Comparison between Cahn and Hilliard's model and a representation of fluid interfaces obtained by a variational method in continuum mechanics (Q1116819)

Principles expressing that the free energy of a system in isothermal equilibrium is minimum are usually used in mathematical physics. By writing that, at a given temperature, the free energy of a non-uniform system with a single component is stationary, \textit{J. W. Cahn} and \textit{J. E. Hilliard}, in some famous and useful papers, obtained an equation which represents distribution of composition (or density) in fluid interfaces in equilibrium [(*) J. Chem. Phys. 28, No. 2, 258--267 (1958; \url{doi:10.1063/1.1744102}); 31, No. 3, 688--699 (1959; \url{doi:10.1063/1.1730447})]. The fluid medium is a continuum that does not offer a surface of discontinuity. This idea is due to \textit{J. D. Van der Waals} [in ``Théorie de la capillarité dans l'hypothèse d'une variation continue de la densité'', Arch. Néerl. 28, 121--209 (1893; JFM 25.1585.01)]. It has been used again in problems of dynamic changes of phase [\textit{J. Serrin} (ed.), New perspectives in thermodynamics. Berlin etc.: Springer-Verlag (1986; Zbl 0591.00028)] and has been generalized by \textit{P. Casal} and the first author for non-isothermal systems where thermodynamic parameters of density and entropy have large variations [e.g. C. R. Acad. Sci., Paris, Ser. II 306, No. 2, 99--104 (1988; Zbl 0637.76108)]. In this paper, we will be considering the same case as Cahn and Hilliard did. The free energy of a small element of a solution having a spatial variation in composition is represented by the sum of two terms, one being the free energy that the element could have if surrounded by material of the same composition as itself, and the other, a term which to a first approximation is proportional to the square of the composition gradient. However, a difficulty is to decide what form of variational principle is considered to be satisfactory. Our purpose is to show that research of extremals for a functional associated with composition (or density) of a fluid requires precautions and only continuum mechanics allows to avoid the difficulties. The equation that Cahn and Hilliard pointed out (*), involves a coefficient \(\kappa\) associated with term of space of composition gradient. Their calculation method is a plain application of the Euler equation for variational calculus and, in fact, deals with a single dimensional physical system. Tensorial nature of density, mathematically represented by a three-form, it not taken into account. Cahn and Hilliard's equation has not a vectorial nature as a balance equation in fluid mechanics does. However, coefficient \(\kappa\) being a function of density only, Cahn and Hilliard's equation is a first integral equivalent to equations obtained with the methods of continuum mechanics. So, all the results of (*) are valid.