Liquid matte, produced during the pyrometallurgical production of copper, is often represented as a mixture of stoichiometric copper and iron sulfide: (Cu2S, FeS). This is a
simplistic approximation, because the sulfur content of mattes depends on the partial pressure of sulfur, so the better representation would be (Cu2±xS, Fe1±yS). Moreover, in the case of iron
sulfide, the liquid phase extends all the way towards metallic liquid iron without a miscibility gap. The miscibility gap between liquid copper and copper sulfide is not very large as well. Thus, it is even better to consider mattes as (Cu, Fe, S) mixture with a high tendency for copper and iron to surround themselves by sulfur. In other words, there is a strong first nearest neighbor (FNN) short-range ordering (SRO) between iron and copper from one side and sulfur from the other side. Furthermore, oxygen is readily dissolved in mattes, as it is shown by many studies [10-15], and metals exibit FNN SRO with oxygen as well as with sulfur. The model for matte must include oxygen: (Cu, Fe, O, S). Later in the present thesis it is shown that, with the introduction of oxygen, copper and iron start to exhibit at least two different oxidation states, which has an impact on the thermodynamic properties. The final choice of components for the liquid metal/matte phase is:
(
I II II III)
Cu , Cu , Fe , Fe , O, S (2.8) Species in brackets occupy the same and only sublattice. Numbers I, II and III are not charges, but rather “valences”, they are used to calculate the metal/nonmetal ratio with the maximum short range ordering (SRO). The components O and S have “valence” of II, thus SRO is maximum at the compositions CuI:O = 2:1, FeII:S = 1:1, FeIII:O = 2:3, etc. The model (2.8) represents liquid from metals to oxides, to sulfides, to nonmetals.
The Bragg-Williams formalism is not suitable to model the thermodynamic properties of such liquid. The strong first-nearest neighbor interaction between metals and nonmetals has a
dramatic effect on the entropy of mixing, so it no longer may be approximated by random mixing. The Modified Quasichemical formalism [16, 17] has been developed to address the problem of SRO.
For the liquid metal/matte phase, the Modified Quasichemical Model in pair approximation (MQMPA) was used. In MQMPA, the following pair exchange reaction is considered:
AB
(A-A) (B-B)+ =2(A-B); ∆g (2.9)
where (i-j) represents a first-nearest-neighbor pair and ∆gAB is the non-configurational Gibbs energy change for the formation of two moles of (A-B) pairs.
When the solution is formed from pure components A and B, some (A-A) and (B-B) pairs break to form (A-B) pairs, so the Gibbs energy of mixing is given by [16]:
config AB A A B B AB ( ) ΔS 2 mix n G G n g n g T g ∆ = − + = − + ∆ (2.10)
where g and A g are the molar Gibbs energies of pure liquid components; B nA, nB and nAB are the numbers of moles of A, B and (A-B) pairs and ΔSconfig is the configurational entropy of
mixing given by randomly distributing the (A-A), (B-B) and (A-B) pairs. Since no exact expression is known for the entropy of this distribution in three dimensions, an approximate equation is used which was shown [16] to be an exact expression for a one dimensional lattice (Ising model) and to correctly reduce to the random mixing point approximation (Bragg Williams) expression when ∆gAB is equal to zero.
AB
g
∆ can be expanded as an empirical polynomial in terms of the mole fractions of pairs [16]: AB AB AB AA BB ( ) 1 ij i j i j g g g X X + ≥ ∆ = ∆ +
∑
(2.11)or in terms of component fractions
AB AB AB A B ( ) 1 ij i j i j g g ω Y Y + ≥ ∆ = ∆ +
∑
(2.12)where ∆gAB and gABij are the parameters of the model which can be functions of temperature. In practice, only the parameters gABi0 and gAB0j need to be included. The definition for “coordination- equivalent” component fraction Y is given in Ref. [17]. m
The composition of maximum short-range ordering is determined by the ratio of the coordination numbers. Let ZA and ZB be the coordination numbers of A and B. These coordination numbers can vary with composition as follows [16]:
AA AB A A A AA AA AB AB AA AB 2 1 1 1 2 2 n n Z Z n n Z n n = + + + (2.13) BB AB B B B BB BB AB BA BB AB 2 1 1 1 2 2 n n Z Z n n Z n n = + + + (2.14) where A AA Z and A AB
Z are the values of ZA when all nearest neighbors of an A are As, and when all nearest neighbors of an A are Bs, respectively, and where ZBBB and ZBAB are defined similarly. For example, in order to set the composition of maximum short-range ordering at the molar ratio
A/ B =2
n n , one can set the ratio B A
BA/ AB =2
Z Z . Values of ZBAB and ZABA are unique to the A-B binary system, while the value of A
AA
Z is common to all systems containing A as a component. Even though the model is sensitive to the ratio of the coordination numbers, it is less sensitive to their absolute values. It was found by experience that selecting values of the coordination numbers which are smaller than actual values often yields better results. This is due to the inaccuracy introduced by an approximate equation for the configurational entropy of mixing which is only exact for a one dimensional lattice. The smaller coordination numbers partially compensate this inaccuracy in the model equations, being more consistent with a one dimensional lattice. Therefore, the “coordination numbers” in our model are essentially treated as model parameters which are used mainly to set a physically reasonable composition of maximum short-range ordering. In general, coordination numbers are not equal to valencies of species in (2.8).
The similar reasoning about the interpolation of binary parameters into a multicomponent system is true for quasichemical parameters as for Bragg-Williams parameters. The latter was given in Section 2.1. The application of geometric models for the Modified Quasichemical
formalism is discussed in details by Pelton and Chartrand [17]. The components of liquid solution (2.8) are divided into two groups: metals (CuI, CuII, FeII and FeIII) and nonmetals (O, S). By this means, in every ternary subsystem, either three components belong to the same group or one belongs to a different group. The “Kohler-like” extrapolation is applied in the first case; the “Toop-like” extrapolation is used in the second case, with the different component considered as an asymmetric one.