By Philip Hofmann
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Extra resources for Solid state physics : an introduction
Eq. 95) n 1/3 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ . For the case of kinetically limited growth (if the conditions 1 for any of the components), we have (+) wn,n+1 = k 4πDeﬀ cβ a2α n2/3 , aβ 1 = k Deﬀ k i=1 2 νiα . xiβ αi Di For diffusion-limited growth of the clusters (if the conditions 1 for any of the components), we similarly get (+) wn,n+1 = d 4πcβ aα Deﬀ n1/3 , 1 = d Deﬀ k i=1 2 νiα . 98) 36 2 Determination of the Coefﬁcients of Emission in Nucleation Theory With ωβ = V /nβ = 1/cβ , we can always make the replacement 4πcβ = 3/a3β in the above equations.
Nj + 1, . . 48) and as a consequence (+) wnj ;nj +1 (−) wnj +1;nj = exp − [ΔG(nj + 1) − ΔG(nj )] kB T . 49) In Eq. 48), μjv is the chemical potential of a particle of the jth component in the ambient vapor phase, while μjd (n1 , n2 , . . , nj + 1, . . , nk ) is its value in a cluster of the speciﬁed composition. As in the one-component case, the value of μjd accounts for surface and other possible ﬁnite size effects. In Eq. 49), the values of ni , i = j, are kept constant, again. By the introduction of an auxiliary function f (∗) (n1 , n2 , .
33) A comparison with the ordinary continuity equation, (∂ρ/∂t) + divj = 0, shows that the deterministic growth rate may be expressed as dn (+) = −wn,n+1 dt ∂ΔG (n) ∂n 1 kB T . 34) This equation allows one to determine the coefﬁcients of aggregation w(+) if the macroscopic growth rates dn/dt of the clusters are known. 4 Generalization to Multicomponent Systems Assume that both ambient and newly evolving phases may be composed of k different components. The distribution function with respect to cluster sizes is then a function of all numbers nj of particles of the different components j (j = 1, 2, .