Heterogeneous nucleation models in solidification have been developed to predict grain size over several decades. Up to now, the models are divided into three categories. The first is a well-known spherical cap nucleus model, where the effectiveness of nucleant, i.e. contact angle, is mainly focused on characterising the nucleants. In this model, the kinetics is determined by activation energy of embryos formation and the adsorption rate of atoms into the embryos. The second is a free growth model, in which active nucleants are determined by the size distribution of nucleants, and the heterogeneous nucleation is considered as an athermal process. The third is a constitutional supercooling model in which active nucleants are triggered by the constitutional undercooling at the front of growing crystals. All models remain uncertain to describe constitutionally the kinetics of heterogenous nucleation. In addition, the crystal growth rate affects significantly the heterogeneous nucleation events since growing crystals release solute atoms and heat. Therefore it is very important to approximate a flux of solute at the solid–liquid interface. In practical aspect, how to predict the grain size in the equiaxed zone is of interest. Equiaxed grains in front of growing columnar block the growth when the volume of equiaxed grains reaches a critical value, and eventually equiaxed zones are formed, i.e. columnar to equiaxed transition (CET). It is believed that the grain size in the equiaxed zone can be predicted using the cooling rate of remained melt at the time when CET occurs. In addition, further investigations on the efficiency variation of grain refiner are also required, although it is agreed that the efficiency decreases significantly with increasing amounts of nucleant.

几十年来，已经开发出凝固中的异质成核模型来预测晶粒尺寸。到目前为止，模型分为三类。第一个是众所周知的球冠核模型，其中成核的有效性，即接触角，主要集中在表征成核。在该模型中，动力学由胚胎形成的活化能和原子在胚胎中的吸附率决定。第二种是自由生长模型，其中活性成核剂由成核剂的尺寸分布决定，异相成核被认为是一个非热过程。第三种是结构过冷模型，其中活性成核剂由生长晶体前部的结构过冷触发。所有模型仍然不确定从结构上描述异质成核的动力学。此外，晶体生长速率显着影响异质成核事件，因为生长的晶体会释放溶质原子和热量。因此，近似固液界面处的溶质通量非常重要。在实践方面，如何预测等轴区的晶粒尺寸是令人感兴趣的。当等轴晶粒的体积达到临界值时，生长柱状前的等轴晶粒阻碍生长，最终形成等轴区，即柱状到等轴转变（CET）。相信在等轴区的晶粒尺寸可以使用CET发生时剩余熔体的冷却速率来预测。此外，