Capillary condensation in pores with rough walls: A density functional approach

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Capillary condensation in pores with rough walls: A density functional approach
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  Journal of Colloid and Interface Science 313 (2007) 41–52www.elsevier.com/locate/jcis Capillary condensation in pores with rough walls:A density functional approach P. Bryk  a , W. R˙zysko a , Al. Malijevsky b , S. Sokołowski a , ∗ a  Department for the Modeling of Physico-Chemical Processes, Maria Curie-Skłodowska University, 20-031 Lublin, Poland  b  E. Hála Laboratory of Thermodynamics, ICPF, Academy of Sciences, 165 02 Prague 6, Suchdol, Czech Republic Received 18 January 2007; accepted 27 March 2007Available online 27 April 2007 Abstract The effect of surface roughness of slit-like pore walls on the capillary condensation of a spherical particles and short chains is studied. The gasmolecules interact with the substrate by a Lennard-Jones  ( 9 , 3 )  potential. The rough layer at each pore wall has a variable thickness and densityand consists of a disordered quenched matrix of spherical particles. The system is described in the framework of a density functional approachand using computer simulations. The contribution due to attractive van der Waals interactions between adsorbate molecules is described by usingfirst-order mean spherical approximation and mean-field approximation. © 2007 Elsevier Inc. All rights reserved. Keywords:  Adsorption; Pore; Capillary condensation; Density functional theory; Quenched-annealed systems 1. Introduction Real adsorbing surfaces are usually rough, so a fluid uponadsorption encounters geometrically and/or energetically het-erogeneous walls. Recently, the effects of heterogeneity onadsorption have been considered theoretically and employingsimulation techniques for several model systems [1–19]. In par- ticular, it has been shown that the surface roughness can in-fluence their wettability [20–24]. To study this effect some of  us proposed a model of a rough wall created by depositing alayer of molecules on an idealized substrate and next imposinga quenching procedure [23,24]. The quenched layer was then exposed to the adsorbing gas. Surface roughness in this modelis a combined effect of the preparation procedure, the densityof the deposited layer and its thickness. To describe the adsorp-tion of a fluid a modified Fischer–Methfessel approach [25] thatis closely related to the so-called “Mark-I” version of the Tara-zona’s density functional theory (DFT) [26] was applied. Thismodel was used to study the effect of surface roughness on thewetting transitions [24], to investigate adsorption of fluids in * Corresponding author.  E-mail address:  stefan.sokolowski@neostrada.pl (S. Sokołowski). pores with rough walls and to examine the behavior of mixturesnear semi-permeable membranes [27–29].Recently, Schmidt and co-workers proposed a DFT to de-scribe quenched-annealed mixtures [30–33]. Their approach is based on the replica trick and allows the treatment of situationswhere the quenched random matrix as well as the annealedfluid are inhomogeneous. Applications of the theory includedinvestigation of the adsorption properties of hard spheres andmodel colloid–polymer mixtures in bulk matrices and at ma-trix surfaces, and the influence of the quenched disorder onphase transitions like fluid demixing, isotropic-nematic order-ing and freezing. Particularly rich wetting behavior was foundfor colloid–polymermixtures adsorbed against porous wall. Forreview of all these applications see Ref. [32].The theory of Schmidt and co-workers treats both the an-nealed and the quenched species on the level of their one-bodydensity distributions. Hence the complicated external potentialthatthequenchedparticlesexertonthefluidneverentersexplic-itly into the theoretical framework. This approach constitutesan enormous simplification as far as practical computations areconcerned,butalsosuffersofseverallimitations,asitwasnotedby Lafuente and Cuesta [34]. These authors derived an alterna- tive DFT for quenched-annealed mixtures from first principles 0021-9797/$ – see front matter  © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2007.03.077  42  P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52 without resorting to the replica trick. They showed that thedisorder-averaged free energy of the fluid is a functional of theaverage density profile of the fluid as well as the pair correlationof the fluid and matrix particles. However, an implementationof this theory is difficult and from this reason it is preferable touse the approach of Schmidt and co-workers.To employ theoretical methods for calculating adsorptionisotherms a model for fluid–fluid and fluid–adsorbent interac-tions must be formulated first. The molecular parameters forfluid–fluid interaction are usually determined from bulk ther-modynamic data, whereas solid–fluid potential parameters areobtained by fitting theoretical results against experimental ad-sorption isotherms. It appears, however, that in a vast major-ity of cases [18,19] the predicted isotherms are stepwise in-dicating the sequential formation of molecular layers, which,however, are not observed experimentally. In the case of someadsorbents, like graphitized carbon, improvement of the pre-dictability of isotherms is achieved by modifying the fluid–fluidinteraction parameters [18,19,35]. The altering of their values is  justified by the importance of multi-body effects [35–37]. Thesuccess is of this procedure is possible with graphitized carbonblack because of the fairly homogeneous nature of the surface,however, such an approach does not enjoy a similar success inthedescriptionofadsorptionisothermforotheradsorbents,e.g.,for silica gel [38–40]. One possible reason is energetic hetero- geneity of adsorbing surfaces (chemical heterogeneity) surfaceroughness (geometrical heterogeneity), which is considered atthe molecular level and attributed to a molecular feature of amorphous solid [18,19].In the case of heterogeneousadsorbents the adsorbing poten-tial is a complicated function that varies in three-dimensionalspace. Apart from the fact that usually only a little is knownabout the spatial dependence of the adsorbing potential forreal substrates, numerical calculations (including simulations)would be extremely time-consuming. One possible route to de-scribe effectively such systems is to replace the energy land-scape by the density of “auxiliary matrix particles” landscapeand to employ the methods developed for quenched-annealedmixtures [41]. According to this treatment a complicated ad-sorbing potential does not enter explicitly theoretical equations.The model developed in our previous works [23,24,27] seemsto be appropriate to handle the effects of surface heterogeneity.Indeed, a similar model was used recently by Ravikovitch andNeimark  [42], who applied DFT of Schmidt [30,32], and Reich and Schmidt [31] to model adsorption on geometrically hetero-geneous surfaces and porous solids with rough pore walls.In this work we present DFT of adsorption of chain mole-cules in pores with rough walls. The theory combines the ap-proach of Schmidt [32] with the theory of Yu and Wu [43] and Cao and Wu [44]. The latter theory can be successfully applied to describe adsorption of such important from experimentalpoint of view [45–49] complex fluids as carbon dioxide, hydro-carbons, polymers and many others [50]. Moreover, this theoryis numerically almost as simple as density functional theoriesof simple fluids [26]. Obviously, the proposed approach can be also used to study adsorption of spherically symmetric (atomic)molecules.Previous applications of the density functional theory havebeenprimarilyfocusedonvariousweighteddensityapproxima-tions for the repulsive part of the Helmholtz energy functionaland on the mean field approximation (MFA) for an attractivecontribution. The major problem of the MFA is that it signifi-cantly underpredicts the critical temperature for uniform fluids;hence the corresponding interfacial properties maybe unphysi-cal. Recently Tang [51] proposed a first-order mean-sphericalapproximation (FMSA) theory for a long ranged attraction.The FMSA theory, with the analytical expression of a radialdistribution function or a direct correlation function as input,significantly improves the MFA for the long ranged attractionand is applicable to both bulk and inhomogeneous systems us-ing a single set of molecular parameters. The DFT for simplenonuniform fluids based on the Fundamental Measure Theory[52–54] for hard-sphere free-energy contribution and on theFMSA for the attractive contribution is probably the most accu-rate approach to the theory of simple confined fluids [55–58].Therefore the aim of this paper is also to extend the FMSAto the case of nonuniform systems with quenched component.In the case of adsorption of spherical symmetric molecules thetheoretical predictions are compared with the results of com-puter simulations. The phase diagrams for model systems havebeen evaluated using hyper-parallel tempering technique [59]. 2. Theory The system under consideration is a fluid composed of chainmolecules confined in a slit-like pore of the width  H  . Thechains are built of   M   segments, each of diameter  σ  ss . We stressthat setting  M   = 1 in all equations presented below the theoryreduces to that for adsorption of spherically symmetric (atomic)adsorbates.The chain connectivity is assured by imposing the bond-ing potential [43],  V  b ( R ) =  M  − 1 j  = 1  v b ( | r j  + 1 − r j  | ) , where  R ≡ ( r 1 , r 2 ,..., r M  )  denotes a set of coordinates describing the po-sitions of all segments and  v b  is the bonding potential betweentwo adjacent segments. We assume that the segments are tan-gentially jointed, so that the binding potential is(1)exp  − βV  b ( R )  = M  − 1  j  = 1 δ  | r j  + 1 − r j  |− σ  ss  / 4 πσ  2ss . We also take into account attractive van der Waals potentialbetween the segments. This potential is the same for all the seg-ments and is given by(2) u ss (r) =  4 ε ss  (σ  ss /r) 12 − (σ  ss /r) 6  ,  for  r  r cut , ss , 0 ,  for  r > r cut , ss . In the above  r cut , ss  is the cut-off distance, and  ε ss  is the energyparameter.Each segment  j   interacts with a single pore wall viaLennard-Jones (9,3) potential(3) v j  (z j  ) = ε gs  (z 0 /z j  ) 9 − (z 0 /z j  ) 3  , where  z j   is the distance between the  j  th segment and the wall.The total external potential,  V  ext ( R ) , imposed on a chain mole-cule is the sum of the potentials acting on individual segments,  P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52  43 due to both pore walls  V  ext ( R )  =  M j  = 1 [ v j  (z j  ) + v j  (H   − z j  ) ] .The parameters  ε gs , and  z 0  are identical for all segments. Thelast assumption can be easily removed.In order to model pores with rough walls, a layer consist-ing of a quenched matrix is placed at each wall [23,24,27]. Weassume that the matrix is built of spherical molecules of diame-ter  σ  mm . However, the presented theory can be also extended tothe case of matrices made of chain particles. The interaction be-tween matrix species and a segment of the chain is given by thepotential  u ms (r) , which can be either a hard-sphere potential(4) u ms (r)  =  ∞ ,  for  r  σ  ms , 0 ,  for  r > σ  ms , or a truncated Lennard-Jones potential(5) u ms (r)  =  4 ε ms  (σ  ms /r) 12 − (σ  ms /r) 6  ,  for  r  r cut , ms , 0 ,  for  r > r cut , ms . Here  σ  ms  =  (σ  ss  +  σ  mm )  and  r cut , ms  is the cut-off distance forthe fluid–matrix interaction.The local density of the matrix is described by the function ρ m (z) . At the moment we do not specify the form of   ρ m (z) , northe process leading to the formation of the matrix distribution atthe pore walls. Of course, if the matrix inhomogeneity is gener-ated in response to an external potential,  V  m (z) , that acts on thematrix particles before the quench, then the matrix configura-tion is drawn from an equilibrium distribution according to theHamiltonian of the matrix particles, in particular an equilibriumDFT can be applied for this purpose [30–32]. In general, how- ever, in the DFT of quenched-annealed systems the distribution ρ m (z)  is an input into the theory and the step of matrix develop-ment is completely separated from the second step, which relieson the adsorption of a fluid.Following Schmidt [30–33] the replica trick is applied towrite down the grand potential functional for the confined fluid Ω  ρ m (z),ρ( R )   =  F  id  ρ( R )   + F  ex  ρ m (z),ρ( R )  (6) +    d R ρ( R )  V  ext  ρ( R )   − µ  , where  ρ( R )  and  µ  are the local density of chain particles andtheir chemical potential, respectively. Moreover,  F  id (ρ( R ))  isthe configurational ideal free energy and the excess free energyfunctional,  F  ex (ρ m (z),ρ( R )) , describes the interparticle inter-actions of adsorbate particles with adsorbate particles and withthe matrix particles. We are aware that this theory has a numberof weak points that should be pointed out. Although the replicatrick is a widely applied method of statistical physics, it intro-duces assumptions that are difficult to justify concerning theanalytic continuation of the grand potential as a function of thenumber of replicas, and the replica symmetry or its breaking.Moreover,contrarytoDFTof fluids,theformulationof theden-sity functional theory of quenched-annealed systems by Eq. (6)makes difficult to derive the set of Ornstein–Zernike equationsforthesystemfromfunctionalrelations.Forexample,atpresentit is not at all clear what the meaning of the second derivativesof   F  ex (ρ m (z),ρ( R ))  is [34]. Nevertheless, we have decided tofollow the ideas of Schmidt [30–32] because at the moment thisapproach is the only one that can be implemented to the consid-ered case in a relatively easy manner.Similarly as in DFT of simple nonuniform fluids we startwith writing down the system free energy. The ideal free energyfunctional is known exactly [43] βF  id  ρ( R )   =  β    d R ρ( R )V  b ( R ) (7) +    d R ρ( R )  ln  ρ( R )   − 1  . The excess free energy,  F  ex , can be split into the hard-spherecontribution,  F  hs , which results from the hard-sphere repulsionbetween segments and between segments and matrix particles,the attractive contribution due to segment–segment interaction, F  att  and, eventually, due to segment–matrix interaction, and thecontribution due to chain connectivity,  F  c . According to theapproach of Yu and Wu, which is based on the FundamentalMeasure Theory [52–54] and on Wertheim [60,61] theory of  association, the contributions  F  hs  and  F  c  are functionals of theweighted densities. In order to write down the definitions of theweighted densities we first introduce the average segment den-sity profiles for chain molecules(8) ρ s ( r )  = M   j  = 1 ρ s ,j  ( r )  = M   j  = 1    d R δ( r  −  r j  )ρ( R ), where  ρ s ,j  ( r )  is the local density of the segment  j   of the chain.The weighted densities are then defined as n α ( r )  =  n α, s ( r ) + n α, m ( r )  =    d r  ρ s ( r  )w α  | r  −  r  | ,d  ss  (9) +    d r  ρ m  r   w α  | r  −  r  | ,d  mm  , where scalar ( α  =  0 , 1 , 2 , 3) and vector ( α  =  V  1 ,V  2) weightfunctions,  w α (r,σ  x ) ,  x  =  ss , ms are given in Ref. [52]. Thequantities  d  ss  and  d  mm  are the effective hard-sphere diametersof the reference hard-sphere fluid.Each of the components,  F  α ,  α  =  hs or c is expressed as avolume integral  F  α  =    Φ α ( r ) d r . The hard-sphere contributionis evaluated from [52–54](10) Φ hs ( r )  = ˜ Φ hs  r ;{ n α, s } , { n α, m }   − ˜ Φ hs  r ;{ n α, s  ≡  0 } , { n α, m }  . ˜ Φ hs  r ;{ n α, s } , { n α, m }  = − n 0 ln ( 1 − n 3 ) + n 1 n 2  − n V  1  · n V  2 1 − n 3 (11) + n 32  1 − ξ  2  3 n 3  + ( 1 − n 3 ) 2 ln ( 1 − n 3 ) 36 πn 23 ( 1 − n 3 ) 2  , where  ξ( r )  = | n V  2 ( r ) | /n 2 ( r ) .The contribution  Φ c  is obtained from Wertheim’s first-orderperturbation theory [60,61](12) Φ c ( r )  = 1 − M M n 0 , s ζ   ln  y hs (d  ss )  ,  44  P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52 where  ζ   =  1 − n V  2 , s  · n V  2 , s /(n 2 , s ) 2 and  y hs  is the contact valueof the hard-sphere radial distribution function for the particlesoftheannealedcomponentinthemixturecontainingmatrixandannealed particles(13) y hs (d  ss )  = 11 − n 3 + n 2 d  ss ζ  4 ( 1 − n 3 ) 2  + (n 2 d  ss ) 2 ζ  72 ( 1 − n 3 ) 3 . The simplest treatment of attractive forces follows from theassumption that  d  ss  =  σ  ss  and  d  mm  =  σ  mm  and from the appli-cation of a MFA. Introducing the Weeks–Chandler–Andersendivision [62] of the Lennard-Jones potentials (Eqs. (2) and (5)) (14) ˜ u x (r)  =  − ε x ,  for  r  2 1 / 6 σ  x , x  =  ss , ms ,u x (r),  for  r >  2 1 / 6 σ  x , x  =  ss , ms , we have F  att  = 12    d r d r  ˜ u ss  | r  −  r  |  ρ s ( r )ρ s ( r  ) (15) +    d r d r  ˜ u ms  | r  −  r  |  ρ s ( r )ρ m ( r  ). If the matrix–fluid interactions are of hard-sphere type, the sec-ond term in Eq. (15) disappears.As we have already noted, the application of FMSA in-stead of MFA improves substantially accuracy of theoreticalpredictions, compared with MFA. However, it is possible onlywhen the fluid–matrix interactions are given by the hard-spherepotential. All the details concerning FMSA can be found in[55–58]. In the case of FMSA the effective hard-sphere diam-eter  d  ss  is used. If the density-dependence of   d  ss  is neglected,its temperature-dependence can be approximated by the expres-sion [63](16) d  ss  = σ  ss   0 d r  1 − exp  − βu ss (r)  . The diameter  d  ss  is used so often that it is very tempting to em-ploy an approximation by a simple analytical expression. Oneof the most widely exploited expression is that due to Cotter-man et al. [64](17) d  ss  = 1 + 0 . 2977 T  ∗ 1 + 0 . 33163 T  ∗ + 0 . 00104771 (T  ∗ ) 2 , where  T  ∗ =  kT/ε ss . A collection of other expressions proposedto approximate Eq. (16) can be found in Silva et al. [65] paper. Sensitivity of thermodynamics of bulk fluids to the Barker–Henderson diameter, which covers both its approximation incalculation and improvement in rationality was discussed byTang [66]. In the case of nonuniform systems, however, a sim- plifying assumption that  d  ss  is just  σ  ss  does not change theoverall picture of surface phase transition occurring in the sys-tem. Obviously,  d  mm  ≡  σ  mm , by definition. The Helmholtz freeenergy functional due to the long ranged attraction is then givenby(18) F  att  = − 12    d r d r  ˜ c ss  | r  −  r  |  ρ s ( r )ρ s ( r  ), where  c ss ( | r  −  r  | )  is the attractive part of the direct correlationfunction, which can be expressed as [55–58](19) c ss (r)  =  c Y  (T  ∗ 1  ,z 1 ,d  s s,r) − c Y  (T  ∗ 2  ,z 2 ,d  s s,r), r < d  ss , 0 , d  ss  < r < σ  ss ,u ss (r), r > σ  ss , where  z 1  =  2 . 9637 /σ  ss ,  z 2  =  14 . 0167 /σ  ss ,  T  ∗ 1  =  T  ∗ d  ss /k 1 , T  ∗ 2  =  T  ∗ d  ss /k 2 ,  k 1  =  k 0 exp [ (σ  ss  −  d  ss )z 1 ]  k 2  =  k 0 exp [ (σ  ss  − d  ss )z 2 ]  and  k 0  =  2 . 1714 σ  ss . The first-order Yukawa direct cor-relation function,  c Y  , in Eq. (19) is given in appendix of Ref. [67].TheequilibriumconfigurationoffluidmoleculessatisfiestheEuler–Lagrange equation(20) ρ( R )  =  exp  βµ − βV  b ( R ) − βv( R ) − βΛ( R )  , where  µ  is the chemical potential and  Λ( R )  =  δF  ex /δρ( R ) ,where  F  ex  =  F  hs  +  F  att  +  F  c  is the effective potential field dueto intra- and intermolecular interactions. The excess free energyfunctional  F  ex  depends on the average segment density  ρ s , thus(21) δF  ex δρ( R ) = M   i = 1 λ i ( r ), λ i ( r )  = δF  ex δρ s ( r ). In this work we consider systems that are inhomogeneousin only one (say  z ) direction. Then one obtains (cf. Refs. [43,68,69])(22) ρ s ,i (z)  =  exp (βµ) exp  − βλ i (z)  G i (z)G M  + 1 − i (z), where the functions  G i (z)  are determined from the recurrencerelation(23) G i (z)  =    d z  exp  − βλ i − 1 (z)  θ(σ   − | z  − z  | ) 2 σ G i − 1 (z  ), for  i  =  2 , 3 ,...,M   and with  G 1 (z)  ≡  1.The confined system is in an equilibrium with the bulk fluid.The bulk fluid is characterized by the same value of the chem-ical potential  µ  as the confined system and its properties aredescribed by the equations given above assuming that the exter-nal field  V  ext  is zero and that the segment densities  ρ s ,j   and thetotal segment density  ρ s  are constant in the entire bulk systemand equal to  ρ bs ,j   and  ρ b , s , respectively. Obviously, the matrixconcentration is zero in the bulk system.The numerical methods for the minimization of the grandpotential are identical with those described in our previous pa-pers [68,69]. In particular, we introduce the appropriate propa- gator functions, as described in details in our previous works.For the sake of brevity these equations are not presented here,but we refer the reader to srcinal works [43,68,69].Finally, we introduce the reduced quantities. All the en-ergy parameters are expressed in  ε ss  units,  ε ∗ gs  =  ε gs /ε ss ,  ε ∗ ms  = ε ms /ε ss . Similarly, the diameter  σ  ss  is used as unit of length.Then  σ  ∗ ms  =  σ  ms /σ  ss ,  H  ∗ =  H/σ  ss ,  z ∗ =  z/σ  ss  and the reduceddensities  ρ ∗ s ,j  (z)  are defined as  ρ ∗ s ,j  (z)  =  ρ s ,j  (z)σ  3ss ,  ρ ∗ bs ,j   = ρ bs ,j  (σ  ss ) 3 ,  ρ ∗ bs  =  ρ bs (σ  ss ) 3 , etc.  P. Bryk et al. / Journal of Colloid and Interface Science 313 (2007) 41–52  45 3. Results and discussion As we have noted, setting  M   =  1 in the relevant equationsgiven above, the theory reduces to that for atomic adsorbates.The aim of our model calculations performed for systems in-volving spherically symmetric molecules is two fold. First, wewould like to compare the results of MFA and FMSA with com-puter simulations. Secondly, we intend to compare two cases:the case of matrix molecules interacting with adsorbate atomsvia repulsive (hard-sphere) interactions and via Lennard-Jones ( 12 , 6 )  potential.TheMFAthatwasusedinavastmajorityofpapersaimingatthe application of DFT to experimental data, is perhaps the sim-plest possible that it is able to capture the behavior of nonuni-form systems with repulsive and attractive forces at quantitativelevel. However, this approximation should not be treated asqualitatively correct. An attractive alternative for MFA providesFMSA. The latter theory, with analytical expression for a directcorrelation function as an input, which makes it numerically assimple as MFA, significantly improves the MFA for the longranged attraction [55–58].At the beginning of our discussion we compare the densityfunctionalpredictionswiththeresultsofgrandcanonicalMonteCarlo simulations. Because the simulation method is quite stan-dard, we omit all the technical details and only note that thephase diagrams for the confined systems were obtained em-ploying hyper-parallel tempering technique [59]. The surface roughness was modelled assuming that the matrix distributionis(24) ρ m (z)  =  ρ m0 ,  for 0  z  z m  and  H   −  z m  z  H, 0 ,  otherwise , where  z m  is the matrix thickness at each pore wall. In the cal-culations below we have assumed that  z m  =  σ  ss . The diameterof matrix molecules was the same as the Lennard-Jones diam-eter of fluid particles,  σ  ∗ mm  =  1. The matrix–fluid interactionswere of hard-sphere type and the cut-off of the fluid–fluid po-tential was  r ∗ cut , ss  =  2 . 5. The pore width was kept constant andequal  H  ∗ =  5 and the fluid–pore wall potential parameters were ε ∗ gs  =  9 and  z ∗ 0  =  0 . 5.Fig. 1 shows the density profiles obtained from computersimulations and from both theories at supercritical temperature, T   ∗ =  1 . 5. For all the systems in Fig. 1 the bulk (reference) sys-tem density was  ρ ∗ b , s  =  0 . 6958, which corresponds to the valueof the chemical potential from the bulk system Grand Ensem-ble Monte Carlo simulations equal to  µ/kT   = − 0 . 5. It is notsurprising that the FMSA improves the agreement with simula-tions in comparison with the data obtained from the mean-fieldtheory. This is particularly well seen for the system without ma-trix. The improvement is also evident when one compares thedependence of the average adsorbate densities in the pore,(25)  ρ  = H    0 ρ s (z) d z/H, plotted versus the bulk density, see Fig. 2. The presence of  quenched molecules causes significant changes in the structure Fig. 1. Density profiles from simulations (symbols), from MFA (solid lines)and from FMSA (remaining lines) inside the pore of   H  ∗ =  5 at  ρ ∗ b , s  =  0 . 6958,at  T   =  1 . 5. The consecutive curves from bottom are for  ρ ∗ m0  =  0 . 5, 0.2959,0.09694 and 0 (i.e., for the pore without the quenched particles). The values of all remaining parameters of the model are given in the text.Fig. 2. Average fluid densities in the pore versus the bulk density. The consec-utive curves from bottom are for  ρ m0  =  0 . 5, 0.2959, 0.09694 and 0 (the porewithout the quenched particles). The results of computer simulations are givenas points, theoretical curves resulting from MFA and from FMSA are given as( ··· ) and (—), respectively. Parameters as the profiles in Fig. 1 and the nomen-clature is the same as in Fig. 1. of adsorbed fluid. Due to pore walls “screening” by the matrixmolecules, the height of the local density peak adjacent to thewall is lowered at higher matrix density  ρ m0 .
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