tailieunhanh - Kinetics of Materials - R. Balluff_ S. Allen_ W. Carter (Wiley_ 2005) Episode 9

Tham khảo tài liệu 'kinetics of materials - r. balluff_ s. allen_ w. carter (wiley_ 2005) episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 14 1 ISOTROPIC SURFACES 345 Therefore the energy decreases continuously with time if the Rayleigh instability condition is satisfied A Acrit Any perturbation with a wavelength less than the circumference of the cylinder will not grow. The particular characteristics of morphological evolution are determined by the dominant transport mechanism their analyses derive from the diffusion potential which depends on the local curvature. For a surface of revolution about the z-axis the curvature is given by Eq. that is d2R dz2 213 2 Substituting Eq. into Eq. and expanding for small e Ro yields k z 1 R o rỊ 2ttRo 2 A 2ttz cos A - 1 Evolution of Perturbed Cylinder by Vapor Transport Suppose that a perturbed cylinder with radius given by Eq. evolves by vapor transport in an environment with an ambient vapor pressure in equilibrium with the unperturbed cylinder _pamb Pe i K 1 RO . Then using Eqs. and Bv -pr - K y Ito J According to Eq. Rcyi Ro so Eq. shows that vn at z 0 is approximately de t dt. Therefore using Eq. de t Bv r 2 1 TH 1- A 1432 Small perturbations therefore evolve according to e t e 0 et TV A where the amplification factor l rv Bv R ị l A 2 . This first-order kinetic result is consistent with the previous Rayleigh result only perturbations with wavelengths longer than Acrit will grow. Evolution of Perturbed Cylinder by Surface Diffusion Suppose that the perturbed cylinder considered above evolves by surface diffusion. A first-order differential equation for the amplitude e t follows from inserting Eq. into the surface diffusion relation Eq. and again setting vn de t dt at z 0 de t _ 4tt2Bs dt R2A2 346 CHAPTER 14 SURFACE EVOLUTION DUE TO CAPILLARY FORCES In addition to the Rayleigh result Eq. predicts that a particular perturbation wavelength Amax grows the fastest and hence dominates the morphology of the .