On the uniqueness of stratification-induced structural transformations in confined films

M. Schoen: On the Uniqueness of Stratification-Induced Structural Transformations in Confined Films 1355

On the Uniqueness of Stratification-Induced Structural Transformations in Confined Films

Martin Schoen Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany

Key Words: Computer Experiments / Interfaces / Phase Transitions / Statistical Mechanics / Thermodynamics

This article is devoted to an investigation of structural transformations which occur in molecularly thin films confined between planar structured and unstructured rigid solid surfaces as the number of fluid strata parallel with the walls changes under conditions of fixed chemical potential ,u and temperature T The transformations become manifest as periodic maxima in the transverse isothermal compressibility K ,, computed in grand canonical ensemble Monte Carlo simulations via fluctuations in film density. The transformation occurs at nearly identical characteristic separations sz of the surfaces regardless of whether these are discrete (i.e., composed of individual atoms) or smooth in transverse dimensions. The transformation is caused mainly by confinement to spaces of molecular dimension and is unique, that is independent of details of the film-wall interaction. Because of the inhomogeneity of a film between structured surfaces in transverse dimensions a definition of K , , in terms of density fluctuations is precluded in general. However, by considering only restricted thermodynamic transformations on a length scale defined in units of the lattice constants characterizing the wall structure, such a definition is possible introducing K , I as a density fluctuation-related quantity in a coarse-grained sense. Variations of the local density of the film around the transition points suggest to perceive the phase transition as stratification- induced, because it happens after the new stratum has begun to form.

1. Introduction

If a fluid film is confined between two solid surfaces (i.e., walls) separated by a microscopic distance s,, film molecules tend to arrange themselves in strata parallel with the confining walls, that is the film is inhomogeneous in the direction normal to the walls because of their symmetry- breaking nature. A quantitative measure of stratification is the local density which is an oscillatory function of position with respect to the walls. Peaks in this function represent strata of film molecules. If sz is microscopically small rang- ing from, say, one to about ten molecular diameters, these peaks are rather narrow and tall indicating that the strata are well localized. Upon increasing s, over the range of one to ten molecular diameters, new strata form rather abruptly (i.e., “pop in”) at characteristic values of s,.

Not surprisingly stratification has a significant impact on material properties of the film. As was first demonstrated experimentally by means of the surface forces apparatus (SFA) [l], the total force exerted by the film on the walls oscillates between repulsion and attraction with increasing sz under conditions of fixed temperature T and chemical potential ,u of the film. The period of oscillations may be related to geometrical factors (i.e., bond length, molecular “diameter”, etc.) of film molecules which suggests stratification even though the local density itself cannot be directly measured [2]. It was left mainly to computer simulations to establish the close relation between stratification manifested in the local density and oscillations of the total force on the walls beyond doubt (see [3] and refs. therein). These simulations demonstrated that stratification is a rather general feature of confined phases, that is it is qualitatively independent of the precise nature of the film- wall interaction potential and should be regarded as a consequence of confinement to spaces of molecular dimensions. This is illustrated by simple, albeit diverse, systems,

such as hard spheres between hard walls [4, 51 or “soft” Lennard-Jones atoms between molecularly structured [3, 6, 71 or smooth Lennard-Jones walls [3, 8- 101, which exhibit stratification and serve as useful models, even if much more complex systems are the objects of ultimate interest.

This generality and the apparent abruptness of the transformation of a film with i strata into one accommodating i k 1 strata now raises the question whether the transformation may be regarded as a phase transition. This question was addressed recently in a study of a film confined between smooth Lennard-Jones walls, which revealed that a change in the number of strata is associated with cusp-like peaks in the transverse isothermal compressibility K ~ I [I I]. The occurrence of such peaks in conjunction with a continuous variation of the (average) film density during the transformation suggested that stratification involves an order-disorder transition (of an order higher than first), during which the packing of film molecules changes. Up to now a parallel analysis for films between molecularly structured walls could not be carried out because K I J could not be expressed in terms of density fluctuations in thermody- namically open systems. The lack of such an expression is due to the lack of a Gibbs-Duhem equation in general, which reflects the reduced symmetry of a film between structured walls. The purpose of the present article is to show how a specialized Gibbs-Duhem equation may be derived for a film between molecularly structured walls if one restricts thermodynamic transformations of the film to a coarse-grained length scale defined in units of the lattice constant(s) characterizing the wall structure. From the spezialized Gibbs-Duhem equation K ~ I may then again be expressed in terms of fluctuations in film density which permits one to correlate stratification of the film between molecularly structured walls with the cusp-like peaks in rcll in a fashion similar to [ I l l . Results of this analysis are presented in Sect. 4. The models are introduced in Sect. 3.

Ber. Bunsenges. Phys. Chem. 100, 1355 -1362 (1996) No. 8 0 VCH Verlagsgesellschaft mbH, 0-69451 Weinheim, 1996 0005-9021/96/0808-1355 $15.00+.25/0

1356 M. Schoen: On the Uniqueness of Stratification-Induced Structural Transformations in Confined Films

However, the paper begins in Sect. 2 with a development of the thermodynamic and statistical-physical basis for a discussion of phase transitions in confined thin films between structured walls.

2. Theory

Fig. 1 displays a diagram of the film part of which is taken to be a lamella of fluid bounded by two walls in z- direction and by segments of the (imaginary) planes x = 0, x = s,, y = 0, and y = sy. The lamella is in thermal and material contact with the remainder of the confined film and a (n infinitely large homogeneous) bulk reservoir which form the surroundings in the thermodynamic sense. The walls are identical and rigid in the laboratory coordinate frame, whose origin is at 0. The crystallographic structure of the walls is described by a rectangular unit cell having transverse dimensions I, x ly and the unit vectors associated to the unit cells in the two walls remain parallel. In general, each wall consists of a number of planes of atoms parallel with the x - y plane. The plane at the film-wall interface shall be referred to as the surface plane. The surface plane of the lower wall is taken to be constrained in the x - y plane. The distance between the surface planes is s,. The walls are in registry, that is the walls are aligned such that corresponding atoms in the two walls are always exactly op- posite each other (see Fig. 1). The coordinates of a given atom (2) in the upper surface plane (z = s,) are related to those of its counterpart (1) in the lower surface plane (z = 0) by

+%+ Fig. 1 Schematic side view of a finite lamella (shaded in gray) of a film between structured walls. The arrows associated with T,, and TY,, point in arbitrarily chosen directions

Thus, the extensive variables characterizing the lamellar system are entropy S, number of film molecules N, s,, sy, and sz. Gibbs’ fundamental equation governing an infini- tesimal, reversible transformation can then we written as

where the mechanical work terms can be expressed succinct- ly as

The second sum runs over Cartesian components (a = x ,y ,z ) , ds, is a displacement of a lamella boundary in a-direction, A , is the area of the a-directed face of the lamella, and Tau is the average of the a-component of the stress applied to A,. Note that if the force exerted by the lamella on the a-directed face points outward, Tau is negative by convention. Thus, d Wmech is the mechanical work done by the lamella on the surroundings. The presence of only diagonal terms of the stress tensor indicates that shearing of the lamella is ignored.

For the subsequent discussion it is convenient to intro- duce the grand potential

as a Legendre transform of the internal energy U in the usual way. Employing Eq. (2) one may write

dQ = - Sd T - N d p + Txxsys,dsx

+ Tyysxszdsy+ Tzzsxsydsz . ( 5 )

In contrast to a homogeneous bulk phase it can be demonstrated here that Txx = f,(s,,sy) and Tyy = fy(s,,sy) under conditions of fixed T, p, and s, [12]. Thus, under these conditions Q is not homogeneous of degree one in the re- maining extensive variables and may only be integrated if the equations of state f,(s,,sy) and fy(s,,sy) are known. These, however, are generally unknown but depend on details of the system as is shown in I121 numerically. From this it follows that a Gibbs-Duhem relation does not exist in general which reflects the inhomogeneity (i.e., the reduced symmetry) of the film in transverse directions. The inhomogeneity is a consequence of the discrete structure of the surface planes. While a Gibbs-Duhem relation does not exist for general transformation ds,-+dsh, a specialized (i.e., “coarse-grained”) Gibbs-Duhem relation may be derived for cases in which the transverse dimensions of the lamella are changed only discretely, that is in such a way that the surface plane at the film-wall interface of the lamella comprises an integer number n of unit cells in both x and y directions, that is

s, = nix sy = nly .

Thus, the exchange of work between the lamella and its surroundings is effected on a coarse-grained length scale defined in units of (r,, I y ] . Elimination of s, and sy in Eq. ( 5 ) in favor of n gives


d Q = -SdT-Ndp+2Tllaszndn+ Tz,an2dsz , (7) where the transverse isothermal compressibility is defined as

where a: = &Iy is the unit cell area and

(8) Thus, from Eqs. (14- 16) one has + Tyy (T, A n I,, n I,, s, 11

is the “mean” stress applied transversely on the n x n lamella. If T, p, and s, are fixed, T,, and Ty,, are periodic in s, and s,, having period I, and I, respectively. Thus, for the restricted class of transformations n-n’ = n -t m(n, m integer) TII is constant provided n and n’ are sufficiently large for intensive properties to be independent of the (microscopic) size of the lamella. Under these conditions Eq. (7) can be integrated to get

(9) 52 = TIIas,n 2 .

Eq. (9) may be differentiated to give

d52=2T11as,ndn+an~d(Tl~s,) . (10)

Equating the expressions for d52 given in Eqs. (7) and (10) and rearranging terms yields the following “coarse- grained” Gibbs-Duhem equation:

(1 1) 0 = Sd T+ N d p + an d ( TI 1 s,) - a n Tzz ds, .

Eq. (17) can also be derived under the more restrictive assumption of homogeneity of the lamella in transverse directions which is approximately correct if s, is sufficiently large (i.e,, for sufficiently thick films) but holds by definition for fluid-like films (with which this article is ex- clusively concerned) between smooth surface planes (see Sect. 3 below). Then one has T, = Tyy = Tll and Eq. (7) can be written as

Eq. (18) may be integrated at fixed T, p, and s, because here TI1 # f(A) on arbitrary length scales. The integration yields

At fixed T and s,, Eq. (11) reduces to

-Ndp = an2d(TIIsz) .

If one may write

Eq. (19) parallels Eq. (9) but differs from the latter because it is valid for continuous variations of transverse dimensions of the lamella. By a derivation parallel to the one between Eqs. (10- 16) one eventually recovers Eq. (17) [ l l ] .

From a statistical-physical perspective one can furthermore show by standard textbook arguments that 52 is related to the grand canonical ensemble partition function Zvia [13, 141

(12)

and is also fixed, then ~ and TI I depend solely on

(13 a) dp = (%)T,n,szm ’ Q = - B - ’ l n z (20)

where 8: = l/(kB T)(kB Boltzmann’s constant). In the clas- sical limit [13, 141 one has (13b)

w A - ~ N Substituting Eqs. (13) into Eq. (12), one gets - - - = c $flN j e-bU(rN)&N (21)

N = O N! VN

where A is the thermal de Broglie wavelength [13, 141 and U ( r N ) is the (model-dependent, see Sect. 3) configurational energy. Relevant macroscopic properties can be expressed

At fixed T, s , and n , N is proportional to n 2 (i.e., to area in terms of microscopic properties through partial A = sxsr = n a). The left hand side of Eq. (14) can there- derivatives of 52 given by Eq. (20). For example, from fore be recast as Eqs. (18) and (20) it follows directly that

(I4) 3 -an2sz ( aN)T,n,sz=N(%) T,n,sz

1


Depending on how the differentiation of the integral in the second line of Eq. (22) is carried out, two alternative expressions for T,, can be derived [15]. The so-called “virial” expression is given by

where the film-film contribution can be written as

N N 2 Tz,,ff = --+-(As,)-’ (N) 1 ( c c utf (r j j )%) (23b)

PAS, 2 i = 1 j # i ‘ij

and the film-wall contribution as

In Eqs. (23) the brackets ( . . . ) denote ensemble averages and u ‘ : = du/dr. The distance between two film molecules or a film molecule and a wall atom located in wall k are given by rij and r r ) , respectively. The alternative “force” expression for T,, also follows directly from Eq. (22) as [I 51

where F:2) is the force exerted by the film on surface plane 2. In other words, A Tzz is the average force that must be applied (externally) to wall 2 so that it remains stationary in the state of thermodynamic equilibrium. Because the entire system is mechanically stable one also has

(Fp) = - ( F p ) (24 b)

and Eqs. (23 - 24) provide a useful check on internal con- sistency of computer simulations to be presented in Sect. 4 where the close relation between (the experimentally accessi- ble) T,, and stratification will be demonstrated.

The description of phase transitions associated with stratification can be facilitated by the following considera- tions. In general, a phase transition occurs at a point in thermodynamic state space at which the m-th derivative of the relevant thermodynamic potential is nonanalytic with respect to its natural variables [14]. The parameter m denotes the order of the phase transition. Under the present conditions (where ,u and T are natural variables of 52)

(25 a)

and

are of principal interest. Combining Eqs. (25 a) and (25 b), a comparison of the resulting expression with Eq. (17) reveals that

where it should be borne in mind that relation (17) is predicated on the coarse-grained length scale. Because of the definition of the order of a phase transition one expects from Eq. (25a) that the average film density n: = ( N ) / ( A s,) changes discontinuously (but remains finite) during a first-order transition. During a second- order transition n will vary continuously, whereas K I I will change discontinuously [ 161.

3. Model Systems

Two different models are considered. In both N spheri- cally symmetric film molecules are located between two surface planes separated by a distance sz along the z-axis of the (Cartesian) coordinate system. Film molecules interact in a pairwise fashion via Lennard-Jones (12, 6) potentials

Films of infinite extent in the x - y plane are mimicked by imposing periodic boundary conditions and the minimum image convention [17]. The side lengths s = s, = sy of the (square) computational cell are chosen large enough so that film properties do not depend sensibly on its size.

In model A film molecules are confined between surface planes, each consisting of N, wall atoms distributed in the face-centered cubic (fcc) (100) configuration. Coordinates of corresponding wall atoms in both surface planes are related through Eq. (1). In model A transverse homogeneity is prevented in general by the discrete structure of the walls. A wall atom interacts with a film molecule in a pairwise fashion where for simplicity

(28 a) A

u fw = uff + u H W (2,’)

and

is a hard-wall background potential introduced formally to render the distance between the walls s, unambiguous. For the states considered here, however, wall atoms are so densely packed that film molecules do not interact with the hard-wall background. The number of collisions of film molecules with the hard wall is monitored during each GCEMC run and turned out to be always zero under the present conditions. In model B each surface plane is again


assumed to consist of N, wall atoms distributed according to the same fcc (100) structure. However, the atoms are now “smeared out” over the surface plane which is achieved by averaging u& over the two translationally invariant dimensions. Thus,

(29)

where d,: = 2/12 because of the fcc (100) structure. In model B the walls lack any distinct crystallographic structure, that is they are smooth on a molecular scale. Thus, under the present conditions the (fluid-like) film is by definition homogeneous in transverse dimensions (see Sect. 2). Equilibrium properties of films in both models such as T,, and K I I are computed as ensemble averages in grand canonical ensemble Monte Carlo (GCEMC) simulations, which are described in detail in [18].

Although a casual look at Eqs. (27 - 29) may give the im- pression that the two models are quite similar, it seems worthwhile to stress that this is truly not the case as far as their impact on film properties is concerned. For example, model A permits a film in thermodynamic equilibrium with a bulk liquid to solidify through the template effect [6, 191, that is the distinct wall structure induces solidification, whereas under the same thermodynamic conditions the walls of model B are incapable of inducing solidification. Transport properties of liquid-like films are also quite different for the two models. For example, in model A, self diffusion of film molecules may be anisotropic in transverse directions [20] and increases toward the film’s center, whereas in model B it is greatest in the immediate vicinity of the walls, because of lack of lateral friction [21]. It should perhaps also be emphasized that in both models the three-dimensional character of the walls is neglected. A semi-infinite number of planes of wall atoms stacked behind the surface plane would give rise to long-range film- wall interactions [22], which are known to be important as far as adsorption and wetting phenomena at single solid surfaces are concerned [23]. Since, on the other hand, short- range packing effects dominate the phenomena of interest here, the employment of single surface-plane models appears to be justifiable.

4. Results

Experimentally stratification was first conjectured from measurements of Tzz(s,) at fixed ,u and T (see [l] and refs. therein). For a film maintained at T* = 1.00 and p * = - 1.20 (in the usual dimensionless (starred) units [17]) this quantity is obtained in GCEMC simulations and plotted in Fig. 2 . The plot shows that Tzz(s,) is a damped oscillatory function with a period of approximately one molecular “diameter”. The oscillations may be taken as signatures of stratification as one may infer from a comparison between Tzz(s,) and the average film density n(s,) (see

2

1.5

1

0.5

+E 0

-0.5

-1

-1.5

-2 1.b ’ 1 ’ 215 ’ ; ‘,3(5 ’ d ’ 415 ’ b 5

sz

Fig. 2 The normal stress TZz ( , left ordinate) and the average film density n * (0 , right ordinate) as functions of wall separation sl for model A at T* = 1 .OO and p* = - 12.0. The lines are obtained by fitting a cubic spline to the GCEMC data and are intended to guide the eye

Eq. (31 b) below) which are also plotted in Fig. 2. The latter increases rather sharply at characteristic values of s,. The locations of maxima of n(s,) may be interpreted as those wall separations at which a given number of molecules can pack most conveniently in individual strata between the walls. That the maxima in n(s,) correspond to minima in the normal stress Tzz(s,) further supports this notion. This argument suggests that the sharp increase of n (s,) cor- responds to the formation of new strata. One also notes from Fig. 2 that the structure in both curves appears to be “washed out” with increasing s,, which complies with one’s physical intuition: as s, increases inner portions of the film are less affected by the film-wall interaction. Hence, these inner portions are expected to be less stratified (see, for example, Fig. 3.5 g in [3]). In fact, if sz is sufficiently large stratification will persist only in the immediate vicinity of the walls leaving the remainder of the film homogeneous at the density of the corresponding bulk fluid [ 191. In this case one expects

lim Tzz(sz) = - P SZ’ m

where P is the pressure of a corresponding bulk fluid in the same thermodynamic state. Under the present conditions the bulk fluid turns out to be a low-density gas at a pressure P* = 0. The plot of Tzz(s,) in Fig. 2 shows that over the range of s, considered here the effects of confinement still prevail so that T,, has not yet reached its limiting value.

Nevertheless, both T,, and n are rather indirect indica- tors of stratification. The local density

on the other hand, is a direct measure of stratification. It is related to the average film density n via


-.-

1.2

0.9 In Eq. (31 a) (N(z)) is the average number of film molecules located in a thin slice of width Az parallel with the walls. Obviously, p( ’ ) (z ) ignores the film’s inhomogeneity in transverse dimensions (model A) and should really depend on the vector argument r . Averaging over transverse dimensions (which leads to Eq. (31 a)) is, however, permissible here if one is mainly interested in effects of confinement. As an example, Fig. 3 shows plots of p(’)(z) as functions of position between the walls and for various wall separations. As can be seen from the figure, p( ’ ) ( z ) is an oscillatory function, its peaks corresponding to individual strata of film molecules. These strata are spatially well localized: peaks are rather tall and narrow and minima between neighboring peaks correspond to regions of low density, that is a low probability of finding a film molecule at a point in space which does not lie in a stratum, regardless of the distribution of all other molecules. The plots in Fig. 3 also reveal a structural transformation between a bi- and a trilayer film occurring over the present range of wall separations.

0 - 0 0

0 0

- 0

2

rl

1.5 - - a,

1

0.5

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

ZIS,

Fig. 3 The local density p ( ‘ ) * ( z ) for model A at T* = 1.00 and p * = - 12.0 as a function of position z/s, between the walls which are located at Z/Sz= +0 .5 ; s f=3 .30 (A) , 3.40(0), 3 . 5 0 ( ~ ) , 3.70(0) , 3.80(A). Note that p ( ’ ) ( z ) results from averaging p(’) ( r ) over transverse dimensions (see text)

r P # I 1.5 -

r l . - - a,

1 -

0.5 -

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ”

ZIS, Fig. 3 The local density p ( ‘ ) * ( z ) for model A at T* = 1.00 and p * = - 12.0 as a function of position z/s, between the walls which are located at Z/Sz= +0 .5 ; s f=3 .30 (A) , 3.40(0), 3 . 5 0 ( ~ ) , 3.70(0) , 3.80(A). Note that p ( ’ ) ( z ) results from averaging p(’) ( r ) over transverse dimensions (see text)

A comparison with plots in Fig. 2 now permits the conclusion that the structure of T,,(s,) and of n(s,) does, in- deed, reflect stratification as conjectured above. Further- more, Fig. 3 indicates that in a fully stratified trilayer film there are two structurally identical strata in the immediate vicinity of the walls (i.e., “contact” strata) which are distinct from the middle stratum in that its peak is about 25% lower than that of a contact stratum. Structural identi- ty of the contact strata is a consequence of the cylindrical symmetry of the model, whereas the different peak height reflects the diminishing impact of the film-wall potential on inner portions of thicker films.

During the transformation from i to i + 1 strata, the film undergoes remarkable changes in its transverse isothermal

0 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8

s,. Fig. 4 The transverse isothermal compressibility K ; as a function of wall separation S:; ( 0 ) model A at T* = 1.00, p* = - 12.0; (0) model B at T* = 1.00, p* = - 10.0

compressibility, as can be seen from a plot of q ( s z ) in Fig. 4, which exhibits cusp-like peaks (given the resolution of the data) at characteristic values of s,. An inspection of p(‘)(z) in the vicinity of these peaks reveals that they can be associated with a structural transformation involving mono- and bilayer films (sf z2.6), bi- and trilayer films (sf = 3.9, and tri- and tetralayer films (sz z 4.4). The separation Asf = 0.9 is close to one molecular diameter, as one would expect for spherical film molecules. Also shown in Fig. 4 is a plot of K~~ (s,) for model B exhibiting two maxima at values of s, for which p( ’ ) ( z ) indicates a transformation from mono- to bilayer (sf G 2.6) and from bi- to trilayer (sf z 3.6). The striking similarity of peak positions in the K I I curves for both models confirms the notion of “generality” of stratification, despite the substan- tially different impact models A and B have on the various macroscopic properties of the film (see Sect. 2): the formation of individual strata is the result of mere confinement of fluid films and does not depend much on details of the interactions. Besides the difference in the thermodynamic state of the film, the substantial difference in intensity of K I I , on the other hand, points to significant differences in the mechanism of the structural transformation (e.g., changes of structural characteristics of the film during the transition) which is strongly system-dependent. The mechanism operating in films between structured walls will be analyzed in depth in a separate publication [24]. For model B a previous analysis led to the conclusion that stratification is associated with an order-disorder phase transition involving films with different packing characteristics in transverse dimensions [ 1 1 ]. Another feature com- mon to both models is a slight shift of peak positions in K I I (s,) to smaller s, compared with the position of minima of T, (s,) corresponding to fully stratified films (see Fig. 5). Such a shift, already observed for model B in [Il l , signifies that the phase transition occurs after the formation of the new stratum has begun. In this sense maxima of K I I (s,) demarcate stratification-induced second-order phase transitions. This notion is furthermore bolstered by a comparison between Figs. 3 and 4, which indicates that the for-


-1.5

I- -0.5

-1

0 - 0 U - 0 0

- 2 - 1 . I . I . I . I . I . I . I . I

*- X -

mation of the new middle stratum is already in progress at the maximum of K I I at s,* 9 3.5. A comparison between K I I (s,) in Fig. 4 and n(s,) in Fig. 2 shows that the latter is apparently a continuous function over regions where the former exhibits cusp-like maxima. From the definition of a phase transition as given at the end of Sect. 2, one may speculate that stratification is a higher- (presumably second-) order phase transition. However, to establish the order of stratification-induced structural phase transitions is beyond the scope of this paper.

5. Concluding Remarks

The present article presents results of an investigation of stratification-induced phase transitions in confined molecularly thin films. These transitions manifest themselves as maxima in the transverse isothermal compressibility of the films which is computed as a function of wall separation (i.e., film thickness) at fixed Tand p. These conditions correspond to those encountered in actual laboratory experiments involving the SFA [I]. Based upon the analyticity of derivatives of order m (rn 2 0) of the grand potential with respect to p it was concluded in [I I] that stratification-induced phase transitions are not of first- but presumably of second-order, and involve structurally different phases as far as model B is concerned: the structural change is continuous across the transition point so that the transition may be viewed as a second-order phase transition according to Landau and Lifshitz who state that “ . . . a second-order phase transition is continuous, in the sense that the state of the body changes continuously.” (p. 431 in [25]).

The transition is unique in the sense that it occurs at nearly the same characteristic wall separations for films accommodating the same number of strata, regardless of the film-wall interaction potential. This seems remarkable because the different film-wall potentials for the two models have a significant impact on the macroscopic pro-

perties of the confined film [6, 19, 201 as well as on the mechanisms of the transition [ l l , 241. The notion of uniqueness is furthermore corroborated by a comparison with earlier work by Antonchenko et al., who computed the isothermal compressibility of a hard-sphere fluid between hard walls [ 5 ] and observed periodic maxima separated by As,* = 1 similar to the ones reported here. However, these authors do not attempt to relate the variation of the isothermal compressibility to stratification. In other words, stratification-induced phase transitions are driven by packing effects due to mere confinement to spaces of molecular dimension and do not depend strongly on details of the film-wall interaction. This is demonstrated here for two rather different models, one in which the walls lack any discrete crystallographic structure, and one in which the walls consist of surface planes composed of rigidly fixed atoms. Although under the present conditions the (fluid- like) film in the former model is homogeneous in transverse dimensions by definition, it will generally be inhomogeneous in these dimensions in the latter model. The impor- tance of packing effects in strongly confined films has also been noted by Tarazona et al. who investigated wetting transitions, capillary condensation and evaporation in molecularly thin films [26].

The reduced symmetry of the film between structured walls has important consequences for its thermodynamic description. For instance, the lack of a Gibbs-Duhem relation in general [I21 precludes a definition of the transverse isothermal compressibility in terms of density fluctuations in open systems. Such a definition becomes possible, however, by considering compressional work done on a finite lamella of the (infinite) film in a restricted way which permits movements of the (imaginary) pistons separating the lamella from its environment on a coarse-grained length scale. This length scale is defined in units of the lattice constant(s) characterizing the structure of the (discrete) walls. Thus, in view of this restriction the so-defined transverse isothermal compressibility is a coarse-grained quantity on a molecular scale. Details of this derivation are presented here, while the interested reader is referred to [12, 271 for a discussion of the lamella-based thermodynamic treatment of confined films.

I am indebted to D. J. Diestler (University of Nebraska) for discus- sions which led to the treatment outlined in Sect. 2 and to an anony- mous referee for comments that helped to improve the manuscript. I am also grateful to the Deutsche Forschungsgemeinschaft (DFG) for a Heisenberg fellowship and to the Scientific Council of the Hochstlei- stungsrechenzentrum (HLRZ) at Forschungszentrum Jiilich for a generous allotment of computer time on their Cray.

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On the uniqueness of stratification-induced structural transformations in confined films