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Modeling short-range ordering (SRO) in solutions

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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.

Bragg-Williams Model

(no short-range ordering)

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.

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.

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.

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.

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.

Example:Al-Sc-Mg

Al-Sc binary liquids exhibit strong SRO

Mg-Sc and Al-Mg binary liquids are less ordered

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

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].

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.

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.

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).

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.)