Confinement of Ar between two identical parallel semi-infinite walls

The confinement of Ar in planar slits of two identical parallel semi-infinite walls of alkali metals, alkaline-earth metal Mg, and CO(2) is investigated within the framework of the density functional theory. It is assumed that (1) the fluid atoms interact via a recently proposed effective attractive pair potential with strength, epsilon(ff), which reproduces the experimental data of the surface tension of the liquid-vapor interface at the bulk coexistence curve, and (2) the adsorption on the walls is described by ab initio potentials characterized by a well depth, W(sf). In this way the systems were studied in the framework of a realistic approach. We found that for small coverages, the slit is always filled by forming two symmetric vapor films, one at each wall. For increasing coverage the behavior depends on the ratio W(sf)/epsilon(ff) and the temperature T. In the case of alkali metals, we found at the triple point, T(t), of the adsorbate a regime of average density rho(av) ( *) in which the ground state exhibits asymmetric density profiles, leading to the so-called spontaneous symmetry breaking (SSB) effect. The SSB appears at an average density rho(sb1) ( *) and disappears at a higher average density rho(sb2) ( *). When T is increased, the range of densities rho(sb1) ( *)<or=rho(av) ( *)<or=rho(sb2) ( *) diminishes and eventually the SSB disappears at a critical temperature, T(sb), which coincides with the critical prewetting temperature T(cpw) observed in the adsorption on a single wall. For T>T(cpw) the slit is filled symmetrically up to the phase transition to capillary condensation. All these features are examined as a function of the strength of the substrate and the width of the slit. Furthermore, no SSB effect was found for Mg and CO(2).


I. INTRODUCTION
The study of the physisorption of fluids on solid substrates led to very fascinating phenomena mainly determined by the relative strengths of fluid-fluid ͑f-f͒ and substratefluid ͑s-f͒ attractions, f f and W sf , respectively. 1 An interesting tour from adsorption on a single wall toward confinement in a slit geometry is described in Ref. 2. In that letter, Rascón and Parry present a simple theoretical model which smoothly connects wetting and capillary condensation, showing the rich sequence of different regimes and phenomena induced by the substrate shape.In the present work we shall mainly refer to properties of fluids confined between two identical planar parallel semi-infinite walls.We consider structureless walls which define a x-y plane.It is usually assumed that the density profile ͑r͒ of the fluid is invariant on this plane depending only on the coordinate z perpendicular to the walls.About 20 years ago, van Leeuwen and co-workers carried out calculations for simple fluids confined in such a slit geometry on the Delf molecular dynamics ͑MD͒ processor, which was designed ad hoc for MD simulations of this kind of system. 3,4It was assumed that the atoms of the fluid interact via standard Lennard-Jones ͑LJ͒ potentials with strength f f and size f f parameters.The width of the slit was L = 29.1 f f , which is sufficiently large to avoid any capillary effect and support solid-liquid ͑s-l͒, solid-vapor ͑s-v͒, and liquid-vapor ͑l-v͒ interfaces.6][7][8][9] These authors found that the behavior depends on the ratio r = W sf / f f .Starting from r Ӎ 0.1 they obtained that for increasing r the profiles of the ground state were sequentially ͑i͒ symmetric with absence of liquid close to the walls ͑drying͒, ͑ii͒ asymmetric with liquid at one wall and vapor at the other ͑incomplete or partial wetting͒, and ͑iii͒ symmetric with liquid at both walls ͑complete wetting͒.These structures can be understood in terms of the balance of surface tensions at the s-l, s-v, and l-v interfaces, i.e., ␥ sl , ␥ sv , and ␥ lv , respectively. 5We shall come back to the latter point in a next section.Now, we shall only mention that the occurrence of the asymmetric profiles of point ͑ii͒ is known in the literature as the spontaneous symmetry breaking ͑SSB͒ effect.
The works of Berim and Ruckenstein 10,11 renewed the interest in searching for the SSB effect in real systems.These authors studied in Ref. 11 the confinement of Ar in a closed slit composed of two identical walls of CO 2 assuming that the s-f interaction is given by the 9-3 potential proposed by Ebner and Saam. 12They analyzed a slit of width L =15 f f and stated that SSB occurs for temperatures below a critical value T sb = 106 K.In a revisited study of this system reported in Ref. 13, we found that the conditions for the occurrence of SSB had been fulfilled in Ref. 11 because the well depth of the s-f attraction was diminished by the presence of an extra hard-wall repulsion.In looking for SSB in other systems, we found that inert gases confined in slits of alkali metals exhibit suitable ratios r for the appearance of this effect.Preliminary results for Ne and Ar confined in such a type of slits have recently been reported elsewhere. 14,15n the present work we shall report on a rather comprehensive investigation about the behavior of Ar confined in slits of two identical planar walls.The calculations have been carried out within the density functional ͑DF͒ theory.The adsorption is described by using the ab initio s-f potentials of Ref. 16 for alkaline walls and of Refs.17 and 18 for CO 2 .In order to study the wetting properties of all the above mentioned substrates, the computational task must be performed over the entire temperature range between T t and T c of the adsorbate.For this purpose, a recently published proposal 19 for the mean-field approximation ͑MFA͒ of the attractive part of the f-f interaction was adopted.This effective potential gives a satisfactory description of the surface tension of the liquid-vapor interface, ␥ lv , of Ar at coexistence for temperatures in the whole range T t ↔ T c . 15,20The behavior of the systems was studied as a function of the width of the slits.It was found that in the cases of alkali metals ͑Cs, Rb, K, Na, and Li͒ the SSB effect occurs at temperatures lower than certain critical values T sb .In the present paper, we shall present detailed results for Rb, Na, and Li sweeping the entire range of attraction strengths of the alkali metals ͑see Table I͒.On the other hand, the SSB effect does not occur for the stronger attractors Mg and CO 2 .
It is known that the isotherms for the adsorption on single planar walls which exert an attraction of moderate strength ͓i.e., 0.7Շ r Շ 2.5͔ exhibit a first-order wetting transition at the point ͓T w , w = 0 ͑T w ͔͒ between T t and T c .T w is characterized by the appearance of coexisting thin and very thick adsorbed fluid films, while for T Ͻ T w the coverage of adsorbed films is finite.Under these conditions for T Ն T w there is an associated prewetting line in the − T plane. 21he locus of these prewetting lines extends away from ͑T w , w ͒ into the region of pressures below the corresponding bulk saturation value P 0 ͑T͒ and terminates at the surface critical point ͑T cpw , cpw ͒.A typical phase diagram of the prewetting phenomenon in the T − plane is depicted in Fig. 5 of Ref. 19.In order to get a deeper insight of the SSB effect, we analyzed the correlation between the values of T sb obtained in the present work and the T cpw determined in a recent study of the adsorption of Ar on single walls performed by utilizing exactly the same MFA. 15,20he 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.Section IV is devoted to the conclusions.

II. THEORETICAL BACKGROUND
Let us first outline the utilized DF formalism and the MFA adopted for describing the attractive part of the f-f interaction.Next we shall give the equations used for determining the density profile.

A. Density functional theory
In a DF theory the Helmholtz free energy F DF of an inhomogeneous fluid embedded in an external potential U sf ͑r͒ is expressed as a functional of the local density ͑r͒ ͑see, e.g., Refs.22 and 23͒, 2 ͵͵ drdrЈ͑r͒͑rЉ͒⌽ attr ͉͑r − rЈ͉͒ + ͵ dr͑r͒U sf ͑r͒.

͑2.1͒
The first term is the ideal gas free energy, where k B is the Boltzmann constant, ⌳ is the de Broglie thermal wavelength,  24 where ¯͑r͒ is an properly averaged density and d HS is the HS diameter.The third term is the attractive f-f interactions, treated in a MFA manner.Finally, the last integral represents the effect of the external potential U sf ͑r͒ exerted on the fluid.
The attractive part of the f-f interaction was described by an effective pair interaction devised in Ref. 19 by using the separation of the LJ potential of Weeks, Chandler, and Andersen, 25 ⌽ attr where r m =2 1/6 ˜ff is the position of the LJ minimum.No cutoff radius was introduced for the pair potential.The well depth ˜ff and the interaction size ˜ff are considered as free parameters because the use of bare values ArAr / k B = 119.76K and ArAr = 3.405 Å, overestimates the experimental result T c = 150.86K.The latter effect may be observed in Fig. 2  In summary, the complete DF formalism has three adjustable parameters ͑namely, id , ˜ff , and ˜ff ͒, which were determined by imposing that at l-v coexistence the pressure as well as the chemical potential of the bulk l and v phases should be equal ͓i.e., P͑ l ͒ = P͑ v ͒ and ͑ l ͒ = ͑ v ͔͒.In practice, we set d HS = ˜ff and at a given temperature T imposed that the coexistence data of l , v , and P͑ l ͒ = P͑ v ͒ = P 0 for Ar quoted in Table X of Ref. 26 be reproduced.The procedure is described in Ref. 19.Close to T t , the values obtained for the parameter id are compatible with unity, while near T c they become about 0.9 accounting for the complicated interactions in the regime where the liquid and vapor densities are of the same order.
The DF formulation outlined above not only provides an adjusted bulk equation of state at saturation, but, in addition, it predicts reasonable values of ␥ lv at liquid-vapor coexistence.As reported in Refs.15 and 20, these results are in good agreement over the entire range of temperatures T t Յ T Յ T c , with the experimental data for Ar tabulated by Wu and Yan. 27Moreover, Fig. 2͑b͒ in Ref. 20 indicates that ͑i͒ the present DF results are similar to the MD values obtained by Trokhymchuk and Alexandre 28 and ͑ii͒ the largest difference with experimental data occurs close to T t amounting 10%.Therefore, the present MFA becomes an appropriate realistic approach for studying the adsorption of Ar.

B. Euler-Lagrange equations
The equilibrium density profile ͑r͒ 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,

͑2.3͒
Here the Lagrange multiplier is the chemical potential of the system.In the case of a planar symmetry where the flat walls exhibit an infinite extent in the x and y directions, we assume that the profile depends only on the coordinate z perpendicular to the substrate.For this geometry, the variation of Eq. ͑2.3͒ yields the following Euler-Lagrange ͑EL͒ equation: where and Here F id / A and F HS / A are free energies per unit area of one wall A and L is the width of the slit.The final EL equation may cast into the form id k B T ln͓⌳ 3 ͑z͔͒ + Q͑z͒ = , ͑2.7͒ where The number of particles N per unit wall's area A is In order to get solutions for ͑z͒ it is useful to rewrite Eq. ͑2.7͒ as ͑2.11͒The relation between and N s is obtained by substituting Eq. ͑2.10͒ into the constraint of Eq. ͑2.9͒, When solving this kind of system it is usual to define dimensionless variables z ‫ء‬ = z / ˜ff for the distance and ‫ء‬ = ˜ff 3 for the densities.In these units the box size becomes L ‫ء‬ = L / ˜ff .

III. RESULTS AND ANALYSIS
The confinement of Ar atoms in a planar slit is governed by the total potential where the contributions U ab initio ͑z͒ are potentials exerted by single walls.For the analysis of confinement between alkaline walls, we adopted in all the cases the ab initio potentials of Chismeshya, Cole, and Zaremba 16 ͑CCZ͒ with parameters listed in Table I therein.The study of the Ar/ CO 2 system was accomplished by utilizing the s-f interaction adopted by Mistura et al. 17 The latter potential was evaluated in terms of ab initio contributions derived by Marshall et al. 18 ͑hereafter referred as M-M potential͒ and is more realistic than the widely used in the literature 9-3 potential proposed by Ebner and Saam 12 ͑ES͒.Figure 1 shows a comparison of the reduced adsorption potentials U ab initio ͑z͒ / ArAr corresponding to the different examined substrates.Although the M-M and ES potentials for Ar/ CO 2 have already been in Fig. 3 of Ref. 17, we find convenient to mention here the main differences.The asymptotic behavior in both cases is proportional to 1 / z These widths guarantee that the pair interaction between two atoms located at different walls is negligible, and that with L ‫ء‬ = 60 it is even larger.The slit with L ‫ء‬ = 10 is wider than twice the radius cutoff r c ‫ء‬ = 2.5 usually adopted in MD and Monte Carlo calculations and was analyzed to search for size effects.The solutions of the EL equations were calculated at fixed average densities defined in terms of N and L as av ‫ء‬ = N s ˜ff 3 / L = N s ‫ء‬ / L ‫ء‬ .The calculations were carried out by setting the reduced step of the discretization at ⌬z ‫ء‬ = ⌬z / ˜ArAr = 0.025.By iterating self-consistently the set of connected Eqs.͑2.7͒-͑2.12͒one always gets the groundstate ͑g-s͒ solution, obtaining a good convergence which may be improved by increasing the number of iterations.Asymmetric g-s occurs in a closed range of average densities sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ .In the regime of av ‫ء‬ where asymmetric profiles exhibit the lowest free energy, one may obtain symmetric solutions by imposing explicitly a symmetric ͑z͒ in the whole chain of equations.

A. Symmetry of density profiles
Let us begin the description of the analysis by reporting the results obtained for Ar confined between walls of Li.For temperatures below T w = 110 K we obtained large ranges of av ‫ء‬ where the asymmetric solutions exhibit a lower free energy than the corresponding symmetric ones.In spite of the fact that there is a general idea that a connection exists between the SSB effect and nonwetting, we found, by contrast, that the SSB behavior extends above the wetting temperature.The adsorption of Ar on a single planar substrate of Li exhibits a prewetting line which extends over about 10 K from T w = 110 to T cpw Շ 119 K; 15,20 this feature allows for a detailed study of asymmetric solutions above T w .
Let us first show the sorts of solutions with lowest free energy which may be obtained for this kind of systems and, subsequently, to describe the main results of our work.Figure 2 shows three examples of g-s solutions determined at T = 115 K ͑larger than T w ͒ for a slit of L ‫ء‬ = 40.These results were obtained by increasing coverage and are the common  pattern for alkali metals.In this geometry, the repulsion at the walls causes that the profiles at contact, ‫ء‬ ͑z ‫ء‬ =0͒ and ‫ء‬ ͑z ‫ء‬ = L ‫ء‬ ͒, both be equal zero.The result labeled by 1 ͑dashdotted curve͒ is a small asymmetric profile and that labeled by 2 ͑solid curve͒ is the largest asymmetric solution at this temperature; both cases are examples of partial wetting.By further increasing av ‫ء‬ the SSB effect disappears and the g-s becomes symmetric ͑complete wetting͒ as indicated by the curve labeled by 3 ͑dashed curve͒.
Figure 3 shows the free energy per particle, f DF = F DF / N in units of k B T, for both symmetric and asymmetric solutions for the Ar/Li system at T = 115 K as a function the average density.According to this picture the g-s exhibits asymmetric profiles between a lower and an upper limit sb1 ‫ء‬ = 0.057Յ av ‫ء‬ Յ sb2 ‫ء‬ = 0.192.Out of this range no asymmetric solutions were from the set of Eqs.͑2.7͒-͑2.12͒.For higher temperatures up to T = 118 K the range sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ diminishes as the temperature increases.For T Ն 119 K the profiles corresponding to the g-s are always symmetric.The same qualitative pattern is exhibited by solutions for slits with other widths.The slits built of alkali metals heavier than Li exert a weaker attraction on Ar atoms.This feature can be seen in Fig. 1 and Table I.The smaller values of r favor the occurrence of asymmetric g-s.In practice, for the remaining alkali metals we obtained a qualitatively similar behavior to that found for Li.The size of the asymmetry will be discussed in the next subsection.
The conditions for the occurrence of a partial ͑one wall͒ wetting or a complete ͑two walls͒ wetting can be expressed in terms of the balance of ␥ sl , ␥ sv , and ␥ lv surface tensions.Since such a behavior was carefully discussed in previous works, 5,8,13 here we shall restrict ourselves to briefly outline the main features.When the liquid is adsorbed symmetrically like in the case of profile 3 in Fig. 2, there are two s-l and two l-v interfaces.Hence, the total surface excess energy may be written as On the other hand, for an asymmetric profile ␥ tot asy becomes ␥ tot asy = ␥ sl + ␥ lv + ␥ sv .

͑3.3͒
The three quantities of the right hand side of this equation are related by Young's law ͓see, e.g., Eq. ͑2.1͒ in Ref. 30͔, where is the contact angle defined as the angle between the wall and the interface between the liquid and the vapor ͑see Fig. 1 in Ref. 30͒.By using Young's law, Eq. ͑3.3͒ may be rewritten as with cos = ͑␥ sv − ␥ sl ͒ / ␥ lv Ͻ 1.If one changes ␥ sl by increasing enough N s ͑as shown in Fig. 3͒, and/or T, and/or the strength of U sf ͑z͒ eventually the equality ␥ sv − ␥ sl = ␥ lv may be reached yielding cos = 1.Then the system would undergo a transition to a symmetric profile where both walls of the slit are wet.
It is important to remark that, indeed, there are two degenerate asymmetric solutions.Besides that shown in Fig. 2 where the profiles exhibit the thicker film adsorbed on the left wall ͓left asymmetric solution ͑LAS͔͒, there is an asymmetric solution with exactly the same free energy but where the thicker film is located near the right wall ͓right asymmetric solution ͑RAS͔͒.
On the other hand, Fig. 1 indicates that the adsorption potentials exerted by walls of Mg and CO 2 are significantly more attractive than that of Li, the strongest of alkali metals.This feature causes that already at T t the filling of slits built of Mg and CO 2 is accomplished by the formation of symmetric profiles.Figure 4 shows a sequence of profiles for increasing average density av ‫ء‬ from 0.0555 to 0.555 for the Ar/ CO 2 system at T t .This behavior is in agreement the wetting at T t found in Ref. 20.

B. Asymmetry coefficient
The asymmetry of density profiles may be measured by the quantity According to this definition, if the profile is completely asymmetrical about the middle of the slit, i.e., for ͑i͒ ͑z Ͻ L / 2͒ 0 and ͑z Ն L / 2͒ =0, or ͑ii͒ ͑z Ͻ L / 2͒ = 0 and ͑z Ն L / 2͒ 0, this quantity becomes +1 or Ϫ1, respectively, while for symmetric solutions it vanishes.The asymmetry coefficients ⌬ N were evaluated for all the determined solutions.We shall first describe in detail the results for slits of Li and, next, refer to slits of other substrates.Figure 5 shows ⌬ N for LAS profiles as a function of av ‫ء‬ .The results for L ‫ء‬ = 10, 20, 40, and 60 are displayed in four different panels.Data for several values of T are included in each panel.The temperatures are T = T t = 83.78K, T = 109 K ͓just below T w ͑Ar/ Li͒ = 110.0K͔, and five values above T w , namely, T = 114, 115, 116, 117, and 118 K.One may observe that in all these panels the range where asymmetric solutions occur, i.e., sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ , diminishes with increasing temperatures.The SSB effect always disappears at T = 119 K, which is the upper limit for T cpw determined in Refs.15 and 20 for the adsorption of Ar on a single wall of Li ͑see Table I͒.In addition, Fig. 5 indicates that the range sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ at a given T, close to T cpw , becomes narrower and moves toward smaller av ‫ء‬ when L ‫ء‬ is increased.It is expected that for even wider slits the data for T = 114 K will approach that of T = 115 K.
In order to illustrate the results obtained for Na and Rb, we report the asymmetry coefficients for slits of L ‫ء‬ = 60. Figure 6 shows ⌬ N for LAS profiles in a slit of Na as a function of av ‫ء‬ .The displayed data correspond to the following temperatures: T = T t , T = 124 K ͓just below T w ͑Ar/ Na͒ = 124.8K͔, and five values above T w , namely, T = 126, 127, 128, 129, and 130 K.As in the case of Li, the range sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ , where the asymmetric solutions occur diminishes for increasing temperatures.Eventually, the SSB effect disappears at T = 131 K, which is the upper limit for T cpw quoted in Table I.
Figure 7 shows ⌬ N for LAS profiles in a slit of Rb with L ‫ء‬ = 60 as a function of av ‫ء‬ .Data evaluated at T = T t , T = 138 K ͓just below T w ͑Ar/ Rb͒ = 138.7 K͔, and three values above T w , namely, T = 139, 140, and 141 K are plotted.As for the substrates analyzed above, the range sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ diminishes when the temperature is increased and the asymmetric solutions disappear at T = 141 K, which is the upper limit for T cpw included in Table I.The results for slits of K are very similar to that for Rb.

C. Critical prewetting temperature
Let us now determine systematically the maximum temperatures T sb for the occurrence of the SSB effect.We shall illustrate the procedure by describing the analysis of the data plotted in Fig. 5

͑3.7͒
The results of these fits are depicted in Fig. 8.One may observe that all the fitting curves are consistent with a common value T ab = 118.4K.
There is a general idea that the existence of g-s asymmetric profiles is a signature for nonwetting, however, we found that the SSB effect occurs for temperatures higher than T w .It is important to emphasize that the isotherms for adsorption on a single wall indicate that on a prewetting line, at a given T, the thick films are stable only above a critical wetting coverage n c ͑cw͒.The coverage is defined as where B = ͑z → ϱ͒ is the asymptotic bulk density which should be equal to v at a given T. Below the critical value n c ͑cw͒ the thick film is unstable and coexists with a thin one at a common chemical potential.This regime of coexistence extends from n c ͑fw͒ to n c ͑cw͒, below n c ͑fw͒ there are stable thin films.The range ⌬n c = n c ͑cw͒ − n c ͑fw͒ diminishes when temperature increases and becomes zero at T cpw .The asymmetric solutions in a slit ͑like that labeled by 2 in Fig. 2͒ are the stable realization of the unstable regime of coexistence in the adsorption on a single wall.Our results indicate that the disappearance of the SSB effect coincides with the end of the prewetting line.As can be seen in Table I, the obtained value of T sb for slits of Li is in agreement with the limits for T cpw determined in a previous work.We are able to perform a meaningful comparison because the slits and the single walls were studied by applying the same DF theory with equal MFA.Therefore, T sb may be associated to T cpw .
The results for ⌬ N ͓ av ‫ء‬ ͑crit͔͒ calculated in the cases of the Ar/Na, Ar/K, and Ar/Rb slits were analyzed in the same way as the Ar/Li system.The obtained T sb are listed in Table I.A glance at this table indicates that the values determined in the present work lay in the interval determined in the analysis of single walls in Ref. 20.We do not report results for Ar/Cs because the adsorption potential for this system is almost equal ͑indistinguishable in the scale of Fig. 1͒ to that for Ar/Rb.

IV.
The confinement of Ar between identical planar structureless walls was studied.A series of substrates with different attractive strengths was analyzed.It was found that the profiles of Ar confined in slits of alkali metals exhibit SSB.This effect is present in a certain range of average densities sb1 ‫ء‬ Յ av ‫ء‬ Յ sb2 ‫ء‬ already at T t of Ar and this range diminishes for increasing temperatures.A crucial output of this work is the finding that SSB occurs above the wetting temperature T w and disappears when a temperature T sb Ͻ T c is reached.Furthermore, it was shown that by examining the evolution of the asymmetry coefficient ⌬ N ͓ av ‫ء‬ ͑crit͔͒ as a function of temperature one can precisely determine T sb .On the other hand, it was found that the SSB effect does not occur for slits of strongly attractive walls of Mg and CO 2 .
A detailed examination of the obtained T sb for all the alkali metals indicates that these values fall between the lower-and upper-limit for the critical prewetting temperatures T cpw established in Ref. 20 from the analysis of the adsorption on single walls.To the best of our knowledge this is the first time that such a correlation between prewetting and SSB is noticed.
Let us put our findings in a general scenario of the physisorption on planar inert walls.In Ref. 20   when a fluid adsorbed on a single wall exhibits a triple-point wetting, like for Ar adsorbed on Mg and on CO 2 , there is no prewetting line.In these cases the analysis of the confinement between two walls yields always symmetric profiles in the entire range of temperatures T t Յ T Յ T c ͑see, e.g., Fig. 4͒.
For systems with T w Ͼ T t , like Ar adsorbed on walls of alkali metals, there appears a prewetting line where there is a coexistence of thin and thick films which cannot be seen together in only one profile.Both these films become equal at T cpw .When studying the confinement of the latter kind of systems below T w in a slit geometry, one always gets a regime of average density where there are asymmetric solutions.The existence of asymmetric g-s extends above T w up to a critical temperature T sb which coincides with T cpw .In this regime of temperature the confined fluid exhibits in one profile the coexistent thin and thick films ͑see profile 2 in Fig. 2͒.We conjecture that the coincidence of T sb and T cpw is not fortuitous, but it is a signature indicating that the same phenomenon is simultaneously occurring in both geometries.Namely, that the conditions for complete wetting are reached at the same temperature.Regarding a future work, let us mention that Berim and Ruckenstein 31 found that under certain circumstances the symmetry on the x-y plane might be broken.We are planning to search for this kind of symmetry breaking in realistic systems.When preparing the final version of the present paper, we became aware that Berim and Ruckenstein 32 found that the SSB effect also occurs in the case of the quantum fluid 4 He confined between identical walls of Cs.
We hope that the present theoretical results could encourage some experimental groups to try to perform measurements of adsorption for the described systems in slit geometries.

that the integral ͐ 0 ϱ r 2
of Ref.22, which shows the phase envelopes for several approaches.This drawback is mainly due to the fact dr⌽ attr WCA ͑r͒ =−͑32 ͱ 2 / 9͒ ArAr ArAr 3 evaluated with the bare parameters becomes too negative.The determined ratios ˜ArAr / ArAr and ˜ArAr / ArAr are slightly smaller than unity.

FIG. 1 .
FIG.1.Reduced adsorption potentials exerted on Ar atoms by the left wall of the slit.Solid curves are CCZ potentials ͑Ref.16͒ for alkali substrates, while the dashed curve stands for the interaction between Ar and a wall of CO 2 from Refs. 17 and 18.

FIG. 3 .
FIG.3.Reduced free energy per particle of Ar confined in a slit of Li with L ‫ء‬ =40 at T = 115 K displayed as a function of the average density.The curve labeled by circles corresponds to symmetric solutions, that labeled by triangles corresponds to asymmetric ones, and the vertical dashed line at av ‫ء‬ = 0.192 indicates the end of ground states with asymmetric profiles.

FIG. 7 .
FIG. 7. Asymmetry parameter for Ar confined between two identical walls of Rb located at a distance L ‫ء‬ = 60 as a function of average density.From outside to inside the curves correspond to temperatures T = 83.78,138, 139, 140, and 141 K.

FIG. 8 .FIG. 6 .
FIG.8.Both LAS and RAS branches of the asymmetry parameter for Ar confined between walls of Li at temperatures T Ն T w .Data for slits of several widths are displayed: L ‫ء‬ =10 ͑up-triangles͒, L ‫ء‬ =20 ͑squares͒, L ‫ء‬ =40 ͑down-triangles͒, and L ‫ء‬ =60 ͑circles͒.Solid curves are fits to Eq. ͑3.7͒ used for determining T sb ϵ T cpw ; the obtained value is indicated by the horizontal line.

TABLE I .
Properties of Ar confined in slits with wall of alkali metals, Mg, and CO 2 .The ratio W sAr / ArAr , wetting T w , and critical prewetting T cpw temperatures are listed.C indicates continuous growth at T Ն T t and PW denotes present work.id is a free parameter introduced in Eq. ͑2͒ of Ref.23͑in the standard theory it is equal unity͒.The second term accounts for the repulsive f-f interaction approximated by a hard-sphere ͑HS͒ functional; we utilized the f HS ͓ ¯͑r͒ ; d HS ͔ developed byKierlik and Rosinberg, and ‫ء‬ = 10, 20, 40, and 60.The slits with walls located at L ‫ء‬ = 20 and 40 are just equal to that utilized by Maciołek, Evans, and Wilding 29 in their study of a capillary condensed LJ fluid confined in slits by means of a DF theory.
3, but the M-M potential exhibits a softer repulsion than the ES one.Moreover, the well depths located at the corresponding z m differ significantly one from the other being U ab initio ͑z m ͒ / ArAr = −3.86 and U ES ͑z m ͒ / ArAr = −2.78.Due to this sizable difference an analysis similar to that performed in Ref. 20 leads to different results for T cpw , the M-M potential yields complete wetting at T t without any prewetting line, while the ES one gives a prewetting line with a T cpw about 15 K above T t .In the present work we shall report results obtained for slits of widths L FIG.4.Density profiles of Ar confined in a slit of CO 2 with L ‫ء‬ =60 at T t .The displayed spectra correspond to average densities av ‫ء‬ ranging from 0.0555 to 0.555 with a constant step.The horizontal line is the liquid density l ‫ء‬ = 0.808 at saturation for T t .