Complex dynamics of phase changes occur when the alloy solution's temperature suddenly drops below a critical value. The well-known Cahn-Hilliard model shows that a system of fourth-order parabolic partial equations controls this intricate process. However, the Cahn-Hilliard equation with more than four component phases in three dimensions has not been solved to our best knowledge. In this work, the negative chemical potential, namely the first variation of free energy, was convoluted with a sixth-order accurate Laplacian kernel and used as the gradient flow in a projected gradient method. Also, we calculated the Lagrangian multiplies of Gibbs n-simplex phase constraint in a nested loop. Numerical examples illustrate that the proposed method can reveal the nucleation, separation, and growth of grains for alloys with up to 16 component phases in three dimensions. When the number of component phases is large than four, we found small grains are usually dissolved and redeposited onto larger ones. However, the separated phases twist into a highly interconnected structure in the binary and ternary alloys.

当合金溶液的温度突然下降到临界值以下时，就会发生复杂的相变动力学。著名的Cahn-Hilliard模型表明，四阶抛物线偏方程组控制着这一复杂过程。但是，就我们所知，尚未解决在三个维度上具有四个以上成分相的Cahn-Hilliard方程。在这项工作中，负化学势，即自由能的第一个变化，被六阶精确的拉普拉斯算子进行了卷积，并用作投影梯度法中的梯度流。同样，我们在嵌套循环中计算了吉布斯n-单相相位约束的拉格朗日乘数。数值算例表明，该方法可以揭示形核，分离，在三个维度上具有多达16个组成相的合金的晶粒生长和生长。当组分相的数量大于四个时，我们发现小晶粒通常会溶解并重新沉积到较大的晶粒上。但是，分离的相在二元和三元合金中扭曲成高度互连的结构。