The Goldstone mode and resonances in the fluid interfacial region


The development of a molecular theory of inhomogeneous fluids and, in particular, of the liquid–gas interface has received enormous interest in recent years; however, long-standing attempts to extend the concept of surface tension in mesoscopic approaches by making it scale dependent, although apparently plausible, have failed to connect with simulation and experimental studies of the interface that probe the detailed properties of density correlations. Here, we show that a fully microscopic theory of correlations in the interfacial region can be developed that overcomes many of the problems associated with simpler mesoscopic ideas. This theory originates from recognizing that the correlation function displays, in addition to a Goldstone mode, an unexpected hierarchy of resonances that constrain severely its structural properties. Indeed, this approach allows us to identify new classes of fully integrable models for which, surprisingly, the tension, density profile and correlation function can all be determined analytically, revealing the microscopic structure of correlations in all generalized van der Waals theories.

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Fig. 1: Comparison of σl(q) and σg(q) with σHD(q).
Fig. 2: Results for model potentials whose S(z;q) displays a single resonance at \(\xi q = \sqrt {n^2 - 1}\).
Fig. 3: G(z, z′;q) for the double-cubic potential for different wave vectors q and fixed κz′ = 3.
Fig. 4: Sullivan model results for S(0;q) using the Carnahan–Starling equation of state for hard spheres.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


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We are grateful to F. Höfling for providing his simulation results and for discussions. A.O.P. acknowledges the EPSRC, UK, for grant EP/L020564/1 (Multiscale Analysis of Complex Interfacial Phenomena). C.R. acknowledges the support of grant FIS2015-66523-P (MINECO/FEDER, UE).

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A.O.P. conceived the problem and subsequent joint theoretical development. C.R. performed joint theoretical development and numerical computations.

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Correspondence to C. Rascón.

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Parry, A.O., Rascón, C. The Goldstone mode and resonances in the fluid interfacial region. Nat. Phys. 15, 287–292 (2019).

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