Surface Self-Diffusion Instability in Electric Fields
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SURFACE SELF-DIFFUSION INSTABILITY IN ELECTRIC FIELDS Michael I. Ojovan*, Pavel P. Poluectov** *Moscow Scientific and Industrial Association “Radon”, RUSSIA, The 7-th Rostovsky Lane, 2/14, Moscow, 119121, RUSSIA, E-mail: [email protected], **State Scientific Centre VNIINM, Moscow, RUSSIA ABSTRACT The plane form is the equilibrium one for surfaces of condensed matter. Deviations can be caused usually by crystal structure. Herein we will describe an effect of surface instability due to selfdiffusion processes of atoms and molecules in the near surface electric field. Self-diffusion processes (as it was shown by Mullins) cause relaxation of any deviation (protuberance) from the plane form due to the increased concentration of surface atoms and its consequent smoothing. This process we studied for the case when there is an electric field near the surface. The near surface electric field can be due to either the location of material in an external (homogeneous or inhomogeneous) electrical field or self-charges on the surface. There is an increasing of electric field intensity near protuberances both in external and self-formed electrical fields: the higher is the curvature of surface the stronger is the intensity of the near surface electrical field. Consequently two competing processes occur during surface molecules mass transfer: both the selfdiffusion smoothing of surface molecule concentration and drawing of molecules in the strong electric field regions. Depending on the initial shape of the protuberance either relaxation or instability occurs. There is a critical wavelength λ 0=RskBT/2Uz, which shows that shorter wavelength deviations decrease their amplitudes and longer wavelength deviations grow in amplitude by time. Here Rs is characteristic of the material, T is temperature, and Uz is the interaction energy of surface molecules with the electric field. Since there are random variations of any surface from the plane form, being placed in an electric field these surfaces will be unstable depending on the intensity of electric field and properties of material. DEVIATIONS FROM PLANE SURFACE Consider a disturbance (perturbation) of the surface plane shape, which we will suppose to have an amplitude much lower than the linear sizes of disturbed surface concerned. Any disturbance of the plane shape of the surface can be decomposed by plane waves:
+
O [ P
=
∑
ξ
N
OLN [ P
(1)
where ξk is the amplitude of given plane wave with the wavelength λ =2π/k. Therefore one can consider only the evolution of the disturbance with given wave number k. We assume that the wavelength is large enough comparable with the free path length. Also it is assumed that the disturbance is not strong so that ξkk D
W
(24)
are growing with time (are unstable). As a consequence that means that there is a threshold electric field above which the surface diffusion of molecules causes a shape instability and leads to the destruction of the film. This threshold on electric field can be found from the equation: D
W
=
/
V
Y
,
(25)
whe
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