Pavel Grinfeld

Towards thermodynamics of elastic electric conductors

Abstract We discuss some problems of equilibrium shapes and possible morphological patterns of crystalline deformable conductors. To this end we propose a master system describing quasistatic evolution of shapes of such conductors owing to rearrangement of their material particles. The driving force of the evolution is the diminishing of the total energy of the system consisting of the elastic, electrostatic and surface energies. The proposed master system is based on the assumption that the characteristic time scale of establishing equilibrium with respect to rearrangement of materials particles is much greater than the time scales of establishing equilibrium with respect to mechanical and electrostatic degrees of freedom. The proposed system is based on principles and notions of the exact nonlinear continuum mechanics in the Eulerian description. We use the exact master system to derive an explicit formula of increment or decrement in the growth of small disturbances of an isotropic, uniformly stressed elastic half-plane in a uniform electrostatic field. In the case of small deviations from the unstressed state we express the coefficients of the dispersion relation in terms of the Lamé and Poisson modules. After establishing the key dispersion equation on the solid basis of the nonlinear theory, we show some modifications of the master system that will allow us to establish the same dispersion equations without any appeal to the notions and concepts of the exact theory. Finally we apply our system to the case of incompressible substances and demonstrate that the suggested master system leads to the same (in)stability criterion as that established long ago for a liquid conductor in an electrostatic field.

The Stress Driven Rearrangement Instabilities in Electronic Materials and in Helium Crystals

At present, there is a consensus that various Stress Driven Rearrangement Instabilities (SDRI) are the implications of the mathematically rigorous theoretical Gibbs thermodynamics. Many applied researchers and practitioners believe that SDRI are also universal physical phenomena occurring over a large range of length scales and applied topics. There is a multitude of publications claiming experimental observation of the SDRI based phenomena. This opinion is challenged by other highly respected scholars claiming theoretical inconsistencies and multiple experimental counterexamples. Such an uncertainty is too costly for further progress on the SDRI topic. The ultimate goal of our project is to resolve this controversy. The project includes experimental, theoretical, and numerical studies. Among various plausible manifestations of SDRI, the authors focused only on two most promising for which the validity of the SDRI has already been claimed by other researchers: a) stress driven corrugations of the solid-melt phase interface in macroscopic quantum 4 He and b) the dislocation-free Stranski-Krastanov pattern of growth of semiconductor quantum dots. We devised a program and experimental set-ups for testing applicability of the SDRI mechanisms using the same physical systems as before but using implications of the SDRI theory for 2D patterning which have never been tested in the past.

A Rigorous Framework for the Landau and Lifshitz Approach to Thomson Electrostatics

Communicated by Gregory L. Naber Abstract. Landau and Lifshitz [7] proposed a novel formulation of the famous Thomson theorem, also known as the Thomson variational principle. In an attempt to explain, rather than postulate, the distribution of electrical charge exclusively on the surface of the conductor, Landau and Lifshitz allow the admissible variations in the electrical charge to penetrate the interior of the conductor. This is a valuable generalization of their predecessors’ work, as well as a step towards basing more of the analysis on first principles. Landau and Lifshitz’ approach has not received the attention it deserves because it was not formulated as a rigorous technique, but rather as a slight of hand to arrive at a known result. In this paper, we construct a rigorous mathematical framework based on the Landau and Lifshitz idea. In particular, we prove that surface distribution of charges corresponds to the absolute minimum of electrostatic energy.