Evolution equations for phase separation and ordering in binary alloys
Evolution equations for phase separation and ordering in binary alloys
复制标题
DOI：
10.1007/bf02188691
复制
发表时间：
1994-08
影响因子：
1.6
通讯作者：
J. Cahn;A. Novick-Cohen
中科院分区：
文献类型：
--
作者：
J. Cahn;A. Novick-Cohen
我们探索两种唯象方法导致耦合Cahn-Hilliard和Cahn-Allen方程描述系统的动力学，可以同时进行一级相分离和有序无序转变，从相同的离散晶格自由能函数。在第一种方法中，这个离散的能量和系统的演化被假定为梯度流的准连续极限。在第二种方法中，一组离散的梯度流演化方程推导出的晶格动力学和准连续极限，然后采取。我们在体心立方Fe-Al二元合金的背景下证明，重要的是要选择适应平均浓度变化的变量，以及潜在的可能共存相的有序结构。只有这样，这两种方法才会导致大致相同的连续统描述。我们推测，一般来说，描述这类系统动力学所需的变量数等于N1 +N2−1，其中N1由描述可能平衡相的对称群所需的不可约表示的基的跨度的维数给出，N2是化学组分的数目。对于Fe-Al合金，这意味着一个守恒序参量和一个非守恒序参量的描述。通过低阶非线性项将Cahn-Hilliard方程耦合到Cahn-Allen方程，给出了所得的描述。两种唯象方法的大致等效性增加了所得演化方程有效性的可信度。类似的描述对于其中竞争相的结构更复杂的合金系统也应该是有效的。
We explore two phenomenological approaches leading to systems of coupled Cahn-Hilliard and Cahn-Allen equations for describing the dynamics of systems which can undergo first-order phase separation and order-disorder transitions simultaneously, starting from the same discrete lattice free energy function. In the first approach, a quasicontinuum limit is taken for this discrete energy and the evolution of the system is then assumed to be given by gradient flow. In the second approach, a discrete set of gradient flow evolution equations is derived for the lattice dynamics and a quasicontinuum limit is then taken. We demonstrate in the context of BCC Fe−Al binary alloys that it is important that variables be chosen that accommodate the variations in the average concentration as well as the underlying ordered structure of the possible coexistent phases. Only then will the two approaches lead to roughly the same continuum descriptions. We conjecture that in general the number of variables necessary to describe the dynamics of such systems is equal toN1+N2−1, whereN1is given by the dimension of the span of the bases of the irreducible representations needed to describe the symmetry groups of the possible equilibrium phases andN2is the number of chemical components.N1of these variables are nonconserved, and the remaining are conserved and represent the average concentrations. For the Fe−Al alloys this implies a description of one conserved order parameter and one nonconserved order parameter. The resultant description is given by a Cahn-Hilliard equation coupled to a Cahn-Allen equation via the lower-order nonlinear terms. The rough equivalence of the two phenomenological methods adds credibility to the validity of the resulting evolution equations. A similar description should also be valid for alloy systems in which the structure of the competing phases is more complicated.