Papers in Physics, vol. 2, art. 020002 (2010) Received: 23 December 2009, Accepted: 24 February 2010 Edited by: D. A. Stariolo Licence: Creative Commons Attribution 3.0 DOI: 10.4279/PIP.020002 www.papersinphysics.org ISSN 1852-4249 Multilayer approximation for a confined fluid in a slit pore G. J. Zarragoicoechea,1, 2∗ A. G. Meyra,1 V. A. Kuz1 A simple Lennard–Jones fluid confined in a slit nanopore with hard walls is studied on the basis of a multilayer structured model. Each layer is homogeneous and parallel to the walls of the pore. The Helmholtz energy of this system is constructed following van der Waals-like approximations, with the advantage that the model geometry permits to obtain analytical expressions for the integrals involved. Being the multilayer system in thermodynamic equilibrium, a system of non-linear equations is obtained for the densities and widths of the layers. A numerical solution of the equations gives the density profile and the longitudinal pressures. The results are compared with Monte Carlo simulations and with experimental data for Nitrogen, showing very good agreement. I. Introduction The effects on phase transition of confined fluids in a slit-like pore have been studied by simulation and different theories [1–11]. In a previous work, we constructed a generalized van der Waals equa- tion for a fluid confined in a nanopore [12, 13]. The shift of the critical parameters was in good agreement with lattice model and numerical sim- ulation results, and the predicted critical temper- ature remarkably reproduced the experiment. In that work, we concluded that the confined van der Waals fluid theory seemed to work better than the bulk one, maybe due to the fact that the higher virial contributions not considered in both theo- ries were less important in the confined fluid than in the bulk. A similar treatment was used previ- ously by Schoen and Diestler [14]. Following that ∗E-mail: vasco@iflysib.unlp.edu.ar 1 IFLYSIB-Instituto de F́ısica de Ĺıquidos y Sistemas Biológicos (CONICET, UNLP, CICPBA), 59 No. 789, 1900 La Plata, Argentina. 2 CICPBA-Comisión de Investigaciones Cient́ıficas de la Prov. de Buenos Aires. line of reasoning, here we study a simple fluid con- fined between two infinite parallel hard walls (slit pore). The walls are at a distance L apart. To study the confined fluid, we propose a multilayer model [15]: the fluid is distributed in n thin lay- ers, one beside the other. Each layer has a uniform density, and can be observed as a non-autonomous phase. A particle in a given layer interacts with its neighbors inside the layer, and with every par- ticle in the other layers. Defay and Prigogine and Murakami et al. have shown that, in a liquid gas interface, the deviation from the Gibbs’ adsorption equation becomes practically negligible in the case of a two layer model [16], and that as the number of transition layers grows, the multilayer model be- comes perfectly consistent with the Gibbs’ equation [17]. The van der Waals-like approximations made in developing this multilayer model theory limit its validity to the low density regime. II. Theory The model system consists of a fluid of N Lennard– Jones particles confined in a slit nanopore. The hard walls of the pore, separated at a distance L (in 020002-1 Papers in Physics, vol. 2, art. 020002 (2010) / G. J. Zarragoicoechea et al. the x direction), have a surface area S (S → ∞). We divided the fluid into n layers, each layer being parallel to the pore walls. The layer i has Ni par- ticles (N = ∑n i=1 Ni), a width Lxi (L= ∑n i=1 Lxi), and a volume Vi = SLxi. Then the Helmholtz en- ergy [18] can be written as A = −kT ln  ZNλ−3Nn∏ i=1 Ni!   . (1) The configuration integral ZN for a pair potential vij may be approximated as ZN = ∫ n∏ i=1 ii NiNjV Ni−1 i V Nj−1 j n∏ k=1 k 6=i,k 6=j V Nk k ∫ Ri ∫ Rj f12dr1dr2 + n∏ i=1 V Nii . The first term in Eq. (3) stands for particles in the layer i . The second term comes from the interac- tion of one particle in layer i with one particle in layer j . In a compact form, and assuming that a layer sees three nearest neighbor layers, ZN = ( n∑ i=1 N2i 2V 2i Ii (4) + n−1∑ i=1 i+3∑ j=i+1 j≤n Ni Vi Nj Vj Iij + 1 ) n∏ i=1 Vi Ni. The integrals Ii and Iij , for the slit pore geome- try and after low density approximations, can be analytically solved to give Ii = ∫∫ Ri f12dr1dr2 ≈− ∫∫ |r1−r2|<σ dr1dr2 − ∫∫ |r1−r2|≥σ v12 kT dr1dr2 = −2Viσ3(b−Bi) − 2Viσ 3ε kT Ai (5) Ii, i+1 = ∫ Ri ∫ Ri+1 f12dr1dr2 ≈− ∫∫ |r1−r2|<σ dr1dr2 − ∫∫ |r1−r2|≥σ v12 kT dr1dr2 = −Viσ3Bi − Viσ 3ε kT Ai, i+1 (6) Ii, i+2 = ∫ Ri ∫ Ri+2 f12dr1dr2 ≈ − ∫∫ |r1−r2|≥σ v12 kT dr1dr2 = −Viσ 3ε kT Ai, i+2 (7) Ii, i+3 = − Viσ 3ε kT Ai, i+3 (8) In the above expressions vij was taken to be the Lennard–Jones pair interaction, being ε and σ the potential parameters. The integrals Ii, i+2 and Ii, i+3 do not contain the excluded volume term be- cause we suppose that the layer widths are Lxi ≥ σ. The expressions for A and B in the preceding equa- tions, functions of Lxi , are given in the Appendix A. The Helmholtz energy, Eq. (1) together with Eq. (4), has the final expression A ≈−kT   n∑ i=1 N2i 2V 2 i Ii + n−1∑ i=1 i+3∑ j=i+1 j≤n Ni Vi Nj Vj Ii,j   − n∑ i=1 NikT ln Vi Ni + NkT (ln λ3 − 1) (9) 020002-2 Papers in Physics, vol. 2, art. 020002 (2010) / G. J. Zarragoicoechea et al. The pressure tensor [12, 13] and chemical poten- tials are obtained from the following equations pxx,i = − 1 LyiLzi ( ∂A ∂Lxi ) T,N pyy,i = pzz,i = − 1 LxiLyi ( ∂A ∂Lzi ) T,N (10) µi = ( ∂A ∂Ni ) T,V,Nj 6=i If the system is in mechanical and chemical equi- librium, the xx components of the pressure tensor and the chemical potentials for each layer must be equal. From these equations, giving as input the wall separation L and the mean density ρ∗ = ρσ3, it is constructed a system of (n − 1) non-linear equa- tions with (n − 1) unknowns (layer densities and widths) to be numerically solved. The low com- putational cost is taken for granted given that the code is easily written and the calculations are car- ried out on a Pentium 4 processor running at 2.66 GHz. At a temperature T∗=kT /ε=1, we have ex- plored the cases with L=10σ and L=15σ, at dif- ferent mean densities. We have also compared the theoretical results with experimental data coming from studies of nitrogen adsorption in graphite slit pores at room temperature [19]. III. Monte Carlo simulation For numerical simulations, N Lennard–Jones par- ticles are confined between hard walls separated at a distance L. The unit cell is build up taken the walls to be of size Ly and Lz in the y and z direc- tions respectively, directions on which the period- ical boundary conditions are applied. The density profiles and pressures were obtained taking average values in fluid slabs parallel to the walls. The pres- sure tensor was used in the simple virial form, as indicated in references [20, 21]. With T∗=1, and for both slit pore widths L=10σ and L=15σ, the size of the unit cell was set to Ly =Lz =30σ, taking the number of particles N to correspond with the mean density. The range of the Lennard–Jones interactions was considered with a cutoff radius of 5σ. Figure 1: Density profiles for a n=9 layer model of a confined fluid in a slit pore (solid symbols). The temperature is T∗=1.0 and the wall separation is L=10σ, with mean densities ρ∗=1/20 (circles), 1/15 (squares), 1/10 (triangles), 1/5 (diamonds), and 1 4 (stars). Open symbols represent the Monte Carlo simulations. Figure 2: zz pressure tensor component. Captions as in Fig. 1. IV. Results In Figs. 1 and 2, the density profiles and zz components of the pressure tensor are shown for T∗=1 and L=10σ. The mean densities studied are ρ∗=1/20, 1/15, 1/10, 1/5, and 1/4. The agreement of the theoretical density profiles with the Monte Carlo simulations is very good. For the pressure there is a rather good correspondence for low den- sities, up to ρ∗=1/10. For the higher densities, dif- 020002-3 Papers in Physics, vol. 2, art. 020002 (2010) / G. J. Zarragoicoechea et al. Figure 3: Density profiles for a n=13 layer model of a confined fluid in a slit pore (solid symbols). The temperature is T∗=1.0 and the wall separa- tion is L=15σ, with mean densities ρ∗=1/10 (tri- angles), and 1/5 (circles). Open symbols represent the Monte Carlo simulations. ferences appear, though the tendencies are similar. The discrepancies come first from the low density approximations done to get the Helmholtz energy. But, while in the simulation slab particles fluctu- ate and at higher densities some clusterization oc- curs, in the theory each layer is supposed to have a homogeneous density which makes it hard for the theoretical pressures to follow those obtained by simulation. For the density profiles, averaging the number of particles in each slab evidently compen- sates the clusterization, and the theory gives good results, at least for the rather low densities studied. The same picture applies to the behavior of the sys- tem for T∗=1 and L=15σ, at mean densities ρ∗= 1/10, and 1/5, represented in Figs. 3 and 4. The results, as expected for hard repulsive walls, show a low density region next to the walls and an increasing density profile, with a maximum at the center of the slit pore. This behavior is also shown with density functional theory [1] and in other Monte Carlo simulations [2]. Finally, the good agreement of the theory with the experiment can be seen in the results shown in Fig. 5. In this figure, the excess number of molecules per unit area of pore surface Γ is plotted in function of the external pressure, at T∗=3.18 and L=4σ. These parameters approximate the ex- perimental values [19] T=303 K and L=1.45 nm, if Figure 4: zz pressure tensor component. Captions as in Fig. 3. ε/k = 95.2 K and σ = 3.75 Å are used to character- ize the nitrogen. In this case, due to the size of the sample, n=3 layers have been used for calculation. Γ is defined as Γ = N −Ng S = (ρ∗ −ρ∗g) L σ3 (11) where Ng/ρ∗g is the number/density of particles which would occupy the slit pore in the absence of the adsorption forces. ρ∗g and the external pressure are determined equating the chemical potential in- side the slit pore (Eq. 10) to the chemical poten- tial coming from the bulk van der Waals equation at the same temperature. The theoretical results presented here are similar to the numerical simu- lation results obtained by the same authors who have done the experiment [19]. They assume that the differences at higher pressures could be a con- sequence of the uncertainty in the determination of the pore geometry. V. Conclusions The application of a simple theory, with van der Waals-like approximations to the Helmholtz en- ergy, to a particular model of spatial distribution makes it possible to obtain analytical expressions for the thermodynamic quantities. The study of a confined fluid in a slit pore geometry with a mul- tilayer approximation produces good results when compared with Monte Carlo simulations at low den- sities. The agreement with a particular experi- 020002-4 Papers in Physics, vol. 2, art. 020002 (2010) / G. J. Zarragoicoechea et al. Figure 5: Excess number of molecules per unit area of pore surface Γ as function of the external pres- sure. The full line represents the experiment (digi- talized from Ref. [19]), and the dots are our theo- retical results. ment on nitrogen confined in a graphite slit pore is remarkable, even though an excess quantity is in study. It may be concluded that the confinement reduces the importance that higher virial contribu- tions have on the equation of the state of the con- fined fluid. Classical density functional theory [22] can also be applied to study the slit pore geometry, with very good agreement with experiments and simulations. Though the theoretical work devel- oped in these pages is not a competitor of density functional theory, it has the advantages of having analytical expressions, and the possibility of eas- ily introducing two immiscible components: for in- stance one or two layer lubricants wetting the walls and a gas or a liquid filling the rest of layers forming the capillary volume. Acknowledgements - This work was partially supported by Universidad Nacional de La Plata and CICPBA. G. J. Z. is member of “Carrera del Inves- tigador Cient́ıfico” CICPBA. Appendix A Expressions of quantities used in Eqs. 5–8: b = 2 3 π; Bi = π4 σ Lxi ; Ai = a1 + a2 Lxi + a3 L3 xi + a4 L9 xi a1 = −169 π; a2 = 3 2 π; a3 = −13π; a4 = 1 90 π (A1) A correction has been made to get good critical parameters for the bulk (L →∞). For Argon a1= -5.7538 and b=1.3538, and for nitrogen a1= -1.5955 and b=1.0349. Ai, i+1 = π90 [ − 1 L9 xi − 1 LxiL 8 xi+1 + 1 Lxi(Lxi+Lxi+1)8 ] −π 3 [ − 1 L3 xi − 1 LxiL 2 xi+1 + 1 Lxi(Lxi+Lxi+1)2 ] −3 2 π Lxi (A2) Ai, i+2 = π90 [ 1 L8 xi+1 − 1 (Lxi+Lxi+1)8 − 1 (Lxi+1+Lxi+2)8 + 1 (Lxi+Lxi+1+Lxi+2)8 ] 1 Lxi −π 3 [ 1 L2 xi+1 − 1 (Lxi+Lxi+1)2 − 1 (Lxi+1+Lxi+2)2 + 1 (Lxi+Lxi+1+Lxi+2)2 ] 1 Lxi (A3) Ai, i+3 = π90 [ 1 (Lxi+1+Lxi+2)8 − 1 (Lxi+Lxi+1+Lxi+2)8 − 1 (Lxi+1+Lxi+2+Lxi+3)8 + 1 (Lxi+Lxi+1+Lxi+2+Lxi+3)8 ] 1 Lxi −π 3 [ 1 (Lxi+1+Lxi+2)2 − 1 (Lxi+Lxi+1+Lxi+2)2 − 1 (Lxi+1+Lxi+2+Lxi+3)2 + 1 (Lxi+Lxi+1+Lxi+2+Lxi+3)2 ] 1 Lxi (A4) [1] S A Sartarelli, L Szybisz, Correlation be- tween asymmetric profiles in slits and standard prewetting lines, Pap. 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