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Processes 2019, 7, 248 4 of 16 2.2. Theory 2.2.1. Classical Nucleation Theory According to classical nucleation theory [14], the nucleation rate (J) can be expressed according to Equation (1), i.e., (kBT) (lnS) where A is the pre-exponential factor, β is the geometric factor, γsl is the interfacial free energy, v is the molecular volume, kB is the Boltzmann constant, T is the absolute temperature, and S is the supersaturation. Generally, the induction time tind consists of three parts: the relaxation time, tr; the time required to generate a stable nucleus, tn; and the time required for the nucleus to grow to be detectable, tg. Making the assumption that the tind is mainly composed of tn, which is inversely proportional to the nucleation rate, we get: J=K·t −1 (2) ind Combining Equations (1) and (2), the relationship between induction time and supersaturation can be given by: βγ3v2 sl J = A exp− 3 2 (1) 32 K βγslv t=exp (3) ind A 32 (kBT) (lnS) Taking the logarithm of both sides of Equation (3) and rearranging gives: βγsl 3 v2 lntind = 3 2 +B (4) (kBT) (lnS) Equation (4) has been proved to be valid by Sohnel and Mullin when the induction time is mainly dominated by the time required to generate a stable nucleus [15]. Supersaturation is an important factor affecting the nucleation rate. In addition to classical nucleation theory, the nucleation rate can also be considered a function of supersaturation, as shown in the following empirical equation, Equation (5). The effect of supersaturation on the nucleation rate can be visually reflected by determining the nucleation order. The empirical equation can be given by: J = k∆Cn (5) Combining Equations (2) and (5), we get: t=K (6) ind k(∆C)n The relationship between relative supersaturation and the supersaturation ratio can be expressed as: σ = S − 1 = ∆C (7) supersaturation can be given by: t=K (8) ind k(σC∗ )n Taking the logarithm of both sides of Equation (8) and rearranging gives: K ln tind = ln k(C∗)n − nln(σ) (9) C∗ Combining Equations (6) and (7), the relationship between induction time and relativePDF Image | Reactive Crystallization Process of Lithium Carbonate
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