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Processes 2019, 7, 248 5 of 16 Equation (9) reveals that there is a linear relationship between ln tind and ln(σ), which can be employed to determine the nucleation order n at a given temperature. 2.2.2. Mechanism of Crystal Growth The growth mode of the Li2CO3 crystal was identified by fitting the induction time data measured from reactive crystallization experiments over a range of supersaturations at different temperatures using expressions derived by Van der Leeden et al. [16], which can be expressed by 1 α n1 tind = JV + anJGn−1 (10) where J is the nucleation rate, V is the volume of the system, α is the volume fraction of the new phase formed, an is a shape factor, G is the growth rate, and n = mv + 1 (m is the dimensionality of growth and 0.5 < ν < 1). As for Equation (10), the first term which originates from the mononuclear mechanism is often negligible in comparison to the second term because of the polynuclear mechanism. For three-dimensional nucleation, the steady-state nucleation rate can be given as B J = KJSexp − 2 (11) ln S where K is the nucleation rate constant and B = βγsl3v2 . J (kBT)3 The relationship between the growth rate and supersaturation is generally described as G = KG f(S) (12) in which KG is the growth rate constant and f (S) is a function of supersaturation depending on the growth mechanism. The expressions of f (S) corresponding to different growth mechanisms are shown in Table 1. Table 1. The expressions of f (S) for different growth mechanisms. where α n1 Au = anKJKGn−1 f(S) (S − 1) (S − 1)2 (S − 1) (S−1)23S13 exp−B2D 3lnS Combining Equations (10)–(12), we get tind =Au[f(S)] n S n exp nln2S F1 F2 F3 F4 Growth Mechanisms Normal growth Spiral growth Volume diffusion-controlled growth 2Dnucleation-mediatedgrowth (1−1) −1 B (13) (14) For normal, spiral, and volume diffusion-controlled growth, Equation (13) can be rearranged to give where Fu(S)=lnAu+ B (15) nln2 S n−11 Fu(S) = ln tind[f(S)] n Sn (16)PDF Image | Reactive Crystallization Process of Lithium Carbonate
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