Comprehensive Determination of Kinetic Parameters in Solid-State

Article pubs.acs.org/crystal

Comprehensive Determination of Kinetic Parameters in Solid-State Phase Transitions: An Extended Jonhson−Mehl−Avrami− Kolomogorov Model with Analytical Solutions Yuepeng Pang,† Dongke Sun,∥ Qinfen Gu,‡ Kuo-Chih Chou,† Xunli Wang,†,# and Qian Li*,†,§ †

State Key Laboratory of Advanced Special Steels & Shanghai Key Laboratory of Advanced Ferrometallurgy & School of Materials Science and Engineering, Shanghai University, Shanghai 200072, China ‡ Australian Synchrotron, 800 Blackburn Road, Clayton 3168, Australia § Materials Genome Institute, Shanghai University, Shanghai 200072, China # Department of Physics & Materials Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China ∥ CompuTherm, LLC, Madison, Wisconsin 53719, United States ABSTRACT: An extended Jonhson−Mehl−Avrami−Kolomogorov model with analytical solutions is developed to achieve the comprehensive determination of kinetic parameters in solid-state phase transitions. A new subexpression is given for the case of adequately small activation energy of nucleation, which is ignored in previous works. Exact or approximate analytical solutions to the expressions in this model for isothermal and non-isothermal conditions are improved by introducing the Euler integral of the first kind, which are proved to be valid, accurate, and simple for possible parameters. On the basis of this model, five kinetic parameters (pre-exponential factor, activation energy of nucleation, activation energy of growth, nucleation index, and growth index) can be comprehensively determined by simultaneous analysis of isothermal and non-isothermal experimental data. Three practical examples, including two simulated examples and one experimental example, are then presented to illustrate and validate the comprehensive determination in detail, in which the kinetic parameters determined by this approach are confirmed to be highly plausible. This is of great importance for in-depth understanding the kinetic mechanisms and taking full advantage of the solid-state phase transitions.

1. INTRODUCTION

ξ = f (ξex )

The kinetics, i.e., the time dependent behaviors, of the solidstate phase transitions draw much attention because not only do they significantly affect the microstructures as well as the properties of the products (e.g., structural materials), but they are also crucial for the functional materials that involve the solid-state phase transitions (e.g., energy storage materials).1−6 Describing the kinetics by appropriate models and therein parameters is an efficient approach for in-depth understanding and to exploit the solid-state phase transitions.7 The Jonhson− Mehl−Avrami−Kolomogorov (JMAK) model considers the solid-state phase transitions as three simultaneous procedures: nucleation, growth, and impingement,8−13 and are widely and successfully applied in the crystallization of amorphous materials,14−16 preparation of nanomaterials,16−18 hydrogenation/dehydrogenation of hydrogen storage materials,19,20 charge/discharge of Li-ion batteries,21,22 and so on.23−25 In the principle of the JMAK model, nucleation, growth, and impingement are modeled as three modules, and the transformed fraction can be described as © 2016 American Chemical Society

(1)

with ξex =

∫0

t

I (T , τ )V (T , τ ∼ t ) d τ

(2)

where f(ξex) is the impingement module expressing the relationship between the real transformed fraction (ξ) and the extended transformed fraction (ξex), I(T, τ) is the nucleation module expressing the nucleation rate at τ time and T temperature, V(T, τ ∼ t) is the growth module expressing the volume of a τ-time formed nucleus at t time. The classical JMAK model describes nucleation as the site saturation or linear continuous mode, describes growth as the interface-controlled or diffusion-controlled mode, and describes impingement as the randomly dispersed nuclei isotropic growth mode (see Appendix A).8−12 Furthermore, Kempen et al.26 Received: February 4, 2016 Revised: March 1, 2016 Published: March 3, 2016 2404

DOI: 10.1021/acs.cgd.6b00187 Cryst. Growth Des. 2016, 16, 2404−2415

Crystal Growth & Design

Article

Table 1. Summary of the Equations in the Extended NI-JMAK Model

a

Proposed in the present work.

managed to give analytical solutions to the expressions in the classical JMAK model (see Appendix B), making the determination of kinetic parameters through experiments much more practical. However, the kinetic parameters determined according to this model is sometimes discrepant with the assumption of the model.5,13 To overcome this disadvantage, the JMAK model is improved by modifying the modules of nucleation, growth, and impingement. For example, the inhomogeneity,27−30 self-catalysis,31−33 and mixture33,34 are further considered in the nucleation module, and the anisotropic growth35,36 and nonrandom nuclei distribution13,37 are considered in the impingement module, which also introduces one or more new kinetic parameters into the classical JMAK model. Another well accepted strategy is to treat the JMAK model semiempirically according to a serious of simulations and experiments.38,39 Among them, the nucleation index-incorporated JMAK (NI-JMAK) model perfectly solves the problem in the classical JMAK model by taking the selfcatalysis of the nucleation into consideration and introducing one new kinetic parameter: the nucleation index. Liu et al.33 describe the index-incorporated nucleation mode as I (T , τ ) =

d⎡ ⎢ dτ ⎣

∫0

τ

a ⎛ −ΔEn ⎞ ⎤ ⎟ dη ⎥ I0 exp⎜ ⎝ RT ⎠ ⎦

Equations A5 and 4 are the original expressions of the NIJMAK model for the site saturation nucleation mode and the index-incorporated nucleation mode, respectively. Five independent kinetic parameters, i.e., the pre-exponential factor (N0G0d/m or aI0aG0a + d/m), activation energy of nucleation (ΔEn), activation energy of growth (ΔEg), nucleation index (a), and growth index (d/m), are involved in this model. When the nucleation index (a) equals to 1, eq 4 reduces to eq A6, which indicates that the NI-JMAK model is compatible with the classical JMAK model. The introduction of the nucleation index greatly enlarges the application range of the model, which, however, also makes the determination of kinetic parameters much more difficult due to the complicated double integrals involved. Following a similar approach of Kempen et al.,26 Liu et al.33 successfully gave the analytical solutions to the expressions in the NI-JMAK model (exact solution for isothermal condition and approximate solution for non-isothermal condition, see Appendix C). Unfortunately, some certain cases still exist (e.g., the activation energy of nucleation (ΔEn) is adequately small and the growth index (d/m) equals to 0.5 and 1.5) where this approach is not adequate to be applied because invalid or unacceptably inaccurate results may be yielded (see Appendix D). As a result, not all the kinetic parameters can be determined simultaneously and precisely using these solutions. In other words, the comprehensive determination of the kinetic parameters cannot be achieved on the basis of the NI-JMAK model yet. To our knowledge, most of the published works applying the NI-JMAK model are focused only on the so-called Avrami exponent (the sum of the nucleation index and growth index, a + d/m) and the apparent activation energy (the weighted sum of the activation energy of nucleation and growth, (aΔEn + d/mΔEg)/(a + d/m)) obtained from isothermal kinetic analysis and the individual values of the five kinetic parameters which contain more mechanistic

(3)

and after substituting eqs 3, A3, and A4 into eqs 1 and 2, the original expression in this model is ⎧ ⎪ ξ = 1 − exp⎨− ⎪ ⎩ ⎡ ⎢G0 ⎢⎣

∫τ

t

∫0

t

d⎡ ⎢ dτ ⎣

∫0

τ

a ⎛ −ΔEn ⎞ ⎤ ⎟ dη ⎥ I0 exp⎜ ⎝ RT ⎠ ⎦

⎛ −ΔEg ⎞ ⎤ exp⎜ ⎟ dη ⎥ ⎝ RT ⎠ ⎥⎦

d/ m

⎫ ⎪ dτ ⎬ ⎪ ⎭

(4) 2405



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information have to be ignored.14−25,40,41 Moreover, the nonisothermal kinetic data are usually analyzed by some model-free methods (e.g., Kissinger−Akahira−Sunose (KAS) method42,43) instead of the NI-JMAK model, and only the apparent activation energy can be obtained.44−48 These significantly prevent the JMAK model from wide applications and hinder the in-depth understanding of the kinetic mechanisms of the solid-state phase transitions. Therefore, it is of great importance to achieve the comprehensive determination of kinetic parameters by developing an extended NI-JMAK model with valid, accurate, and simple analytical solutions. In this work, the NI-JMAK model is extended and its applicability is broadened to a much wider context. The analytical solutions to the expressions for isothermal and nonisothermal conditions in this model are developed, and their validity, accuracy, and simplicity are demonstrated for possible parameters, including adequately small activation energy of nucleation (ΔEn) and certain values of growth index (d/m = 0.5, 1.5). On the basis of this model, the comprehensive determination of kinetic parameters is achieved by simultaneous analysis of isothermal and non-isothermal kinetic behaviors, and three practical examples are given to illustrate and validate the process in detail.

when d/m is not an integer. The following equation shows the integral in eq D2, ID2 =

∫0

t

⎡ aI0aτ a − 1⎢G0 ⎢⎣

∫τ

(7)

∫0

1

z x − 1(1 − z) y − 1 dz

(8)

Equation 7 can be transformed into ID2 = t a + d / m

∫0

1 ⎛ τ ⎞a − 1⎛ ⎜

⎟

⎝t ⎠

⎜

⎝

1−

τ ⎟⎞ t⎠

d/m

⎛τ⎞ d⎜ ⎟ ⎝t ⎠

⎛ d ⎞ = beta⎜a , + 1⎟t a + d / m ⎝ m ⎠

(9)

where the domains of both a and d/m + 1 are positive real numbers. Then, eq D2 becomes ⎡ ⎛ d ⎞ ξ = 1 − exp⎢⎢ −aI0 aG0 d / m beta⎜a , + 1⎟ ⎝ m ⎠ ⎣ ⎛ −aΔE − n exp⎜ ⎜ RT ⎝

d ΔEg m

⎤ ⎞ ⎟t a + d / m ⎥ ⎥ ⎟ ⎠ ⎦

(10)

This is an analytical solution of eq D2 without any approximation. More importantly, it is valid not only for d/m of 1, 2, and 3, but also for d/m of 0.5 and 1.5. So eq 10 can perfectly solve the essential problem in analytical solutions to the isothermal expressions. 2.3. Analytical Approximate Solutions for NonIsothermal Condition. The analytical approximate solution (eq B3) to the expression with the site saturation nucleation mode for non-isothermal condition (eq A5) has been proved to be accurate enough.26 Thus the analytical approximate solutions to the expressions with the index-incorporated nucleation mode for non-isothermal condition (eqs D4 and 6) are developed in this work, respectively. For eq D4, how to find an accurate and simple approximation of the following double integral ID4 is the key issue,

(5)

t

τ a − 1(t − τ )d / m dτ

beta(x , y) =

After substituting eqs 5, A3, and A4 into eqs 1 and 2, the following expression can be obtained to describe the nonisothermal kinetic behavior with ΔEn/RT ≈ 0, ⎧ ⎪ ξ = 1 − exp⎨ ⎪ ⎩

t

Inspired by the Euler integral of the first kind49,50

2. EXTENDED NI-JMAK MODEL WITH ANALYTICAL SOLUTIONS 2.1. A New Subexpression. When the activation energy of nucleation (ΔEn) is adequately small (the definition of “adequately small” is presented in Section 2.5), ΔEn/RT can be regarded as zero without significant error (ΔEn/RT ≈ 0). So the following expression of nucleation is proposed for the case of non-isothermal condition and adequately small ΔE n according to eq 3, I(T , τ ) = aI0 aτ a − 1

∫0

d/m ⎫ ⎛ −ΔEg ⎞ ⎤ ⎪ exp⎜ ⎟ dη ⎥ dτ ⎬ ⎪ ⎝ RT ⎠ ⎥⎦ ⎭

(6)

As summarized in Table 1, the extended NI-JMAK model contains two original expressions (eqs A5 and 4) and five subexpressions for different cases (eqs A5, D2, D4, and 6). In this table, kinetic behaviors for the isothermal condition are described by two subexpressions (eqs A5 and D2), which correspond to the site saturation nucleation mode and indexincorporated nucleation mode, respectively. The non-isothermal kinetic behaviors are described by three subexpressions. Equation A5 corresponds to the site saturation nucleation mode; eqs D4 and 6 correspond to the index-incorporated nucleation mode with ΔEn/RT ≫ 0 and ΔEn/RT ≈ 0, respectively. 2.2. Analytical Exact Solutions for Isothermal Condition. Because the expression with the site saturation nucleation mode for isothermal condition (eq A5) has been well solved to eq B1,26 the remaining issue here is that how to solve the expression with the index-incorporated nucleation mode for isothermal condition (eq D2) analytically, especially

ID4 =

∫0

t

⎡ −aΔE ⎤ n (βτ + T0)2(a − 1) exp⎢ ⎥ ⎣ R(βτ + T0) ⎦

⎧ ⎨ ⎩

⎪

∫τ

t

⎪

d/m ⎡ −ΔEg ⎤ ⎫ ⎥ dη ⎬ dτ exp⎢ ⎣ R(βη + T )0 ⎦ ⎭ ⎪

⎪

(11)

Following the principle of the Euler integral of the first kind, let F (x ) =

∫0

x

⎡ −ΔEg ⎤ ⎥ dη exp⎢ ⎣ R(βη + T0) ⎦

(12)

Then, ID4 can be transformed to 2406



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⎡ −aΔE ⎤ n (βτ + T0)2(a − 1) exp⎢ ⎥ ⎣ R(βτ + T0) ⎦ [F(t ) − F(τ )]d / m dτ

∫0

ID4 =

∫0

=

t

⎡

t

aΔEn / ΔEg − 1

F (τ )

(βτ + T0)

⎢⎣

1 − aΔEn / ΔEg

T

⎡ R(βt + T )2 ⎤ 0 ⎥ = (βt + T0)2(a − 1)⎢ ΔEg ⎢⎣ ⎥⎦

1 − aΔEn / ΔEg

⎡

ID4 = (βt + T0)

∫0

t

⎢⎣

(14)

+ T0)2 ⎤ ⎥ ΔEg ⎥⎦

1 − aΔEn / ΔEg

F(τ )aΔEn / ΔEg − 1[F(t ) − F(τ )]d / m dF(τ ) ⎡

+ T0)2 ⎤ ⎥ =(βt + T0) ΔEg ⎢⎣ ⎥⎦ ⎛ aΔE d ⎞ n beta⎜⎜ , + 1⎟⎟F(t )aΔEn / ΔEg + d / m + 1 Δ E m ⎝ ⎠ g 2(a − 1) ⎢ R(βt

1 − aΔEn / ΔEg

⎤ ⎞⎛ 2 ⎞ a + d / m ⎥ ⎟⎜ RT ⎟ ⎥ ⎟⎝ β ⎠ ⎥ ⎠ ⎥⎦

∫0

t

⎧ τ a − 1⎨ ⎩ ⎪

∫τ

⎪

t

d/m ⎡ −ΔEg ⎤ ⎫ ⎥ dη ⎬ dτ exp⎢ ⎣ R(βη + T0) ⎦ ⎭

(20)

x is an integer

(21)

Both beta and gamma functions, especially gamma function, are easy to calculate. The values of beta and gamma functions can be worked out by most of the computing software, and they can also be found in the mathematical handbooks. Figure 1 shows the three-dimensional (3D) plots of beta(x, y) in the range of 0.2 < x < 5 and 0.2 < y < 5. The value of beta(x, y) monotonically decreases with increasing x and/or y, from infinity with x and/or y = 0 to zero with x and/or y = ∞. In addition, the beta function is plane symmetry with respect to the x = y plane. In this work, a and aΔEn/ΔEg can be all positive real numbers and d/m + 1 can only be five discrete values. Thus, the beta function reduces to five lines (marked by green balls) in this work. 2.5. Verification of the Solutions. Equation 10 is the newly developed analytical exact solution to the expression with the index-incorporated nucleation mode for isothermal condition in this work, and its validity is investigated below. The numerical calculations (NC, by MathCAD 14, PTC Software, Needham, MA, USA, TOL = 10−13) of the original expression (eq 4) are performed with four sets of parameters, including different values of T, a, d/m, ΔEn/R, and ΔEg/R, as shown in Figure 2 (symbols). These parameters are then substituted into eq 10, and the results are also shown in Figure 2 (lines). As expected, it is observed that the lines perfectly

(16)

⎪

⎪

(19)

gamma(x)gamma(y) gamma(x + y)

gamma(x) ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ (x − 1)!

Apparently, eq 16 is simpler and more compact than eq C2. In this solution, two mathematical approximations are made. For the zero-order approximation of eq 14, the accuracy is high enough when ΔEn/RT ≫ 0. This is satisfied because ΔEn/RT ≫ 0 is one of the assumptions for eq D4. For the first order approximation of the temperature integral (eq D5), ΔEg/RT ≫ 0 is needed to obtain a high accuracy, and this is always satisfied due to the difficulty for diffusion and/or interfacial process in the solid-state phase transitions.3 As for eq 6, finding an approximation of the following double integral is a crucial issue. I6 =

(T − T0) ⎥ ⎦

Gamma function is an extension of the factorial function. If x is a positive integer,

⎡ ⎛ aΔE d ⎞ beta⎜ ΔE n , m + 1⎟ ⎢ ⎝ ⎠ g ξ = 1 − exp⎢ −aI0 aG0d / m d/m+1 a−1 ⎢ ΔEg ΔEn ⎢⎣ d ΔEg m

a⎥

2d / m

beta(x , y) =

(15)

After applying the first order approximation of the temperature integral (eq D5)51 for F(t), we can finally get

⎛ −aΔE − n exp⎜ ⎜ RT ⎝

⎡ −ΔEg ⎤ ⎥ dη exp⎢ ⎣ R(βη + T0) ⎦

This approximation is accurate when T in the interest region is larger than T0 and ΔEg/RT ≫ 0, and these are always satisfied for most cases. Equations 16 and 19 are the analytical approximate solutions to the expressions for non-isothermal condition with ΔEn/RT ≫ 0 and ΔEn/RT ≈ 0, respectively. They are simple and compact, and are also proven to be accurate in Section 2.5. 2.4. Beta Function. Compared to the previous reports, the beta function beta(x, y), with x standing for a or aΔEn/ΔEg and y standing for d/m + 1, is the major difference. The properties of the beta function beta(x, y) in this model is briefly discussed below. Beta function can be calculated by gamma function through the following equation,

ID4 can be obtained as 2(a − 1) ⎢ R(βt

t

⎤

+ T0)2 ⎤ ⎥ ΔEg ⎥⎦

2(a − 1) ⎢ R(βτ

∫0

⎡ ⎛ − d ΔE ⎞ Rd / m ⎜ m g⎟ ξ = 1 − exp⎢⎢ −aI0 aG0d / m exp d/m a+d/m ⎜ RT ⎟ aΔEg β ⎠ ⎝ ⎣

(13)

After using a zero-order approximation in the integral as following ⎡

⎡ −ΔEg ⎤ ⎥ dη = exp⎢ ⎣ R(βη + T0) ⎦

and the first order approximation of the temperature integral (eq D5), I6 can be solved to an analytical approximate solution and the final result is

1 − aΔEn / ΔEg

⎢⎣ [F(t ) − F(τ )]d / m dF(τ )

t

(18)

+ T0)2 ⎤ ⎥ ΔEg ⎥⎦

2(a − 1) ⎢ R(βτ

(βτ + T0)

∫τ

(17)

By applying a zero-order approximation as following 2407



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Figure 3. Demarcation of the analytical approximate solutions for nonisothermal condition with different ΔEn. Symbols: eq 4-NC, lines: eq 16 and dashes: eq 19.

Figure 1. 3D plots of the beta function beta(x, y) with x standing for a or aΔEn/ΔEg and y standing for d/m + 1. The green balls indicates the lines when d/m + 1 equals five discrete values.

unacceptably inaccurate (ΔTmax/T > 10%) when ΔEn/R = 100 K. Its fitness to eq 4-NC becomes much better (ΔTmax/T < 0.5%) when ΔEn/R increases from 100 to 3000 K. As for eq 19 (dashes), it remains unchanged with different ΔEn/R values and perfectly matches with eq 4-NC (ΔTmax/T < 0.3%) when ΔEn/R = 0. Another finding is that both eqs 16 and 19 show lower temperatures than eq 4-NC for all ΔEn/R values. From the above results, we can conclude that there does exist a certain ΔEnc/R which can be calculated by uniting eqs 16 and 19. When ΔEn/R > ΔEnc/R, eq 16 should be applied and when ΔEn/R < ΔEnc/R, eq 19 should be applied. Generally speaking, ΔEnc/RT is in the range of 0.1−5, so when ΔEn/RT > 5 or ΔEn/RT < 0.1, eqs 16 or 19 can be applied without further calculations, respectively. Then the accuracy of eqs 16 and 19 is further demonstrated by comparing them with eq 4-NC with different kinetic parameters. For ΔEn/RT ≫ 0, it is found that when d/m is an integer, eq 16 is identical to eq C2 (previously proposed in ref 33). Figure 4 presents the comparison of eqs 4-NC, 16 and C2

Figure 2. Verification of the analytical exact solution for isothermal condition with different kinetic parameters. Symbols: eq 4-NC and lines: eq 10.

match with the symbols no matter whether d/m is an integer or not. It is noteworthy that when d/m is an integer, ⎛ d ⎞ d / m is an integer d / m i ⎜ beta a , + 1⎟ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ∏ ⎝ m ⎠ a+i i=1

(22)

which reduces eq 10 to eq C1 (previously proposed in ref 33) and further confirms the validity of eq 10. Equations 16 and 19 are the newly developed analytical approximate solutions to the expression with the indexincorporated nucleation mode for non-isothermal condition (ΔEn/RT ≫ 0 and ΔEn/RT ≈ 0, respectively). The demarcation of the two equations is first illustrated in Figure 3, which shows the comparison of eqs 4-NC, 16 and 19 with different ΔEn/R values. It is observed that eq 4-NC (symbols) shifts to higher temperatures with increasing ΔEn/R. In addition, eq 16 (lines) is invalid when ΔEn/R = 0 and is

Figure 4. Verification of the analytical approximate solutions for nonisothermal condition and ΔEn/RT ≫ 0 with d/m of 1, 2, and 3. Symbols: eq 4-NC, lines: eqs 16 and C2. 2408



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parameters, and the maximum error of T is negligible (ΔTmax/T < 0.3%). This indicates the accuracy of eq 19. We therefore believe that all the solutions in this model are valid and accurate for possible parameters, which is of great importance for practical applications, especially for determination of kinetic parameters.

with different d/m values of integer. In Figure 4, eq 16 are totally the same as that of eq C2, and they match well with eq 4-NC (ΔTmax/T < 0.2%). When d/m equals to 0.5 and 1.5, eqs 16 and C2 show large difference. Figure 5 presents the comparison of the two

3. COMPREHENSIVE DETERMINATION OF KINETIC PARAMETERS 3.1. Five Kinetic Parameters. Determination of kinetic parameters is crucial to understand the kinetic behaviors since

Figure 5. Verification of the analytical approximate solutions for nonisothermal condition and ΔEn/RT ≫ 0 with different kinetic parameters. (a) d/m of 0.5 and (b) d/m of 1.5. Symbols: eq 4-NC, lines: eq 16 and dashes: eq C2. Figure 7. Simulated isothermal and non-isothermal data with the preset parameters in Table 2. Panel (a) and (b): example I, (c) and (d): example II.

approximate solutions as well as eq 4-NC with different values of a and ΔEn. It is obvious that eq 16 fits well with eq 4-NC (ΔTmax/T < 0.2%), while relatively larger discrepancy is observed between eqs 4-NC and C2, especially for d/m = 0.5 (ΔTmax/T > 9%). This result reveals that eq 16 is more accurate than eq C2. Therefore, the accuracy of eq 16 is confirmed from the above comparisons. For ΔEn/RT ≈ 0, Figure 6 shows the comparison of eqs 4NC and 19 with different values of a, d/m, and ΔEg. This figure shows that eq 19 fits very well with eq 4-NC with different

Table 2. Preset Parameters of the Practical Examples example Ia

IIa

IIIb

preset kinetic parameters

preset experimental parameters isothermal

a

3.0 × 1027 min−a‑d/m 1.7

d/m

1.5

ΔEn/R ΔEg/R aI0aG0d/m a

10000 K 15000 K 3.0 × 1027 min−a‑d/m 2

nonisothermal T0 β isothermal

d/m

1

ΔEn/R ΔEg/R N/A

100 K 40000 K

aI0aG0d/m

T

483, 493, 503, 513, 523 K

T

500, 510, 520, 530, 540 K

nonisothermal T0 β isothermal T nonisothermal T0 β

a

Figure 6. Verification of the analytical approximate solution for nonisothermal condition and ΔEn/RT ≈ 0 with different kinetic parameters. Symbols: eq 4-NC and lines: eq 19.

273 K 0.5, 1, 2, 4, 8 K/min

273 K 0.5, 1, 2, 4, 8 K/min 609.9, 614.9, 619.9, 624.9, 629.9 K

273 K 2.5, 5, 10, 20, 40 K/min

Simulated examples. bExperimental example.14

kinetic parameters can give insights into the kinetic mechanisms of the solid-state phase transitions. 2409



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referring to handbooks for common materials. (4) The nucleation index (a) reveals the degree of self-catalysis of nucleation. a < 1 means that the existing nuclei inhibit the nucleation, and a > 1 means the existing nuclei accelerate the nucleation. So a can only be qualitatively determined from the degree of self-catalysis of nucleation. (5) The growth index (d/ m) contains the dimensionality of growth (d) and the mode of growth (m). Sometimes, d can be directly determined by microstructure observations (e.g., scanning and transmission electron microscopy) before and after phase transitions. Information of m can be obtained (though not easy) from the comparison between the diffusion and interfacial process of the solid-state phase transitions. Analyzing the experimental data on the basis of analytical kinetic model is one of the most successful approaches to determine the kinetic parameters. However, in the previous investigations.14−25,40,41 only part of the parameters can be determined due to the imperfection of the JMAK model. In this work, based on the extended NI-JMAK model and its valid, accurate and simple analytical solutions, all the five kinetic parameters can be comprehensively determined by simultaneous analysis of isothermal and non-isothermal kinetic behaviors. 3.2. Linear Functions. To guarantee feasibility and stability of the regressions, the analytical solutions are transformed into linear functions,

Figure 8. Plots of ln[−ln(1 − ξ)] against ln(t) according to eq 23 and as-acquired ln Kiso against 1/T and their linear regressions. (a) and (b): example I, (c) and (d): example II.

The extended NI-JMAK model has five independent kinetic parameters. (1) The pre-exponential factor (N0G0d/m for the site saturation nucleation mode, aI0aG0a + d/m for the indexincorporated nucleation mode) reveals the intrinsic properties of the solid-state phase transitions. It can hardly be directly determined due to its complexity. (2) The activation energy of nucleation (ΔEn) and (3) the activation energy of growth (ΔEg) stand for the energy barriers for nucleation and growth, respectively. These may be semiquantitatively determined by

⎡ ⎛ −ΔEsum ⎞⎤ ⎟⎥ ln[− ln(1 − ξ)] = n ln t + ln⎢K iso exp⎜ ⎝ RT ⎠⎦ ⎣

(23)

Figure 9. Plots of ln(βn/T2n)according to eq 24 and ln{βa + d/m/[T2d/m(T − T0)a]} against 1/T according to eq 25 with all possible values of a and d/ m and their linear regressions. (a) and (b): example I, (c) and (d): example II, (e) and (f): example III. 2410



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Table 3. Determination of the Candidate Sets of Kinetic Parameters eq 27 a

d/m

ln(Kiso/min‑a‑d/m)

2.7 2.2 1.7 1.2 0.2 2.5 2 1.5 1 0 2.3 1.8 1.3 0.8

0.5 1 1.5 2 3 0.5 1 1.5 2 3 0.5 1 1.5 2

61.41

sxample I

II

III

a

ln(Kniso1/min‑a‑d/m)

ln(Kniso2/min‑a‑d/m)

39500

31.35

61.48

40200

37.78

50.24 46.75 43.25 39.75 32.75 55.27 51.77 48.27 44.77

175.2a

113600a

ΔEsum/R (K)

eq 26

ΔEn/R (K)

ΔEg/R (K)

ineq 28

11920 11870 11780 11630 9450 1835 1210 609 145

14620 13400 12980 12770 12540 71230 37780 26190 20030

yes yes yes yes yes yes yes no no

44430 45080 46210 48700

22810 32450 35680 37320

yes yes yes yes

no 145.3

161.4 157.9 154.4 150.9

Obtained from ref 14 directly.

Table 4. Selection of the Correct Sets of Kinetic Parameters from the Candidates example

aI0aG0d/m (min‑a‑d/m)

a

d/m

ΔEn/R (K)

ΔEg/R (K)

ΔEn (kJ/mol)

ΔEg (kJ/mol)

× × × × × × × × × × ×

2.7 2.2 1.7 1.2 0.2 2.5 2 2.3 1.8 1.3 0.8

0.5 1 1.5 2 3 0.5 1 0.5 1 1.5 2

11920 11870 11780 11630 9450 1835 1210 44430 45080 46210 48700

14620 13400 12980 12770 12540 71230 37780 22810 32450 35680 37320

99.1 98.7 97.9 96.7 78.6 15.3 10.1 369.4 374.8 384.2 404.9

121.6 111.4 107.9 106.2 104.3 592.2 314.1 189.6 269.8 296.6 310.3

I

2.7 3.3 3.0 2.0 1.3 2.6 3.0 5.6 6.2 4.8 2.5

II III

ln

ln

1027 1027 1027 1027 1026 1027 1027 1076 1076 1076 1076

−ΔEsum βn = + ln K niso1 2n RT T

β a+d/m T

2d / m

a

(T − T0)

=

−ΔEsum + ln K niso2 RT

selected

yes

yes

yes

regression. The slopes and intercepts correspond to n and ln[Kiso exp(ΔEsum/RT)], respectively. Then plot ln[Kiso exp(ΔEsum/RT)] against 1/T and fit the plots by linear regression. The slope and intercept correspond to −ΔEsum/R and ln Kiso, respectively. (2) Analyze the non-isothermal data for different heating rates. List all possible values of a and d/m and find the temperatures when the transformed fraction reaches to a fixed value (ξf). Then plot ln(βn/T2n) and ln{βa + d/m/[T2d/m(T − T0)a]} against 1/T for possible a and d/m values and fit them with a fixed slope (as-acquired −ΔEsum/R) by linear regression. The intercepts correspond to ln Kniso1 and ln Kniso2, respectively. (3) Determine the candidate sets of kinetic parameters. For a = 0, which corresponds to the site saturation nucleation mode, N0G0d/m, ΔEg/R, and d/m can be calculated. If

(24)

(25)

in which n, ΔEsum, Kiso, Kniso1, and Kniso2 can all be unambiguously expressed by the five kinetic parameters as shown in Table 1, making the simultaneous analysis of isothermal and non-isothermal kinetic behaviors possible. Equation 23 is the linear function for isothermal condition, which is quite similar to that of the classical JMAK model. Equation 24 is the linear function that describes the nonisothermal kinetic behaviors for the site saturation nucleation mode and index-incorporated nucleation mode with ΔEn/RT ≫ 0. Interestingly, eq 24 is compatible to the KAS equation, which indicates that the KAS equation can be applied in these cases. Equation 25 describes the non-isothermal kinetics for the index-incorporated nucleation mode with ΔEn/RT ≈ 0. As there are obvious differences in the left side of the equation between eq 25 and the equations in the model-free methods,44−47 the model-free methods are not suitable for this case. 3.3. Process of the Determination. Comprehensive determination of the kinetic parameters is performed by the following steps successively. (1) Analyze the isothermal data for different temperatures. Plot ln[−ln(1−ξ)] against ln(t) and fit these plots by linear

ln K iso

⎡ ⎛ ΔEg ⎞d / m⎤ ⎢ = ln K niso1 + ln −ln(1 − ξf )⎜ ⎟ ⎥ ⎢ R ⎠ ⎥⎦ ⎝ ⎣

(26)

this set of kinetic parameters can be determined as a candidate. For a > 0, which corresponds to the index-incorporated nucleation mode, aI0aG0a + d/m can be calculated for each a and d/m values. If ΔEn and ΔEg can be figured out from the following equation 2411



(

−ln(1 − ξf )beta a , ln

d m

Article

⎛ aΔE beta⎜ ΔE n , ⎝ g

d m

40 K/min, respectively. As seen in Figure 9, the fitness of the plots is acceptable, and their intercepts (ln Kniso1 and ln Kniso2) are listed in Table 3. For example I, all possible a values are larger than zero. Five sets of ΔEn/R and ΔEg/R can be figured out for possible a and d/m values from eq 27, and inequation 28 holds simultaneously. For example II, when a = 0 and d/m = 3, eq 26 does not hold. When a > 0, four sets of ΔEn/R and ΔEg/R can be figured out for possible a and d/m values from eq 27, and inequation 28 holds only for a = 2.5, d/m = 0.5 and a = 2, d/m = 1. For example III, all possible a values are larger than zero. Four sets of ΔEn/R and ΔEg/R can be figured out for possible a and d/m values from eq 27, and inequation 28 holds simultaneously. Therefore, five, two, and four candidate sets are determined for examples I, II, and III, respectively, as shown in Table 4. Assume that 3D growth (d = 3) and positive self-catalysis of nucleation (a > 1) can be determined for example I, and twodimensional (2D) growth (d = 2) can be determined for example II, by further measurements and analysis. Then the third and second sets of candidates can be selected as the correct sets for examples I and II, respectively. As for example III, 3D growth (d = 3) and diffusion-controlled growth (m = 2) can be determined from the SEM observations and the nature of amorphous Pd40Ni10P20Cu30.14 So the third candidate set can be selected as the correct set. The results are shown in Table 4. For examples I and II, after a comparison between the preset kinetic parameters and the determined kinetic parameters, it is found that the kinetic parameters determined by this model match well with the preset ones. Particularly, the preexponential factor (N0G0d/m or aI0aG0a + d/m), nucleation index (a) and growth index (d/m) are highly accurate, while the activation energy of nucleation (ΔEn) is a little larger and the activation energy of growth (ΔEg) is a little smaller than the preset ones. As for the experimental example (example III), all the kinetic parameters are highly plausible according to relative theories and experiments.52,53 So we can conclude that this comprehensive determination of kinetic parameters is practical for solid-state phase transitions. By this approach, we can explore the differences in kinetic parameters of materials with different composition and structure, such as before and after doping, heat treatment, and so on, which can definitely provide insights into the kinetic mechanisms of the materials. These investigations will be carried out in our future work.

ΔEg d / m + 1 ΔEn a − 1 R

)( R )

+1

( )

⎞ + 1⎟ ⎠

= ln K iso − ln K niso1

(27)

and inequation d/m ⎡ ⎛ d ⎞ ⎛ mΔEsum ⎞ ⎤ ⎟ ⎥ ln⎢ −ln(1 − ξf )beta⎜a , + 1⎟a⎜ ⎝ m ⎠ ⎝ Rd ⎠ ⎥⎦ ⎢⎣

< ln K iso − ln K niso2

(28)

for possible a and d/m values, these sets of kinetic parameters can be considered as candidates (see Appendix E). After the implementation, one or more sets (maximum five) of kinetic parameters may be determined as candidates. (4) Select the correct set of kinetic parameters. If more than one candidate sets of kinetic parameters exist, they cannot be distinguished by only isothermal and nonisothermal experiments. Therefore, further selecting of the kinetic parameters should be performed by some direct measurements and/or theoretical analysis as mentioned in Section 3.1. Because candidates of the kinetic parameters have been given, little information from these measurements and/or analysis is required to finally select the correct kinetic parameters. 3.4. Practical Examples. Three practical examples (I, II, and III) are given to further illustrate and validate the comprehensive determination of kinetic parameters. Examples I and II use the simulated data obtained by eq 4-NC with two different sets of preset parameters as kinetic data. Isothermal curves at different temperatures and non-isothermal curves at different heating rates are demonstrated as shown in Figure 7. Example III use the experimental kinetic data of the crystallization of the amorphous Pd40Ni10P20Cu30 after preannealing 600 s at 624.9 K reported in ref 14, where isothermal curves at different temperatures and non-isothermal curves at different heating rates are presented. All the preset parameters are listed in Table 2. Then, the following steps are performed. Figure 8 shows the analysis results of the isothermal data for examples I and II. A good linearity is observed for all the plots in Figure 8a,c. Here, n is determined to be 3.2 and 3.0 for examples I and II, respectively. The values of ln[K iso exp(−ΔEsum/RT)] are calculated to be −14.12, −15.59, −17.12, −18.71, −20.37 at 523, 513, 503, 493, 483 K for example I, and −12.97, −14.37, −15.83, −17.35, −18.92 at 540, 530, 520, 510, 500 K for example II . As shown in Figure 8b,d, these plots show good linearity and the slopes (−39500 for example I and −40200 for example II) and the intercepts (61.41 for example I and 61.48 for example II) of the fitted lines correspond to -ΔEsum/R and ln Kiso, respectively. For example III, the isothermal data have been analyzed by a similar approach in ref 14, and the values of n, −ΔEsum/R, and ln Kiso are calculated to be 2.8, −113600, and 175.2, respectively, from their results. Figure 9 presents the analysis results of the non-isothermal data for the three examples. In these cases, ξf = 63.2% (−ln(1 − ξf) = 1) is selected and the temperatures are 535.4, 550.5, 566.7, 583.7, 601.7 K for example I, 519.4, 532.8, 546.8, 561.5, 577 K for example II at β = 0.5, 1, 2, 4, 8 K/min, and 634.0, 642.1, 648.6, 656.4, 664.1 K for example III at β = 2.5, 5, 10, 20,

4. CONCLUSIONS In the present work, a new subexpression is proposed to extend the applicability of the NI-JMAK model. This extended model can be used to describe the solid-state phase transitions with adequately small activation energy of nucleation, which was ignored in the previous works. In addition, analytical solutions to the isothermal and non-isothermal expressions in this model are also developed. These exact or approximate solutions are proved to be valid, accurate and simple for possible parameters due to the introduction of the Euler integral of the first kind. Furthermore, after transforming the solutions into linear functions, a comprehensive determination of kinetic parameters can be achieved by linear regressions of the isothermal and non-isothermal experimental data and further analysis. By applying this comprehensive determination to two simulated examples, it is found that the determined kinetic parameters match well with the preset ones. In addition, for an experimental example, the kinetic parameters determined by 2412



Article

⎡ ⎤ ⎛ − d ΔE ⎞ g ⎟ d / m⎥ ⎢ m d/m ⎜ ξ = 1 − exp⎢ −N0G0 exp t ⎥ ⎜ RT ⎟ ⎝ ⎠ ⎣ ⎦

this approach are compatible to relative theories and experiments. These indicate that this approach is highly plausible. We therefore believe that this model would like to facilitate the understanding and further exploitation of the solid-state phase transitions.

and for non-isothermal condition, i.e.,

■

T = βt + T0

APPENDIX A The site saturation nucleation mode describes the nucleation as I(T , τ ) = N0δ(τ − 0)

d / m⎤ ⎡ ⎛ − d ΔE ⎞⎛ 2 ⎞ g ⎟ RT ⎢ m d/m ⎜ ⎜ ⎟ ⎥ ξ = 1 − exp⎢ −N0G0 exp ⎜ RT ⎟⎜ β ΔE ⎟ ⎥ g⎠ ⎝ ⎠⎝ ⎣ ⎦

where N0 is a constant standing for the nucleus number per volume in the site saturation nucleation mode, δ(τ − 0) denotes Dirac function. The linear continuous nucleation mode describes the nucleation as

(B3)

The analytical exact solution to eq A6 for isothermal condition is ⎡ 1 ξ = 1 − exp⎢⎢ − I0G0d / m 1+ ⎣

(A2)

where I0 is a constant standing for the intrinsic nucleation rate in the linear continuous nucleation mode, ΔEn means the activation energy of nucleation and R is the gas constant. As for growth, a compact expression of V(T, τ ∼ t) is given to describe both the interface-controlled growth and diffusion-controlled growth modes, ⎡ Vs(T , τ ∼ t ) = ⎢G0 ⎢⎣

∫τ

t

d/m ⎛ −ΔEg ⎞ ⎤ exp⎜ ⎟ dη ⎥ ⎝ RT ⎠ ⎥⎦

and the analytical approximate solution to eq A6 for nonisothermal condition is ⎡ ⎛ −ΔE − d ΔE ⎞ n g⎟ ⎢ m d / m Cc1∗ exp⎜ ξ = 1 − exp⎢ −I0G0 d ⎜ ⎟ RT +1 ⎝ ⎠ ⎣ m

(A3)

⎤ ⎛ RT 2 ⎞1 + d / m⎥ ⎟ ⎜ ⎥ ⎝ β ⎠ ⎦

■

APPENDIX C The analytical exact solution to eq 4 for isothermal condition isand the analytical approximate solution to eq 4 for nonisothermal condition is

(A4)

∫0

t

d / m⎫ ⎛ −ΔEg ⎞ ⎤ ⎪ exp⎜ ⎟ dη ⎥ ⎬ ⎝ RT ⎠ ⎥⎦ ⎪ ⎭

⎡ ⎛ −aΔE − C ∗ n ξ = 1 − exp⎢⎢ −I0 aG0 d / m d ca exp⎜ ⎜ RT +1 ⎝ ⎣ m ⎤ ⎛ RT 2 ⎞a + d / m⎥ ⎜ ⎟ ⎥ ⎝ β ⎠ ⎦

(A5)

for the site saturation nucleation mode, and ⎧ ⎪ ξ = 1 − exp⎨− ⎪ ⎩ ⎡ ⎢G0 ⎢⎣

∫τ

t

∫0

t

d ΔEg m

⎞ ⎟ ⎟ ⎠

(C2)

where Cca* is a correction factor, which is independent of t and T, and is defined by ΔEn, ΔEg, a, and d/m. Liu et al.33 give five approximate expressions of Cca* for different d/m.

⎛ −ΔEn ⎞ ⎟ I0 exp⎜ ⎝ RT ⎠

d/m ⎫ ⎛ −ΔEg ⎞ ⎤ ⎪ exp⎜ ⎟ dη ⎥ dτ ⎬ ⎪ ⎝ RT ⎠ ⎥⎦ ⎭

(B5)

where Cc1* is a correction factor, which is independent of t and T, and is defined by ΔEn, ΔEg, and d/m. Kempen et al.26 give five approximate expressions of Cc1* for different d/m.

Therefore, the classical JMAK model contains two expressions, i.e., ⎧ ⎡ ⎪ ξ = 1 − exp⎨−N0⎢G0 ⎢⎣ ⎪ ⎩

d m

⎤ ⎛ −ΔE − d ΔE ⎞ n g ⎟ 1 + d / m⎥ m ⎜ exp⎜ ⎥ ⎟t RT ⎝ ⎠ ⎦ (B4)

Here, G0 is the intrinsic growth rate that only depends on the growth geometry of the nuclei and the thermodynamic properties of the growth, ΔEg is the activation energy of growth, d is the dimensionality of the growth (d = 1, 2, 3), m is the growth mode parameter (m = 1 for the interface-controlled growth mode; m = 2 for the diffusion-controlled growth mode), and d/m is usually called the growth index. The impingement mode of randomly dispersed nuclei isotropic growth is expressed as f (ξex ) = 1 − exp(ξex )

(B2)

with β as the heating rate and T0 as the starting temperature, the analytical approximate solution to eq A5 is

(A1)

⎛ −ΔEn ⎞ ⎟ I(T , τ ) = I0 exp⎜ ⎝ RT ⎠

(B1)

■

APPENDIX D In ref 33, the integral in the expression of nucleation is first solved for isothermal and non-isothermal conditions, respectively. For isothermal conditions, an exact solution to eq 3 can be easily found as

(A6)

for the linear continuous nucleation mode.

■

⎛ −aΔEn ⎞ ⎟ I(T , τ ) = aI0 aτ a − 1 exp⎜ ⎝ RT ⎠

APPENDIX B For isothermal condition, the analytical exact solution to eq A5 is

(D1)

and eq 4 becomes 2413


Crystal Growth & Design ⎧ ⎪ ξ = 1 − exp⎨− ⎪ ⎩ ⎡ ⎢G0 ⎢⎣

∫τ

t

∫0

t

Article

⎛ −aΔEn ⎞ ⎟ aI0 aτ a − 1 exp⎜ ⎝ RT ⎠

d/m ⎫ ⎛ −ΔEg ⎞ ⎤ ⎪ exp⎜ ⎟d η ⎥ d τ ⎬ ⎪ ⎝ RT ⎠ ⎥⎦ ⎭

(D2)

For non-isothermal condition, an approximate solution to eq 3 is given as ⎛ R ⎞a − 1 2(a − 1) ⎛ −aΔEn ⎞ ⎟ I(T , τ ) = aI0 a⎜ exp⎜ ⎟ T ⎝ RT ⎠ ⎝ ΔEnβ ⎠

(D3)

and eq 4 becomes ⎧ ⎪ ξ = 1 − exp⎨− ⎪ ⎩ ⎡ ⎢G0 ⎢⎣

∫τ

t

∫0

t

⎛ R ⎞a − 1 2(a − 1) ⎛ −aΔEn ⎞ ⎟ exp⎜ aI0 a⎜ ⎟ T ⎝ RT ⎠ ⎝ ΔEnβ ⎠

⎛ −ΔEg ⎞ ⎤ exp⎜ ⎟d η ⎥ ⎝ RT ⎠ ⎥⎦

d/m

⎫ ⎪ dτ ⎬ ⎪ ⎭

Figure 10. Demarcation of the analytical approximate solutions for non-isothermal condition with different ΔEn when ΔEsum is a constant. Symbols: eq 4-NC, lines: eq 16 and dashes: eq 19.

■

(D4)

*Address: No. 275 Mailbox, 149 Yanchang Road, Shanghai 200072 P. R. China; Fax & Tel +86-21-5633-8065; e-mail: [email protected], [email protected]

It should be noted that the first order approximation of the temperature integral51

∫0

t

⎛ −ΔE ⎞ ⎛ −ΔE ⎞ RT 2 ⎟ dη = ⎟ exp⎜ exp⎜ ⎝ RT ⎠ ⎝ RT ⎠ β ΔE

Notes

The authors declare no competing financial interest.

■

(D5)

ACKNOWLEDGMENTS This work was financially sponsored by the National Natural Science Foundation of China (51222402, 51501107), “Shu Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (13SG39), Science and Technology Committee of Shanghai (14521100603) and China Postdoctoral Science Foundation (2015M571541). Q.G. acknowledges the research support from Australian Synchrotron.

was used in eq D4, which is accurate enough only when ΔE/RT ≫ 0 in the temperature range of interest. So ΔEn/RT ≫ 0 is the premise for using eq D4 as the nucleation expression. However, as a matter of fact, it is found that the activation energy of nucleation for some materials can be adequately small, even close to zero.5 So the present NI-JMAK model cannot be applied for describing the kinetic behaviors of the solid-state phase transitions for the case of adequately small ΔEn. Furthermore, for the analytical solution (eq C1) to eq D2, it is not valid when the upper bound (d/m) of the product of sequences is not an integer. As for the analytical solution (eq C2) to eq D4, in addition to the approximation of eq D5, an approximation of Cca* is also applied, and five expressions of Cca* are given for different d/m values. But the approximation of Cca* is not accurate enough, especially for d/m = 0.5 and 1.5 (see Figure 5). These are key disadvantages that prevent the NI-JMAK model from wide application.

■

AUTHOR INFORMATION

Corresponding Author

■

REFERENCES

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APPENDIX E

As seen in Figure 10, when ΔEsum/R is a constant, eq 4-NC has a lower limit when ΔEn/R = 0, and increasing ΔEn/R make eq 4-NC shift to higher temperatures monotonously. eq 16 also increases with ΔEn/R, and exhibits lower (more or less) temperatures in comparison with eq 4-NC, and the discrepancy becomes huge for adequately small ΔEn/R. So the determined results may be very inaccurate when ΔEn is close to 0. Equation 19 is accurate when ΔEn/R = 0 and thus represents the lower limit of eq 4-NC. Therefore, eq 27 and inequation 28 should hold simultaneously. 2414



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Comprehensive Determination of Kinetic Parameters in Solid-State

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