Distribution coefficients in dilute binary and ternary alloys
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T o analyze heat and solute flows during the solidification of an alloy, the b a s i c parameters needed are t h e r m a l conductivities, heat capacities, heats of m e l t ing, diffusion constants, and distribution coefficients. Compared t o the o t h e r parameters, relatively little work has been done on the determination of distribution coefficients. A b r i e f outline of methods for t h e i r measurement follows. THE DETERMINATION OF DISTRIBUTION COEFFICIENTS BY INDIRECT MEANS 1) From Thermodynamic Data The distribution coefficient can be expressed in thermodynamic t e r m s by the following equation, g i v e n by Hillert.1 ko(i) = Z ( i , S ) / X ( i , L ) = E X P { I / R T [ ° G ( i , L ) - ° G ( i , S) + E G ( i , L ) - E - G ( i , S) ]}
[1]
A list of symbols is g i v e n at the end of the paper.
accurate determination of ko. For s e v e r a l systems t h e r e is information on activities in both the s o l i d and in the liquid, but the v a l u e s were obtained over different temperature r a n g e s . Some inaccuracy is then introduced in transposing the data to a c o m m o n temperature. Additional e r r o r s in the calculated v a l u e for ko a r i s e from the determination of ° G ( i , L ) - ° G ( i , S ) , w h e r e the latent heat of melting of the solute, and its specific heat in b o t h the solid and liquid states, must be ascertained. The measurement of activity in alloys, from e l e c trode potentials or vapor pressures, involves more complex operations than a r e needed for a d i r e c t determination of ko. Indeed, the thermodynamic properties of the solid at the phase boundary are often best obtained by combining thermodynamic data for the liquid phase with m e a s u r e d distribution coefficients. In g e n e r a l , the relationships between distribution coefficients and activities in the solid and liquid phases provide a useful c h e c k for self-consistency in m e a s ured values, but one cannot be used for the determination of the other.
Now E-G(i, S) = R T In y(i, S) and E - G ( i , L ) = R T In y(i, L) Eq. [1] can therefore be modified to
2) From Experimental Liquidus Slopes For a binary alloy, in the r e g i o n over which the solvent acts in an i d e a l m a n n e r , Eq. [1] can be rewritten as follows (component 1 is the m a j o r constituent), k0(1) = [1 - X ( 2 , S ) ] / [ 1 - X ( 2 , L)]
ko(i) = [ 7 ( i , L ) / 7 ( i , S)] E X P { 1 / R T [ ° G ( i , L ) - ° G ( i , S ) ]} In systems w h e r e the activity coefficient at the phase boundaries is known for one solute in b o t h the solid and liquid phases at the same temperature, ko(i) for that solute can be found. T h e r e are, unfortunately, few systems for which the necessary data are available, and even in t h e s e the values for activity coefficient are usually insufficiently precise t o allow K. G. DAVIS is ResearchScientist, Physical Metallurgy Division, Department ofEnergy, Mines, and Resources, Ottawa, Canada. Manuscript submittedMay 4, 1971. METALLURGICAL TRANSACTIONS
= E X P { 1 / R T [ ° G ( 1 , L ) - °G(1,
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