Modelingshort-range ordering(SRO) in solutions

Arthur D. Pelton and Youn-Bae Kang

Centre de Recherche en Calcul Thermochimique,Département de Génie Chimique,

École PolytechniqueP.O. Box 6079, Station "Downtown"

Montréal, Québec H3C 3A7Canada

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.

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Binary solution A-B

Bragg-Williams Model(no short-range ordering)

ln lnconfigurationalA A B B

A B BW

iiBW A B

EA B BW

S R X X X X

H X X

L X X

S X X

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.

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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.

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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.

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Associate ModelA + B = AB ; AS

AB “associates” and unassociated A and B are randomly distributed over the lattice sites.Per mole of solution:

,

ln ln ln

exp

A A AB

B B AB

AB

A B

configA A B B AB AB

A A A B AB

AB AS

AB A B AS

X n n

X n n

n

n n A B

S R n X n X n X

X n n n n

H X

K X X X RT

where : moles of associates

moles of unassociated and

where :

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

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

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Quasichemical Model (pair approximation)A and B distributed non-randomly on lattice sites

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

ZXA = 2 nAA + nAB

ZXB = 2 nBB + nAB

Z = coordination numbernij = moles of pairs

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

The pairs are distributed randomly over “pair sites”

2 2ln ln ln 2

ln ln

configAA AA A BB BB B AB AB A B

A A B B

S R X X X X X X X X X X

R X X X X

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

2

2

" " 4exp

AB QM

AB AA BB QM

H X

K X X X RT

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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.

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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.

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

Combining the quasichemical and Bragg-Williams models

2" " 4expAB AA BB QMK X X X RT

Sconfig as for quasichemical model

2QM AB BW A BH X X X

Term for nearest-neighbor interactions

Term for remaining lattice interactions

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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.

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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.

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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.

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

This also results in H and Sconfig functions with minima which are less sharp.

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

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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.

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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.

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Displacing the composition of maximum short-range ordering

Associate Model:– Let associates be “Al2Ca”– Problem arises that partial

no longer obeys Raoult’s Law as XCa 1.

Quasichemical Model:

Let ZCa = 2 ZAl

ZAXA = 2 nAA + nAB

ZBXB = 2 nBB + nAB

Raoult’s Law is obeyed as XCa 1.

configCaS

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Prediction of ternary properties from binary parameters

Example: Al-Sc-MgAl-Sc binary liquids exhibit strong SRO

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

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

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Bragg-Williams Model

0 0

0

A B B C C ABW A B BW B C BW C A

BW A B BW B C

BW C A

H X X X X X X

If while

positive deviations result along the AB-C join.

The Bragg-Williams model overestimates these deviations because it neglects SRO.

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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].

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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.

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Quasichemical Model

AB BC CAQM A B QM B C QM C AH X X X

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.

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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.

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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.

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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.)

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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).

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Quasichemical Model

With Z = 2 and QM > 0, positive

deviations are predicted, but

immiscibility never results.

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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.

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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.

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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).

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

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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.)