7146
Langmuir 1997, 13, 7146-7150
Effect of Boundary Conditions on Fluctuations and Solid-Liquid Transition in Confined Films A. Tkachenko and Y. Rabin* Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Received July 7, 1997. In Final Form: October 6, 1997
X
A simple model of confinement-induced freezing in thin liquid layers is presented. Under boundary conditions of vanishing displacement field at the walls, confinement suppresses the amplitude of fluctuations in a solid film below a critical value determined by the Lindemann criterion of melting. The resulting increase of the melting temperature compared to that of a bulk solid is estimated, and the connection with experiments is discussed. The effects of other boundary conditions are also studied.
1. Introduction Recent experimental and theoretical studies1-6 show that when the thickness of a thin liquid film confined between two solid plates is reduced to the point where only a few molecular layers can fit between the plates, the film acquires shear rigidity characteristic of a solid. In particular, an abrupt transition into a solid phase when some critical film thickness (six to eight molecular layers, depending on the molecular constituents) was reported.3 Computer simulations and some simple models suggest that this effect is related to the existence of wall-induced epitaxial ordering in the confined film.4-6 Although this mechanism appears plausible, it normally requires commensurability of the crystal lattices of the wall and of the confined film. Experiments3 have detected the solidification in various substances, whose lattices are far from being commensurate to the wall (the molecular diameter of the molecules of the film is typically much larger than the size of the constituents of the wall). Thus, the epitaxial mechanism should be ruled out, and the explanation to the observed behavior is expected to be of a more universal type. Instead of attempting to describe the phenomenon on a detailed molecular level, we will construct a “minimal” phenomenological model which, we believe, captures the essential physics of the problem. We will not try to address directly the difficult problem of confinement effects on the freezing of a liquid (since even the easier problem of bulk freezing is not well-understood). Instead, we will consider the easier problem of melting and try to understand how the presence of boundaries affects the melting of a solid and then argue that the melting and the freezing phenomena must be strongly related. In a recent communication7 we presented a mechanism of melting which depends only weakly on the microscopic properties of the system, via the boundary conditions imposed on the displacement field in the confined solid. We showed that freezing under confinement can be understood in terms of the empirical Lindemann criterion of melting,8 according to which melting takes place when the amplitude of thermal fluctuations of the molecules in X Abstract published in Advance ACS Abstracts, December 1, 1997.
(1) Israelashvili, J. N.; McGuigan, P. M.; Homola, A. M. Science 1988, 240, 189. (2) Van Alsten, J.; Granick, S. Phys. Rev. Lett. 1988, 61, 2570. (3) Klein, J.; Kumacheva, E. Science 1995, 269, 819. (4) Schoen, M.; Rhykerd, C. L.; Diestler, D.; Cushman, J. H. Science 1989, 245, 1223. (5) Thompson, P. A.; Robbins, M. O. Science 1990, 250, 792. (6) Thompson, P. A.; Robbins, M. O.; Grest, G. S. Isr. J. Chem. 1995, 35, 93. (7) Tkachenko, A.; Rabin, Y. Solid State Commun. 1997, 103, 361.
S0743-7463(97)00741-5 CCC: $14.00
the solid exceeds a certain critical level. Using a rather formal approach (Green’s function formalism), we evaluated the confinement-induced correction to the fluctuation amplitude of the bulk solid and then used the Lindemann criterion to estimate the associated shift of the melting temperature. In the present work we explore the simple and intuitive physical picture underlying the above mechanism, by examining the effect of different boundary conditions on the fluctuations and melting in solid films. In section 2 we discuss the generic case of a semi-infinite solid. We construct the normal modes of the elastic Hamiltonian of a solid subjected to various boundary conditions on the displacement field and use the equipartition theorem to calculate the amplitude of thermal fluctuations in the solid, as a function of distance from the surface. We show that if all the components of the displacement field vanish at the boundary, the fluctuation amplitude at finite distances from the surface is smaller than that in a bulk solid. While the above result conforms with intuitive expectations, we find that, contrary to intuition, if the zero-displacement boundary condition is imposed only on the normal component, fluctuations are actually enhanced compared to the bulk. In section 3 we apply these results to thin solid films. Using the appropriate generalization of the Lindemann criterion, we estimate the confinement-associated shift of the melting temperature and suggest that this increase is responsible for the observed solidification of normal liquids under confinement. We discuss the possible physical origin of the zero-displacement boundary conditions (“surface pinning”). 2. Fluctuations in a Semi-infinite Solid We begin the discussion of the effect of various boundary conditions on the thermal fluctuations of the molecules about their average position by considering the simplest case of a semi-infinite solid (at z > 0). We introduce the displacement field u which describes arbitrary deviations (e.g., due to thermal fluctuations) of the molecules from their average positions in the sample. We assume the displacement field to be governed by a standard elastic “Hamiltonian” of an isotropic solid:
H)
∫
µ 1 dr ∂iuj∂iuj + (∂iui)2 2 1 - 2σ
[
]
(1)
Here µ is the shear modulus and σ is the Poisson ratio of the solid. (8) Tabor, D. Gases, Liquids and Solids, 3rd ed.; Cambridge University Press: Cambridge, 1993.
© 1997 American Chemical Society
Solid-Liquid Transition in Confined Films
Langmuir, Vol. 13, No. 26, 1997 7147
We now perform the normal mode analysis of the above Hamiltonian for various boundary conditions imposed on u at z ) 0. It is easy to show that as long as we consider only those field configurations, which satisfy the zerodisplacement boundary conditions
u|z)0 ) 0
Thus, the eigenvalues for the above surface modes are
hy(q) )
µq 2
(7)
hL(q) )
µq 2
(8)
(2)
(
hR(q) ) 1 + the Hamiltonian is diagonalized by the following family of the standing-wave-type modes:
u ∼ sin(kz) exp(iq‚r)
(3)
Here r is the two-dimensional in-plane coordinate, q is its Fourier-conjugated wave vector, and the z axis is taken to be normal to the surface of the solid. In a more general case, when the displacement field does not vanish at the surface of the solid, the family of the above “standing wave”-type normal modes is not complete. In order to account for the degrees of freedom associated with the displacement at the boundary, additional modes should be introduced. These surface modes (u) must satisfy the following condition of “orthogonality” with respect to the Hamiltonian, eq 1, to the “standing wave”-type modes (u′)
H(u + u′) ) H(u) + H(u′)
(4)
Substitution into eq 1 and a simple calculation show that the surface modes are the solutions of the equation
∆u +
1 ∇(∇‚u) ) 0 1 - 2σ
(5)
subject to appropriate boundary conditions. There are three independent branches of such surface modes, corresponding to three possible polarizations of the vector field u at the boundary. The general expression for the qth component of the displacement field associated with the surface modes can be written as:
[
(
2qz e × 3 - 4σ L exp(-qz) exp(iqx) (6)
)]
uqs ) uq(y)ey + uq(L)eL + uq(R) eR -
where we have chosen the x-axis along the direction of the vector q. Here ei (i ) x,y,z) values are the corresponding unit vectors, eL ) (ex + iez)/x2, and eR ) eL* ) (ex - iez)/ x2. Note that the mode with the amplitude uq(R) is not a pure right-handed mode (the right-left symmetry is broken in our half-space geometry) but becomes one at the surface (at z ) 0). The above modes satisfy the condition of orthogonality to the “standing waves”, eq 5, and are mutually orthogonal in the sense of eq 4. For any surface mode v(r), the eigenvalue of the Hamiltonian is given by the following general form:
]|
1 µ h ) - v∂zv* + v (∇‚v*) 2 1 - 2σ z
[
z)0
2 µq 3 - 4σ 2
)
(9)
The total mean square fluctuation amplitude at a given point is given by the sum of the contributions of the “standing waves” and those surface modes which are specified by the imposed boundary conditions.
〈u2(z)〉 ) 〈u2(z)〉zero +
(
∑q 〈|uq(y)|2〉 + 〈|uq(L)|2〉 +
[ (
〈|uq(R)|2〉 1 +
) ])
2qz
2
3 - 4σ
exp(-2qz) (10)
Here 〈u2(z)〉zero is the contribution of Fourier-type “standing waves”, eq 3, which would be the only contribution for zero-displacement boundary conditions, eq 2, when all the surface modes are suppressed. The thermal averages in the above expression can be calculated using the equipartition theorem
hi(q)〈|uq(i)|2〉 )
kT 2
(1)
The difficulty with direct summation of the Fourier modes is that the result is sensitive to the nonuniversal shortscale behavior of the elastic Hamiltonian, which is the general situation for three-dimensional solids. In order to overcome this problem, we will calculate the surfaceinduced correction to the bulk value of the mean square fluctuation amplitude. Remarkably, this can be done by the analysis of the surface modes only. For doing so we consider the reference bulk system as being composed of two semi-infinite solid samples (“up”, z > 0, and “down”, z < 0). We then “glue” them together by demanding that the displacement field is continuous at the boundary, i.e.
udown|z)0 ) uup|z)0
(12)
The above normal mode analysis can be applied to both semispaces. The Fourier-type “standing waves” are not sensitive to the boundary conditions at z ) 0, while the surface modes in the two samples become coupled. Symmetry implies that surface modes in one semispace are related by mirror reflection (exchange of eR and eL) to those in the other one:
u(y)down ) u(y)up u(L)down ) u(R)up u(R)down ) u(L)up The coupling between the two semispaces results in the increase of the eigenvalues of the surface modes:
h ˜ 1(q) ) 2h1(q) ) µq
(13)
h ˜ 2(q) ) h ˜ 3(q) ) h2(q) + h3(q) )
4(1 - σ) µq (14) 3 - 4σ
7148 Langmuir, Vol. 13, No. 26, 1997
Tkachenko et al.
By construction, the fluctuations in the system of the two bounded semispaces are identical to those in the bulk (i.e., are independent of z), and therefore, eq 10 can be rewritten as:
〈u 〉bulk ) 〈u (z)〉zero + 2
2
(
∑q 〈|u˜ q
[ (
〈|u˜ q(R)|2〉 1 +
(L) 2
) ])
2qz
2
3 - 4σ
kT
∑ 2µ q
(
[ ( ) ])
(3 - 4σ)
2
2qz
2+
[
(
exp(-2qz)
3 - 4σ
4(1 - σ)
(16)
q
) ]
1
(
)
1 + (3 - 4σ)2 kT 1+ δ〈u (z)〉zero ) 8πµz 2(1 - σ)(3 - 4σ)
δ〈u2(z)〉⊥ ≡ 〈u2(z)〉⊥ - 〈u2〉bulk )
(17)
The above expression becomes valid for z > 1/qmax, i.e., already at a distance of several molecular diameters from the surface. As expected, the zero-displacement boundary condition at the surface results in suppression of fluctuations at a distance z from the boundary, the magnitude of the effect decreasing as 1/z. This result for the thermal fluctuations in a semi-infinite solid subjected to the zero-displacement boundary conditions can be generalized for the case of a free boundary, with unrestricted surface fluctuations:
δ〈u2(z)〉zero +
[ (
∑q 〈|uq(y)|2〉 + 〈|uq(L)|2〉 + 〈|uq(R)|2〉 1 +
(
[( ) ])
1
∑ 2 + 4(1 - σ) 2µ q
2qz
3 - 4σ
2
) ])
2qz
× 3 - 4σ exp(-2qz) (18)
q
This yields
δ〈u2(z)〉free )
(
[ (
[ (
kT )
(
) ])
kT 1 1 2+ 1+ 8πµz 3 4σ 2(1 - σ)
2
(19)
We conclude that the presence of a free surface results in enhancement of fluctuations compared to the bulk and that the enhancement effect decays with the distance from the surface as 1/z. In the spirit of the Lindemann criterion, one could speculate that the enhancement of fluctuations near a free surface is responsible for the well-known phenomenon of surface melting. In addition to the above-considered limits of completely suppressed and completely unrestricted surface fluctuations, one can discuss an important intermediate case when only normal fluctuations are suppressed on the surface (its simplest physical realization would be a solid
qz
)
∑ 1 - (1 - σ) 2µ q
2qz
) ]) 2
3 - 4σ
exp(-2qz)
exp(-2qz) q
(23)
and thus
δ〈u2(z)〉⊥ )
2
exp(-2qz)
+2
(
∑q 〈|uq(y)|2〉 + 〈|uq+|2〉 × 1+ 1-
δ〈u2(z)〉free ) 〈u2(z)〉free - 〈u2(z)〉bulk ) δ〈u2(z)〉zero +
kT
4(1 - σ) µq 3 - 4σ
Taking into account the contribution of the two nonvanishing surface modes (uq(y) and uq+), we obtain the correction to the mean square fluctuation in the bulk
we obtain
)
(21)
2qz e exp(-qz) exp(iqx) 3 - 4σ L (22)
h+ ) h1 + h2 )
∞ ∑q f 2π∫0 q dq
(
(20)
The corresponding eigenvalue of the Hamiltonian is
Replacing the sum by an integral
2
uy|y)0 * 0
In other words, we should replace these two modes with their linear combination satisfying the boundary condition, eq 20
uq+ ) uq+ eR + 1 -
δ〈u2(z)〉zero ≡ 〈u2(z)〉zero - 〈u2〉bulk )
1+
ux|z)0,
u(L) ) u(R)
exp(-2qz) (15)
Here u˜ (i) denote the field amplitudes subjected to the “glued” boundary conditions, eq 12. Using (15) and the equipartition theorem yields
-
uz|z)0 ) 0,
This boundary condition has no effect on the “standing waves” (3) and on one of the surface modes, u(y). In order to satisfy eq 20 we must have
| 〉 + 〈|u˜ q | 〉 +
(y) 2
in contact with an ideal structureless hard wall):
1 kT(1 - 2σ) 16π µz (1 - σ)
(24)
We conclude that when only normal displacements are forbidden at the surface, the amplitude of fluctuations away from it is actually enhanced compared to its value in the bulk solid (the two amplitudes coincide for incompressible solids, σ ) 1/2). Again the effect decays as 1/z with distance from the surface. In order to estimate the effect of the different boundary conditions on the confinement-induced shift of the bulk melting temperature, we have to compare the corrections, eqs 17, 19, and 24, with the mean square fluctuation amplitude in the reference bulk solid, 〈u2〉bulk. An accurate calculation of the latter requires, in general, knowledge of the molecular-scale behavior of the underlying Hamiltonian and is beyond the scope of this work. In the spirit of this work, we will estimate 〈u2〉0 using the longwavelength asymptotic form of the true Hamiltonian, eq 1, with a cut-off at some length scale λ ) π/kmax of order of molecular size
〈u2〉bulk =
∫k
