Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces

Abstract

-The limit n → ∞ of the classical O(n) φ 4 model on a 3d film with free surfaces is studied. Its exact solution involves a selfconsistent 1d Schrödinger equation, which is solved numerically for a partially discretized as well as for a fully discrete lattice model. Extremely precise results are obtained for the scaled Casimir force at all temperatures. Obtained via a single framework, they exhibit all relevant qualitative features of the thermodynamic Casimir force known from wetting experiments on 4 He and Monte Carlo simulations, including a pronounced minimum below the bulk critical point.A celebrated example of fluctuation-induced forces is the Casimir force between two metallic, grounded plates in vacuum [1].1 Such forces caused by the confinement of quantum electrodynamics (QED) vacuum fluctuations of the electromagnetic fields are expected to have considerable technological relevance. This has made them the focus of much ongoing research activity. During the past two decades, it has become increasingly clear that a wealth of similarly interesting classical analogs of such effective forces, induced by thermal rather than quantum fluctuations, exist [3].2 Two important classes of such "thermodynamic Casimir forces" 3 are forces induced by fluctuations in nearly (multi)critical media between immersed macroscopic bodies or boundaries, and forces due to confined Goldstone modes [6]. Clear experimental evidence for the existence of such thermodynamic Casimir 1 For a review of the Casimir effect in QED and an extensive lists of references, see [2] 2 For reviews of the thermodynamic Casimir effect and extensive lists of references, see [4] 3 Following established conventions we use the term "thermodynamic Casimir forces" for forces induced by thermal fluctuations, in particular, also for near-critical Casimir forces, reserving the name critical Casimir forces to those where the medium is at a critical point. This topic must not be confused with those of thermal effects on QED Casimir forces and thermal Casimir-Polder forces, which are less universal since material properties of the media and confining objects matter; see, e.g., [5] forces was provided first indirectly by measurements of the thinning of 4 He wetting films at the λ-point as the temperature T is lowered below the bulk critical temperature T c [7]. Subsequently, direct measurements of the thermodynamic Casimir force on colloidal particles in binary liquids near the consolute point could be achieved [8].Despite obvious analogies, crucial qualitative differences between thermodynamic and QED Casimir forces exist. First, the latter usually can be studied in terms of effective free field theories in confined geometries where the interaction of the electromagnetic field with the material boundaries is taken into account through boundary conditions. By contrast, investigations of thermodynamic Casimir forces at (multi)critical points necessarily involve interacting field theories. Second, whereas electromagnetic fields average to zero in the groun...

Fluctuation-induced forces in confined ideal and imperfect Bose gases

“…(4.47) yield corrections to scaling to ξ D /D of the form g −1 D −(4−d) with the familiar n = ∞ exponent ω g = 4 − d. As is discussed in some detail in [26] and [27], they can be eliminated by taking the limit g → ∞ [51]. In fact, it follows from Eq.…”

“…Of particular interest is the (d = 3)-dimensional case on which we now focus. For it, a number of exact analytic results have been obtained for the classical theory with DDBCs [26,27,[30][31][32][33][34][35], which are known to apply to this theory with free BCs at d = 3 asymptotically in the large length scale limit. Using a combination of techniques such as direct solutions of the self-consistent equations [32], short-distance and boundary-operator expansions [27], trace formulas [34], inverse scattering methods for the semi-infinite case D = ∞ and matched semiclassical expansions for D < ∞ [33], exact analytic results for several series expansion coefficients of the self-consistent potential v * (z) and for the asymptotic x → ∞ behaviors of the eigenvalues ε DD ν , eigenfunctions h DD ν , and the classical scaling functions of the residual free energy and the Casimir force have been determined.…”

“…However, the computation of these scaling functions for all values x 0 of the scaling variable x introduced in Eq. (4.54) required the use of numerical methods [26,27,29].…”

“…We introduce such an imperfect Bose gas model with free BCs in Sec. IV D. Its scaling function Θ DD ∞,3 (tD) may be obtained for all values of the scaling field t ∝ µ c − µ 0 from the numerical solution of the O(∞) φ 4 model determined in [26][27][28][29]. Furthermore, a variety of exact analytical results may be inferred from those known for the O(∞) φ 4 model subject to DDBCs [30][31][32][33][34][35].…”

Critical and near-critical phase behavior and interplay between the thermodynamic Casimir and van der Waals forces in a confined nonpolar fluid medium with competing surface and substrate potentials

“…[47] one has performed measurements of the Casimir force in thin wetting films of binary liquid mixture. One the theoretical side, the effect has been studied via exact calculations in the two-dimensional Ising model [48][49][50][51][52][53][54][55][56][57], the three dimensional spherical model [58][59][60][61][62][63][64][65][66], via conformal-theoretical methods [67][68][69][70][71][72][73], within mean-field type calculations on Ising type [74][75][76][77][78] and XY models [80], through renormalization-group studies via ε-expansion [81][82][83][84][85][86][87][88] and via fixed dimension d techniques [87][88][89] of O(n) models, as well as via MonteCarlo calculations [79,[90][91]…”