Stress Driven Instability in Non-Hydrostatically Stressed Crystals and its Role in the Problems of Crystalline Thin Film
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STRESS DRIVEN INSTABILITY IN NON-HYDROSTATICALLY STRESSED CRYSTALS AND ITS ROLE IN THE PROBLEMS OF CRYSTALLINE THIN FILMS
MICHAEL A. GRINFELD Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
ABSTRACT
It was demonstrated in [1] that, in the absence of surface tension aflat boundary of non-hydrostatically stressed elastic solids is always unstable with respect to "mass rearrangement". The physical mechanisms of the rearrangement can be different, for instance, a)melting-freezing or vaporization-sublimation processes at liquid-solid or vapor-solid phase boundaries, b)surface diffusion of particles along free or interfacial boundaries, c)adsorption-desorbtion of the atoms in epitaxial crystal growth, etc.. .We discuss the role of this instability in the problems of epitaxy and, in particular, the opportunities delivered by this instability for explanation of the recently discovered phenomena of the dislocation-free Stranski-Krastanow pattern of growth [2]. These phenomena cannot be interpreted in the framework of traditional viewpoints since, according to the classical theory, the Stranski-Krastanow pattern is a result of proliferation of the misfit dislocation appearing on the interface "crystalline film-substratum" [3].
Mechanism of the instability, the bct-numbers, critical wave-length and thickness
Suppose that a solid plate of the shape of rectangular L x Lz with traction-free horizontal edges is stretched horizontally. We denote as 2T the value of uniform uniaxial stresses, then, T is equal to the maximal shear stresses acting at the bisectional cross-sections. This deformation reg results in accumulation of a certain amount of the elastic energy E Now, imagine that the migration of "atoms" along the upper external boundary causes corrugations of this surface. Then the same displacements of the vertical edges will bring the "damaged" plate into the oon-uniform equilibrium state with a certain stored elastic energy Eirre. The following inequality is valid for the solid body of any symmetry [1]:
Eirreg
J
(K)
; J"(K) = 2K
+ (3
- 4)
sh2K
-
(6)
K + 2(1 - 0) (3 - 40) sh K ch K
ne 1.) ne H A neutral dimensionless wave-number K obeys the equation J (Kn) = For K >> 1 (i.e., for the relatively short horizontal wave-lengths) J (K) approaches K/(1 -0), and (6) gives us a criterion equivalent to (4). In the opposite asymptotic case of small values of dimensionless wavenumber K
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