Скачать презентацию Modeling short-range ordering SRO in solutions Arthur D Скачать презентацию Modeling short-range ordering SRO in solutions Arthur D

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Modeling short-range ordering (SRO) in solutions Arthur D. Pelton and Youn-Bae Kang Centre de 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 H 3 C 3 A 7 Canada

Enthalpy of mixing in liquid Al-Ca solutions. Experimental points at 680° and 765°C from 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. 2

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

Enthalpy of mixing in liquid Al-Sc solutions at 1600°C. Experimental points from [5]. Thick 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. 4

Partial enthalpies of mixing in liquid Al-Sc solutions at 1600°C. Experimental points from [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. 5

Calculated entropy of mixing in liquid Al-Sc solutions at 1600°C, from the quasichemical 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. 6

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

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

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

Quasichemical Model (pair approximation) A and B distributed non-randomly on lattice sites (A-A)pair + Quasichemical Model (pair approximation) A and B distributed non-randomly on lattice sites (A-A)pair + (B-B)pair = 2(A-B)pair ; w. QM ZXA = 2 n. AA + n. AB ZXB = 2 n. BB + n. AB Z = coordination number nij = moles of pairs Xij = pair fraction = nij /(n. AA + n. BB + n. AB) The pairs are distributed randomly over “pair sites” This expression for DSconfig is: § mathematically exact in one dimension (Z = 2) § approximate in three dimensions 10

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

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

The quasichemical model with Z = 2 tends to give DH and DSconfig functions 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 Term for nearestneighbor interactions Term for remaining lattice interactions DSconfig as for quasichemical model 13

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

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

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

The quasichemical model with Z > 2 (and w. BW = 0) This also The quasichemical model with Z > 2 (and w. BW = 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. 17

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

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

Displacing the composition of maximum short-range ordering Associate Model: – Let associates be “Al 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 XCa 1. Quasichemical Model: Let ZCa = 2 ZAl ZAXA = 2 n. AA + n. AB ZBXB = 2 n. BB + n. AB Raoult’s Law is obeyed as XCa 1. 20

Prediction of ternary properties from binary parameters Example: Al-Sc-Mg Al-Sc binary liquids exhibit strong 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 21

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

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

Al 2 Sc-Mg join in the Al-Mg-Sc phase diagram. Experimental liquidus points [19] compared 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]. 24

Associate Model Taking SRO into account with the associate model makes things worse! Now 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. 25

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

Miscibility gaps calculated for an A-B-C system at 1100°C from the quasichemical model when 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 k. J mol-1 at the equimolar composition. Calculations for various ratios (w. BW /w. QM) for the A-B solution with Z = 2. Tie-lines are aligned with the AB-C join. 27

Miscibility gaps calculated for an A-B-C system at 1100°C from the quasichemical model when 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 k. J mol-1 at the equimolar composition. Calculations for various values of Z. Tie-lines are aligned with the AB-C join. 28

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

Ga-Pb phase diagram showing miscibility gap. Experimental points from [14]. Curves calculated from the 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 (k. J mol-1). 30

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

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

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

Ga-Pb phase diagram showing miscibility gap. Experimental points from [14]. Curves calculated from the 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 (k. J mol-1). 34

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

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