We solve the standard Lifshitz-Slyozov (LS) model with conservation of total particles in the limit of small super-saturation. The new element is an effective initial condition that follows from the initial exhaustion of nucleation as described in a previous paper [Farjoun and Neu, Phys. Rev. E 78]. The effective initial condition is characterized by a narrow distribution of cluster-sizes, all much larger than critical. In the subsequent solution, one of the LS similarity solutions emerges as the long-time limit, as expected. But our solution tells more. In particular, there is a "growth" era prior to what is usually called "coarsening". During "growth" the clusters (all of nearly the same size much larger than critical) eventually exhaust the super-saturation (the exhaustion of nucleation in the previous era results from only a small decrease in super-saturation). This allows the critical size to catch up to the clusters, and the traditional "coarsening" begins: Subcritical clusters dissolve and fuel the growth of the remaining super-critical clusters. Our analysis tracks the evolution of cluster sizes during growth and coarsening by complimentary use of asymptotic and numerical methods. We establish characteristic times and cluster sizes associated with growth and coarsening from physical parameters and the initial super-saturation. The emerging distribution is discontinuous at the largest cluster size, and our model selects the discontinuous LS similarity solution as the long time limit. There are strong indications that the smooth similarity solution proposed in the original LS paper emerges on a, yet longer, late-coarsening time-scale.
我们确定的集群大小的分布，出现从一个初始阶段的均匀聚集与保守的总粒子密度。预测背后的物理成分基本上是经典的：超临界核是以Zeldovich速率产生的，在单体耗尽之前，特征团簇尺寸如此之大，以至于团簇经历扩散限制生长。在数学上，集群大小的分布满足“大小空间”中的平流偏微分方程。在这个创建阶段，集群成核，然后增长到一个比临界尺寸大得多的尺寸，所以在Zeldovich率的超临界集群的成核是由零尺寸的有效边界条件表示。对流偏微分方程的有效边界条件构成了一个“创建信令问题”的集群大小的演变分布在创建时代。应用于平流信号问题的主导平衡论点表明，在初始成核自由能垒G * 中，创生时代的特征时间和团簇大小呈指数级大。特征时间与exp（2 G */ 5 k B T）成正比，团簇中特征单体数与exp（3 G */5 k B T）成正比。指数大的特征时间和集群大小给出了数学信号问题的后验验证。在一个简短的说明中，马尔琴科得到了这些指数和数值前因子2/5和3/5。我们的工作增加了这些缩放所暗示的动力学模型的实际解决方案，以及连接到创造时代之后的聚集过程的后续阶段的基础。
Morphological, nanostructural, and compositional evolution during phase separation of a model Ni-Al-Mo superalloy: Atom-probe tomographic experiments and lattice-kinetic Monte Carlo simulations
