Maxwell and Van Der Waals Revisited
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MAXWELL AND VAN DER WAALS REVISITED
E.C. Aifantis Michigan Technological University, Houghton, MI and University of Minnesota, Minneapolis, MN 55455,
49931,
USA
USA
ABSTRACT We utilize a modern continuum mechanics framework to reconsider an old problem for fluid interfaces, also addressed by Maxwell and van der Waals. We prove that their results need not be valid necessarily. This conclusion is arrived at as a consequence of questioning the existence of thermodynamic potentials and the validity of usual thermodynamic ,relations within unstable (spinodal) regions. One central result is that Maxwell's equal area rule needs not be valid and certain statistical models are shown to be internally inconsistent. Precise conditions for the validity of Maxwell's rule and the variational theory of van der Waals are established in terms of the coefficients defining the interfacial stress. Finally, a generalized continuum thermodynamics framework is developed which provides an alternative derivation of van der Waals variational theory and properly extends it to dynamic situations. However, other possibilities exist which allow a thermodynamics of fluid interfaces not necessarily restricted by the conditions of Maxwell and van der Waals. The results of the paper could be viewed as a convincing argument to utilize this framework (with the necessary modifications) to interpret more complex phenomena of phase transformations.
INTRODUCTION The purpose of this presentation is to indicate that there are situations where use of standard variational thermodynamics may lead to restrictive results. This is illustrated by confining attention to the simplest problem of phase transformations: an equilibrium liquid-vapor transition. Standard approaches to this problem by Gibbs, Maxwell and van der Waals and later by Cahn-Hilliard and Davis-Scriven have assumed the existence of usual thermodynamic functions and validity of thermodynamic relations within the spinodal (unstable region of the p-n diagrams). This practice has led, among other things, to a rather questionable derivation of Maxwell's rule. Moreover, it has produced expressions for the interfacial stress which could probably be of limiting validity. To overcome these difficulties an alternative framework is proposed. This proposal is delivered in two parts. In the first part of the paper we dispense with thermodynamics and consider the liquid-vapor interface within a mechanical framework only. The interface is viewed as a continuum supporting a stress tensor whose constitutive form (expressed in terms of a gradient approximation) is restricted to obey Cauchy's differential equations of equilibrium. This gives rise to an overdetermined system of differential equations which in one dimension yields a non-linear second-order differential equation which can be integrated to give an analytic expression for the interfacial density profile. Existence conditions for this analytic solution are established. These relations are essentially the condition of mechanical equilibrium (equality
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