Papers in Physics, vol. 1, art. 010001 (2009) Received: 17 June 2009, Accepted: 28 August 2009 Edited by: S. A. Cannas Licence: Creative Commons Attribution 3.0 DOI: 10.4279/PIP.010001 www.papersinphysics.org ISSN 1852-4249 Correlation between asymmetric profiles in slits and standard prewetting lines Salvador A. Sartarelli,1∗ Leszek Szybisz2−4† The adsorption of Ar on substrates of Li is investigated within the framework of a den- sity functional theory which includes an effective pair potential recently proposed. This approach yields good results for the surface tension of the liquid-vapor interface over the entire range of temperatures, T, from the triple point, Tt, to the critical point, Tc. The be- havior of the adsorbate in the cases of a single planar wall and a slit geometry is analyzed as a function of temperature. Asymmetric density profiles are found for fluid confined in a slit built up of two identical planar walls leading to the spontaneous symmetry breaking (SSB) effect. We found that the asymmetric solutions occur even above the wetting temperature Tw in a range of average densities ρ ∗ ssb1 ≤ ρ ∗ av ≤ ρ∗ssb2, which diminishes with increasing temperatures until its disappearance at the critical prewetting point Tcpw. In this way a correlation between the disappearance of the SSB effect and the end of prewetting lines observed in the adsorption on a one-wall planar substrate is established. In addition, it is shown that a value for Tcpw can be precisely determined by analyzing the asymmetry coefficients. I. Introduction The study of physisorption of fluids on solid sub- strates had led to very fascinating phenomena mainly determined by the relative strengths of ∗E-mail: asarta@ungs.edu.ar †E-mail: szybisz@tandar.cnea.gov.ar 1 Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Gutierrez 1150, RA–1663 San Miguel, Argentina. 2 Laboratorio TANDAR, Departamento de F́ısica, Comisión Nacional de Enerǵıa Atómica, Av. del Libertador 8250, RA–1429 Buenos Aires, Argentina. 3 Departamento de F́ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Univer- sitaria, RA–1428 Buenos Aires, Argentina. 4 Consejo Nacional de Investigaciones Cient́ıficas y Técnicas, Av. Rivadavia 1917, RA–1033 Buenos Aires, Argentina. fluid-fluid (f-f) and substrate-fluid (s-f) attrac- tions. In the present work we shall refer to two of such features. One is the prewetting curve iden- tified in the study of fluids adsorbed on planar sur- faces above the wetting temperature Tw (see, e.g., Pandit, Schick, and Wortis [1]) and the other is the occurrence of asymmetric profiles of fluids confined in a slit of identical walls found by van Leeuwen and collaborators in molecular dynamics calcula- tions [2, 3]. It is known that for a strong substrate (i.e., when the s-f attraction dominates over the f-f one) the adsorbed film builds up continuously showing a complete wetting.In such a case, neither prewetting transitions nor spontaneous symmetry breaking (SSB) of the profiles are observed, both these phenomena appear for substrates of moder- ate strength. The prewetting has been widely analyzed for ad- sorption of quantum as well as classical fluids. A 010001-1 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. summary of experimental data and theoretical cal- culations for 4He may be found in Ref. [4]. Studies of other fluids are mentioned in Ref. [5]. These investigations indicated that prewetting is present in real systems such as 4He, H2, and inert gases adsorbed on alkali metals. On the other hand, after a recent work of Berim and Ruckenstein [6] there is a renewal of the inter- est in searching for the SSB effect in real systems. These authors utilized a density functional (DF) theory to study the confinement of Ar in a slit com- posed of two identical walls of CO2 and concluded that SSB occurs in a certain domain of tempera- tures. In a revised analysis of this case, reported in Ref. [7], we found that the conditions for the SSB were fulfilled because the authors of Ref. [6] had diminished the s-f attraction by locating an extra hard-wall repulsion. However, it was found that inert gases adsorbed on alkali metals exhibit SSB. Results for Ne confined by such substrates were re- cently reported [8]. The aim of the present investigation is to study the relation between the range of temperatures where the SSB occurs and the temperature depen- dence of the wetting properties. In this paper we illustrate our findings describing the results for Ar adsorbed on Li. Previous DF calculations of An- cilotto and Toigo [9] as well as Grand Canonical Monte Carlo (GCMC) simulations carried out by Curtarolo et al. [10] suggest that Ar wets Li at a temperature significantly below Tc. So, this sys- tem should exhibit a large locus of the prewetting line and this feature makes it very convenient for our study as it was already communicated during a recent workshop [11]. The paper is organized in the following way. The theoretical background is summarized in Sec. II.. The results, together with their analysis, are given in Sec. III.. Sec. IV. is devoted to the conclusions. II. Theoretical background In a DF theory, the Helmholtz free energy FDF[ρ(r)] of an inhomogeneous fluid embedded in an external potential Usf (r) is expressed as a functional of the local density ρ(r) (see, e.g., Ref. [12]) FDF[ρ(r)] = νid kB T ∫ dr ρ(r){ln[Λ3ρ(r)] − 1} + ∫ dr ρ(r) fHS[ρ̄(r); dHS] + 1 2 ∫ ∫ dr dr′ρ(r) ρ(r′′) Φattr(| r − r′ |) + ∫ dr ρ(r) Usf (r) . (1) The first term is the ideal gas free energy, where kB is the Boltzmann constant and Λ =√ 2 π h̄2/mkB T the de Broglie thermal wavelength of the molecule of mass m. Quantity νid is a pa- rameter introduced in Eq. (2) of [13] (in the stan- dard theory it is equal unity). The second term accounts for the repulsive f-f interaction approxi- mated by a hard-sphere (HS) functional with a cer- tain choice for the HS diameter dHS. In the present work we have used for fHS[ρ̄(r); dHS] the expression provided by the nonlocal DF (NLDF) formalism de- veloped by Kierlik and Rosinberg [14] (KR), where ρ̄(r) is a properly averaged density. The third term is the attractive f-f interactions treated in a mean field approximation (MFA). Finally, the last inte- gral represents the effect of the external potential Usf (r) exerted on the fluid. In the present work, for the analysis of ph- ysisorption we adopted the ab initio potential of Chismeshya, Cole, and Zaremba (CCZ) [15] with the parameters listed in Table 1 therein. i. Effective pair attraction The attractive part of the f-f interaction was de- scribed by an effective pair interaction devised in Ref. [5], where the separation of the Lennard-Jones (LJ) potential introduced by Weeks, Chandler and Andersen (WCA) [16] is adopted ΦWCAattr (r) =   −ε̃ff , r ≤ rm 4ε̃ff [( σ̃f f r )12 − ( σ̃f f r )6] , r > rm . (2) 010001-2 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. Here rm = 21/6σ̃ff is the position of the LJ mini- mum. No cutoff for the pair potential was intro- duced. The well depth ε̃ff and the interaction size σ̃ff are considered as free parameters because the use of the bare values εff/kB = 119.76 K and σff = 3.405 Å overestimates Tc. So, the complete DF formalism has three ad- justable parameters (namely, νid, ε̃ff , and σ̃ff ), which were determined by imposing that at l-v co- existence, the pressure as well as the chemical po- tential of the bulk l and v phases should be equal [i.e., P(ρl) = P(ρv) and µ(ρl) = µ(ρv)]. The pro- cedure is described in Ref. [5]. In practice, we set dHS = σ̃ff and imposed the coexistence data of ρl, ρv, and P(ρl) = P(ρv) = P0 for Ar quoted in Ta- ble X of Ref. [17] to be reproduced in the entire range of temperatures T between Tt = 83.78 K and Tc = 150.86 K. ii. Euler-Lagrange equation The equilibrium density profile ρ(r) of the adsorbed fluid is determined by a minimization of the free energy with respect to density variations with the constraint of a fixed number of particles N δ δρ(r) [ FDF[ρ(r)] −µ ∫ dr ρ(r) ] = 0 . (3) Here the Lagrange multiplier µ is the chemical po- tential of the system. In the case of a planar sym- metry where the flat walls exhibit an infinite extent in the x and y directions, the profile depends only on the coordinate z perpendicular to the substrate. For this geometry, the variation of Eq. (3) yields the following Euler-Lagrange (E-L) equation δ[(Fid + FHS)/A] δρ(z) + ∫ L 0 dz′ρ(z′)Φ̄attr(| z −z′ |) + Usf (z) = µ , (4) where δ(Fid/A) δρ(z) = νid kB T ln [Λ 3 ρ(z)] , (5) and δ(FHS/A) δρ(z) = fHS[ρ̄(z); dHS] + ∫ L 0 dz′ρ(z′) δfHS[ρ̄(z′); dHS] δρ̄(z′) δρ̄(z′) δρ(z) . (6) Here Fid/A and FHS/A are free energies per unit of one wall area A. L is the size of the box adopted for solving the E-L equations. The boundary condi- tions for the one-wall and slit systems are different and will be given below. The final E-L equation may cast into the form νid kB T ln [Λ 3 ρ(z)] + Q(z) = µ , (7) where Q(z) = fHS[ρ̄(z); dHS] + ∫ L 0 dz′ρ(z′) δfHS[ρ̄(z′); dHS] δρ̄(z′) δρ̄(z′) δρ(z) + ∫ L 0 dz′ρ(z′) Φ̄attr(| z −z′ |) + Usf (z) . (8) The number of particles Ns per unit area, A, of the wall is Ns = N A = ∫ L 0 ρ(z) dz . (9) In order to get solutions for ρ(z), it is useful to rewrite Eq. (7) as ρ(z) = ρ0 exp ( − Q(z) νid kB T ) , (10) with ρ0 = 1 Λ3 exp ( µ νid kB T ) . (11) The relation between µ and Ns is obtained by sub- stituting Eq. (10) into the constraint of Eq. (9) µ = −νid kB T × ln [ 1 NsΛ3 ∫ L 0 dz exp ( − Q(z) νidkBT )] .(12) When solving this kind of systems, it is usual to define dimensionless variables z∗ = z/σ̃ff for the distance and ρ∗ = ρσ̃3ff for the densities. In these units the box size becomes L∗ = L/σ̃ff . 010001-3 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. III. Results and Analysis In order to quantitatively study the adsorption of fluids within any theoretical approach,one must re- quire the experimental surface tension of the bulk liquid-vapor interface, γlv, to be reproduced sat- isfactorily over the entire Tt ≤ T ≤ Tc tempera- ture range. Therefore, we shall first examine the prediction for this observable before studying the adsorption phenomena. i. Surface tension of the bulk liquid-vapor interface Figure 1 shows the experimental data of γlv taken from Table II of Ref. [18]. In order to theoretically evaluate this quantity the E-L equations for free slabs of Ar, i.e. setting Usf (z) = 0 , (13) were solved imposing periodic boundary conditions ρ(z = 0) = ρ(z = L). At a given temperature T , for a sufficiently large system one must obtain a wide central region with ρ(z ' L/2) = ρl(T) and tails with density ρv(T), where the values of ρl(T) and ρv(T) should be those of the liquid-vapor coexis- tence curve. The surface tension of the liquid-vapor interface is calculated according to the thermody- namic definition γlv = (Ω + P0 V )/A = Ω/A + P0 L , (14) where Ω = FDF −µN is the grand potential of the system and P0 the pressure at liquid-vapor coexis- tence previously introduced. We solved a box with L∗ = 40. The obtained results are plotted in Fig. 1 together with the prediction of the fluctuation the- ory of critical phenomena γlv = γ0lv(1 − T/Tc) 1.26 with γ0lv = 17.4 K/Å 2 (see, e.g., [19]). One may realize that our values are in satisfactory agree- ment with experimental data and the renormaliza- tion theory over the entire range of temperatures Tt ≤ T ≤ Tc, showing a small deviation near Tt. ii. Adsorption on one planar wall It is assumed that the physisorption of Ar on a one wall substrate of Li is driven by the CCZ potential, i.e., Figure 1: Surface tension of Ar as a function of temperature. Squares are experimental data taken from Table II of Ref. [18]. The solid curve corre- sponds to the fluctuation theory of critical phenom- ena and the circles are present DF results. Figure 2: Adsorption isotherms for the Ar/Li sys- tem, i.e., ∆µ as a function of coverage Γ`. Up- triangles correspond to T = 119 K; circles to T = 118 K; diamonds to T = 117 K; squares to T = 116 K; down-triangles to T = 114 K and stars to T = 112 K. Usf (z) = UCCZ(z) . (15) The E-L equations were solved in a box of size L∗ = 40 by imposing ρ(z > L) = ρ(z = L). The solution gives a density profile ρ(z) and the corresponding chemical potential µ. Adsorption 010001-4 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. isotherms at a given temperature were calculated as function of the excess surface density. This quan- tity, also termed coverage, is often expressed in nominal layers ` Γ` = (1/ρ 2/3 l ) ∫ ∞ 0 dz[ρ(z) −ρB] , (16) where ρB = ρ(z →∞) is the asymptotic bulk den- sity and ρl the liquid density at saturation for a given temperature. By utilizing the results for µ obtained from the E-L equation and the value µ0 corresponding to saturation at a given tempera- ture T , the difference ∆µ = µ−µ0 was evaluated. Figure 2 shows the adsorption isotherms for tem- peratures above Tw, where an equal area Maxwell construction is feasible. This is just the prewet- ting region characterized by a jump in coverage Γ`. The size of this jump depends on temperate. The largest jump occurs at Tw and diminishes for in- creasing T until its disappearance at Tcpw. Density profiles just below and above the coverage jump for T = 114 K are displayed in Fig. 3, in that case Γ` jumps from 0.5 to 3.6. Therefore, the formation of the fourth layer may be observed in the plot. Figure 3: Examples of density profiles of Ar ad- sorbed on a surface of Li at T = 114 K displayed as a function of the distance from the wall located at z∗ = 0. Dashed curves are profiles for Γ` below the coverage jump, while solid curves are stable films above this jump. The wetting temperature Tw can be obtained from the analysis of the values of ∆µ/kB at which Figure 4: Prewetting line for Ar adsorbed on Li. The solid curve is the fit to Eq. (17) and reaches the ∆µpw/kB = 0 line at Tw = 110.1 K. the jump in coverage occurs at each considered tem- perature. The behavior ∆µpw/kB vs T is displayed in Fig. 4. A useful form for determining the tem- perature Tw was derived from thermodynamic ar- guments [20] ∆µpw(T) = µpw(T) −µ0(T) = apw (T −Tw)3/2 . (17) Here apw is a model parameter and the exponent 3/2 is fixed by the power of the van der Walls tail of the adsorption potential Usf (z) '−C3/z3. The fit of the data of ∆µ/kB to Eq. (17) yielded Tw = 110.1 K and apw/kB = −0.16 K−1/2. On the other hand, according to Fig. 2, the criti- cal prewetting point Tcpw lies between T = 118 and 119 K. At the latter temperature, the film already presents a continuous growth. Our values of Tw and Tcpw are smaller than those obtained from prior DF calculations [9] (Tw = 123 K and Tcpw ' 130 K) and GCMC simulations [10] (Tw = 130 K). The difference with the DF evaluation of Ref. [9] is due to the use of different effective pair potentials as we explain in Ref. [5], where the adsorption of Ne is studied. The present approach gives a reasonable γlv, while that of Ref. [9] fails dramatically close to Tt. The difference with the GCMC results cannot be interpreted in a straightforward way. 010001-5 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. iii. Confinement in a planar slit In the slit geometry, where the Ar atoms are con- fined by two identical walls of Li the s-f potential becomes Usf (z) = UCCZ(z) + UCCZ(L−z) . (18) The walls were located at a distance L∗ = 40, this width guarantees that the pair interaction between two atoms located at different walls is negligible. In fact, this width is wider than L∗ = 29.1, which was utilized in the pioneering molecular dynamics calculations [2, 3]. Accordingly, the E-L equations were solved in a box of size L∗ = 40. In this geom- etry, the repulsion at the walls causes the profiles ρ(z = 0) and ρ(z = L) to be equal to zero. The solutions were obtained at a fixed dimensionless av- erage density defined in terms of N, A, and L as ρ∗av = N σ̃ 3 ff/AL = N ∗ s /L ∗. Figure 5: Free energy per particle (in units of kB T) for Ar confined in a slit of Li with L∗ = 40 at T = 115 K displayed as a function of the average density. The curve labeled by circles corresponds to symmetric solutions, while that labeled by triangles corresponds to asymmetric ones. The SSB occurs in a certain range of average density ρ∗ssb1 ≤ ρ ∗ av ≤ ρ∗ssb2. For temperatures below Tw = 110.1 K, we ob- tained large ranges of ρ∗av where the asymmetric solutions exhibit a lower free energy than the cor- responding symmetric ones. In spite of the fact that there is a general idea that a connection ex- ists between the SSB effect and nonwetting, we have found, by contrast, that SSB behavior extends above the wetting temperature. Furthermore, we have also found a relation between prewetting and SSB. Figure 5 shows the free energy per particle, fDF = FDF/N, for both symmetric and asymmetric solutions for the Ar/Li system at T = 115 K > Tw as a function of the average density. According to this picture, the ground state (g-s) exhibits asym- metric profiles between a lower and an upper limit ρ∗ssb1 = 0.057 ≤ ρ ∗ av ≤ ρ∗ssb2 = 0.192. Out of this range no asymmetric solutions were obtained form the set of Eqs. (7)-(12). Similar features were ob- tained for higher temperatures until T = 118 K, above this value the profiles corresponding to the g-s are always symmetric. Figure 6 shows three examples of solutions determined at T = 115 K. The result labeled 1 is a small asymmetric profile, that labeled 2 is the largest asymmetric solution at this temperature. So, by further increasing ρ∗av, the SSB effect disappears and the g-s becomes symmet- ric, as indicated by the curve labeled 3. When the asymmetric profiles occur, the situation is denoted as partial (or one wall) wetting. The symmetric solutions account for a complete (two wall) wet- ting. These different situations can be interpreted in terms of the balance of γsl, γsv and γlv sur- face tensions, carefully discussed in previous works [2, 3, 7]. Here we shall restrict ourselves to briefly outline the main features. When the liquid is ad- sorbed symmetrically like in the case of profile 3 in Fig. 6, there are two s-l and two l-v interfaces. Hence, the total surface excess energy may be writ- ten as γ sym tot = 2 γsl + 2 γlv . (19) On the other hand, for a asymmetric profile γasytot becomes γ asy tot = γsl + γlv + γsv . (20) The three quantities of the r.h.s. of this equation are related by Young’s law (see, e.g., Eq. (2.1) in Ref. [21]) γsv = γsl + γlv cos θ , (21) where θ is the contact angle defined as the angle be- tween the wall and the interface between the liquid and the vapor (see Fig. 1 in Ref. [21]). By using Young’s law, the Eq. (20) may be rewritten as 010001-6 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. γ asy tot = 2 γsl + γlv (1 + cos θ) , (22) with cos θ = (γsv − γsl)/γlv < 1. If one changes γsl by increasing enough Ns (as shown in Fig. 5), and/or T , and/or the strength of Usf (z), eventually the equality γsv −γsl = γlv may be reached yield- ing cos θ = 1. Then, the system would undergo a transition to a symmetric profile where both walls of the slit are wet. Figure 6: Density profiles of Ar confined in a slit of Li with L∗ = 40 at T = 115 K. The displayed spectra denoted by 1, 2 and 3 correspond to average densities ρ∗av = 0.074, 0.192 and 0.218, respectively. It is important to remark that, indeed, there are two degenerate asymmetric solutions. Besides that one shown in Fig. 6 where the profiles exhibit the thicker film adsorbed on the left wall (left asym- metric solutions - LAS), there is an asymmetric so- lution with exactly the same free energy but where the thicker film is located near the right wall (right asymmetric solutions - RAS). The asymmetry of density profiles may be mea- sured by the quantity ∆N = 1 Ns ∫ L/2 0 dz [ρ(z) −ρ(L−z)] . (23) According to this definition, if the profile is com- pletely asymmetrical about the middle of the slit, i.e. for: (i) ρ(z < L/2) 6= 0 and ρ(z ≥ L/2) = 0; or (ii) ρ(z < L/2) = 0 and ρ(z ≥ L/2) 6= 0 this Figure 7: Asymmetry parameter for Ar confined by two Li walls separated by a distance of L∗ = 40 as a function of average density. From outside to inside the curves correspond to temperatures T = 112, 114, 115, 116, 117 and 118 K. The asymmetric solutions occur for different ranges ρ∗ssb1 ≤ ρ ∗ av ≤ ρ∗ssb2. Figure 8: Circles stand for both branches of the asymmetry parameter for Ar confined in an L∗ = 40 slit of Li walls for temperatures between Tw and Tcpw. The solid curve is the fit to Eq. (24) used to determine Tcpw. quantity becomes +1 or −1, respectively, while for symmetric solutions it vanishes. We evaluated the asymmetry coefficients of so- lutions obtained for increasing temperatures up to T = 118 K. The results for LAS profiles at tem- 010001-7 Papers in Physics, vol. 1, art. 010001 (2009) / S. A. Sartarelli et al. peratures larger that Tw are displayed in Fig. 7 as a function of the average density. One may ob- serve how the range ρ∗ssb1 ≤ ρ ∗ av ≤ ρ∗ssb2 dimin- ishes under increasing temperatures. The SSB ef- fect persists at most for the critical ρ∗av(crit) = (17/24) σ̃2ff × 10 −2 ' 0.074 with σ̃ff expressed in Å. We shall demonstrate that by analyzing the data of ∆N for ρ∗av(crit) it is possible to determine the critical prewetting point. Figure 8 shows these values for both the LAS and RAS profiles, calcu- lated at different temperatures, suggesting a rather parabolic shape. So, we propose a fit to the follow- ing quartic polynomial T = Tcpw + a2∆ 2 N + a4∆ 4 N . (24) This procedure yielded Tcpw = 118.4 K, a2 = −14.14 K, and a4 = −16.63 K. The obtained value of Tcpw is in agreement with the limits established when analyzing the adsorption isotherms of the one-wall systems displayed in Fig. 2. These results indicate that the disappearance of the SSB effect coincides with the end of the prewetting line. IV. Conclusions We have performed a consistent study within the same DF approach of free slabs of Ar, the adsorp- tion of these atoms on a single planar wall of Li and its confinement in slits of this alkali metal. Good results were obtained for the surface ten- sion of the liquid-vapor interface. The analysis of the physisorption on a planar surface indicates that Ar wets surfaces of Li in agreement with previous investigations. The isotherms for the adsorption on one planar wall exhibit a locus of prewetting in the µ − T plane. A fit of such data yielded a wetting temperature Tw = 110.1 K. In addition, these isotherms also show that the critical prewet- ting point Tcpw lies between T = 118 and 119 K. These results for Tw and Tcpw are slightly below the values obtained in Refs. [9, 10], the discrepancy is discussed in the text. On the other hand, this investigation shows that the profiles of Ar confined in a slit of Li present SSB. This effect occurs in a certain range of average densities ρ∗ssb1 ≤ ρ ∗ av ≤ ρ∗ssb2, which diminishes for increasing temperatures. The main output of this work is the finding that above the wetting temper- ature the SSB occurs until Tcpw is reached. To the best of our knowledge this is the first time that such a correlation is reported. Furthermore, it is shown that by examining the evolution of the asymmetry coefficient one can precisely determine Tcpw. The obtained value Tcpw = 118.4 K lies in the inter- val established when analyzing the adsorption on a single wall. Acknowledgements - This work was supported in part by the Grants PICT 31980/5 from Agencia Nacional de Promoción Cient́ıfica y Tecnológica, and X099 from Universidad de Buenos Aires, Ar- gentina. [1] R Pandit, M Schick, M Wortis, Systematics of multilayer adsorption phenomena on attrac- tive substrates Phys. Rev. B 26, 5112 (1982). 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