Analysis of Interfacial Stability

Interfacial stability has been studied by analytical means [4, 5, 9, 11-15] and also numerically [19-21], In the following, we present some of the analytical results of most immediate interest. Many other important results may be found in the 

Spherical Particle during Diffusion-Limited Growth in an Isothermal Binary Solid. This problem was analyzed by Mullins and Sekerka who found expressions for the rate of growth or decay of shape perturbations to a spherical H-rich /3-phase particle of fixed composition growing in an a matrix as in Section 20.2.1 [9]. Perturbations are written in the form of spherical harmonics. Steps to solve this problem are  

Spherical harmonics are derived from solutions of Laplace s equation ih spherical coordinates using the method of separation of variables—i.e., a solution of the form 

The 4 ) function turns out to be an exponential and the ( ) function consists of Legendre polynomials. Their product ( / ) ( ) gives the spherical harmonic functions which Arfken writes as Y 6, fi) and which Mullins and Sekerka write as Y[m (6, f ). Then, from Eq. 20.56, 

The functions Y 6, j ) are tabulated and can be represented as in Fig. 20.13. A series of spherical harmonics can be used to represent an arbitrary perturbation of a sphere, much the same as a Fourier series can represent an arbitrary function of a single variable. 

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