Determination of the Origin and Magnitude of Logarithmic Finite-Size Effects on Interfacial Tension: Role of Interfacial Fluctuations and Domain Breathing

Fabian Schmitz, Peter Virnau, and Kurt Binder

Phys. Rev. Lett. 112, 125701 – Published 26 March 2014; Erratum Phys. Rev. Lett. 112, 239902 (2014)

Abstract

The ensemble-switch method for computing wall excess free energies of condensed matter is extended to estimate the interface free energies between coexisting phases very accurately. By this method, system geometries with linear dimensions $L$ parallel and $L_{z}$ perpendicular to the interface with various boundary conditions in the canonical or grand canonical ensemble can be studied. Using two- and three-dimensional Ising models, the nature of the occurring logarithmic finite-size corrections is studied. It is found crucial to include interfacial fluctuations due to “domain breathing.”

Received 30 January 2014

DOI:https://doi.org/10.1103/PhysRevLett.112.125701

Erratum

Erratum: Determination of the Origin and Magnitude of Logarithmic Finite-Size Effects on Interfacial Tension: Role of Interfacial Fluctuations and Domain Breathing [Phys. Rev. Lett. 112, 125701 (2014)]

Fabian Schmitz, Peter Virnau, and Kurt Binder

Phys. Rev. Lett. 112, 239902 (2014)

Authors & Affiliations

Fabian Schmitz, Peter Virnau, and Kurt Binder

Institut für Physik, Johannes Gutenberg-Universität, D-55099 Mainz, Staudinger Weg 7, Germany

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Vol. 112, Iss. 12 — 28 March 2014

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Figure 1
Useful boundary conditions to study interfaces. For simplicity, we specialize to a $d$ -dimensional Ising system in a box of linear dimension(s) $L$ parallel to the interface(s) [shown by thick wavy lines], and a linear dimension $L_{z}$ perpendicular to the interface. In parallel directions, periodic boundary conditions (PBC) are applied throughout. The double arrows indicate the sign of the magnetization ( $m_{+} > 0$ , $m_{-} < 0$ ) in the domains. $⟨ m_{+} ⟩ = m_{coex}$ , $⟨ m_{-} ⟩ = - m_{coex}$ , $m_{coex}$ being the spontaneous magnetization of the Ising model. Left side shows antiperiodic boundary condition (APBC) in the $z$ direction; right side shows PBC in the $z$ direction (then necessarily two interfaces must occur).
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Figure 2
(a) Schematic explanation of the “ensemble switch method” to find the interfacial free energy. A system is constructed as a linear combination of two Hamiltonians $H (κ) = κ H_{1} + (1 - κ) H_{0}$ , where $H_{1}$ is the desired system of interest (containing interfaces). $H_{0}$ consists of two separate systems of half the linear dimension $L_{z}$ each, and with PBC each so that no interfaces occur, and $0 \leq κ \leq 1$ . The free energy difference between the states with $H (κ = 0)$ and $H (κ = 1)$ yields twice the interfacial free energy $γ$ . (b) Sampling the probability distribution $P_{L, L_{z}} (m)$ where $m$ is the total magnetization per spin of a system with PBC throughout, one finds two sharp peaks (of height $P_{\max}$ ) and a flat minimum (of height $P_{\min}$ ) in between, with $γ = \ln P_{\min} / (2 k_{B} T L^{(d - 1)})$ . Data are for the case $d = 3$ , $L = 20$ , $k_{B} T / J = 3.0$ Ising model.
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Figure 3
Interfacial tension $γ_{L, L_{z}}$ for the $d = 2$ (a),(b) and $d = 3$ (c) Ising model, plotted vs the scaling variable $L^{- (d - 1)} \ln L_{z}$ at fixed $L$ . (a) compares the cases APBC(c), PBC(c) and APBC(gc), for two temperatures, $k_{B} T / J = 1.2$ , $L = 10$ , upper three straight lines, and $k_{B} T / J = 2.0$ , $L = 20$ , lower three straight lines. The slopes are theoretical values $x_{⊥} = 1 / 2$ , $3 / 4$ and 1, respectively. (b) shows data for PBC(c) in $d = 2$ at three temperatures; $k_{B} T / J = 1.2$ , three top lines; $k_{B} T / J = 1.6$ , three middle lines; $k_{B} T / J = 2.0$ , three lower lines; at three choices of $L$ ( $L = 10$ , 20, 40, from top to bottom). These data show that the slope ( $x_{L} = 3 / 4$ ) neither depends on $L$ nor on temperature. (c) shows data for PBC(c) with $d = 3$ , $k_{B} T / J = 3$ , and 5 choices of $L$ , as indicated. (c) also shows preliminary data for a LJ fluid (see text).
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Figure 4
Interfacial tension $γ_{L, L_{z}}$ plotted vs $L$ . (a) shows data for $d = 2$ , $k_{B} T / J = 1.2$ , $L_{z} = 60$ , 120 while (b) shows $d = 3$ , $k_{B} T / J = 3.0$ , $L_{z} = 20$ , 40, 80. The horizontal straight line shows known values of $γ_{\infty}$ [22, 25], while the curves are fits of Eq. (2) to the data (symbols) for the cases APBC(c), top set of curves; PBC(c), middle set; APBC(gc), bottom set; in each set, $L_{z}$ increases from top to bottom. The theoretical values of $x_{⊥}$ , $x_{∥}$ from Table 1 were used.
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