) potential. The model for the adsorption potential is an idealization, in comparison to real situations. However, principal focus of our work is to reveal some general trends of the phase behavior of the confined symmetric mixture rather than to deal with specific systems. According to this objective, the number of the model parameters has been reduced to a minimum, as it has been mentioned above. The local densities, ri(r), of the species is1, 2 are computed according to a density functional approach. In the present study, we use the fundamental measure density functional method, originally derived by Rosen-

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

feld w27x, which is known as one of the most accurate to describe non-uniform fluid mixtures. Because this theory has been described and used in several publications, see e.g. Refs. w27–29x, we write down only final equations. The density profile equation, obtained by minimizing the excess grand canonical potential,

|µF r Žr.,r Žr. yFwr ,r x∂dr y8|drµm r Žr.yr yr Žr.VŽr.∂ w y 1

Vexs

x

2

z ~ |

b1

b2

2

w iy i x

z bi~ |

i

is1 2

|

q8 drµwyriŽr.lnriŽr.yriŽr.z~ywrbilnrbiyrbix∂ is1

q

x

|

1 2 8 dr dr9wrjŽr9.riŽr.yrbjrbixuijŽatt.Ž)ryr9)., 2 i,js1

| |

(4)

reads, lnwyriŽr.yrbiz~s x

1 3 8 kT as1

B ≠F E ≠F F yC D ≠na Gµr(r9)sr V ≠naŽr9. S

|

T

U T

1 2 y 8 kT as1

|

(7b)

nvaŽr1.s8 dr2riŽr1qr2.wvaŽ)r2)., as1,2 is1

where wa(r), as0, 1, 2, 3 (scalar quantities) and wva(r), as1, 2 (vector quantities) are the weight functions. The vectorial contribution to the free energy given by Eq. (6) is zero for a homogeneous fluid. Explicit expressions for the weight functions are given in Refs. w27,28x, Moreover, uijŽatt.Žr. denotes the attractive part of the Lennard–Jones potential (1), defined as common according to the Weeks–Chandler–Andersen scheme w30x, Sy´ , r-rmin T ij uijŽatt.Žr.sU T VuijŽr. rGrmin

(8)

,

where rmins21y6s Moreover, the hard sphere density functional has been evaluated assuming the hard sphere diameters equal to s. The relationship between the mi and rbi’s is: mi ykTsylnŽ1yn3.qŽsy2.n2yŽ1yn3.

|

y

2

83

W

|

T

qŽsy2. xyn1yŽ1yn3.qŽ1y8p.

waŽ)ryr9).dr9

X T

Y

B ≠F E ≠F U F yC T D ≠nva Gµr(r9)sr V ≠nvaŽr9. S

2w

T

bi∂

z

w

3 =n22yŽŽ1yn3.2.|~ qŽsy2. yxn0yŽ1yn3.

W T

waŽ)ryr9).dr9

X T

bi∂

Y

z

qn1n2yŽ1yn3.2qŽ1y12p.n32yŽ1yn3.3 |~

yV(r)ykT q

1 2 dr9wyrjŽr9.yrbjz~uijŽatt.Ž)ryr9).. kT 8 js1

|

x

|

In the above equation, mi is the configurational chemical potential of the component i, rbi is the density of the i-th component in a bulk fluid in equilibrium with the confined system, F is the free energy functional of hard spheres w27–29x, FykTsyn0lnŽ1yn3. 2

n1n2ynv1Ønv2 n2 wn2ynv2Ønv2x q q , 24p Ž1yn3.2 Ž1yn3.

(6)

and the quantities na and nva are the averaged densities, given by the following equations: 2

|

naŽr1.s8 dr2riŽr1qr2.waŽ)r12)., as0,1,2,3 is1

|

q 8 rbj dr uijŽatt.Žr.,

(5)

js1,2

(9)

where the average densities are calculated according to Eqs. (7a) and (7b) with local densities equal to the bulk densities. We also introduce the symbol x to abbreviate the bulk fluid composition, xsrb1 y rb, where rbsrb1q rb2 is the total bulk fluid density. The dimensionless U 3 3 bulk densities are rU b srbs and rbisrbis , is1,2. The method of solution of the density profile equation, Eq. (5), applied by us, is based on a standard iteration procedure. All the integrations have been carried out by employing the fast Fourier transform technique. The knowledge of the density profiles allows to calculate the excess grand thermodynamic potential, Vex, as well as the adsorption isotherm of species i, Gi, as functions of chemical potentials of two components of a binary mixture in question,

(7a) Gis

1 R2c

|

Rc

r2drriŽr..

0

(10)

84

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

The total isotherm is the sum of individual adsorptions, GsG1qG2. We also define the average density 4 in the spherical pore, NrMsGyVc, where Vcs pRc3 is 3 the cavity volume and the selectivity, SsG1 yG. The reduced adsorption and average density are G*sGs2 and Nr*MsNrMs3 All the results presented below have been obtained for an equimolar bulk fluid composition, i.e. for xs0.5. In other words, the chemical potential of both species have been identical, m1sm2sm, and thus bm is the only independent variable. 3. Results and discussion Similar to our recent works w23–25x, we would like to comment briefly on the phase diagrams of the binary symmetric model mixtures considered in this work, i.e. M75, M65 and M55. The knowledge of the bulk phase diagrams prior to the studies of adsorption is necessary, because the consecutive calculations have been carried out assuming that the adsorption takes place from a gas phase at an equimolar composition. In other words, the bulk gas density (chemical potential) has been always lower than the corresponding value at the bulk gas– liquid coexistence point. In each case the bulk phase diagram has been calculated according to the same procedure as that used for confined fluid (i.e. by minimizing V with respect to the total density of the mixture and to its composition, x, with external field switched off, however). The position of the l-line for second-order transitions between mixed and demixed fluids line, with respect to the envelope of the first order transitions, mainly depends on the energy of the cross interaction. For the system with the highest energy of the cross interaction (i.e. for the model M75), the lline enters the coexistence envelope quite far below the liquid–vapor critical point. Then the critical end point appears. At the temperatures below the critical end point, a mixed vapor coexists with a demixed liquid. In the case of the strongest demixing, i.e. for the model M55, the coexistence between mixed vapor phase and demixed fluid terminates at a tricritical point. For an intermediate value of the cross interaction, i.e. for the system M65, the phase diagram is characterized by the presence of a triple point and of a tricritical point as well. A complete discussion of those three types of topology of the bulk phase diagrams can be found in the recent work of Wilding et al. w21x. We begin the discussion of non-uniform systems with the model fluid M75, confined to the spherical pores of U different radii, RU c sRcyss7 and Rc s14 with attractive U walls, characterized by ´gss´gsy´s15. The construction of the phase diagram requires the calculation of the isotherms and of the excess grand thermodynamic potential, Vex (as functions of bm). The location of the first order transition at each temperature is determined by

Fig. 1. Illustration of the method of construction of the phase diagrams for confined systems. Part a shows the adsorption isotherms and part b – the dependence of the excess grand potential per unit surface area, A, on the configurational chemical potential. Solid and dashed lines denote adsorption and desorption branches, respectively. The equilibrium capillary condensation is indicated by vertical dotted line in part a and by the arrow in part b. Parts c and d shows density profiles of both components just before (part c) and just after (part b) capillary condensation. The profiles before capillary condensation are identical for both components. The calculations have been carried out for M75 fluid in the pore of R*c s14, ´*gss14, at T*s0.9.

the crossing point of two branches of Vex, corresponding to the two coexisting phases. One example of the plots of adsorption isotherm for the pore RU c s14 is shown in Fig. 1a while the corresponding bVex curves are given in Fig. 1b. The calculations have been carried out at T*s0.9 In general, the adsorption isotherms exhibit hystereses. The metastable parts of this particular isotherm are also presented in Fig. 1a, together with a jump resulting from the first order transition (capillary condensation of the mixture in the pore). Two solutions of the equation for the density profiles describe two phases of different density and composition. An example of the changes of the density profiles, r*i Žr.sriŽr.s3, during the transition is shown in panels c and d of Fig. 1. Before the first order transition the fluid particles are

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

Fig. 2. Part a shows the adsorption–desorption isotherm. Line decorated by black circles denotes thermodynamically stable states. Part b presents the density profiles. Line 1 denotes the profiles just before the capillary condensation, bmsy4.03 (the profiles are identical for both components). Line 2 indicates the profiles (the same for both components) just after the capillary condensation, bmsy4.01. Finally, lines 3 and 4 are the profiles of the components 1 and 2 at the chemical potential bmsy3.586 slightly higher than the chemical potential of the demixing transition. The calculations have been carried out for M75 fluid in the pore of, R*c s14, ´*gss14 and at T*s 1.0.

concentrated only in the vicinity of the solid surface covering at most three layers at this temperature (Fig. 1c). Profiles of both species before the transition are identical, due to equimolarity. The condensation yields all the pore volume filled by fluid particles and is accompanied by a demixing (Fig. 1d). The second example corresponds to the same model system, but at higher temperature, T*s1.0, see Fig. 2. We observe two phase transitions in the spherical pore. Specifically, the confined fluid undergoes capillary condensation between two mixed phases and also the second-order demixing transition. Concerning demixing transition, it is worth mentioning that the branch of

85

bVex describing states with S/0.5 is stable with respect to the branch for Ss0.5, these curves are not shown for economy of space. However, the corresponding part of the adsorption isotherm shows higher adsorption for the selectivity S/0.5 (Fig. 2a). It is important to mention that the branches of the isotherm corresponding to S/0.5 and Ss0.5 meet tangentially at the onset of the l-line. The density profiles before the first-order capillary condensation transition and after it is shown in Fig. 2b (the solid and dotted line, respectively). Also we included the density profiles of species at the chemical potential value very slightly after the demixing transition occurred. The phase diagrams, in the temperature-density plane, resulting from the above described analysis are displayed in Fig. 3a, which also presents the bulk phase diagram for the model M75 as a reference. The decreasing of the cavity radius results in shrinking the coexistence envelope and lowering the critical temperature of the capillary condensation. Also, the critical end point temperature decreases. (The latter point is marked by a star). However, the inclination of the l-line remains practically unchanged for the two confined systems and for the bulk fluid. The vapor branch of the coexistence envelope exhibits very big changes under spherical confinement. Nevertheless, the shape of the temperaturedensity projections of the phase diagrams for two confined systems is similar to the bulk coexistence envelope. Another insight into the phase transformations provides the T*ybm projection of the phase diagram, Fig. 3b. The branches for capillary condensation for confined systems are shifted to the lower values of the chemical potential comparing to the bulk, whereas the branches for the second order demixing transition are shifted to higher values of the chemical potential. The value for the chemical potential corresponding to the critical end point is lower in confined fluids in comparison to the bulk model. Stronger confinement leads to more pronounced shifts. It is important to mention that the difference between the critical point temperature and the critical end point temperature has a strong tendency to decrease with increasing confinement effects. Finally, a comparison of the composition changes during capillary condensation (at temperatures below the critical end point) in confined systems with respect to the bulk model is shown in Fig. 3c. Narrower coexistence envelope in the temperature-density plane is accompanied by a smaller jump in composition during that transition. Next figures, Fig. 4a and b, demonstrate the phase diagrams for the model M65 confined in spherical pores with strongly attractive walls Ž´*gss15.. The pore radius has been taken the same as in previously discussed model, i.e. R*c s7 and R*c s14. The model M65 is characterized by stronger demixing, in comparison to the model M75. We observe that for both confined

86

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

Fig. 3. Phase diagrams in the temperature–density plane (part a), in the temperature-chemical potential plane (part b) and in the temperatureselectivity plane (part c). Stars in parts a and b denote the critical end points. All calculations are for the model M75 and for ´*gss15.

systems the type of the phase diagram changes from the second to the third class w21x, characterized by the absence of a critical point and by the presence of the tricritical point. At temperatures lower than the tricritical point the system undergoes first-order phase transition with jump of density upon condensation whereas at higher temperatures only the second-order continuous demixing is observed. The tricritical temperature becomes lower if the confinement is stronger, i.e. for narrower spherical pores. However, the tricritical density increases with increasing confinement. Evaluation of the tricritical point precisely is quite demanding from the computational point of view. Changes of composition, from equimolar to out of equimolarity, are described in Fig. 4b. Previously, we have also observed changes of the topology of the phase diagram from the second to the third class in narrow slit-like pores. In other words, such changes can be observed if the critical temperature

decreases substantially. Spherical pore geometry provides stronger confinement effects in comparison to the slit-like pore if one considers pores with equal slit width and sphere radius. The critical point of the first-order condensation transition between two mixed phases is strongly depressed in the spherical pore at R*c s14 and even more depressed in smaller cavities. Therefore, we observe changes of the type of the phase diagram in these cavities. Moreover, our additional calculations have shown that even in larger cavities of the order R*c s20 and 25, condensation as the transformation between mixed phases does not occur. Rather, capillary condensation as the first-order transformation between dilute fluid phase of equimolar composition and dense phase with non-equimolar composition takes place. Possibly, one can obtain change of the type of the topology of the phase diagram even for the model M75 in cavities smaller than R*c s7, but we have not explored this problem.

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

Fig. 4. Phase diagrams in the temperature–density plane (part a) and in the temperature-selectivity plane (part b). All calculations are for the model M65 and for ´*gss15.

The last part of our study has been carried out for the model M55. This fluid exhibits the strongest demixing and its bulk phase diagram belongs to the third class. Fig. 5a shows an example of the phase diagram obtained for the spherical pore of the radius R*c s7. The phase diagram for the confined fluid is topologically the same as for the bulk. The envelope ends at the tricritical point, its temperature is lower in comparison with the bulk counterpart. Also, the coexistence envelope for the confined model is much narrower than the one for the bulk fluid mixture. In spite of a well pronounced difference between temperatures of the tricritical point for the bulk model and for the confined one, our

87

discussion above shows that confinement has a much more pronounced effect on the condensation first-order transition than on the demixing transition. In the case of the bulk M55 model it is known that the coexistence envelope between mixed phases describes metastable transitions. It is narrower and lies at lower temperatures comparing to the envelope shown in Fig. 5a for the bulk model. In the case of confined model, such an envelope would terminate even at lower temperatures than we discuss. A comparison of the temperature vs. composition envelopes (Fig. 5c) shows that the bulk and confined model behave qualitatively similar in this aspect. However, at a given temperature confined condensed mixture is characterized by a lower degree of demixing. Such a behavior can be attributed to a lower average density of confined condensate in comparison to the bulk model. Our principal findings can be summarized in the following way. We have investigated the adsorption in spherical pores of symmetric binary mixtures, which exhibit three types of bulk phase behavior. Our observations regarding the types of the phase behavior are the following. In a model with weak demixing trends, we observe mixed gas to mixed liquid first-order transition at higher temperatures, that terminates at the critical point and mixed gas-like to demixed fluid-like first-order transition at lower temperatures, ending at a tricritical point. There exists a triple point temperature, separating the two above regimes. This type of behavior can be observed in spherical pores of different diameter. However in narrower pores and for the models with stronger demixing trends things can become different. Namely, the first order transition between two mixed phases disappears and the corresponding triple point disappears. We have then the first order transition between mixed gas phase and demixed liquid with a tricritical point. Such phase behavior exists in narrow, as well as in wide spherical pores. Changes of the type of topology of the phase coexistence envelope can be induced by spherical confinement. No such kind of behavior has been observed for slit-like pores w23,25x of the width comparable with the cavity diameter. Therefore, confinement effects in the phase diagrams and adsorption due to spherical geometry are much stronger in comparison to the effects induced by slitlike pore walls. Assessing significance of the above results one should remember that they have been obtained within the density functional theory that is based on a mean field approximation for attractive interactions. More refined approaches, which take into account molecular correlations are worth to extend for systems considered in this study. Other peculiarities of the phase behavior are possible to obtain for the model mixtures considered in this work. In particular, we have not explored possible influence of the strength of adsorbing potential on phase behavior

88

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89

Fig. 5. Phase diagrams in the temperature–density plane (part a), in the temperature-chemical potential plane (part b) and in the temperatureselectivity plane (part c). All calculations are for the model M55 and for ´*gss15.

of a fluid confined in a spherical pore. Here we only state that in the case of adsorbing wall studied in the present work, the phase diagrams plotted in the temperature-selectivity plane are similar for all investigated pores. In this case, the effect of the confinement is quite usual and a decrease of the pore width causes the shift of the entire phase diagram down along the temperature axis. However, varying strength and range for the adsorbing potential for each species of the mixture the phase diagram can be changed. Acknowledgments This work has been supported by the CONACyT of Mexico under grant 37323-E and by the National University of Mexico under grant IN-113201. A.M. has been partially supported by the Ministry of Education, Youth and Sport of Czech Republic under grant LN00A032.

References w1x L.D. Gelb, K.E. Gubbins, R. Radhakrishnan, M. SliwinskaBartkowiak, Rep. Prog. Phys. 62 (1999) 1572. w2x S. Dietrich, M. Schick, Phys. Rev. B 33 (1986) 4952. w3x (a) K. Grabowski, A. Patrykiejew, S. Sokolowski, { Thin Solid Films 304 (1997) 344 (b) K. Grabowski, A. Patrykiejew, S. Sokolowski, Thin Solid { Films 342 (1999) 259. w4x M. Sliwinska-Bartkowiak, S.L. Sowers, K.E. Gubbins, Langmuir 13 (1996) 1182. w5x N. Choudhury, S.K. Ghosh, Phys. Rev. E 64 (2001) 021206. w6x F. Schmid, N.B. Wilding, Phys. Rev. E 63 (2001) 031201. w7x Y.K. Tovbin, L.K. Zhidkova, V.N. Komarov, Russ. J. Chem. B 50 (2001) 786. w8x A.V. Shevade, S. Jiang, K.E. Gubbins, J. Chem. Phys. 113 (2000) 6933. w9x F. Bulnes, A.J. Ramirez-Pastor, V.D. Pereyra, J. Mol. Catal. A: Chem. 167 (2001) 129. w10x R.F. Cracknell, D. Nicholson, N. Quirke, Mol. Phys. 80 (1993) 885.

A. Malijevsky et al. / Journal of Molecular Liquids 112 (2004) 81–89 w11x G.S. Heffelfinger, Z. Tan, K.E. Gubbins, U.M.B. Marconi, F. van Swol, Mol. Simul. 2 (1989) 393. w12x G.S. Heffelfinger, F. van Swol, K.E. Gubbins, J. Chem. Phys. 89 (1988) 5202. w13x S. Jiang, K.E. Gubbins, P.B. Balbuena, J. Phys. Chem. 98 (1994) 2403. w14x W.T. Gozdz, ´ ´ ´ K.E. Gubbins, A.Z. Panagiotopoulos, Mol. Phys. 84 (1995) 825. w15x S. Sokolowski, { J. Fischer, Mol. Phys. 71 (1990) 393. w16x Z. Tan, K.E. Gubbins, J. Phys. Chem. 96 (1992) 845. w17x J.S. Rowlinson, F.L. Swinton, Liquids and Liquid Mixtures, Butterworth, London, 1982. w18x A. Vidales, A.L. Benavides, A. Gil-Villegas, Mol. Phys. 99 (2001) 703. w19x J.-L. Wang, G.-W. Wu, R.J. Sadus, Mol. Phys. 98 (2000) 715. w20x G. Kahl, E. Scholl-Paschinger, ¨ A. Lang, Monatsch. Chem. 132 (2001) 1413.

89

w21x N.B. Wilding, F. Schmid, P. Nielaba, Phys. Rev. E 58 (1998) 2201. w22x P.H. van Konynenburg, R.L. Scott, Philos. Trans. R. Soc. Lond. Ser. A 298 (1980) 495. w23x A. Martinez, O. Pizio, S. Sokolowski, J. Chem. Phys. 118 { (2003) 6008. w24x K. Bucior, A. Patrykiejew, O. Pizio, S. Sokolowski, Z. Soko{ {lowska, Mol. Phys. 101 (2003) 721. w25x A. Martinez, O. Pizio, A. Patrykiejew, S. Sokolowski, J. Phys. { Condens. Matt. 15 (2003) 2269. w26x D. Henderson, S. Sokolowski, Phys. Rev. E 52 (1995) 758. { w27x Y. Rosenfeld, Phys. Rev. Lett. 63 (1989) 980. w28x S. Roth, S. Dietrich, Phys. Rev. E 62 (2000) 6926. w29x M. Schmidt, Phys. Rev. E 62 (2000) 3799. w30x J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. 54 (1971) 5237.