Validation with Exact Solution

To test the multiphase PSG model by comparison with the exact population-balance model equations, a test problem was made by increasing the diffusion coefficient of phase B via setting the ratio DB/DA=1.1 and keeping other parameters unchanged as in test problem of section 4.3.

The results of the PSG method and exact solution for this problem of multiphase precipitation are shown in Figure 4.6 and 4.7. As shown in Figure 4.6, the total volume of each precipitate predicted with the PSG method (i.e. the dimensionless number density of pseudomolecules) matches the exact population-balance solution and furthermore conserves mass by matching the input function (after scaling with n1,eq to make dimensionless), which increases with time according to Eqs. (4.14)-(4.15). In addition, a good agreement for the total particle number density, N , is also found, which decreases with time once precipitation starts. _T^*

The histories of number density and molar fraction from both methods are compared in Figure 4.7. The particle size distribution evolves in a similar manner to that of single precipitates in Figure 3.10, and the number density of pseudomolecules of each precipitate phase evolves with time with similar values for both methods.

Because the logarithmic distribution of molar fraction with each size group, defined in Eq.

(4.26), was causing instability, the molar fractions were simply fixed to have the center value throughout the size group. This solved the stability, but likely caused the accuracy problem with matching the molar fractions in Fig. 4.7(b). The trends of smaller fractions at small-size and large-size and larger fractions for intermediate size of phase A are observed in both methods, but PSG method underestimates the values at peaks and bottoms, which is caused by a bad choice of

“border molar fractions” here. A good agreement is expected to be available from implicit scheme to avoid stability, and better estimations of border molar fractions In future.

133 4.6 Solution Details

In order to avoid stability problems due to dissociation exceeding diffusion growth, and to improve the time-efficiency of the calculation, the implicit Euler schemes are always adopted here to integrate these differentiate equations through time. Because mass conservation has been validated with test problems, the number density of single pseudomolecules of each precipitate phase is then computed with Eq. (4.28), instead of the more time-consuming Eq. (4.20) used previously:

The ordinary differential equations (4.18) are discretized using implicit backward Euler scheme, which gives the similar results with Eq. (3.109) for each precipitate phase. Starting from known values at time step i, Gauss-Seidel method is used to calculate the number densities of a certain precipitate phase at time step i+1 from the smallest size group to largest size group.

The same calculation will be done after moving to the next precipitate phase, and continues until all phases are calculated. The positive number density and molar fractions are always predicted now.

The suggested multiphase PSG model is most suitable for mutually-soluble precipitates.

All precipitates can be divided into several groups of completely mutually-exclusive precipitates.

The precipitation of different groups can be modeled as a mutually-exclusive extreme case by Eqs. (4.11)-(4.12). Inside each group, the precipitates are mutually soluble, which can be modeled by the suggested new multiphase PSG model. Thus the models described in this chapter can give a simplified approach to estimate the complete multiphase precipitation behaviors in alloys.

Whether the precipitates are mutually exclusive and soluble can be determined by difference of the crystal structures and lattice parameters of precipitate phases as in chapter 2.

The underlying physics of this criterion is to compare the interface energies between matrix and difference precipitates, and between precipitates of different phases. If the interface energies between precipitates are much larger, it will cause the different precipitate phases to occur separately and goes to mutually-exclusive extreme. On the other hand, the precipitates of different phases tend to nucleate and grow on each other, if the interface energies between precipitates are much smaller. This causes heterogeneous precipitation, which is the

mutually-134

soluble extreme. In fact, a real precipitation happens between these two extremes, and each precipitate has a certain potential to attract or repel other precipitates. A true simulation of this requires arranging a size distribution of each precipitate phase, which can interact with precipitates of all phases.

135 4.7 Tables and Figures

(a). Particle size distributions

(b). Number densities of single pseudomolecules

Figure 4.1: Comparison for the mutually-exclusive precipitates at different time by multiphase and single-phase precipitation models

136

(a). Particle size distributions

(b). Number densities of single pseudomolecules

Figure 4.2: Comparison for the mutually-soluble precipitates at different time by multiphase and single-phase precipitation models

137

Figure 4.3: Comparison of total particle size distributions for the mutually-exclusive and mutually-soluble precipitates at different time

138

1 10 100 1000 10000

0.4 0.5 0.6 0.7 0.8

D_B/D_A=0.98

D_B/D_A=0.95 D_B/D_A=1 DB/D

A=1.02

Molar fraction of precipitate phase A

Containing number of pseudomolecules, i D_B/D_A=1.05

Figure 4.4: Influence of changing ratios of diffusion coefficients on molar fractions of phase A for mutually-soluble precipitates at t^*=100

139

(a). Global picture

(b). Zoom-in picture

Figure 4.5: Influence of changing ratios of interface energies on molar fractions of phase A for mutually-soluble precipitates at t^*=100

1 10 100 1000 10000

Molar fraction of precipitate phase A

Containing number of pseudomolecules, i

Molar fraction of precipitate phase A

Containing number of pseudomolecules, i

140

0.01 0.1 1 10 100

0 1 2 3 4 5 6

R_V=2

Dimensionless number density

Dimensionless time Exact solution

N^*_M^A N^*_M^B N^*_T PSG method

N^*_M^A N^*_M^B N^*

T

Figure 4.6: Comparison of multiphase diffusion curves calculated by PSG method with exact solution for RV=2

141

Molar fraction of precipitate phase A

Number of size group, j t^*=50

(b). Molar fraction of precipitate phase A

Figure 4.7: Comparison of multiphase diffusion curves of each size group calculated by PSG method with exact solution for RV=2

142 CHAPTER 5