3.1 Conventional solidstate kinetic reaction model
In solidphase dynamics, a rational dynamic model was adopted to describe the reaction mechanism, which can be also intuitively reflected from the corresponding dynamics equation [28][29]. The common equations of dynamic mechanism and model of solidstate reaction were listed in Table 1.
Table 1
Reaction mechanism and dynamics equations for Different Reaction Models [22, 30, 31]
Model

Dynamics equation

Reaction mechanism

Model type

Avrami─Erofeyev (A2)

[− ln(1 − α)]1/2 =kt

Nucleation and growth


Avrami─Erofeyev (A3)

[− ln(1 − α)]1/3=kt


Avrami─Erofeyev (A4)

[− ln(1 − α)]1/4=kt


2D diffusion (D2)

(1− α) ln(1 − α) + α = kt

Diffusion Mechanism

Cylindrical model

3D diffusion─Jander (D3)

[1 − (1 − α)1/3]2=kt

Spherical model

Ginstling─Brounshtein (D4)

[1−(2/3)α]−(1− α)2/3=kt

Spherical model

Firstorder (F1)

ln(1 α)=kt



Zeroorder (R1)

α = kt

Interfacial Chemical Reaction Mechanism

Reactionorder models

Contracting area (R2)

1 − (1 − α)1/2=kt


Contracting volume (R3)

1(1 − α)1/3=kt


k is the slope of the solidstate kinetic equation, t is the reaction time, and α is conversion rate. 
3.2 Determination of solidstate kinetics conversion rate
Figure 1 showed the XRD patterns of specimens mixed with ZnO at 1500, 1510, 1520, 1530, 1540, 1550, and 1560°C for 2, 4, 6, and 8 h respectively. As can be seen from the Fig. 1, the intensity of diffraction peak of (C,B)8A6P at 15001530°C was very low, and had changed a little with the increase of holding time. And the diffraction peak of CA was obvious, suggesting that the transformation from CA to (C,B)8A6P was difficult within the range of calcination temperatures. However, as the calcination temperature increased, the peak intensity of (C,B)8A6P increased significantly while the peak intensity of CA decreased dramatically with the calcination temperature ranging from 1540 to 1560°C, which indicated the calcination temperature range of 15401560°C was more beneficial to the formation of (C,B)8A6P.
Standard curves should be established for quantitative analysis using XRD internal standard method. In this experiment, specimen calcined at 1560 °C for 8 h was regarded as standard specimen with 100% conversion rate. Then, 20%, 40%, 60% and 80% standard specimen was mixed with 40% ZnO for XRD analysis, respectively. The results were shown in Fig. 2. The diffraction peak of (C,B)8A6P mixed with ZnO at 23.62° was compared with that of ZnO at 36.3°, then the standard curves was established according to the ratio of the two diffraction peaks intensity for the subsequent quantitative analysis. The results were shown in Fig. 3.
The conversion rate of specimens at elevated temperatures ranged from1500 to 1560°C with the holding time of 2, 4, 6 and 8 h was given in Table 2 and Fig. 4. By comparing the characteristic peak area of XRD patterns, the conversion rate of specimens at different temperature for different holding time was determined. It can be seen that the conversion rate of (C,B)8A6P changed a little with an increasing holding time, and it was not exceed 0.5 with the holding time of 8 h and the calcination temperature ranging from 1500 to 1530°C. Furthermore, as a transition calcination temperature region of 1530 to 1540°C, the conversion rate increased sharply at elevated temperatures ranged from 1530 to 1540°C. In addition, the conversion rate was more than 0.8 with the holding time of only 2 h when the temperature ranged from 1540 to 1560°C, and it reached 0.995 with the holding time of 8 h, which indicated that higher calcination temperature as well as holding time had a significant effect on the formation of (C,B)8A6P mineral.
Table 2
Conversion rate of specimens at 15001560°C for different heat preservation times

1500°C

1510°C

1520°C

1530°C

1540°C

1550°C

1560°C

2h

0.2

0.236

0.29

0.284

0.802

0.833

0.901

4h

0.236

0.255

0.36

0.372

0.843

0.875

0.95

6h

0.246

0.271

0.39

0.402

0.875

0.899

0.974

8h

0.27

0.297

0.43

0.455

0.904

0.931

0.995

3.3 Dynamic model fitting
The conversion rate α determined by internal standard method at different temperature for different heat preservation times was substituted into the solidstate reaction kinetic mechanism and model formula given in Table 1. Then, the relationship between conversion rate α and holding time T was obtained and the least square method principle was adopted for linear regression analysis. The line regression coefficient reflects the degree of linear correlation, so the determination coefficient more closed to 1 meant the better fitting results. The data of all temperatures was fitted in Origin and the results was shown in Fig. 5. The slopes of lines were the reaction rate constants.
The determination coefficient R2 of fitting lines was given in Fig. 6. It can be seen that the highest determination coefficient can be observed for model D2 (2D diffusion), D3 (Jander equation) and D4 (Ginstling equation) at 1500, 1510, 1520 and 1530°C, which indicated the regression model fitting results was optimal in this temperature range. Meanwhile, in terms of model F1, R1, R2 and R3, the determination coefficient in this temperature range was also higher. However, the low conversion rate indicated the difficulty in the formation of (C,B)8A6P mineral. The reaction may be controlled by multiple processes simultaneously, and the diffusion rate was lowest. The (C,B)8A6P mineral was formed on the surface of CATCP (interfacial chemical reaction), and then crystal nucleus was formed and grow into small grains (nucleation and growth) or reactant diffused through the product layer and penetrated into the unreacted region (diffusion process). In comparison, the reaction was mainly controlled by diffusion process and had the best fitting results with the Jander equation. The kinetic model of Jander equation [32] assumes that the reactants are the equaldiameter spheres. One reactant is surrounded by the other, and they are in full contact with the products. The reaction proceeds from the surface to the center and contacts in parallelplate mode, and the thickness of products layer is proportional to the time. This model is suitable for low conversion rates.
At 1540°C, the Ginstling equation (D4) had the highest determination coefficient (R2=0.99531) and the interfacial chemical reaction (R3) had a higher determination coefficient (R2=0.99423). The same case also applied to 1550 and 1560°C. The conversion rate of (C,B)8A6P mineral was higher, and it was more than 90% with the holding time of 8 h at 15401560°C. The reaction kinetics of the formation of (C,B)8A6P mineral at this stage should be also controlled by diffusion, but it had the best fitting results with the Ginstling equation, which assumed that the product layer was a spherical shell instead of a plane after the reaction began.
The corresponding reaction rate at each temperature was shown in Fig. 7. It can be observed that the reaction rate was no more than 9.4 × 10−7 at 15001530°C. Especially, the reaction rate was only 2.1 × 10−7 at 1500°C, and the reaction rate had changed a little with the increase of temperature in this range. Nevertheless, the reaction rate increased sharply at elevated temperatures ranged from 1530°C to 1540°C. Subsequently, as the calcination temperature increased, the reaction rate increased gradually and reached 5.5 × 10−6 at 1560°C, increasing by more than an order of magnitude compared with initial reaction rate. This was consistent with what the conversion rate performed at different temperature ranges.
3.4 Solidstate reaction activation energy
The relationship between reaction rate and thermodynamic temperature is usually described by the Arrhenius equation (The natural logarithmic form of Arrhenius equation is presented here):
lnk = lnA ─ Ea/RT
where k is the slope of the kinetic equation, A is the preexponential factor, Ea is apparent activation energy (KJ · mol−1), R is the ideal gas constant (8.314J / (mol · K)), and T is the thermodynamic temperature.
Due to the solidstate reaction mechanism between 15001530°C and 15401560°C was different, the activation energy in two temperature ranges should be calculated respectively. The reaction rate constant k and thermodynamic temperature T were listed in Table 3. It can be concluded from the slope of two lines (Fig. 8) that the activation energy of (C,B)8A6P mineral was 1310 and 324 KJ · mol−1 at 15001530°C and 15401560°C, respectively.
Table 3
Reaction rate k and thermodynamic temperature T at 15001560°C
t (°C)

T (K)

k (S−1)

1500

1773.15

2.1×10−7

1510

1783.15

4.8×10−7

1520

1793.15

8.0×10−7

1530

1803.15

9.4×10−7

1540

1813.15

4.4×10−6

1550

1823.15

4.8×10−6

1560

1833.15

5.5×10−6
