Fluctuations of the critical Casimir force

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Fluctuations of the critical Casimir force

Markus Gross, Andrea Gambassi, and S. Dietrich

Phys. Rev. E 103, 062118 – Published 14 June 2021

Abstract

The critical Casimir force (CCF) arises from confining fluctuations in a critical fluid and thus it is a fluctuating quantity itself. While the mean CCF is universal, its (static) variance has previously been found to depend on the microscopic details of the system which effectively set a large-momentum cutoff in the underlying field theory, rendering it potentially large. This raises the question how the properties of the force variance are reflected in experimentally observable quantities, such as the thickness of a wetting film or the position of a suspended colloidal particle. Here, based on a rigorous definition of the instantaneous force, we analyze static and dynamic correlations of the CCF for a conserved fluid in film geometry for various boundary conditions within the Gaussian approximation. We find that the dynamic correlation function of the CCF is independent of the momentum cutoff and decays algebraically in time. Within the Gaussian approximation, the associated exponent depends only on the dynamic universality class but not on the boundary conditions. We furthermore consider a fluid film, the thickness of which can fluctuate under the influence of the time-dependent CCF. The latter gives rise to an effective non-Markovian noise in the equation of motion of the film boundary and induces a distinct contribution to the position variance. Within the approximations used here, at short times, this contribution grows algebraically in time whereas, at long times, it saturates and contributes to the steady-state variance of the film thickness.

Received 24 November 2020
Accepted 20 May 2021

DOI:https://doi.org/10.1103/PhysRevE.103.062118

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Casimir effect Critical phenomena Fluctuations & noise Statistical field theory

Classical fluids Colloids

Statistical Physics & Thermodynamics

Authors & Affiliations

Markus Gross^1,2,*, Andrea Gambassi³, and S. Dietrich^1,2

¹Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany
²IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
³SISSA–International School for Advanced Studies and INFN, via Bonomea 265, 34136 Trieste, Italy

^*gross@is.mpg.de

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Issue

Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
Sketch of the situation considered in this study. A film of thickness $L$ is enclosed by two boundaries (thick black lines) each of surface area $A$ (spanning the plane perpendicular to the figure). The right boundary is enclosed by a thin imaginary volume $V_{s}$ . The CCF acting on the right boundary is obtained from an integration of the stress tensor over the surface of $V_{s}$ , which consists of an inner ( $\partial V_{s, i}$ ) and an outer part ( $\partial V_{s, o}$ ) [see Eq. (2.18)]. The film and the bulk consist of the same liquid, whereas the gray area corresponds to a solid.
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Figure 2
Dynamic correlations $〈 Δ P_{f} (t) Δ P_{f} (0) 〉$ of the film pressure for (a) periodic BCs [Eq. (3.38)] and (b) Neumann BCs [Eq. (3.46)] for $d = 3$ and various temperatures [ $L^{2} τ = 0$ (blue), 10 (red), $10^{5}$ (green), from top to bottom]. The solid curves are obtained by numerically evaluating Eqs. (3.38) and (3.46), while the dashed lines represent the asymptotic laws reported in Eqs. (3.39), (3.40), and (3.43)–(3.45) for (a) and in Eq. (3.47) for (b). The gray dotted line corresponds to a decay $\propto t^{- 3 / 4}$ (applying to $d = 3$ ), while the gray dashed-dotted line represents Eqs. (3.40) and (3.47d). These lines are drawn in order to highlight the influence of the logarithmic correction for $d = 3$ [see Eqs. (3.42) and (3.47e)]. For the evaluation of $〈 Δ P_{f} (t) Δ P_{f} (0) 〉$ at bulk criticality ( $τ = 0$ ) we used a low momentum cutoff $λ ≃ 10^{- 9} / L$ (this specific value is chosen for illustrative purposes). For times shorter than those covered by the plot, the curves corresponding to $L^{2} τ = 10^{5}$ (green) and 10 (red) approach the asymptotic critical ( $τ = 0$ ) short-time behavior given in Eq. (3.43). Due to the smaller value of $L^{2} τ$ , the red curve lies closer to this asymptote and crosses over to the off-critical asymptote [Eqs. (3.44) and (3.47b)] later than the green one. Except for the case of the power law $\propto t^{- 3 / 4}$ , the asymptotic laws reported in these plots do not involve a fitting parameter. We note that $〈 Δ P_{f} (t) Δ P_{f} (0) 〉$ has the same dimension as $A^{- 1} L^{- (d + 1)}$ [see the comment after Eq. (3.10)].
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Figure 3
Contributions $M_{b, f} (t)$ [Eqs. (4.14) and (4.21)] to the mean-squared displacement of the movable film boundary at position $R (t)$ [see Eqs. (4.23) and (4.12)] in $d = 3$ for (a) a bulklike system $[R (t) ≫ L]$ , (b) a film with periodic BCs, and (c) a film with Neumann BCs. The solid curves are the results of the numerical calculation of Eq. (4.40) (with analogous expressions for the bulk and the film with Neumann BCs, see the main text), while the dashed-dotted curves (which practically overlap with the solid curves) represent the short-time scaling prediction reported in Eqs. (4.42), (4.45), and (4.46). The dotted lines represent the function $0.37 \times t$ , obtained from a numerical evaluation of Eq. (4.48), which describes the long-time behavior of $M_{b}$ . The slight deviations from the behavior $\propto t$ at short times are due to the logarithmic correction in Eqs. (4.44) and (4.47). For illustrative purposes, we have used the value $Λ \approx 1000 / L$ for the large momentum cutoff and $D = 0.1 / L^{2}$ for the bare diffusion constant. The curves depend only very weakly on the choices for these values.
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