Phase Behavior of Lennard-Jones Fluids in Slit-like Pores with Walls

J. Phys. Chem. C 2007, 111, 15743-15751

15743

Phase Behavior of Lennard-Jones Fluids in Slit-like Pores with Walls Modified by Preadsorbed Molecules: A Density Functional Approach† O. Pizio,*,‡ A. Patrykiejew,§ and S. Sokołowski§ Instituto de Quı´mica de la UNAM, Coyoaca´ n 04510, Me´ xico, and Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska UniVersity, 20-031 Lublin, Poland ReceiVed: May 14, 2007; In Final Form: July 9, 2007

We have studied the liquid-vapor coexistence of Lennard-Jones fluids in slit-like pores with the walls modified by preadsorbed molecules in the framework of the density functional approach. During the preadsorption step, the grafting of monomers, trimers, or chains consisting of spherical segments is performed. The model of tangentially bonded spheres is used for chain molecules. It is shown that the presence of grafted species may induce condensation and layering phase transitions. The crossover between evaporation and condensation is also influenced by grafted species. Phase transitions in the adsorbed fluid affect the microscopic structure of grafted chains and can result in the peculiarities of their thermodynamic behavior. The entire calculations are performed by using the semi-grand canonical ensemble.

1. Introduction The study of thermodynamics of adsorption of fluids and mixtures in pores and porous media is one of several important problems in chemical physics and related sciences. Of particular interest is the investigation of phase transitions in such physical systems.1-4 Academic research in this area is motivated to a great extent by the development of applications concerned with more-efficient and novel technological processes involving adsorbents. Several techniques and areas of applied science, for example, chromatography, heterogeneous catalysis, oil recovery, gas storage, membrane separation, lubricant developments, and some important biological phenomena and processes require the knowledge and control of the thermodynamic behavior of fluids in pores. In narrow pores (with the pore width on the order of a few molecular diameters), fluids exhibit significantly different physical behavior compared to bulk systems. The competition between fluid-pore walls and fluid-fluid interactions leads, under certain thermodynamic conditions, to surface-driven phase transitions, such as layering, wetting, and capillary condensation. Computer simulation techniques, as well as theoretical methods, including density functional theory and integral equations are commonly used to understand these phenomena.1-6 In a vast majority of theoretical works, adsorption is studied in a single pore, and the pore geometry is assumed to be welldefined. The interaction between pore walls and particles of an adsorbed fluid is usually described by simple potentials that vary only in the direction perpendicular to the surface. However, real porous systems are geometrically and energetically heterogeneous at various scales. The effects of pore heterogeneity on surface phase transitions have been studied in several works.7-17 Different models, for example, with spatially varying adsorbing potentials and those employing the concept of quenchedannealed mixtures, have been used to describe surfaces whose roughness varies on the scale of molecular dimensions. †

Part of the “Keith E. Gubbins Festschrift”. Instituto de Quı´mica de la UNAM. § Maria Curie-Skłodowska University. ‡

The obtained results indicate that adsorption and the surface phase transitions are essentially influenced by the heterogeneity effects. Alternatively, the adsorbing properties of porous solids can be changed by their physical or chemical modification. This possibility is of practical importance and has been explored for several specific purposes. However, modeling and theoretical description of structurally and energetically modified adsorbents is not well-developed so far. Physical modification of the surface of pores may be reached by preadsorption (i.e., at the step preceding adsorption process) of certain molecules.18-24 Experimentally important and theoretically interesting are the adsorbents in which the pore walls are modified by preadsorption of complex molecules, specifically of chains and colloidal particles. In particular, the understanding and description of the microscopic structure and resulting macroscopic properties of a layer of chain molecules grafted to the solid has attracted much attention.25-33 Intrinsically, this type of physical system involves not only grafted species under confinement but also solvent particles that adsorb on the solid surface and on the grafted species. Density functional theory is a powerful methodology for the modeling of inhomogeneous systems that include chain molecules. A few years ago, McCoy, Curro, and co-workers proposed a version of the density functional theory to study the density profiles of chains, tethered to a surface.34-38 The presence of a solvent has been considered at different levels of modeling. Principally, these authors focused on the microscopic structure of tethered chains on a single solid surface in an implicit solvent and analyzed the relation between their density functional approach with the previously developed selfconsistent field theories for tethered chains in a continuum solvent. Also, these authors have attempted to account for solvent species explicitly. However, the method intrinsically cannot describe surface-induced phase transitions in tethered polymer-explicit solvent systems. Several alternative density functional formulations exist for nonuniform chain fluids and their mixtures.39-45 Some of these

10.1021/jp0736847 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/18/2007

15744 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Pizio et al.

theories seem to be promising candidates to describe the thermodynamics of chains tethered to the pore walls in the presence of a solvent explicitly. In particular, it has been demonstrated in recent works46-49 that for relatively simple models of polymeric systems, for example, for the models of chain molecules, the density functional approach developed by Yu and Wu40 is able to accurately describe the segment-level microscopic structure and thermodynamic properties quantitatively. Thus, it provides a method to explore the phase equilibria in nonuniform systems involving chains. However, our primary interest is to study phase transitions in pores with tethered molecules in the presence of explicit solvents and how the model parameters influence phase equilibria. Such a problem has not been explored so far to the best of our knowledge. The model employed in this work is simple but has a sound physical background. Moreover, sophistication of the model may put it closer to the systems of experimental research. 2. Theory We begin with a brief description of the physical systems of the present work. Formally, we are interested in the adsorption of spherical particles, S, in a slit-like pore with identical walls. However, let us assume that this pore has been modified by performing preadsorption of chain molecules, C. During preadsorption, a layer chain of molecules pinned to the pore walls via the first segment is formed. Moreover, the number of the first segments of chains per unit area of each wall is equal. This surface density does not change during posterior adsorption of species S in the range of temperatures in question. Only the distribution of segments with respect to the pore walls can change in the course of adsorption of species S. Also, this distribution changes if the adsorbate S undergoes phase transitions. Alternatively, these phase transitions, for example, capillary condensation of species S, are influenced by the presence of a pinned layer of chain molecules. Let us proceed now to the modeling of the system. We use the model of tangentially jointed spherical segments for chain molecules. Each of the molecules consists of M identical segments with the diameter σ(C). The connectivity of a chain is provided by the bonding potential between two adjacent M-1 segments, Vb. The total bonding potential Vb(R) ) ∑j)1 40 Vb(rj+1 - rj) satisfies the relation M-1

exp[-βVb(R)] )

δ(|ri+1 - ri| - σ(C))/4π(σ(C))2 ∏ i)1

(1)

where R ≡ (r1, r2, ‚‚‚, rM) denotes a set of segment positions. The chain species are distributed in the slit-like pore of width H. All of the segments, but the single terminal segment (the first segment) of each molecule, interact with the pore walls via the potential

V(C) i (z)

{

∞, z e σ(C)/2, z g H - σ(C)/2 ) 0, otherwise

(2)

where i ) 2, 3, ..., M. The first segment called next the surfaceactive segment, however, interacts with the pore walls via a short-range “sticky” potential, V(C) (z), such that the corre1 sponding Boltzmann factor is (C) (C) exp[-βV(C) 1 (z)] ) [δ(z - σ /2) + δ(z - H + σ /2)]

(3)

where is a positive constant. Our approach requires that the number of preadsorbed chain molecules remains constant and that their distribution inside the pore is symmetric with respect to the pore center. The exact value of is irrelevant because the density profiles of chain segments must satisfy an appropriate normalization condition in order to ensure that the number of chain particles in the pore (measured per unit pore volume) is fixed and yields FM, see eq 8 below. However, to be specific, we have taken such that the Boltzmann multiplier at z ) 0.5σ(C) is 10 in the numerical procedure. Consider next the adsorption of spherical molecules of diameter σ(S) into such a slit-like pore with modified walls. The adsorption of species S (denominated as fluid in what follows) is performed at constant temperature; it takes place in a reservoir that fixes the chemical potential, µS. Adsorption of spherical molecules relies on the establishment of chemical equilibrium between the molecules in the pore and in a reservoir of a solely one-component fluid, S, at chemical potential µ (µ ) µS). This process influences the structure of the preadsorbed chain molecules but does not change their amount inside the pore. The spherical molecules interact with the pore walls via the hard-wall potential

V(S) 1 (z) )

{

∞, z e σ(S)/2, z g H - σ(S)/2 0, otherwise

(4)

Moreover, all of the spherical species (i.e., segments and species S) interact via the Lennard-Jones potentials, u(ij)(r), i, j ) S, C, which depend only on the center-to-center distance, r

u(i j)(r) ) 4(i j)[(σ(i j)/r)12 - (σ(i j)/r)6]

for r e r(icutj) (5)

where r(ij) cut is the cutoff distance. We assume additivity of the diameters, that is, σ(ij) ) (σ(i) + σ(j))/2. To present the theory, let us introduce the notation F(C)(R) and F(S)(r) for the density distribution of chains and spherical species, respectively, and define the segment densities, (C) , and the total segment density of chains, F(C) Fs,j s , via the relation M

F(C) s (r) )

∑ j)1

M

(C) Fs,j (r) )

∫ dRδ(r - rj)F(C)(R) ∑ j)1

(6)

To simplify the equations below, we also use the notation (S) (S) F(S) s (r) ≡ Fs,1(r) ≡ F (r). The statistical ensemble appropriate for studying the physical system described above is a semigrand canonical ensemble. The semigrand ensemble thermodynamic potential is

y ) F[F(C)(R), F(S)(r)] +

∫ dRF(S)(r)(V(S)1 (r) - µ)

(7)

where F[F(C)(R), F(S)(r)] is the Helmholtz free-energy functional. Functional y is evaluated under the following constraint

∫V

p

(C) drFs,j (r)/

∫V

p

dr ) FM

(8)

where the integration is carried out over the entire pore volume, Vp. The expression for F[F(C)(R), F(S)(r)] is taken from Yu and Wu. 40 The free-energy functional is divided into ideal free

Lennard-Jones Fluids in Slit-like Pores

J. Phys. Chem. C, Vol. 111, No. 43, 2007 15745

energy, Fid, and the excess term, Fex, due to the hard-sphere contribution, (hs), due to the connectivity of chain molecules, (c), and due to attractive interactions between all of the species, (att), F ) Fid + Fex, Fex ) Fhs + Fc + Fatt. The configurational ideal part of the free-energy functional, Fid, is

βFid ) β

∫ dRF (R)Vb(R) + ∫ dRF(C)(R) (ln(F(C)(R)) - 1] + ∫ drF(S)(r) (ln(F(S)(r)) - 1]

where u(iattj)(|r - r′|) is the attractive part of the Lennard-Jones potential, defined according to the Weeks-Chandler-Andersen scheme57

u(iattj)(r) )

(C)

(9)

The excess free energy due to hard-sphere interactions results from the fundamental measure theory50-52

Fhs )

∫ Φhs(r) dr

(i j) r e 21/6σ(i j), (i j) u (r) r > 21/6σ(i j),

(17)

and the effective hard-sphere diameter is set to be d (i) ) σ(i), i ) S, C. The equilibrium structure is obtained by minimizing y under constraint 8. Using the technique described in previous works, see, for example, ref 47, the density profiles are calculated from the equations

F(S)(r) ) exp{βµ - βλ(S) 1 (r)}

(10)

where

{

(18)

and

Φhs ) -n0 ln(1 - n3) +

n1n2 - n1n2 + 1 - n3

n2 (1 - ξ ) 3

2 3

˜ exp{βVb(R) - β F(C)(R) ) C

n3 + (1 - n3)2 ln(1 - n3) 36π n32(1 - n3)2

(11)

(C) (S) (S) ∫ dr′F(C) s (r′)wR (r - r′) + ∫ dr′Fs (r′)wR (r - r′)

(12)

and

λ(i) j (rj) )

(C) ∫ dr′F(C) s (r′)wR (r - r′) + ∫ dr′F(S)s (r′)w(S)R (r - r′)

w(i) R

(20)

M

M

j)1

l)1

(C) (R) Fs,i (z) ) C ˜ exp[-βλi(z)]G(L) i (z)GM+1-i(z)

(21)

(C) 2

(15)

with ζ ) 1 - n2(n2/(n2)2). Finally, the attractive contribution results from the meanfield approximation applied to the attractive interactions between all the spherical segments

(16)

(22)

where the functions G(P) i (z), P ) L, R are determined from the recurrence relation

G(L) i (z) )

(14)

(C) (C) 2 (C) where ζ(C) ) 1 - n(C) 2 (n2 /(n2 ) ), with y(hs) given by the Boublik-Mansoori-Carnahan-Starling-Leland55,56 expression for the contact value of the radial distribution function of spherical segments C in a mixture of hard spheres containing spheres of C and of S type

(j) (i j) dr dr′F(i) ∫ ∑ s (r) Fs (r′) uatt (|r - r′|) 2 i, j)S,C

+ V(i) j (rj)

∫ dR ∑ δ(r - rj) exp[-βVb(R) - β ∑ λ(C) l (rl)]

(13)

For the definitions of the weight functions and see refs 50-52. The contribution Fc ) ∫ Φc(r) dr is evaluated from Wertheim’s first-order perturbation theory53,54

n2σ ζ (n2σ ) ζ 1 (C) y(C) + + hs (σ ) ) 1 - n3 4(1 - n )2 72(1 - n )3 3 3

δF(i) s (rj)

This equation simplifies to

w(i) R,

1 - M (C) (C) (C) n0 ζ ln[y(C) Φ(c) ) (hs)(σ )] M

δFex

(C) (r) ) Fs,j

C ˜

(S) nR(r) ) n(C) R (r) + nR (r) )

1

(19)

for i ) C, S. From eqs 8 and 19, we obtain the following equation for the segment densities of chain molecules

(S) nR(r) ) n(C) R (r) + nR (r) )

Fatt )

λ(C) j (rj)}

where λ(i) j (rj) is

In the above, ξ(r) ) |n2(r)|/n2(r). The scalar, nR, for R ) 0, 1, 2, 3 and vector, nR, for R ) 1, 2 weighted densities are given by

(C)

∑

j)1,M

(C) (z)] ∫ dz′ exp[-βλi-1

θ(σ(C) - |z - z′|) 2σ(C)

(L) Gi-1 (z′) (23)

and

G(R) i (z) ) (C) (z)] ∫ dz′ exp[-βλM-i+2

θ(σ(C) - |z - z′|) 2σ

(C)

(L) Gi-1 (z′) (24)

(R) for i ) 2, 3, ..., M and with G(L) 1 (z) ) G1 (z) ≡ 1. The constant C ˜ is evaluated by using eq 8. It is worth recalling that the fluid in the reservoir is the onecomponent bulk fluid of Lennard-Jones molecules, S. The configurational chemical potential, µ, in terms of the bulk density, Fb, of species S is

βµ ) lnFb + βµ(hs) + βFb

∫ u(SS) att (r) dr

(25)


Pizio et al.

where µ(hs) is the hard-sphere contribution to the excess chemical potential of a fluid at density Fb resulting from the CarnahanStarling equation of state. 3. Results and Discussion To reduce the number of model parameters, we carried out the calculations assuming that the diameters of all spherical species (segments of chains and of species S) are equal, σ(C) ) σ(S) ) σ. The diameters σ and (SS) have been used as units of length and energy, respectively. Thus, the dimensionless pore width is H* ) H/σ and z* ) z/σ. The density profiles of species are normalized accordingly. The dimensionless density is F*b ) Fbσ3, and temperature is defined in terms of energy for S species, T* ) kT/(SS), µ* ) µ/kT. To characterize the amount of grafted preadsorbed species, we will use the parameter of reduced surface density, NM ) σ2HFM, rather than FM itself. We have investigated the adsorption and phase behavior of several model systems in slit-like pores of different widths. For the sake of convenience, the results are presented first for the model of grafted monomers (M ) 1), next grafted trimers (M ) 3), and finally for the model in which grafted species are chains consisting of six segments (M ) 6). The model (M ) 1) in spite of its simplicity provides a set of useful insight. The two panels of Figure 1 show the effect of grafted density and of the pore width on the vapor-liquid phase diagram for Lennard-Jones fluid confined in the pore ((CC) ) 0, (SC) ) 0). The presence of grafted monomers on the pore walls in some sense is equivalent to the reduction of the pore width or in other words leads to stronger confinement effects. The critical temperature in the pore is reduced in comparison to the pore without grafted particles, and the critical density is slightly lower in the presence of grafted species. In the absence of all attractions but between fluid species, the phase diagrams describe capillary evaporation; that is, the chemical potential of transition is higher in comparison with its value for the bulk fluid at the same temperature. One example of the density profiles at coexistence is given in Figure 1c. Additional confinement effects due to the presence of grafted particles yield the expulsion of a fluid from the vicinity of the pore walls and provide a more “layered” liquid phase in the pore. The next group of figures illustrates the phase behavior of the same model (M ) 1) in the presence of attraction between grafted particles and species filling the pore. Still, we assume that (CC) ) 0. Let us comment first on the case of a wider pore, H* ) 6, Figure 2a. The phase diagrams show that at weaker attraction, (SC)/(SS) ) 1.0, a single envelope is observed, whereas at stronger attraction, (SC)/(SS) ) 2.5, the envelope splits into two parts. One of them describes the layering transition in such a pore, whereas the second corresponds to capillary condensation of a fluid in the entire pore. The triple point may exist at a lower temperature. At intermediate energy, the single envelope is observed but of peculiar form. In the narrower pore, H* ) 6, the layering transition is not observed. In a wider pore, the effect of increasing grafted density is rather small. Dependent on the choice of the ratio (SC)/(SS), either evaporation or condensation in the pore can be observed, that is, the chemical potential of the transition is either higher or lower compared to its value for the bulk coexistence at the same temperature. It is seen in Figure 2b that at low temperatures evaporation takes place (H* ) 6, (SC)/(SS) ) 1), whereas at high temperatures we see the capillary condensation. The coexistence line of the fluid in the pore crosses the bulk coexistence line at certain temperatures. One example of the shape of the density profiles at coexistence for both layering

Figure 1. (a) Liquid-vapor coexistence envelopes for the LennardJones fluid confined in the slit-like pores of the width H* in the density-temperature plane, (CC) ) 0 and (SC)/(SS) ) 0. The solid line is for the bulk fluid. The dashed and dash-dotted lines are for H* ) 10, M ) 1 with NM ) 0.0 and NM ) 0.48, respectively. The same lines with symbols are for the pore H* ) 6. (b) Chemical potentialtemperature projections of the phase diagrams. The nomenclature of lines and symbols is the same as that in part a. (c) Fluid density profiles at coexistence for H* ) 6, T* ) 1.0. The solid and dashed lines are for NM ) 0.0 and NM ) 0.48, respectively.

and condensation is shown in Figure 2c. At the layering transition, the density drastically changes in the vicinity of the



Figure 2. (a) Phase diagram for adsorbed fluid in the density-temperature plane. The solid line is for the bulk fluid coexistence. The dashed, dash-dotted and dotted lines are for (SC)/(SS) ) 1.0, 2.0, and 2.5, respectively (H* ) 10, M ) 1, NM ) 0.48). The dash-double dotted line is for (SC)/(SS) ) 1.0, H* ) 10, NM ) 0.63. The dashed and dash-dotted lines with symbols are for (SC)/(SS) ) 1.0 and 2.5, respectively; H* ) 6 and NM ) 0.48. In all cases, (CC) ) 0. (b) Same phase diagrams as in the part a but in the chemical potential-temperature plane. The same nomenclature of lines is used; just the case of NM ) 0.63 is marked by triangles. (c) Density profiles at coexistence. The profiles describing layering transition and capillary condensation are shown by the dashed and solid lines, respectively (H* ) 10, T* ) 0.72, (SC)/(SS) ) 2.5, NM ) 0.48). (d) Average fluid density in the pore vs chemical potential for the model and conditions as in part c of this figure.

pore walls, and in fact this transition is nothing else but the condensation of a fluid on the grafted species. During true capillary condensation, the fluid fills the entire pore. The behavior of the average density in the pore during these two transitions is shown in Figure 2d. To summarize this group of results, we would like to mention that grafting of pore walls by monomer species may induce condensation and layering even if the pore walls are just impermeable barriers. Alternatively, one can observe either evaporation or condensation dependent on the temperature of observation and on the energetic parameters characterizing grafted species. However, the phase transition in the pore does not alter the density profile of grafted monomer species, F(C)(z). This feature will appear only for more-complex grafted molecules. We proceed to the brief discussion of the model in which the grafted particles are trimers (M ) 3). Some selected phase diagrams in the -T* and T*-µ* plane are given in Figure 3a and 3b, respectively. One can see that the effect of connectivity of segments into chains (those are grafted to the surface) leads to a slightly lower critical temperature in comparison to the case of grafted monomers. This comparison

is performed in the same pore, H* ) 6, at the same nominal density of grafted species (NM ) 0.48 for M ) 1 and NM ) 0.16 for M ) 3) and at the same energetic parameters, (CC) ) 0, (SC)/(SS) ) 0. The presence of attraction between segments and/or between grafted species and fluid particles does not change the critical parameters much. However, the effect of grafted density is strong. In close similarity to the case of the pore with grafted monomers, grafting by trimers can induce crossover between condensation and evaporation in the pore along the temperature axis. There is one effect necessary to mention, however. In order to demonstrate it, we present the density profiles of the second and third segments of grafted trimers at transition in the same pore, H* ) 6, and at similar conditions, T* ) 0.8, F(C) ) 0.16, (SC)/(SS) ) 1 (Figure 4a and b). In part a, however, (CC) ) 0, whereas in the part b (CC)/ (SS) ) 1. It can be seen clearly that the maximum of the distribution of the third segment is affected by the presence/ absence of attraction between grafted particles, reflecting conformational changes in the structure of grafted particles upon the phase transition of adsorbed fluid. In the presence of such attraction, the maxima of the density profiles of the second and third segments are at z* ) 1.5 when the fluid is in the liquid


Figure 3. Average density-temperature and chemical potentialtemperature projections of the phase diagrams for the model of grafted species M ) 3, parts a and b, respectively. All of the lines are for H* ) 6 but the one marked with triangles for the bulk fluid in part b. The solid line for the model of grafted monomers (M ) 1) with NM ) 0.48 is given as a reference ((CC) ) 0, (SC)/(SS) ) 0). The dashed line and solid line with circles are for (CC) ) 0, (SC)/(SS) ) 0.0 and 1.0, respectively (NM ) 0.16). The line with diamonds and dash-dotted line are for the same energetic set of parameters, (CC)/(SS) ) (SC)/(SS) ) 1.0, but for NM ) 0.16 and NM ) 0.32, respectively. The nomenclature of lines and symbols in part b is the same as that in part a.

phase. In the absence of attraction between segments belonging to grafted chains, the third segment is more exposed and its maximum on the density distribution is at z* ≈ 2.0. One can expect that such trends will be more-pronounced for the models of grafted chains with a larger number of segments. Therefore, we proceed to the last group of results concerning the model of grafted chains with M ) 6. Our first illustration in this part of the work concerns the effect of grafting and of pore width on the phase diagrams, Figure 5. The energetic parameters are the following: (CC) ) 0, (SC)/(SS) ) 0. The principal conclusion at this point is that grafting leads to a lower critical temperature of evaporation, in comparison to the pore without grafted particles. The effects of connectivity of segments into chains also leads to a lower critical temperature compared to the case when grafting concerns monomers at the corresponding density. Reduction of the pore width results in a narrower coexistence envelope and lower critical temperature, as is common for stronger confinement.

Pizio et al.

Figure 4. Density profiles of the second (solid line) and third segment (dashed line) in the grafted model M ) 3, H* ) 6, T* ) 0.8, NM ) 0.16, and (SC)/(SS) ) 1. Part a is for (CC) ) 0, whereas part b is for (CC)/(SS) ) 1. The fluid is at coexistence. The lines show segments distribution for a fluid vapor phase, whereas the same lines with symbols are for a fluid liquid phase.

Figure 5. Density-temperature projections of the fluid phase diagrams for the model of grafted chains M ) 6, NM ) 0.08 with respect to the bulk coexistence (dotted line), to the nongrafted pore with H* ) 10 (solid line with circles) and to the pore with H* ) 10 grafted by monomers at NM ) 0.48 (solid line with diamonds). The dash-dotted line, dash-double dotted line, and solid line refer to the model with M ) 6 confined to the pores with H* ) 6, 8, and 10, respectively. In all cases, (CC) ) (SC) ) 0.

The presence of cross attraction between two species (segments of chains and fluid particles) yields certain modifications of the behavior of the phase diagrams. We consider now the



Figure 6. p*-T* projections of the fluid phase diagrams with respect to the bulk coexistence (dotted line). The solid lines with squares, diamonds, and circles are for H* ) 6, 10, and 15, respectively. The model M ) 6 is at the following conditions: NM ) 0.08, (CC) ) 0, (SC)/(SS) ) 1.

case (CC) ) 0, (SC)/(SS) ) 1, Figure 6. The shape of the coexistence envelopes for H* ) 10 and H* ) 6 is as common. However, for a wider pore, H* ) 15, the vapor branch is slightly peculiar. In fact, it comes from the contribution of conformational changes of chains in a quite-wide pore compared to the chain length. In two narrower pores, H* ) 10 and H* ) 6, the conformational changes are hindered and the presence of grafted species seemingly just enhances the confinement effects. We have not attempted to investigate very-wide pores with grafted long chains in the present study, postponing it to a future work. Our next issue is the effect of cross attraction on the phase behavior of the model for a pore of the width H* ) 10, that is, of the width comparable to the nominal chain length, Mσ, Figure 7a. Again, as in the case of pores with grafted monomers, we observe that for a lower strength of this attraction, (SC)/(SS) ) 1, only the single coexistence envelope is observed. Stronger attraction, (SC)/(SS) ) 2.5, induces a layering phase transition. Then the envelope for layering transition connects the principal envelope for condensation at a triple point. Changes of attraction strength influence the shape of the coexistence envelope, and the critical density changes substantially. However, the critical temperature is not very strongly affected by the change of cross attraction. The T*-µ* projection of the phase diagrams is shown in Figure 7b. The overall phase behavior comes from the density profiles that deserve to be discussed in more detail because of their intrinsic importance. In particular, it is interesting to gain insight into the conformational changes of grafted chains upon condensation. For illustrative purposes, we have chosen the density profiles at coexistence for the following set of parameters: H* ) 15, T* ) 1.0 (SC)/(SS) ) 1. However, two cases are presented, in the first case (CC)/(SS) ) 1 (Figure 8a), whereas in the second case (CC)/(SS) ) 0 (Figure 8b). In each of two cases, we show the density profiles of the second, fourth, and sixth segments of grafted chains when the fluid in the pore is at coexistence. The coexistence corresponds to capillary condensation in these two cases. In the presence of attraction between segments of chains, that is, for (CC)/(SS) ) 1 (Figure 8a), we observe that the maxima describing the fourth and sixth segment distribution are at approximately the same position, z* ≈ 2.5, on the liquid branch. Upon fluid condensation, grafted chains extend further from the wall. In contrast, from the curves given in Figure 8b

Figure 7. Density-temperature (part a) and chemical potentialtemperature (part b) projections of the fluid phase diagrams for the model M ) 6, NM ) 0.08, and H* ) 10 with respect to the bulk coexistence (dotted line). In all cases, (CC) ) 0. The solid line, the solid lines with circles, diamonds, and with triangles are for (SC)/(SS) ) 0.0, 1.0, 2.0, and 2.5, respectively. In part b, the same nomenclature is used.

((CC)/(SS) ) 0), we learn that the maximum of the distribution of the fourth and the sixth segment is at z* ≈ 3.0 and at z* ≈ 4.1 on the liquid branch. The grafted chains in the latter case are more “erect” comparied to the case discussed in Figure 8a. One can conclude that the attraction between segments is a competing factor working against the attractions between segments and fluid species. The fluid density (Figure 8c) behaves along trends discussed just now. In the second case, the fluid density in the vapor phase is higher in the vicinity of the grafted pore walls compared to the first case when the segments attract each other. In other words, the segments attract each other and effectively form a slightly denser layer on each of the walls; as a result, fluid particles are unable to penetrate efficiently this layer. In the condensed phase, the fluid density is higher on the surface of the layer formed by the grafted species. To summarize, we have applied the density functional approach to the model of adsorption of a simple fluid in the slit-like pores with walls modified during preadsorption. This modification is in the grafting of monomers, trimers, or chains consisting of spherical segments. The model of tangentially bonded spheres is used for chain molecules. We have studied how the liquid-vapor coexistence envelope for confined Lennard-Jones fluids is influenced by the grafting of particles on the pore walls. The pore walls solely provide impermeability barriers for fluid species. However, grafted species may induce condensation and layering phase transitions. The crossover between evaporation and condensation is also influenced by


Pizio et al. The model used in this work requires additional studies. In particular, it is desirable to investigate in more detail possible phase transformations in the subsystem of long chains in pores or on the single surface. This would require certain modification of the method of calculations. However, the model opens some new possibilities. Namely, one can intend to study adsorption of a fluid in pillared adsorbents. It is interesting to investigate the influence of grafted species on immiscibility lines and loops in the case of adsorption of various mixtures. Another possibility is to extend the proposed treatments to the case of chains with charged and selected charged segments and explore adsorption of ionic species. Some of these problems are currently under study in our laboratory. Acknowledgment. The work of A.P. and S.S. has been partially supported by EU as TOK contract 509249. It is our honor and pleasure to contribute to this special issue dedicated to K. E. Gubbins. For many years, we have admired his scientific activity and share his interest in research of adsorption phenomena. References and Notes

Figure 8. Density distribution of the second (solid line/solid line with circles), fourth (dashed line/dashed line with diamonds), and sixth (dash-dotted line/dash-dots with triangles) segments of the grafted chains in the model M ) 6, H* ) 15, NM ) 0.32, and (SC)/(SS) ) 1. The fluid phase is at coexistence. The lines and the same lines with symbols correspond to the fluid vapor and fluid liquid phase, respectively. In part a (CC)/(SS) ) 1, whereas in the part b (CC) ) 0. (c) The fluid density distribution at coexistence. The solid line and solid line with circles is for the system described in part a of this figure. The dashed line and dashed line with diamonds correspond to the system given in part b. The vapor phase (lines) and liquid phase (lines with symbols) are at T* ) 1.0.

grafted species. The entire calculations are performed by using the semigrand canonical ensemble.

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Phase Behavior of Lennard-Jones Fluids in Slit-like Pores with Walls

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