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On the Gibbs adsorption equation and diffuse interface models

On the Gibbs adsorption equation and diffuse interface models
On the Gibbs adsorption equation and diffuse interface models
In this paper we discuss some applications of the classical Gibbs adsorption equation to specific diffuse interface models that are based on conserved and non-conserved order parameters. Such models are natural examples of the general methodology developed by J. W. Gibbs in his treatment of the thermodynamics of surfaces. We employ the methodology of J. W. Cahn, which avoids the use of conventional dividing surfaces to define surface excess quantities. We show that the Gibbs adsorption equation holds for systems with gradient energy coefficients, provided the appropriate definitions of surface excess quantities are used. We consider, in particular, the phase-field model of a binary alloy with gradient energy coefficients for solute and the phase field. We derive a solute surface excess quantity that is independent of a dividing surface convention, and find that the adsorption in this model is influenced by the surface free energies of the pure components of the binary alloy as well as the solute gradient energy coefficient. We present one-dimensional numerical solutions for this model corresponding to a stationary planar interface and show the consistency of the numerical results with the Gibbs adsorption equation. We also discuss the Gibbs adsorption equation in the context of other diffuse interface models that arise in spinodal decomposition and order-disorder transitions.
adsorption phase-field model diffuse interface model binary alloy gibbs adsorption equation surface free energy
1364-5021
1129-1149
McFadden, G.B.
56b0d29e-1cfb-4775-96d1-d32d50ea08d2
Wheeler, A.A.
eb831100-6e51-4674-878a-a2936ad04d73
McFadden, G.B.
56b0d29e-1cfb-4775-96d1-d32d50ea08d2
Wheeler, A.A.
eb831100-6e51-4674-878a-a2936ad04d73

McFadden, G.B. and Wheeler, A.A. (2002) On the Gibbs adsorption equation and diffuse interface models. Proceedings of the Royal Society A, 458 (2021), 1129-1149. (doi:10.1098/rspa.2001.0908).

Record type: Article

Abstract

In this paper we discuss some applications of the classical Gibbs adsorption equation to specific diffuse interface models that are based on conserved and non-conserved order parameters. Such models are natural examples of the general methodology developed by J. W. Gibbs in his treatment of the thermodynamics of surfaces. We employ the methodology of J. W. Cahn, which avoids the use of conventional dividing surfaces to define surface excess quantities. We show that the Gibbs adsorption equation holds for systems with gradient energy coefficients, provided the appropriate definitions of surface excess quantities are used. We consider, in particular, the phase-field model of a binary alloy with gradient energy coefficients for solute and the phase field. We derive a solute surface excess quantity that is independent of a dividing surface convention, and find that the adsorption in this model is influenced by the surface free energies of the pure components of the binary alloy as well as the solute gradient energy coefficient. We present one-dimensional numerical solutions for this model corresponding to a stationary planar interface and show the consistency of the numerical results with the Gibbs adsorption equation. We also discuss the Gibbs adsorption equation in the context of other diffuse interface models that arise in spinodal decomposition and order-disorder transitions.

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More information

Published date: May 2002
Keywords: adsorption phase-field model diffuse interface model binary alloy gibbs adsorption equation surface free energy

Identifiers

Local EPrints ID: 415
URI: http://eprints.soton.ac.uk/id/eprint/415
ISSN: 1364-5021
PURE UUID: 96446bd8-d830-44e9-b39a-45c513c26da2

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Date deposited: 25 Mar 2004
Last modified: 08 Jan 2022 15:44

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Contributors

Author: G.B. McFadden
Author: A.A. Wheeler

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