Abstract
According to the equilibrium state diagrams, when the melt is cooled to a certain temperature below liquidus, compositions of liquid and solid phases are uniquely determined by corresponding curves in the diagram. However, it does not happen in reality. For equilibrium (which the diagram describes), it is necessary that the melt is maintained indefinitely at each temperature, or thermal conductivity of liquid and solid phases, and the diffusion coefficients of their components, are infinitely large. We made an attempt to find out how these processes occur in reality. In this work, we consider the growth of individual crystal during cooling of a two-component melt. Mathematical model is constructed based on the following. 1. The melt area with volume corresponding to one grain, the periphery of which is cooled according to a certain law, is considered. 2. At the initial instant of time, a crystal nucleus of a certain minimum size is in the liquid. 3. At the surface of crystal, compositions of liquid and solid phases correspond to equilibrium state diagram at a given temperature on its surface. 4. Changes in temperature and composition in liquid and solid phases occur according to the laws of heat conduction and diffusion, respectively. As the melt gets cold and the crystal grows, the liquid phase is enriched in one component and depleted in another, the solid phase – on the contrary. Since the diffusion coefficients of the components in the solid phase are small, the composition of the crystal does not have time to completely equalize its cross section. The model proposed in the work allows us to study this phenomenon, to calculate for each cooling mode how the composition of the crystal will vary over its cross section. The calculations have shown that the temperature equalization occurs almost instantly, and composition of the liquid phase equalizes much slower. Equalization of the solid phase composition does not occur in the foreseeable time. The results of the work will help to improve technology of generation of alloys with an optimal structure.
Highlights
According to the equilibrium state diagrams, when the melt is cooled to a certain temperature below liquidus, compositions of liquid and solid phases are uniquely determined by corresponding curves in the diagram
For equilibrium, it is necessary that the melt is maintained indefinitely at each temperature, or thermal conductivity of liqu id and solid phases, and the diffusion coefficients of their components, are infinitely large
At the surface of crystal, compositions of liquid and solid phases correspond to equilibrium state diagram at a given temperature on its surface
Summary
Согласно равновесным диаграммам состояния при охлаждении расплава до температуры ниже температуры ликвидуса составы жидкой и твердой фаз однозначно определяются соответствующими кривыми на диаграмме. Необходимо, чтобы расплав выдерживался бесконечно долго при каждой температуре или коэффициенты теплопроводности жидкой и твердой фаз, а также коэффициенты диффузии их компонентов были бесконечно велики. Построена математическ ая модель, базирующаяся на следующих положениях: выделена область расплава с объемом, приходящимся на одно зерно, периферия которого охлаждается по определенному закону; в начальный момент времени в жидкости находится зародыш кристалла некоторого минимального размера; у поверхности кристалла составы жидкой и твердой фаз соответствуют диаграмме состояния для рассматриваемой температуры на его поверхности; изменение температуры и состава в жидкой и твердой фазах происходят по законам теплопроводности и диффузии соответственно. Коэффициенты диффузии компонентов в твердой фазе малы, поэтому не происходит полного выравнивания состава по его сечению. Ключевые слова: диаграмма состояния, ликвация, рост кристалла, фазовый переход, математическая модель, диффузия, теплопроводность
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