Derivation of the Casimir contribution to the binding potential for 3D wetting

ABSTRACT

The renormalisation group theory of critical and tri-critical wetting transitions in three-dimensional systems with short-ranged forces, based on analysis of an effective Hamiltonian with an interfacial binding potential $w\left(\ell \right)$, predicts very strong non-universal critical singularities. These, however, have famously not been observed in extensive Monte Carlo simulations of the transitions in the simple cubic Ising model. Here, we show that previous treatments have missed an entropic, or low-temperature Casimir, contribution to the binding potential, arising from the many different microscopic configurations which correspond to a given interfacial one. We derive the full binding potential, including the Casimir correction term, starting from a microscopic Landau–Ginzburg–Wilson Hamiltonian, using a continuum transfer-matrix (path-integral) method. This is illustrated first in one dimension before generalising to arbitrary dimension. The Casimir contribution is qualitatively different for first-order, critical and tri-critical wetting transitions and substantially alters previous predictions for critical singularities bringing them much closer to the simulation results.

GRAPHICAL ABSTRACT

KEYWORDS:

• Wetting
• fluctuations
• Casimir forces

Acknowledgments

This paper is dedicated to the memory of three brilliant people who influenced us, and many, many others, deeply, both professionally and personally. Luis F. Rull, Kurt Binder and Michael E. Fisher.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Ministerio de Ciencia e Innovación (Spain) [grant number PID2021-126348NB-I00]; and Consejería de Economía, Conocimiento, Empresas y Universidad (Junta de Andalucía, Spain) [grants number US-1380729 and P20_00816], cofunded by EU FEDER. A.S. acknowledges FWF Der Wissenschaftsfonds (Austrian Science Fund) for funding through the Lise-Meitner Fellowship [grant number M 3300-N].