modeling short-range ordering (sro) in solutions arthur d. pelton and youn-bae kang centre de...
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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 PolytechniqueP.O. Box 6079, Station "Downtown"
Montréal, Québec H3C 3A7Canada
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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
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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.)