Criterion for judging the homogeneous and heterogeneous nucleation
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RODUCTION
1⫽
DISPERSING bulk samples into grains of diameter 10
to 100 m, 18 kinds of metals (Ag, Ni, Cu, etc.) were undercooled to 0.18 Tm (where Tm is the melting temperature) by Turnbull and Cech.[1] On the basis of the experimental results, it was believed that the undercooling for homogeneous nucleation was 0.18 Tm. However, with the improvement of techniques for undercooling, the achieved degree of undercooling and the volume of liquid that can be highly undercooled have increased significantly.[2–12] At present, the maximum undercooling is obtained in gallium by using the method of emulsification. Perepezko has undercooled the emulsified grains of gallium with a diameter of 20 m to 0.58 Tm.[2] Encased in inorganic glasses, bulk samples can be highly undercooled. Iron- and nickel-base alloys weighing up to 4 pounds were readily undercooled by as much as 300 K.[6] Since a number of liquids have been undercooled over the believed degree for homogeneous nucleation, it is doubtful that the undercooling for homogeneous nucleation was 0.18 Tm. But until now, a method to judge whether a nucleation process is homogeneous has not been achieved. This article deals with how to judge that the nucleation form of a undercooled liquid is homogeneous or heterogeneous. II. THE CRITERION FOR JUDGING THE NUCLEATION FORM A. Criterion Derived from Nucleation of Liquid Suppose that the volume and cooling rate of a liquid are V and Rc , respectively; then, the nondimensional undercooling at which the first nucleus forms in the liquid, *, can be calculated by the following equation:
Z.Y. JIAN, Special Research Fellow, and W.Q. JIE, Professor, are with the State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China; Z.Y. JIAN is also Associate Professor, Department of Materials Science and Engineering, Xi’an Institute of Technology, Xi’an 710032, People’s Republic of China. Manuscript submitted April 4, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS A
兰
*
0
Iv ⭈ V ⭈
Tm ⭈ d Rc
[1]
where is the nondimensional undercooling—it is equal to the ratio of undercooling (⌬T ) to melting point (Tm); and Iv is the homogeneous nucleation frequency per unit volume— it is given by[14]
冋
册
⌬GA Iv ⫽ Av exp ⫺ k(Tm ⫺ ⌬T )
冋
册
[2]
␣ 3Tm2 exp ⫺ 2 k⌬Hv (Tm ⫺ ⌬T )⌬T 2
where Av is a constant of homogeneous nucleation frequency (Av ⫽ 1041⫾1 m⫺3 s⫺1), ⌬GA is the free energy of activation for transporting an atom across the liquid-solid interface, k is the Boltzmann’s constant, ␣ is a factor determined by the shape of the nucleus (for spherical nucleus, ␣ ⫽ 16 /3), is the interface energy between solid and liquid, and ⌬Hv , is the latent heat of fusion per unit volume. The integral term in Eq. [1] is related to the shaded area in Figure 1, but it cannot be worked out directly. In order to calculate the shaded area, a tangent line to the Iv curve at * in Figure 1 is drawn. Thus, the shaded area is divided into two parts. Obviously, the left shaded area is less
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