Modeling short-range ordering (SRO) in solutions Arthur D. Pelton and Youn-Bae Kang Centre de Recherche en Calcul Thermochimique, Département de Génie Chimique, École Polytechnique P.O. Box 6079, Station "Downtown" Montréal, Québec H3C 3A7 Canada

2 Enthalpy of mixing in liquid Al-Ca solutions. Experimental points at 680° and 765°C from [2]. Other points from [3]. Dashed line from the optimization of [4] using a Bragg-Williams model.

3 Binary solution A-B Bragg-Williams Model (no short-range ordering)

4 Enthalpy of mixing in liquid Al-Sc solutions at 1600°C. Experimental points from [5]. Thick line optimized [6] with the quasichemical model. Dashed line from the optimization of [7] using a BW model.

5 Partial enthalpies of mixing in liquid Al-Sc solutions at 1600°C. Experimental points from [5]. Thick line optimized [6] with the quasichemical model. Dashed line from the optimization of [7] using a BW model.

6 Calculated entropy of mixing in liquid Al-Sc solutions at 1600°C, from the quasichemical model for different sets of parameters and optimized [6] from experimental data.

7 Associate Model A + B = AB;  AS AB “associates” and unassociated A and B are randomly distributed over the lattice sites. Per mole of solution:

8 Enthalpy of mixing for a solution A-B at 1000°C calculated from the associate model with the constant values of  AS shown.

9 Configurational entropy of mixing for a solution A-B at 1000°C calculated from the associate model with the constant values of  AS shown.

10 Quasichemical Model (pair approximation) A and B distributed non-randomly on lattice sites (A-A) pair + (B-B) pair = 2(A-B) pair ;  QM Z X A = 2 n AA + n AB Z X B = 2 n BB + n AB Z = coordination number n ij = moles of pairs X ij = pair fraction = n ij /( n AA + n BB + n AB ) The pairs are distributed randomly over “pair sites” This expression for  S config is:  mathematically exact in one dimension (Z = 2)  approximate in three dimensions

11 Enthalpy of mixing for a solution A-B at 1000°C calculated from the quasichemical model with the constant values of  QM shown with Z = 2.

12 Configurational entropy of mixing for a solution A-B at 1000°C calculated from the quasichemical model with the constant values of  QM shown with Z = 2.

13 The quasichemical model with Z = 2 tends to give  H and  S config functions with minima which are too sharp. (The associate model also has this problem.) Combining the quasichemical and Bragg-Williams models  S config as for quasichemical model Term for nearest- neighbor interactions Term for remaining lattice interactions

14 Enthalpy of mixing in liquid Al-Sc solutions at 1600°C. Experimental points from [5]. Curves calculated from the quasichemical model for various ratios (  BW /  QM ) with Z = 2, and for various values of with Z = 0.

15 Enthalpy of mixing for a solution A-B at 1000°C calculated from the quasichemical model with the constant parameters  BW and  QM in the ratios shown.

16 Configurational entropy of mixing for a solution A-B at 1000°C calculated from the quasichemical model with the constant parameters  BW and  QM in the ratios shown.

17 The quasichemical model with Z > 2 (and  BW = 0) This also results in  H and  S config functions with minima which are less sharp. The drawback is that the entropy expression is now only approximate.

18 Enthalpy of mixing for a solution A-B at 1000°C calculated from the quasichemical model with various constant parameters  QM for different values of Z.

19 Configurational entropy mixing for a solution A-B at 1000°C calculated from the quasichemical model with various constant parameters  QM for different values of Z.

20 Displacing the composition of maximum short-range ordering Associate Model: –Let associates be “Al 2 Ca” –Problem arises that partial no longer obeys Raoult’s Law as X Ca  1. Quasichemical Model: Let Z Ca = 2 Z Al Z A X A = 2 n AA + n AB Z B X B = 2 n BB + n AB Raoult’s Law is obeyed as X Ca  1.

21 Prediction of ternary properties from binary parameters Example:Al-Sc-Mg Al-Sc binary liquids exhibit strong SRO Mg-Sc and Al-Mg binary liquids are less ordered

22 Optimized polythermal liquidus projection of Al-Sc-Mg system [18].

23 Bragg-Williams Model positive deviations result along the AB-C join. The Bragg-Williams model overestimates these deviations because it neglects SRO.

24 Al 2 Sc-Mg join in the Al-Mg-Sc phase diagram. Experimental liquidus points [19] compared to calculations from optimized binary parameters with various models [18].

25 Associate Model Taking SRO into account with the associate model makes things worse! Now the positive deviations along the AB-C join are not predicted at all. Along this join the model predicts a random mixture of AB associates and C atoms.

26 Quasichemical Model Correct predictions are obtained but these depend upon the choice of the ratio (  BW /  QM ) with Z = 2, or alternatively, upon the choice of Z if  BW = 0.

27 Miscibility gaps calculated for an A-B-C system at 1100°C from the quasichemical model when the B-C and C-A binary solutions are ideal and the A-B binary solution has a minimum enthalpy of -40 kJ mol -1 at the equimolar composition. Calculations for various ratios (  BW /  QM ) for the A-B solution with Z = 2. Tie-lines are aligned with the AB-C join.

28 Miscibility gaps calculated for an A-B-C system at 1100°C from the quasichemical model when the B-C and C-A binary solutions are ideal and the A-B binary solution has a minimum enthalpy of -40 kJ mol -1 at the equimolar composition. Calculations for various values of Z. Tie-lines are aligned with the AB-C join.

29 Binary Systems Short-range ordering with positive deviations from ideality (clustering) Bragg-Williams model with  BW > 0 gives miscibility gaps which often are too rounded. (Experimental gaps have flatter tops.)

30 Ga-Pb phase diagram showing miscibility gap. Experimental points from [14]. Curves calculated from the quasichemical model and the BW model for various sets of parameters as shown (kJ mol -1 ).

31 Quasichemical Model With Z = 2 and  QM > 0, positive deviations are predicted, but immiscibility never results.

32 Gibbs energy of mixing for a solution A-B at 1000°C calculated from the quasichemical model with Z = 2 with positive values of  QM.

33 With proper choice of a ratio (  BW /  QM ) with Z = 2, or alternatively, with the proper choice of Z (with  BW = 0), flattened miscibility gaps can be reproduced which are in good agreement with measurements.

34 Ga-Pb phase diagram showing miscibility gap. Experimental points from [14]. Curves calculated from the quasichemical model and the BW model for various sets of parameters as shown (kJ mol -1 ).

35 Enthalpy of mixing curves calculated at 700°C for the two quasichemical model equations shown compared with experimental points [15-17].

36 Miscibility gaps calculated for an A-B-C system at 1000°C from the quasichemical model when the B-C and C-A binary solutions are ideal and the A-B solution exhibits a binary miscibility gap. Calculations for various ratios (  BW(A-B) /  QM(A-B) ) with positive parameters  BW(A-B) and  QM(A-B) chosen in each case to give the same width of the gap in the A-B binary system. (Tie-lines are aligned with the A-B edge of the composition triangle.)