SciELO - Scientific Electronic Library Online

 
vol.1 número1A note on the consensus time of mean-field majority-rule dynamics índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

  • No hay articulos citadosCitado por SciELO

Links relacionados

  • No hay articulos similaresSimilares en SciELO

Compartir


Papers in physics

versión On-line ISSN 1852-4249

Pap. Phys. vol.1 no.1 La Plata jun. 2009

 

Correlation between asymmetric profiles in slits and standard prewetting lines

 

Salvador A. Sartarelli,1* Leszek Szybisz2-4

*E-mail: asarta@ungs.edu.ar
E-mail: szybisz@tandar.cnea.gov.ar

1 Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Gutiérrez 1150, RA-1663 San Miguel, Argentina.
2 Laboratorio TANDAR, Departamento de Física, Comisión Nacional de Energía Atómica, Av. del Libertador 8250, RA-1429 Buenos Aires, Argentina.
3 Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, RA-1428 Buenos Aires, Argentina.
4 Consejo Nacional de Investigaciones Científicas y Técnicas, Av. Rivadavia 1917, RA-1033 Buenos Aires, Argentina.

 

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 tensión of the liquid-vapor interface over the entire range of temperatures, T, from the triple point, Tí, to the critical point, Tc. The be-havior of the adsórbate 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 densitieswhich 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 valué for Tcpw can be precisely determined by analyzing the asymmetry coeffcients.

Keywords Density functional calculations; Surface tension; Wetting

 

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 fluid-fluid (f-f) and substrate-fluid (s-f) attractions. 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 moderate strength.

The prewetting has been widely analyzed for ad-sorption of quantum as well as classical fluids. A 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 metáis.

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 conñnement 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 fulñlled because the authors of Ref. [6] had diminished the s-f attraction by locating an extra hard-wall repulsión. However, it was found that inert gases adsorbed on alkali metáis exhibit SSB. Results for Ne conñned 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 ñndings describing the results for Ar adsorbed on Li. Previous DF calculations of An-cilotto and Toigo [9] as well as Grand Canonical Monte Cario (GCMC) simulations carried out by Curtarolo et al. [10] suggest that Ar wets Li at a temperature signiñcantly 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.

The ñrst term is the ideal gas free energy, where ks is the Boltzmann constant and the de Broghe thermal wavelength of the molecule of mass m. Quantityis a parameter introduced in Eq. (2) of [13] (in the standard theory it is equal unity). The second term accounts for the repulsive /-/ interaction approxi-mated by a hard-sphere (HS) functional with a certain choice for the HS diameter <¿hs- In the present work we have used for /Hs[p(r); ¿hs] the expression provided by the nonlocal DF (NLDF) formalism de-veloped by Kierlik and Rosinberg [14] (KR), where p{r) is a properly averaged density. The third term is the attractive /-/ interactions treated in a mean ñeld approximation (MFA). Finally, the last integral 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, Colé, and Zaremba (CCZ) [15] with the parameters listed in Table 1 therein.

i. Effective pair attraction

The attractive part of the /-/ 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

II. Theoretical background

In a DF theory, the Helmholtz free energy Ídf [p(r)\ of an inhomogeneous fluid embedded in an external potential Usf(r) is expressed as a functional of the local density p(r) (see, e.g., Ref. [12])

Here rm = 21'6aff is the position of the LJ mínimum. No cutoff for the pair potential was intro-duced. The well depth éff and the interaction size aff are considered as free parameters because the use of the bare valúes Eff/ks = 119.76 K and a ff = 3.405 A overestimates Tc.

So, the complete DF formalism has three ad-justable parameters (namely, v^, s¡f, and 07/), which were determined by imposing that at l-v co-existence, the pressure as well as the chemical potential of the bulk / and v phases should be equal [Le., P(pi) = P(pv) and p(p¡) = p(pv)]. The pro-cedure is described in Ref. [5]. In practice, we set <¿hs = &ff an(i imposed the coexistence data of p¡, pv, and P(p¡) = P(pv) = Po 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 proñle p(r) of the adsorbed fluid is determined by a minimization of the free energy with respect to density variations with the constraint of a ñxed number of partióles N

Here the Lagrange multiplier p is the chemical potential of the system. In the case of a planar sym-metry where the fíat walls exhibit an infinite extent in the x and y directions, the proñle 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

Here Fid/A and Fus/A are free energies per unit of one wall área 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

III. Results and Analysis

In order to quantitatively study the adsorption of fluids within any theoretical approach,one must re-quire the experimental surface tensión of the bulk liquid-vapor interface, 7^, to be reproduced sat-isfactorily over the entire Tt < T < Tc tempera-ture range. Therefore, we shall ñrst examine the prediction for this observable before studying the adsorption phenomena.

i. Surface tensión of the bulk liquid-vapor interface

Figure 1 shows the experimental data of 7;^ taken from Table II of Ref. [18]. In order to theoretically evalúate this quantity the E-L equations for free slabs of Ar, i.e. setting

were solved imposing periodic boundary conditions p(z = 0) = p(z = L). At a given temperature T, for a sufñciently large system one must obtain a wide central región with p(z ~ L/2) = pi(T) and tails with density pv(T), where the valúes of pi(T) and pv(T) should be those of the liquid-vapor coexis-tence curve. The surface tensión of the liquid-vapor interface is calculated according to the thermody-namic deñnition

where Q = Pdf - p N is the grand potential of the system and Po 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 7;^ = 7^(1 - T/Tc)1'26 with 7^ = 17.4 K/Á (see, e.g., [19]). One may realize that our valúes are in satisfactory agree-ment with experimental data and the renormaliza-tion theory over the entire range of temperatures Tt <T<TCy 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 system, i.e., Áp as a function of coverage T¿. 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.

The E-L equations were solved in a box of size L* = 40 by imposing p(z > L) = p(z = L). The solution gives a density proñle p(z) and the corresponding chemical potential p. Adsorption 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 l

where pg = p(z -> oo) is the asymptotic bulk density and pi the liquid density at saturation for a given temperature. By utilizing the results for p obtained from the E-L equation and the valué po corresponding to saturation at a given tempera-ture T, the difference Ap = p - po was evaluated. Figure 2 shows the adsorption isotherms for tem-peratures above TWl where an equal área Maxwell construction is feasible. This is just the prewet-ting región characterized by a jump in coverage T¿. The size of this jump depends on températe. The largest jump occurs at Tw and diminishes for in-creasing T until its disappearance at Tcpw. Density proñles just below and above the coverage jump for T = 114 K are displayed in Fig. 3, in that case Y¿ jumps from 0.5 to 3.6. Therefore, the formation of the fourth layer may be observed in the plot.


Figure 4: Prewetting line for Ar adsorbed on Li. The solid curve is the ñt to Eq. (17) and reaches the Appw/kg = 0 line at Tw = 110.1 K.

the jump in coverage occurs at each considered temperature. The behavior Appw/kB vsT is displayed in Fig. 4. A useful form for determining the temperature Tw was derived from thermodynamic ar-guments [20]


Figure 3: Examples of density proñles of Ar adsorbed 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 proñles for T¿ below the coverage jump, while solid curves are stable ñlms above this jump.

The wetting temperature Tw can be obtained from the analysis of the valúes of Ap/ks at which

Here apw is a model parameter and the exponent 3/2 is ñxed by the power of the van der Walls tail of the adsorption potential Usf(z) ~ -C¡/z3. The ñt of the data of Ap/ks 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 critical prewetting point Tcpw lies between T = 118 and 119 K. At the latter temperature, the ñlm already presents a continuous growth.

Our valúes 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 ^/¡v, while that of Ref. [9] fails dramatically cióse to Tt. The difference with the GCMC results cannot be interpreted in a straightforward way.

iii. Confinement in a planar slit

In the slit geometry, where the Ar atoms are con-ñned by two identical walls of Li the s-f potential becomes

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 geometry, the repulsión at the walls causes the proñles p(z = 0) and p(z = L) to be equal to zero. The solutions were obtained at a ñxed dimensionless av-erage density deñned in terms of N, A, and L as Pav = N a3,, IA L = N* IL*.


Figure 5: Free energy per partióle (in units of ks T) for Ar conñned 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.

 

For temperatures below Tw = 110.1 K, we obtained large ranges of p*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 partióle, /df = 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 asymmetric proñles between a lower and an upper limit P*sbi = 0-057 < p*av < p*sb2 = 0.192. Out of this range no asymmetric solutions were obtained form the set of Eqs. (7)-(12). Similar features were obtained for higher temperatures until T = 118 K, above this valué the proñles 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 proñle, that labeled 2 is the largest asymmetric solution at this temperature. So, by further increasing p*avl the SSB effect disappears and the g-s becomes symmetric, as indicated by the curve labeled 3. When the asymmetric proñles occur, the situation is denoted as partial (or one wall) wetting. The symmetric solutions account for a complete (two wall) wetting. These different situations can be interpreted in terms of the balance of 7S¡, ^sv and ^/¡v 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 adsorbed symmetrically like in the case of proñle 3 in Fig. 6, there are two s-l and two l-v interfaces. Henee, the total surface excess energy may be written as

with cosí? = (7stI - 7sí)/7ít, < 1. If one changes 7S; by increasing enough _/Vs (as shown in Fig. 5), and/or T, and/or the strength of Usf(z), eventually the equality ~/sv - 7S¡ = 7^ may be reached yield-ing cosí? = 1. Then, the system would undergo a transition to a symmetric proñle where both walls of the slit are wet.


Figure 6: Density proñles of Ar conñned 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 p*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 proñles exhibit the thicker ñlm adsorbed on the left wall (left asymmetric solutions - LAS), there is an asymmetric so-lution with exactly the same free energy but where the thicker ñlm is located near the right wall (right asymmetric solutions - RAS).

The asymmetry of density proñles may be mea-sured by the quantity

According to this deñnition, if the proñle is com-pletely asymmetrical about the middle of the slit, i.e. for: (i) p(z < L/2) ^ 0 and p(z > L/2) = 0; or (ii) p(z < L/2) = 0 and p(z > L/2) = 0 this


Figure 7: Asymmetry parameter for Ar conñned 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 oceur for different ranges yO*s61 < p*av <


Figure 8: Circles stand for both branches of the asymmetry parameter for Ar conñned in an L* = 40 slit of Li walls for temperatures between Tw and Tcpw The solid curve is the ñt to Eq. (24) used to determine Tcpw. quantity becomes +1 or -1, respectively, while for symmetric solutions it vanishes.

We evaluated the asymmetry coefñcients of solutions obtained for increasing temperatures up to T = 118 K. The results for LAS proñles at temperatures larger that Tw are displayed in Fig. 7 as a function of the average density. One may observe how the range y0*s61 < p*av < 9*^2 dimin-ishes under increasing temperatures. The SSB ef-fect persists at most for the critical p*av(crit) = (17/24) ¿r?, x 10-2 c; 0.074 with a¡j expressed in Á.

We shall demónstrate that by analyzing the data of Ajv for p*av(crit) it is possible to determine the critical prewetting point. Figure 8 shows these valúes for both the LAS and RAS proñles, calcu-lated at different temperatures, suggesting a rather parabolic shape. So, we propose a ñt to the follow-ing quartic polynomial

This procedure yielded Tcpw = 118.4 K, a2 = - 14.14 K, and 0,4 = -16.63 K. The obtained valué 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 adsorption of these atoms on a single planar wall of Li and its conñnement in slits of this alkali metal. Good results were obtained for the surface tensión 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 fj, - T plañe. A ñt of such data yielded a wetting temperature Tw = 110.1 K. In addition, these isotherms also show that the critical prewetting point Tcpw lies between T = 118 and 119 K. These results for Tw and Tcpw are slightly below the valúes obtained in Refs. [9, 10], the discrepancy is discussed in the text.

On the other hand, this investigation shows that the proñles of Ar conñned in a slit of Li present SSB. This effect occurs in a certain range of average densities p*sbí < p*av < P*sb2y which diminishes for increasing temperatures. The main output of this work is the ñnding that above the wetting temperature the SSB occurs until Tcpw is reached. To the best of our knowledge this is the ñrst time that such a correlation is reported. Furthermore, it is shown that by examining the evolution of the asymmetry coefñcient one can precisely determine Tcpw. The obtained valué 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 Científica y Tecnológica, and X099 from Universidad de Buenos Aires, Argentina.

[1] R Pandit, M Schick, M Wortis, Systematics of multilayer adsorption phenomena on attrac-tive substrates Phys. Rev. B 26, 5112 (1982).

[2] J H Sikkenk, J O Indekeu, J M J van Leeuwen, E O Vossnack, Molecular-dynamics simulation of wetting and drying at solid-fluid interfaces Phys. Rev. Lett. 59, 98 (1987).

[3] M J P Nijmeijer, C Bruin, A F Bakker, J M J van Leeuwen, Wetting and drying of an inert wall by a fluid in a molecular-dynamics simu-lation, Phys. Rev. A 42, 6052 (1990).

[4] L Szybisz, Adsorption of superfluid 4He films on planar heavy-alkali metals studied with the Orsay-Trento density functional, Phys. Rev. B 67, 132505 (2003).

[5] S A Sartarelli, L Szybisz, I Urrutia, Adsorption of Ne on alkali surfaces studied with a den-sity functional theory, Phys. Rev. E 79, 011603 (2009).

[6] G O Berim, E Ruckenstein, Symmetry break-ing of the fluid density profiles in closed nanoslits, J. Chem. Phys. 126, 124503 (2007).

[7] L Szybisz, S A Sartarelli, Density profiles of Ar adsorbed in slits of CO2: Spontaneous sym-metry breaking revisited, J. Chem. Phys. 128, 124702 (2008).

[8] S A Sartarelli, L Szybisz, I Urrutia, Sponta-neous symmetry breaking and first-order phase transitions of adsorbed fluids, Int. J. Bifurca-tion Chaos (in press).

[9] F Ancilotto, F Toigo, Prewetting transitions of Ar and Ne on alkali-metal surfaces surface, Phys. Rev. B 60, 9019 (1999).

[10] S Curtarolo, G Stan, M J Bojan, M W Cole, W A Steele, Threshold criterion for wetting at the triple point, Phys. Rev. E 61, 1670 (2000).

[11] L Szybisz and S A Sartarelli, Adsorci´on de gases nobles sobre sustratos planos de metales alcalinos, Communication at the Workshop TREFEMAC09 held at the Univerisidad Nacional de La Pampa, Santa Rosa, Argentina, May 4-6 (2009).

[12] P I Ravikovitch, A Vishnyakov, A V Neimark, Density functional theories and molecular sim-ulations of adsorption and phase transitions in nanopores, Phys. Rev. E 64, 011602 (2001).

[13] F Ancilotto, S Curtarolo, F Toigo, M W Cole, Evidence concerning drying behavior of Ne near a Ce surface, Phys. Rev. Lett. 87, 206103 (2001).

[14] E Kierlik, M L Rosinberg, Free-energy den-sity functional for the inhomogeneous hard-sphere fluid: Application to interfacial adsorp-tion, Phys. Rev. A 42, 3382 (1990).

[15] A Chizmeshya, M W Cole, E. Zaremba, Weak biding potentials and wetting transitions, J. Low Temp. Phys. 110, 677 (1998).

[16] J D Weeks, D Chandler, H C Andersen, Role of repulsive forces in determining the equilibrium structure of simple fluids, J. Chem. Phys. 54, 5237 (1971).

[17] V A Rabinovich, A A Vasserman, V I Ne-dostup, L S Veksler, Thermophysical proper-ties of neon, argon, krypton and xenon, Hemi-sphere, Washington DC (1988).

[18] S-T Wu, G-S Yan, Surface tensions of simple liquids, J. Chem. Phys. 77, 5799 (1982).

[19] J Vrabec, G K Kedia, G Fuchs, H Hasse, Vapour-liquid coexistence of the truncated and shifted Lennard-Jones fluid, Mol. Phys. 104, 1509 (2006).

[20] E Cheng, G Mistura, H C Lee, M H W Chan, M W Cole, C Carraro, W F Saam, F Toigo, Wetting transitions of liquid hydrogen films, Phys. Rev. Lett. 70, 1854 (1993).

[21] P G de Gennes, Wetting: statics and dynam-ics, Rev. Mod. Phys. 57, 827 (1985).

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons