Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of nucleation, i.e., the steady-state nucleation rate and the steady-state cluster-size distribution. These two quantities depend on the work of formation, ΔG(n)=−nΔμ+γn2/3, of crystal clusters of size n and, in particular, on the work of critical cluster formation, ΔG(nc). The first term in the expression for ΔG(n) describes changes in the bulk contributions (expressed by the chemical potential difference, Δμ) to the Gibbs free energy caused by cluster formation, whereas the second one reflects surface contributions (expressed by the surface tension, σ: γ=Ωd02σ, Ω=4π(3/4π)2/3, where d0 is a parameter describing the size of the particles in the liquid undergoing crystallization), n is the number of particles (atoms or molecules) in a crystallite, and n=nc defines the size of the critical crystallite, corresponding to the maximum (in general, a saddle point) of the Gibbs free energy, G. The work of cluster formation is commonly identified with the difference between the Gibbs free energy of a system containing a cluster with n particles and the homogeneous initial state. For the formation of a “cluster” of size n=1, no work is required. However, the commonly used relation for ΔG(n) given above leads to a finite value for n=1. By this reason, for a correct determination of the work of cluster formation, a self-consistency correction should be introduced employing instead of ΔG(n) an expression of the form ΔG˜(n)=ΔG(n)−ΔG(1). Such self-consistency correction is usually omitted assuming that the inequality ΔG(n)≫ΔG(1) holds. In the present paper, we show that: (i) This inequality is frequently not fulfilled in crystal nucleation processes. (ii) The form and the results of the numerical solution of the set of kinetic equations are not affected by self-consistency corrections. However, (iii) the predictions of the analytical relations for the steady-state nucleation rate and the steady-state cluster-size distribution differ considerably in dependence of whether such correction is introduced or not. In particular, neglecting the self-consistency correction overestimates the work of critical cluster formation and leads, consequently, to far too low theoretical values for the steady-state nucleation rates. For the system studied here as a typical example (lithium disilicate, Li2O·2SiO2), the resulting deviations from the correct values may reach 20 orders of magnitude. Consequently, neglecting self-consistency corrections may result in severe errors in the interpretation of experimental data if, as it is usually done, the analytical relations for the steady-state nucleation rate or the steady-state cluster-size distribution are employed for their determination.

中文翻译：

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过冷液体的结晶：稳态成核率的自洽校正

晶体成核可以通过一组动力学方程来描述，这些方程适当地解释了控制该过程的热力学和动力学因素。通过对这组方程的数学分析，人们可以制定成核基本特征的解析表达式，即稳态成核率和稳态簇大小分布。这两个量取决于大小为 n 的晶体簇的形成功 ΔG(n)=-nΔμ+γn2/3，特别是取决于临界簇形成功 ΔG(nc)。ΔG(n) 表达式中的第一项描述了由簇形成引起的对吉布斯自由能的体积贡献（由化学势差 Δμ 表示）的变化，而第二项反映了表面贡献（由表面张力表示） , σ: γ=Ωd02σ, Ω=4π(3/4π)2/3，其中 d0 是描述正在结晶的液体中粒子大小的参数），n 是微晶中粒子（原子或分子）的数量，以及n=nc 定义了临界微晶的大小，对应于吉布斯自由能 G 的最大值（通常是鞍点）。簇形成功通常被定义为系统吉布斯自由能之间的差异包含一个具有 n 个粒子的簇和均匀的初始状态。对于形成大小为 n=1 的“集群”，不需要任何工作。然而，上面给出的 ΔG(n) 的常用关系导致 n=1 的有限值。因此，为了正确确定集群形成的工作，应该引入自洽校正，而不是使用 ΔG(n) 形式的表达式 ΔG~(n)=ΔG(n)-ΔG(1)。假设不等式 ΔG(n)≫ΔG(1) 成立，这种自洽校正通常被省略。在本文中，我们表明：（i）这种不等式在晶体成核过程中经常不满足。(ii) 动力学方程组数值解的形式和结果不受自洽修正的影响。然而，（iii）稳态成核率和稳态簇大小分布的解析关系的预测根据是否引入这种校正而有很大差异。特别是，忽略自洽修正会高估临界簇形成和导联的工作，因此，稳态成核率的理论值太低。对于这里作为典型示例研究的系统（二硅酸锂，Li2O·2SiO2），由此产生的与正确值的偏差可能达到 20 个数量级。因此，如果像通常那样使用稳态成核率或稳态簇大小分布的解析关系来确定它们，则忽略自洽校正可能会导致实验数据解释中的严重错误.