Thermodynamic description for concentrated metallic solutions using interaction parameters
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INTRODUCTION AND DISCUSSION
THE metal industry has achieved great advances in previous decades in response to the production of highquality metals. Thermodynamic data such as equilibrium constants and interaction parameters effectively solidify metallurgical fundamentals through an accurate description of the thermodynamics of metallic solutions. This contribution of thermodynamic data to metallurgical fundamentals leads to permanently incremental improvements of metallurgical processes. The present study aims to accurately describe the thermodynamics of metallic solutions using the available thermodynamic parameters. The interaction-parameter formalism called “Wagner formalism” is used frequently to express the thermodynamic properties of solutions, particularly of metals, in terms of a MacLaurin series expansion for the logarithm of the activity coefficients of solutes.[1,2] Considering a nonreacting solution containing N components, the MacLaurin series expansion for ln ␥i , in terms of interaction parameters, is (subscript 1 represents the solvent) ln ␥i ⫽ ln ␥ 0i ⫹
N
兺
ijx j ⫹
j⫽2
⫹
N⫺1
N
兺
ij x 2j
j⫽2
[1]
N
兺 兺 jkix j xk ⫹ R(x3) j⫽2 k⫽j⫹1
The commonly used Wagner formalism is the simplest case of a truncated MacLaurin series, i.e., the first-order truncation, because experimental data are not accurate enough to assess interaction parameters beyond the second order. However, a truncated MacLaurin series expansion is thermodynamically inconsistent.[3,4,5] ln ␥i ⫽ ln ␥ ⫹ 0 i
N
兺
ij x j
[2]
j⫽2
Equation [2] is for a model where ln ␥i maintains a linear relation with the composition of the solutes. Numerous experiZHONGTING MA, Associate Professor, formerly with the Beijing University of Science and Technology, Beijing, P.R. China, is with the Institute of Iron and Steel Technology, Freiberg University of Mining and Technology, Freiberg, Germany. Manuscript submitted January 31, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS B
ments on metallic solutions demonstrate that this linear relation between ln ␥i and x j (i and j ⫽ 2, 3, . . . , N ) holds true over only a modest composition range. For concentrated solutions, the relation is nonlinear. Since higher-order interaction parameters are rarely available, the activity coefficients of solutes at high concentrations have to be calculated using the first-order Wagner formalism. Significant errors occur between experimental and evaluated data at higher concentrations.[4,5] Therefore, the Wagner formalism is limited to dilute solutions. The thermodynamics of a metallic solution showing a nonlinear relation with its composition has, so far, been not well understood. In view of this, many attempts have been made to develop different approaches to describe the thermodynamics of metallic solutions at both dilute and concentrated levels. The available approaches using interaction parameters are discussed as follows. A. Quadratic Formalism and Quasi-Chemical Model Darken[3] and Srikanth and Jacob[6] proposed a quadratic formalism that is
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