A modified regular-solution model for terminal solutions
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Darken's approach has never been extensively used. In systems where the M values are small, it has been possible to obtain a reasonable representation of the properties over the whole system with a single expression like Eq. [1] after adding a few more terms, for instance by use of the so-called Redlich-Kister polynomial /-.o + L,(XA - x . )
Gm = XA°GA + xs°GB + RT(XA lnXA + x ~ lnxB) + XAXBL [1] °GA and °GB are the Gibbs energy values for reference states of pure A and B, chosen as the same phase as the solution under consideration. This model is supposed to apply over the whole range of composition in a binary system. When examining many binary, metallic systems Darken I found that this model seldom applies. He discovered a more useful model which can be derived by adding a term to Eq. [1],
Gm= xA°GA + xs°GB + RT(XA In xA + x8 In xs) + XAXBL ~ + xB Ms
[2]
This model he applied successfully to A-rich terminal solutions. For the B-rich terminal solutions he applied a similar model obtained by adding a term XAMA. By standard methods we can derive expressions for the partial Gibbs energies. Equation [2] yields GA = °Ga 'I- RT In XA + X~LAB
[3]
G8 = °Gs + RT In xB + x ] L ~ + M~
[4]
The latter expression may be given in terms of x~, Gn = °GB + RT In xs + RT In 3/] + LAS(--2XB + X~)
[5] where RT In 3~] has been inserted instead ofL:~ + Ms. The quantity ~/] is here the activity coefficient at infinite dilution. Equation [5] is identical to Darken's Eq. [4c]. It contains two adjustable parameters, LAB and RT In ~/] (or MB). When applying these models to the two terminal solutions, Darken found that the transition region between the two models was often rather thin and he suggested a special model for that region, which, after fitting to the two terminal models, contains an additional, adjustable parameter. Altogether he thus needed five parameters for the description of a binary system. It should be emphasized that information on the activities of both components in one of the terminal solutions allows a check of the model for that region since LAB appears in both Eqs. [3] and [4]. When agreement is not obtained Darken suggested the addition of a cubic term. Formally, Eq. [2] can be obtained from the regular solution model, Eq. [1], by changing the reference state for B from °Gs to °GB + Ms. In Darken's words such a change can be regarded as due to the fact that "the solvent (major component) dominates the nature of the bonding." He furthermore said that "the detailed nature of the bonding is different for each of the pure components" and "as we traverse the entire composition range, the transition must be made somehow." MATS HILLERT is Professor of Physical Metallurgy, Royal Institute of Technology (K.T.H.), S-10044 Stockholm, Sweden. Manuscript submitted November 27, 1985. 1878--VOLUME 17A, OCTOBER 1986
+ L2(XA -- X . )
+ ---
[6]
instead of L. This method is not successful if the M values are large. Darken himself compared that case with an ideal gas system involving a chemical reaction by which a compound
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