Evaluation of the maximum transformation rate for determination of impingement mode upon near-equilibrium solid-state phase transformation

Thermochimica Acta 561 (2013) 54–62

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Evaluation of the maximum transformation rate for determination of impingement mode upon near-equilibrium solid-state phase transformation Yi-Hui Jiang a , Feng Liu a,b,∗ , Shao-Jie Song a , Bao Sun a a b

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China School of Materials and Chemical Engineering, Xi’an Technological University, Xi’an, Shaanxi 710032, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 20 January 2013 Received in revised form 15 March 2013 Accepted 16 March 2013 Available online 25 March 2013 Keywords: Phase transformation Kinetics Thermodynamics Impingement mode

a b s t r a c t Considering the role of hard impingement mode in analyzing phase transformation kinetics, a maximum transformation rate analysis method was developed for pre-selection of impingement mode. Based on an extended analytical model, this method proposed for transformations with very large undercooling is reevaluated for transformations in the vicinity of equilibrium transformation temperature. Accordingly, a new peak maximum analysis method for both isothermal and non-isothermal transformations conducting near or far from the equilibrium state is proposed to determine the prevailing mode of impingement. The results obtained by application of the new method to numerically calculated transformations are in good agreement with the input values. Then, the method was applied to the isothermal ferrite/austenite transformation in Fe–3.28at.%Mn alloy as measured by dilatometry and non-isothermal hcp/bcc phase transformation in pure Zr as measured by differential scanning calorimetry. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Microstructure which affords a great degree of controlling the property of material with a certain chemical composition is always adjusted purposely for engineering application. Since solid-state phase transformation involving nucleation, growth and impingement is an important mean for the adjustment of the microstructure, its kinetics has been studied extensively [1–18]. One of the most interesting theoretical topics in kinetic analysis is extraction of kinetic information from either model fitting [2–14] or recipe analyzing [7,13,15–18]. Since fitting a proper model to experimental data can give reliable values of kinetic parameters, much effort has been spent on modeling the overall transformation kinetics, e.g., the isothermal Jonhson–Mehl–Avrami (JMA) model [2–6], the non-isothermal JMA-like model [7–11], the analytical model [12,13] and the extended analytical model [14]. The development of the kinetic models always comes along with the increase in the efficiency of kinetic recipes. And the recipes (e.g., the Kissinger method [7,15–17], the Avrami plot [13], and the Ozawa plot [18]) for directly extracting kinetic parameters were proposed on the basis of these kinetic models.

∗ Corresponding author at: State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China. Tel.: +86 29 88460374; fax: +86 29 88491000. E-mail address: [email protected] (F. Liu). 0040-6031/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tca.2013.03.023

In many cases, a breakdown of assumptions made in the derivation of the JMA model (e.g., the anisotropic growth and the non-randomly dispersed nuclei) will results in the deviations from JMA equation. Accordingly, some phenomenological equations by introducing impingement factors are proposed (see Section 2.1) to approximately describe the transformation kinetics [13]. It should be emphasized that the phenomenological equations do not match the correlation equations stemming from rigorous statistical treatments [19–22]. However, the impingement factors can explain the transformation mechanism to some extent. Therefore, these simple equations are often employed in kinetic analysis. Without estimating the validity of JMA assumption, the kinetic analysis for real transformation is often performed by fitting the JMA (-like) equation or applying the recipes based on JMA (-like) model to experiment data, and sometimes, untrustworthy results are obtained. Therefore, the maximum transformation rate analysis [23] was proposed for transformations occurring far from equilibrium state (FES) to identify the impingement mode. With the pre-selected impingement factor, the experimental data can be well described by the phenomenal equation [23]. However, due to the effect of thermodynamic term, it remains an open question whether the maximum transformation rate analysis is valid for transformations occurring near equilibrium state (NES). In this work, based on the extended analytical model, the effect of thermodynamic term on the maximum transformation rate analysis is evaluated, and a new straightforward recipe for identification of impingement mode is proposed for transformations occurring either FES or NES. Accordingly, the prevailing impingement modes

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of isothermal ferrite/austenite transformation in Fe–3.28at.%Mn alloy as measured by dilatometry (DIL) and non-isothermal hcp/bcc phase transformation in pure Zr as measured by differential scanning calorimetry (DSC) are studied by the present recipe. Note that the present recipe can only be valid for transformations with constant kinetic parameters. 2. Evaluation of kinetic analysis 2.1. The role of impingement mode The extended transformed fraction, xe , which do not account for the overlap of growing particles can be given in an analytical form [13], n

xe = (k˛) =



 Q  n

K0 exp −

RT

˛

(1)

where ˛ represents t and RT2 /˚ respectively for isothermal and non-isothermal transformations,1 k is the rate constant, K0 the pre-exponent factor of rate constant, R the gas constant, T the temperature, t the time and ˚ the constant heating/cooling rate. In terms of the analytical model, the impingement mode will give the real transformed fraction, f. For classical JMA (-like) equation [1–11], f = 1 − exp (−xe ) and

df =I =1−f dxe

(2a)

Due to blocking effects [19], the impingement of anisotropically growing particles is more severe than isotropically ones. Accordingly, a phenomenological description for anisotropic growth can be given as [13,24],





 −1/(−1)

f = 1 − 1 +  − 1 xe

and

df  = I = (1 − f ) dxe

(2b)

with  (≥1) as impingement factor. The rigorous statistical treatments [19–22,25] indicate that Eq. (2b) is a proper approximation for the anisotropic growth, and it can also be used to the case of nonrandom dispersed nuclei. Here, in order to distinguish between them, another phenomenological equation is proposed for nonrandomly dispersed nuclei [13,26], df = I = 1−fε dxe

(2c)

with ε (≥1) as another impingement factor. Note that the impingement according to Eq. (2c) is less distinct than the one according to Eq. (2a). If  = ε = 1, Eqs. (2b) and (2c) reduce to the JMA (-like) equation. Accordingly, the role of impingement mode in kinetic analysis is demonstrated as follows. 2.1.1. Model fitting By using the numerical approach described in Ref. [13],2 the transformed fraction can be calculated. Then kinetic models can be fitted to this transformed fraction as a function of time/temperature. Thereby the following steps are performed successively [10].

1

Fig. 1. Evolutions of transformed fraction, f, with T (a) and transformation rate, df/dT, with T (b) for non-isothermal FES transformation with constant heating rate (i.e., ˚ = 5, 10 and 20 K min−1 ) for the case of site saturation, interface-controlled growth and impingement due to non-randomly dispersed nuclei with ε = 3, as numerically calculated (lines) using values for model parameters gathered in Table 4 of Ref. [13], and as fitted (symbols) by JMA-like equation.

Compared with isothermal case, K0 in non-isothermal model is modified by a term involving nucleation and growth activation energies. Therefore, ˛ has dimensions of time and time×energy for isothermal and non-isothermal, respectively. 2 To evaluate the extended transformed fraction, a double integration is needed to be executed. With modern computers, it can be solved by sophisticated algorithm, which is called numerical approach. Since the numerical solution is sufficiently accurate, it often serves as a “simulated” transformation to test the validity of the kinetic model.

i) For a case of site saturation, three-dimensional interfacecontrolled growth (n = 3) and impingement due to nonrandomly dispersed nuclei with ε = 3, numerical calculation is performed for three different heating rates (i.e., ˚ = 5, 10 and 20 K min−1 ), with the model parameters gathered in Table 4 of Ref. [13]. The numerically calculated non-isothermal FES transformations are indicated in Fig. 1a (lines). ii) The JMA equation (i.e., Eqs. (1) and (2a) with n, Q and K0 as the fitting parameters) is fitted to the transformed fraction as obtained in step i simultaneously for the three different heating rates. Fitting is performed by minimization of the sum of the squares of the residuals, employing a downhill simplex fitting procedure. The fitting results are shown in Fig. 1a (symbols). Although the assumed impingement mode is incorrect, the fitting results agree well with the numerically calculated transformations (see Fig. 1a). Moreover, the fitted value of Q (=299.7 kJ mol−1 ) is quite close to the input one (=300 kJ mol−1 ). But the deviation of fitted value of n (=3.82) from the input one (=3) is too large to give the correct nucleation mechanism, e.g., n = 3 for site saturation and n = 4 for continuous nucleation. However, if the procedure described in step ii is carried out with a proper

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model (i.e., Eqs. (1) and (2c) with ε = 3), all the fitted values of kinetic parameters are nearly equal to the input ones. 2.1.2. Recipe analyzing Recipe analyzing is an efficient tool for the determination of kinetic parameters because of its simplicity. In the following, the recipes to determine Q (e.g., the “Kissinger plot” [15] and the “GaoWang plot” [17]) are demonstrated by applying them to the nonisothermal transformation indicated in Fig. 1a, and the recipes to determine n (e.g., “Avrami plot”) are shown by applying them to a newly calculated isothermal transformation. According to Fig. 1a, the transformations rate, df/dT (or df/dt), can be calculated for different heating rates (see Fig. 1b). If the temperature at maximum transformation rate, Tp , is known for each heating rate, the value of Q can be quickly obtained from the “Kissinger plot” [15],

 

 Q  =

d ln Tp2 /˚ d 1/Tp

(3a)

R

And the “Gao-Wang plot” [17] requires both Tp and the values of maximum transformation rate, (df/dt)p ,



d ln df/dt



d 1/Tp





p

=−

Q R

(3b)

Since these recipes are independent on kinetic model, Eqs. (3a) and (3b) are valid for various impingement modes. According to Fig. 1b, Eqs. (3a) and (3b) give the values of Q for the transformation (i.e., from Eq. (3a), Q = 299.0 ± 1.8 kJ mol−1 ; from Eq. (3b), Q = 298.9 ± 3.4 kJ mol−1 ), which agree well with the input value (Q = 300 kJ mol−1 ). The numerical approach can also gives the evolutions of f with t for isothermal transformations (T = 800 K) assuming continuous nucleation and three-dimensional interface-controlled growth (n = 4), with the model parameters given in Table 4 of Ref. [13]. The numerical calculated transformations with different impingement modes (Eq. (2a), Eq. (2b) with  = 2 and Eq. (2c) with ε = 2) are shown in Fig. 2a, respectively. Without identifying the impingement mode, n can be determined by directly applying the “Avrami plot” to each curve indicated in Fig. 2a, d ln (− ln (1 − f )) =n d ln t

(4a)

The results are shown in Fig. 2b. It can be seen from Fig. 2b that, only if the impingement mode is consistent with the JMA equation, a reasonable value of n (≈4) that corresponds to “real” transformation mechanisms results, otherwise, misleading values occur. According to the analytical model, for different impingement modes, the “Avrami plot” should be modified, correspondingly, for anisotropic growth [13], d ln

1− −1 (1−f ) −1

d ln t

=n

(4b)

for non-randomly dispersed nuclei [13], d ln  (f ) =n d ln t

(4c)

with (f), see Ref. [13]. By selecting a proper recipe according to impingement mode, the reasonable value of n (≈4) for each transformation indicated in Fig. 2a can be obtained. It follows from the above that the values of n determined by both recipe analyzing and model fitting depend on the assumed impingement mode, whereas the values of Q obtained from the kinetic analysis are independent of the impingement mode. This may explains that, by performing kinetic analysis for the crystallization of amorphous alloys, the strange values of local Avrami

Fig. 2. (a) Evolution of transformed fraction, f, with t for isothermal transformation (T = 800 K) for the case of continuous nucleation, interface-controlled growth and impingement due to anisotropic growth with  = 2, isotropic growth of randomly dispersed nuclei and non-randomly dispersed nuclei with ε = 2, respectively, with values for model parameters gathered in Table 4 of Ref. [13]; (b) Avrami exponent, n, as function of f by applying the “Avrami plot” to this transformations.

exponent are often obtained, but the unreasonable values of effective activation energy are rare. It is generally accepted that the abnormal values of n may be due to transformation mechanism changing, nucleation rate accelerating or transient nucleation, etc. In the present treatment, the studies focus on the transformation with invariant transformation mechanism and specific nucleation mode (i.e., site saturation and continuous nucleation). Even in this case, the incorrect impingement mode will also results in misleading values of n. Therefore, it is necessary to identify the prevailing mode of impingement before performing kinetic analysis.

2.2. Evaluation of the maximum transformation rate In the case of FES, the transformation is controlled by pure kinetic terms, i.e., the kinetics obeys Arrhenius relationship. It has been realized that the transformed fraction at maximum transformation rate, fp , only depends on impingement mode [23], for isothermal transformation, fp = 1 − exp

1 n



−1

(5a)

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n−1 −1 fp = 1 − 1 − n 

1/−1 (5b)

fp = 1 −

fp = 1 − e−1

 ε−1

ε fp

(6a)

1/(1− )

(6b)

xe |f =fp = 1

(6c)

Accordingly, identification of the impingement factor from the experimentally measured values at maximum transformation rate can be carried out with Eqs. (5) and (6) [23]. In the case of NES, the thermodynamic term affects significantly the transformation kinetics. Subject to its effect, the relationship between fp and impingement factor proposed for FES case may not hold for NES case. Since the temperature-dependent thermodynamic term is constant during a whole isothermal transformation, Eq. (5) is also valid for NES case. In the following, Eq. (6) for nonisothermal transformation is reevaluated for NES case. On the basis of the extended analytical model, the general nonisothermal rate equation for both FES and NES transformations can be expressed as [14], df nQH = Ixe dT RT 2

(7)

where H is a model parameter involving the differential form of K0 , H=

RT 2 dK0 /dT +1 Q K0

(8)

For FES transformations during continuous heating, K0 reduces to a constant during the whole transformation (c.f., the JMA-like model), H is equal to unit, and Eq. (7) reduces to the non-isothermal JMA rate equation described in Ref. [11]. In other cases, K0 is a positive temperature-dependent parameter (c.f., the extended analytical model), i.e., for FES transformations during continuous cooling, the transformation initial temperature must be involved, and K0 increases with the decrease of temperature; for NES transformations, the thermodynamic terms should be incorporated, and K0 increases with the degree of temperature deviation from the equilibrium transformation temperature. Therefore, the sign of H is positive during continuous heating. But during continuous cooling, numerical calculation shows that the value of H depends mainly on the first term of the right-hand side of Eq. (8), so, the sign of H is negative. Differentiation of Eq. (7) with respect to temperature, and subdf dI dI df e e df with dx = 1I dT and dT with df lead sequent substitution dx dT df dT dT to,

 dI

df d2 f = dT dT 2

df

xe

1 dH nQH 2 nQH + − + T H dT RT 2 RT 2



(9)

The maximum transformation rate satisfies d2 f/dT2 = 0, and it follows that,

dI df



xe + 1

f =fp

=

2 T

−

1 dH H dT



nQH 

f =fp

/

RT 2

f =fp

(10)

where the subscript fp denotes the value concerned at maximum transformation rate. Assume that a parameter L represents the right-hand side of Eq. (10), L=



2RT  nQH



(5c)

for non-isothermal transformation,

fp = 1 − 

For three cases of impingement, respectively, it then follows from Eq. (10), fp = 1 − exp(L) ∗ e−1

n−1 = nε

fpε−1 xe |f =fp

f =fp

−

RT 2

dH nQH 2 dT

57



(11) f =fp

1 −1 (L) +  

(12a)

1/(−1)

 

εfpε−1 xe fp = 1 − L

(12b)

(12c)

Compared with Eq. (6), the relationship between fp and impingement factor is modified subject to the effect of thermodynamic term. In the following, the effects of the different terms in L on fp are studied separately. The first term in right-hand side of Eq. (11), L1 , is arising from the transformation temperature, L1 =

2RT  nQH

(13)

f =fp

From Eq. (13), the sign of L1 depends only on that of H. Therefore, according to Eq. (12), L1 (<0) leads to a larger fp for transformations during continuous cooling (i.e., H < 0), and vice versa for transformations during continuous heating. For FES transformations, the value of L1 is so small (i.e., Q (∼105 ) is always two orders of magnitude larger than RT (∼103 )) that can be neglected during the maximum transformation rate analysis, see also Eq. (23) in Ref. [23]. Actually, the value of L1 is small for both FES and NES transformations, so it affects slightly the value of fp , which can be neglected compared with the experimental uncertainty [27]. The second term in right-hand side of Eq. (11), L2 , can be regarded as the contribution of thermodynamic term,



L2 = −

RT 2 dH nQH 2 dT



(14) f =fp

Since the differential form of H cannot be given analytically, it is difficult to determine the sign of dH/dT. Numerical calculations have shown that the values for |H| decrease with the degree of temperature deviation from the equilibrium transformation temperature. Therefore, the sign of dH/dT is always negative for transformations during either continuous cooling or heating. Accordingly, L2 (>0) always causes a smaller fp . For the case of FES transformations during continuous heating, L2 arising from the contribution of thermodynamic term is zero (i.e., dH/dT = 0), so, Eq. (12) reduces to Eq. (6), and impingement factor can be identified from the maximum transformation rate analysis. But, in other cases, L2 is large enough to affect the value of fp , and performing the maximum transformation rate analysis will leads to unrealistic impingement factor. It follows from the above that the non-isothermal maximum transformation rate analysis ceases to be valid for NES transformations. This is consistent with the experimental facts. For examples, regardless of impingement mode, the transformations in Fe–2.96at.%Ni alloy during continuous cooling and heating obtain the maximum transformation rate at small value of fp , e.g., the ferrite to austenite transformation [28] and the austenite to ferrite transformation [29]; the small value of fp of the austenite to ferrite transformation in Fe–3.28at.%Mn alloy during continuous cooling cannot be completely attributed to the effect of anisotropic growth [14]. 3. Identification of the impingement mode In general case, the prevailing impingement mode of isothermal transformations can be determined by performing isothermal maximum transformation rate analysis described in Ref. [23]. In this method, the Avrami exponent should be known to determine

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impingement factor (see Eq. (5)), but a proper impingement factor is a guarantee of extracting the correct Avrami exponent (see Eq. (4)). Thus, isothermal maximum transformation rate analysis requires many iteration procedures before the set of proper values is found. Besides, the non-isothermal maximum transformation rate analysis cannot hold any more for general case. Considering the role of impingement in kinetic analysis, a simple recipe to identify the impingement factor for both isothermal and non-isothermal transformations in general case is proposed in this section. 3.1. Theoretical deduction The general isothermal rate equation can often be expressed as, 1 df = knI(xe )1− n dt

(15a)

For the expressions of I and xe , see Eq. (2). In combination with Eq. (1), Eq. (15a) leads to, nIxe df = t dt

(15b)

And Eq. (7) represents the general non-isothermal rate equation. Here, in order to make the kinetic analysis simple, alternative rate equations can be constructed by rewriting Eqs. (7) and (15b), for isothermal transformations, df t = nIxe dt

z=

(16a)

for non-isothermal transformations, df RT 2 = nIxe dT QH

z=

Fig. 3. Dependence of the position of function z maximum on the impingement factors,  or ε.

3.2. Numerical examination 3.2.1. Isothermal FES transformation For the transformations described in Fig. 2a, the evolutions of df/dt vs. f and z vs. f are shown in Fig. 4a and b, respectively. In the following, the prevailing impingement mode of the transformations is identified by performing both the maximum transformation rate analysis and the new recipe analysis.

(16b)

Noting that, firstly, function z was only used to test whether the FES transformation follows the classical JMA equation or not [27]. Differentiation of Eq. (16) with respect to f, and subsequent substie tution of dx with 1I lead to, df



dI dz =n 1+ xe df df



(17)

So, the maximum of function z vs. f plot satisfies dz/df = 0, and it follows that, 1+

dI  df

xe

f =fpz

=0

(18)

where fp z represents the transformed fraction corresponding to the maximum of function z. For three cases of impingement, respectively, it then follows from Eq. (18): fpz = 1 − e−1

(19a)

fpz = 1 −  1/(1−)

(19b)

 ε−1

ε fpz

xe |f =f z = 1 p

(19c)

Note that Eq. (19) has the same form as Eq. (6). If impingement mode is due to isotropic growth of randomly dispersed nuclei, it follows from Eq. (19a), Eq. (19b) with  = 1 or Eq. (19c) with ε = 1 that fp z = 0.632. If anisotropic growth prevails, it follows from Eq. (19b) with  > 1 that fp z < 0.632, and if nonrandomly dispersed nuclei provides a reasonable description of reality, it follows from Eq. (19c) with ε > 1 that fp z > 0.632. Hence, for both isothermal and non-isothermal transformations in general case, the value of fp z provides a direct indication of prevailing mode of impingement by the application of Eq. (19), see also Fig. 3.

Fig. 4. Evolutions of: (a) transformation rate, df/dt, with transformed fraction, f, and (b) function z with f for the transformations described in Fig. 1a.

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First, the isothermal maximum transformation rate analysis is performed. If an assumed impingement factor is randomly selected and the value of fp is given, the Avrami exponent can be determined from both Eqs. (4) and (5). For different assumed impingement factors, a lot of pairs of values of n can be obtained. Only if the assumed impingement factor approaches to the real one, the result obtained from Eq. (4) agrees approximately with that from Eq. (5). Accordingly, in combination with the values of fp indicated in Fig. 4a, the impingement mode of the transformations described in Fig. 2a is identified (see Fig. 5), i.e., for fp = 0.374 ± 0.003, it is anisotropic growth with  = 2.01 ± 0.01; for fp = 0.526 ± 0.003, isotropic growth of randomly dispersed nuclei; for fp = 0.575 ± 0.003, non-randomly dispersed nuclei with ε = 2.03 ± 0.05. Alternatively, the impingement factors can be straightforwardly determined from values of fp z by the application of Eq. (19). According to the values of fp z indicated in Fig. 4b, the impingement mode of the transformations described in Fig. 2a is also identified, i.e., for fp z = 0.500 ± 0.003, it is anisotropic growth with  = 2.00 ± 0.01; for fp z = 0.632 ± 0.003, isotropic growth of randomly dispersed nuclei; for fp z = 0.649 ± 0.003, non-randomly dispersed nuclei with ε = 2.04 ± 0.05. Both the results of the two recipes agree well with the input values. But the newly proposed analysis is more direct, since the procedure to determine the value of n is no longer needed. 3.2.2. Non-isothermal NES transformation The evolution of f vs. T (see Fig. 6a) is numerically calculated for the case of non-isothermal NES transformation with continuous nucleation, interface-controlled growth and impingement due to isotropic growth of randomly dispersed nuclei (i.e.  = ε = 1), for a series of cooling rates (i.e., ˚ = −10, −20 and −40 K min−1 ), with the values for model parameters gathered in Table 3 of Ref. [14]. Consequently, the evolutions of df/dT vs. f and z vs. f are shown in Fig. 6b and c, respectively. It can be seen that the values of fp are varying with cooling rate, but the values of fp z almost remain constant. First, the non-isothermal maximum transformation rate analysis is performed. Following the values of fp indicated in Fig. 6b (fp = 0.572 ± 0.003, 0.556 ± 0.003 and 0.523 ± 0.003, respectively, for ˚ = −10, −20 and −40 K min−1 ), the impingement modes can be determined by the application of Eq. (6), i.e., they are anisotropic growth with  = 1.38 ± 0.01, 1.50 ± 0.01 and 1.78 ± 0.01, respectively, for ˚ = −10, −20 and −40 K min−1 . Then, following the values of fp z indicated in Fig. 6c (fp z ≈ 0.632), Eq. (19) also gives the impingement mode, i.e., it is isotropic growth of randomly dispersed nuclei for each cooling rate. Subject to the effect of thermodynamic term, the degree of the deviation of fp from 1 − e−1 varies with cooling rate for this transformation, see Fig. 6b. Therefore, the maximum transformation rate analysis cannot give the correct impingement factor. However, the impingement factor obtained from the new recipe is in good agreement with the input value. 4. Experimental example 4.1. Isothermal ferrite/austenite transformation in Fe–3.28at.%Mn alloy Experimental details regarding DIL specimen preparation and the subsequence DIL analysis are given in Ref. [14]. The temperature-time program in the DIL experiment was that the specimen was heated up from room temperature up to 1068 K at a rate of 20 K min−1 and kept at this temperature for 80 min. Then following the method in Ref. [30], the austenite fraction, f , can be obtained from the DIL experimental data. Accordingly,

Fig. 5. Avrami exponent at the maximum of transformation rate, np , determined from Eq. (4) (lines) and Eq. (5) (lines + symbols) as function of the assumed impingement factor for the transformations described in Fig. 1a: (a) isotropic growth of randomly dispersed nuclei, (b) anisotropic growth and (c) non-randomly dispersed nuclei.

the evolutions of austenite transformation rate, df /dt, vs. f , and function z vs. f are shown in Fig. 7a and b, respectively. From Fig. 7, the austenite fraction corresponding to the maximum of df /dt and function z can be given as: fp = 0.315 ± 0.010, fp z = 0.489 ± 0.010. According to the value of fp z and Eq. (19b), the identified impingement mode of the isothermal ferrite to austenite transformation is anisotropic growth with  = 2.12 ± 0.11 (c.f. Fig. 3).

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Fig. 7. Evolution of: (a) austenite transformation rate, df /dt, with austenite fraction, f (b) function z with f for the isothermal (annealing at 1068 K) ferrite/austenite phase transformation in Fe–3.28at.%Mn alloy as measured by DIL.

4.2. Non-isothermal hcp/bcc phase transformation in pure Zr

Fig. 6. Evolutions of: (a) transformed fraction, f, with T, (b) transformation rate, df/dT, with f and (c) function z with f for non-isothermal NES transformation with constant cooling rate (i.e., ˚ = −10, −20 and −40 K min−1 ) for the case of continuous nucleation, interface-controlled growth and impingement due to isotropic growth of randomly dispersed nuclei, as numerically calculated using values for model parameters gathered in Table 3 of Ref. [14].

In combination with the identified impingement factor and value of fp , the Avrami exponent can be determined by applying Eq. (5b), i.e., n = 2.89 ± 0.09. Generally, the growth mode of massive transformation is three-dimensional interface-controlled, which corresponds with a growth exponent between 3 and 4. The determined n for the transformation, which is close to 3, implies that the identified impingement mode is reasonable.

The hcp/bcc phase transformation in pure Zr was investigated by recording non-isothermal DSC scans at different heating rates. DSC samples were cut from crystal bar zirconium (purity 99.9%) containing < 50 wt. ppm O. DSC measurements were carried out using Netzsch STA 449 C thermal analysis instrument. Upon a certain temperature-time program, the heat flow difference between the pure Zr sample and the reference was recorded as function of time. The temperature-time programs in the DSC experiment was non-isothermal heating with constant heating rates from 5 to 30 K min−1 . A protective gas atmosphere of pure argon was employed. The sample and reference pan are made of Al2 O3 . Following the method in Ref. [31], the evolution of bcc phase fraction, fˇ , vs. T and bcc phase transformation rate, dfˇ /dt, vs. fˇ can be obtained from the enthalpy change rates (see Fig. 8a and Fig. 8b). Since the transformation conducts in the vicinity of the equilibrium transformation temperature, the values of fp are varying with heating rates (c.f. Fig. 8b). Applying the “Kissinger plot” to the transformations as measured at different heating rates (i.e., Fig. 8a), the values for QH at the heating rate of 15 K min−1 can be obtained3 . Consequently, from Eq. (15b), the function z corresponding to ˚ = 15 K min−1 can be constructed, which is shown in Fig. 8c.

3 For the FES transformations during continuous heating, the “Kissinger plot” leads to the value of Q. However, for NES transformations, due to the effect of thermodynamic term, it leads to a modified value, i.e., QH [14].

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samples. Therefore, it is reasonable to assume that the nucleation sites (i.e. the grain boundaries) are non-randomly distribution for this transformation. 5. Conclusions The impingement mode affects significantly the values for Avrami exponent obtained from kinetic analysis. Therefore, the maximum transformation rate analysis, which was proposed to identify the impingement mode of FES transformations, was reevaluated for general case, i.e., both FES and NES transformations. Due to the effect of thermodynamic term, the non-isothermal maximum transformation rate analysis cannot be applied in FES transformations. Consequently, a new and simple method, which can be applied to both isothermal and non-isothermal general case, was proposed for identification of impingement factor. The newly proposed method has been applied to the real transformation. Following from the new recipe analyzing, the impingement mode of the isothermal ferrite/austenite transformation in Fe–3.28at.%Mn alloy may be due to the anisotropic growth with  = 2.12 ± 0.11. For the non-isothermal hcp/bcc phase transformation in pure Zr, it indicated that the transformation may be due to non-randomly dispersed nuclei with ε = 3.47 ± 0.26. Acknowledgments The authors are grateful to the financial support of the National Basic Research Program of China (973 Program, No. 2011CB610403), the Natural Science Foundation of China (Nos. 51071127 and 51134011), the Fundamental Research Fund of Northwestern Polytechnical University (No. JC20120223), the Doctorate Foundation of Northwestern Polytechnical University (No. CX201311) and the China National Funds for Distinguished Young Scientists (No. 51125002). References

Fig. 8. Experimental values of: (a) bcc phase fraction, fˇ , with T and (b) bcc phase transformation rate, dfˇ /dT, with fˇ for the non-isothermal (annealing with ˚ = 5, 10, 15, 20 and 30) hcp/bcc phase transformation in pure Zr as measured by DSC; (c) evolution of function z with fˇ for the transformation annealing with ˚ = 15 K min−1 .

It follows from Fig. 8c that fp z = 0.685 ± 0.010. This corresponds with non-randomly dispersed nuclei with ε = 3.47 ± 0.26 (c.f. Eq. (19c) and Fig. 3). The result is in agreement with experimental fact. For massive transformation, the grain boundary which can provide the additional energy for nucleation always serves as the nucleation site. The optical microscopy analysis indicates that the hcp phase grain size is rather coarse compared with the size of DSC

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