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### Modelingshort-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

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.

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.

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.

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.

Associate Model

A + B = AB ; wAS

AB “associates” and unassociated A and B are randomly distributed over the lattice sites.

Per mole of solution:

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

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

Quasichemical Model (pair approximation)

A and B distributed non-randomly on lattice sites

(A-A)pair + (B-B)pair = 2(A-B)pair ; wQM

ZXA = 2 nAA + nAB

ZXB = 2 nBB + nAB

Z = coordination number

nij= moles of pairs

Xij= pair fraction = nij /(nAA + nBB + nAB)

The pairs are distributed randomly over “pair sites”

- This expression for DSconfig is:
- mathematically exact in one dimension (Z = 2)
- approximate in three dimensions

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

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

Term for nearest-neighbor interactions

Term for remaining lattice interactions

The quasichemical model with Z = 2 tends to give DH and DSconfig functions with minima which are too sharp. (The associate model also has this problem.)

Combining the quasichemical and Bragg-Williams models

DSconfig as for quasichemical model

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

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

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

The quasichemical model with Z > 2 (and wBW = 0)

This also results in DH and DSconfig functions with minima which are less sharp.

The drawback is that the entropy expression is now only approximate.

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

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

Displacing the composition of maximum short-range ordering

Associate Model:

- Let associates be “Al2Ca”
- Problem arises that partialno longer obeys Raoult’s Law as XCa1.

Quasichemical Model:

Let ZCa = 2 ZAl

ZAXA = 2 nAA + nAB

ZBXB = 2 nBB + nAB

Raoult’s Law is obeyed as XCa1.

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

Bragg-Williams Model

positive deviations result along the AB-C join.

The Bragg-Williams modeloverestimatesthese deviations because it neglects SRO.

Al2Sc-Mg join in the Al-Mg-Sc phase diagram. Experimental liquidus points [19] compared to calculations from optimized binary parameters with various models [18].

Associate Model

Taking SRO into account with the associate model makes thingsworse!

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.

Quasichemical Model

Correct predictions are obtained but these depend upon the choice of the ratio (wBW /wQM) with Z = 2, or alternatively, upon the choice of Z if wBW= 0.

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 (wBW /wQM) for the A-B solution with Z = 2. Tie-lines are aligned with the AB-C join.

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.

Binary Systems

Short-range ordering with positive deviations from ideality (clustering)

Bragg-Williams model with wBW > 0 gives miscibility gaps which often are too rounded. (Experimental gaps have flatter tops.)

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).

Quasichemical Model

With Z = 2 and wQM > 0, positive deviations are predicted, but immiscibility never results.

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

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

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).

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

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 (wBW(A-B)/wQM(A-B)) with positive parameters wBW(A-B)and wQM(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.)

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