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Thin-Thick Film Transitions on a Planar Solid Surface: a Density Functional Study *

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Thin-Thick Film Transitions on a Planar Solid Surface: a Density Functional Study * ISSN: 0256-307X 中国物理快报 Chinese Physics Letters A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://www.iop.org/journals/cpl http://cpl.iphy.ac.cn CHINESE PHYSICAL SOCIETY CHIN. PHYS. LETT. Vol. 27, No. 3 (2010) 037101 Thin-Thick Film Transitions on a Planar Solid Surface: A Density Functional Study * YU Yang-Xin(u养&)**, LI Ying-Feng(o=¸), ZHENG Yuan-Xiang(x远翔) State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 100084 (Received 29 October 2009) A weighted density functional theory is proposed to predict the surface tension and thin-thick film transition of a Lennard{Jones fluid on a planar solid surface. The underlying density functional theory for theLennard{ Jones fluid at low temperature is based on a modified fundamental measure theory for the hard-core repulsion, a Taylor expansion around zero-bulk-density for attraction, and a correlation term evaluated by the weighted density approximation with a weight function of the Heaviside step function. The predicted surface tension and thin-thick film transition agree well with the results from the Monte Carlo simulations, better than thosefrom alternative approaches. For the Ar/CO2 system, the prewetting line has been calculated. The predicted reduced surface critical temperature is about 0.97, and the calculated wetting temperature is below the triple-point temperature. This is in agreement with the experimental observation. PACS: 71. 15. Mb, 68. 35. Rh, 68. 43. −h DOI: 10.1088/0256-307X/27/3/037101 Wetting is a common phenomenon of a liquid ad- to the Helmholtz energy functional F via a Legendre sorbed on a solid substrate surrounded by a vapor transform phase. It is an area where chemistry, physics and Z ext engineering intersect.[1] The theoretical investigations Ω[휌(r)] = F [휌(r)] + [V (r) − 휇]휌(r)dr; (2) for fluid-solid interfaces indicate that the first-order ext wetting transition is probably the more frequent case, where 휌(r) is the density distribution, V (r) is the so that the prewetting (thin-thick film) transition external potential resulting from the solid surface, and should appear in many of these systems.[2] However, 휇 is the chemical potential of the fluid. Without loss up to date, seldom has the prewetting phase transition of generality, we divide the intrinsic Helmholtz energy been observed in experiment. It is also very difficult functional into an ideal gas, a correlation term, a re- to search for the prewetting line with the molecular pulsive and an attractive contribution, i.e., [3] simulations. F = F id + F rep + F cor + F att: (3) Prewetting phase transition was first investigated independently by Cahn[4] using van der Waals' square- The ideal gas contribution to the Helmholtz energy gradient theory and by Ebner and Samm[5] us- functional is exactly known,[9] ing a gradient expansion of the Helmholtz energy Z id X 3 functional. The theoretical understanding of wet- F [휌(r)] = kBT dr휌i(r)[lnf휌i(r)휆i g − 1]; (4) ting transition has been developed considerably since i then. Regrettably, the developed theories are either where 휆i is the de Broglie thermal wavelength, T is [2;6] [7] qualitative or most complicated. the absolute temperature, and kB is the Boltzmann In this Letter, we propose a density functional constant. theory that is accurate at low temperature for the The hard-sphere repulsive contribution F rep can Lennard{Jones (LJ) fluid. The theory is tested by be obtained from the modified fundamental measure [10] the Ar/CO2 system modeled with the truncated 12- theory, 6 LJ potential.[8] The truncated 12-6 LJ potential is Z rep hs[︀ ]︀ expressed as F = kBT Φ fn훼(r)g dr; (5) ( 12 6 4"[︀(︀ 휎 )︀ − (︀ 휎 )︀ ]︀; r ≤ r ; where Φhs[fn (r)g] is the excess Helmholtz free- u(r) = r r c (1) 훼 energy density due to hard-core repulsion. It consists 0; r > rc; of scalar (S) and vector (V) parts where 휎 and " are, respectively, the size and energy Φhs[︀fn (r)g]︀ = Φhs(S) + Φhs(V): (6) parameters, r is the center-to-center distance of the 훼 two interacting molecules, and rc is the cutoff dis- According to the modified fundamental measure tance. The grand potential functional Ω is related theory of Yu et al.,[10] the scalar part is given *Supported by the National Natural Science Foundation of China under Grant Nos 20876083 and 20736003. **To whom correspondence should be addressed. Email: [email protected] ○c 2010 Chinese Physical Society and IOP Publishing Ltd 037101-1 CHIN. PHYS. LETT. Vol. 27, No. 3 (2010) 037101 by the Boublik{Mansoori{Carnahan{Starling{Leland where 훽 = 1=kBT . If we neglect all higher-order terms equation of state, c(n) (n > 2) in Eq. (14), and let the reference bulk den- sity 휌0 ! 0, F att becomes 3 b hs(S) n1n2 n2 ln(1 − n3) Φ = − n0 ln(1 − n3) + + 2 1 ZZ 1 − n3 36휋n3 훽F att = − drdr0c(2)(jr0 − rj)휌(r)휌(r0): (17) 3 2 n2 + 2 ; (7) 36휋n3(1 − n3) For a fluid at bulk density limit 휌b ! 0, the second- order direct correlation function can be exactly ex- and the vector part is expressed as pressed as hs(V) nV 1 · nV 2 n2nV 2 · nV 2 ln(1 − n3) c(2)(r) = exp [︀−훽uatt(r)]︀ − 1; (18) Φ = − − 2 1 − n3 12휋n3 att n2nV 2 · nV 2 where u (r) is the attractive part of LJ potential, − 2 : (8) 12휋n3(1 − n3) i.e., 8 The weighted densities in Eqs. (7) and (8) are defined 0; r < 휎; <> as uatt(r) = [︀(︀ 휎 )︀12 (︀ 휎 )︀6]︀ (19) Z 4" r − r ; 휎 ≤ r ≤ rc; 0 0 (훼) 0 > n훼(r) = dr 휌(r )w (r − r ); (9) : 0; r > rc: where 훼 = 0, 1, 2, 3, V1 and V2. The weight functions For numerical convenience, we expand c(2)(r) in Tay- w(훼)(r) are defined by[10] lor series since j훽uj is a small quantity. Generally the first-order expansion is accurate enough, but because (2) 2 (0) (1) w (r) = 휋d w (r) = 2휋dw (r) = 훿(d=2 − r); temperature is low at thin-thick film transition and (10) the expansion converges slowly, we truncate the ex- (3) w (r) = Θ(d=2 − r); (11) pansion at the second-order term, i.e., (V 2) (V 1) w (r) = 2휋dw (r) = (r=r)훿(d=2 − r): (12) 1 c(2)(r) ≈ −훽uatt(r) + 훽2[uatt(r)]2: (20) In Eqs. (10){(12), Θ(r) is the Heaviside step function, 2 훿(r) denotes the Dirac delta function, and d is the From Eqs. (17) and (20) we can derive the expression hard-core diameter which is a function of tempera- of the attractive contribution to the Helmholtz energy [11] ture. Khedr et al. discussed different approaches functional, which can be approximated by to calculate the ratio 푑/휎. Because we use an accu- rate equation of state which is independent of the ratio 1 ZZ {︂ F att = drdr0 uatt(jr0 − rj) 푑/휎 and the results of present theory are insensitive 2 to the ratio 푑/휎, we select the simple expression of }︂ 훽 [︀ att 0 ]︀2 0 [12] − u (jr − rj) 휌(r)휌(r ): (21) the ratio 푑/휎 given by Cotterman et al. When the 2 * reduced temperature T = kBT=" = 0{15, d can be accurately expressed by[12] It should be pointed out that if we truncate the Tay- lor expansion of c(2)(r) at the first-order term, Eq. (21) d 1 + 0:2977T * [13] = : (13) will reduce to the result of the mean-field theory. 휎 1 + 0:33163T * + 1:0477 × 10−3T *2 In this work, we use a weighted density approxi- mation to evaluate the correlation contribution to the The attractive contribution to the Helmholtz energy Helmholtz energy functional, i.e., functional is derived from a truncated second-order Z functional Taylor expansion around a reference bulk cor cor fluid, F = 휌¯(r) [¯휌(r)]dr; (22) Z 훿F att where cor[¯휌(r)] is the correlation contribution to the F att = F att(휌0) + ∆휌(r)dr b 훿휌(r) Helmholtz free energy per particle of a bulk fluid with ZZ 훿F att density휌 ¯(r). The weighted density for the correlation + ∆휌(r)∆휌(r0)drdr0 + ··· ; (14) 훿휌(r)훿휌(r0) part is defined by Z 0 0 0 (cor) 0 0 where ∆휌(r) = 휌(r) − 휌b , and 휌b is the reference bulk 휌¯(r) = 휌(r )w (jr − rj)dr ; (23) density. The direct correlation functions due to the attractive interaction are defined as where w(cor)(r) is a normalized weight function, and (1) att ex 0 we use the Heaviside step function Θ(r) to express it, c (r) = −훽훿F /훿휌(r) = −훽휇 (휌b ); (15) i.e., (cor) 3 c(2)(jr0 − rj) = −훽훿2F att/훿휌(r)훿휌(r0); (16) w (r) = Θ(d − r): (24) 4휋d3 037101-2 CHIN. PHYS. LETT. Vol. 27, No. 3 (2010) 037101 The excess Helmholtz free-energy per particle due to is in good agreement with the Monte Carlo simulation correlation effect is obtained from the MBWR equa- results.[3] tion of state with the parameters given by Johnson et We compare the various theoretical results with [14] al., the Monte Carlo simulation data in Fig. 2 at T * = 0:88. The vertical lines in Fig. 2 denote the location of 8 * i 6 X ai(휌 ) X cor(휌) = " +" b G − CS − att; (25) the prewetting phase transition.
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