Comment on “Renormalization-Group Calculation of the Dependence on Gravity of the Surface Tension and Bending Rigidity of a Fluid Interface”

Abstract

In a recent Letter [1] a renormalization-group (RG) calculation of the gravity dependence of properties of a liquid-vapor interface is presented whose central point is the use of the (bare) capillary-wave-like Hamiltonian2 |∇ζ| 2 + a 0 (g) 2 ζ 2 + λ 4! ζ 4 (1)to study the interface in the limit of vanishing gravitational acceleration g. Here ζ( R) is the local height of the interface configuration [ζ] at position R relative to the plane z = 0. Recognizing that (1) is just the usual ϕ 4 Hamiltonian and that a 0 (g) vanishes as g → 0, the authors followed the standard RG analysis for the critical Ising system, arrived at the fixed point Hamiltonianand reached a remarkable conclusion: interfacial properties (in d dimensions) for g = 0 should be identical to those in a critical, d − 1 dimensional bulk Ising system. In order for this analysis and its conclusions to be valid, the (renormalized) coupling constant λ must remain positive in this limit, i.e., lim g→0 λ(g) = λ (0) > 0. However, in the present interface case (in contrast to that of critical bulk systems), there are fundamental symmetry requirements that enforce λ (0) = 0. We believe that the authors' introduction of a nontrivial λ(g) is justifiable. But as we shall show in the sequel, their assumption of λ (0) > 0 is untenable, so that their subsequent analysis is irrelevant and the conclusion unreliable.Consider a simple liquid-vapor system in d dimensions, under the influence of a potential g z. Let (x i ) = (R α , x d ≡ z) ∈ R d be the coordinates of a point relative to a fixed frame with Euclidean unit vectors e i . In the absence of gravity, the system respects the Euclidean symmetries and hence is invariant under translations and rotationsis a rotation matrix. In the two-phase region (of the g = 0 system), these symmetries are broken spontaneously by the presence of an interface, so that nonlinear realizations of the Euclidean group are necessary. For g = 0, these symmetries are broken explicitly. Under a rotation, the system transforms into one with a rotated field: g e d → R id g e i .Let us write possible (bare) interface Hamiltonians aswhere ∂ζ, ∂ 2 ζ,. . . stand for first, second, and higher partial derivatives ∂ζ/∂R α etc. H (0) [ζ] must be invariant under the transformations discussed above. We consider separately the consequences of this trivial observation.(i) Translations along the z axis: This symmetry is realized by ζ → ζ ′ = ζ − A d . The invariance of H (0) means that the energy cost of such a translation vanishes as g → 0. Hence L (0) cannot depend explicitly on ζ: we must have L (g) → L (0) (∂ζ, ∂ 2 ζ, . . .). This rules out terms like λ ζ 4 , as well as (2), in the limit g → 0. Yet, λ (0) > 0 is crucial for the approach in [1], the presence of an upper critical dimension of 5, and the fixed point (2).(ii) Rotations about e α : As discussed in Ref.[2], H (0) must realize this symmetry in a nonlinear fashion. Specifically, if only first derivatives ∂ζ are taken into account, then H (0) is simply proportional to the surface area, i.e.,...

Segovia-López and Romero-Rochín Reply:

“…The predictions of all those mean-field models are at odds with what is known and expected as shown, for instance, by the classical capillary wave model of Buff, Lovett, and Stillinger (Ref. [1] of our Letter). Our results are along the lines of that work and some of our predictions are in agreement both with simulations and experiments (Refs.…”