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Scripta Materialia 124 (2016) 164–168
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Thin film phase transformation kinetics: From theory to experiment
M.M. Moghadam ⁎, P.W. Voorhees Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA
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http://dx.doi.org/10.1016/j.scriptamat.2016.07.010 1359-6462/© 2016 Acta Materialia Inc. Published by Elsev
a b s t r a c ta r t i c l e i n f o
Article history: Received 23 May 2016 Accepted 8 July 2016 Available online xxxx
The Level-set method simulation is used to address the effect of finite size on kinetics of thin film phase transfor- mations. The results arefirst interpretedusing the classic Johnson-Mehl-Avrami-Kolmogorov (JMAK) description of a nucleation and growth phase transformation that yields the average Avrami exponent and rate constant as a function of film thickness. The analysis reveals that the JMAK framework can yield a spurious thickness depen- dent activation energy for the transformation. To overcome this problem, we propose an analysis that allows all the kinetic parameters, including the nucleation rate and interface growth velocity in films to be determined from experiment.
© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Phase transformation Kinetics Thin films Activation analysis JMAK theory
Many first order transformations such as crystallization from amor- phous state, relaxation of polarized domain in ferroelectrics, spherulitic crystallization in polymers, cellular precipitation and recrystallization take place through nucleation and interface-limited growth [1–11]. Ki- netics of these transformations is commonly addressed through the classic theory of Johnson-Mehl- Avrami-Kolmogorov (JMAK) [12–16], due to its simplicity and effectiveness. However, the assumptions used in this theory are often violated, which makes it necessary to consider the validity of the approach for a desired system. In particular, the valid- ity of JMAK theory in thin films has been questioned, owing to the as- sumption used in the theory of an infinite system. There have been many successful attempts removing the restrictive assumptions of the JMAK approach [17–20]. However, majority of these studies use sophis- ticated analytical methods and find complicated models as a result, in contrast to simplicity of the original JMAK equation. Hence, they are dif- ficult to implement andmost of the experimental data are still being an- alyzed using the JMAK equation. Here, we consider isothermal isotropic phase transformations in thin films obtained by level-set simulations. The virtue of level-set method is that it is possible to quantify the effect of film thickness on the coefficients of the JMAK theory and thuswe de- velop amethod to extract kinetic parameters such as the nucleation rate and interface velocity from experimental data.
JMAK theory is based on the geometrical evolution of an extended volume of transformations that occur via nucleation and growth. Nucle- ation mechanisms are often categorized based on how frequent nuclei appear in the system. If nucleation happens only at the beginning and growth takes place afterwards, nucleation mechanismwould be identi- fied as a Site-Saturate Nucleation (SSN). While if the system keeps
u (M.M. Moghadam).
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adding nuclei with the same rate over the entire transformation period, it would be defined as constant nucleation rate (CNR). The conventional format of the JMAK equation is shown in Eq. (1).
f V ¼ 1− exp −ktnð Þ ð1Þ
where, fV is the transformed volume fraction, t is time, n is the Avrami exponent and k is the reaction rate constant. The Avrami exponent con- tains information about growth dimensionality (D). In particular, for constant nucleation rate, the Avrami exponent is n=D+1, while for site-saturated nucleation n=D. The rate constant, on the other hand, is computed using the extended volume evolution (regardless of im- pingement). For constant nucleation ratek ¼ ðπ=3Þv3 _NV and for site sat- urated nucleation k=(4π/3)v3NV, where, _NV is constant nucleation rate, NV is nucleation density and v is the constant interface velocity. A sys- tem that has undergone a phase transformation via nucleation and growth processes can be described by two characteristic parameters. The first is a characteristic length (λ), which is related to the final aver-
age grain size and can be defined asλ ¼ ð _NV=vÞ−1=4 for a constant nucle- ation rate and λ ¼ NV−1=3 for site-saturated nucleation. The second parameter is a characteristic time (τ), which is the time scale of the transformation process and defined as τ=λ/v. Clearly both λ and τ are temperature dependent. In addition to fundamental information that is embedded in these two parameters, they can be used to scale all other system's quantities in a way to be comparable with other sys- tems represented by different nucleation rates/densities or interface ve- locities. In this paper we use * symbol to denote dimensionless parameters, t⁎= t/τ and h⁎=h/λ, where h is the film thickness. In these dimensionless variables, _NV
� ¼ 1,NV � ¼ 1 and v⁎=1. We also re- write Eq. (1) based on the characteristic parameters for constant
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Fig. 1.Avrami plot for transformation in different film thicknesses. (a) constant nucleation rate, (b) site-saturated nucleation. Solid lines demonstrate the slope associated with a given n.
165M.M. Moghadam, P.W. Voorhees / Scripta Materialia 124 (2016) 164–168
nucleation rate (Eq. (2a)) and site-saturated nucleation (Eq. (2b)) as follows [18–21]:
f V ¼ 1− exp − π 3 t�4
h i ð2aÞ
f V ¼ 1− exp − 4π 3
t�3 � �
Phase transformations via nucleation and interface-limited growth are an ideal case to be simulated by the level-set method. It can follow grain coalescence as well as account for the effects of contact with the external boundary. Evolution on experimental length and time scales is another advantage of this method, which makes its results directly comparablewith experiments. A detailed explanation of level-setmeth- od and its verification for this application can be found in the literature [22–25]. All presented results are the average values over 40 runs with different random initial configurations of the nuclei. Some of the data appearing in this paper is presented in order to provide a comprehen- sive analysis of different nucleation mechanisms.
The JMAK equation is commonly used in the form of Eq. (3) to con- struct the Avrami plot (Fig. 1).
ln − ln 1− f V½ �½ � ¼ ln k�½ � þ n ln t�½ � ð3Þ
Presenting phase transformation results in this format can be quite informative and makes a comparison of the kinetic parameters much easier. As is evident from Eq. (3), in the Avrami plot the slope of each line gives the Avrami exponent while y-intercept at ln[t⁎]=0 gives log- arithm of rate constant.
Fig. 1 shows Avrami plots under constant nucleation rate and site- saturated nucleation for different film thicknesses. As it is shown, the Avrami exponent (n) depends on the thickness of the film. Moreover, the value of rate constant changes with film thickness, as all lines inter- cept the line ln[t⁎]=0 at different values, whereas according to the JMAK theory in the bulk system it is expected to be constant for all runs at the same temperature .
Fig. 2 illustrates the effect of the film thickness on the JMAK coef- ficients. As the thickness of film increases, the rate constant value as- ymptotes to the value of π/3 for constant nucleation rate, consistent with Eq. (2a). Clearly, the value of the rate constant that is found using the classical JMAK 3D treatment does not hold in thin films, es- pecially when the film thickness is smaller than characteristic length of the system [24,25]. For very thin films (h⁎b1) where one assumes that most of the growth occurs after surface impingement it yields a rate constant of k⁎=πh⁎/3, shown by the dotted line in Fig. 2a . A similar trend can be observed for site-saturated nucleation. As the film thickness becomes large, k⁎ approaches 4π/3 in agreement with Eq. (2b). By contrast, for thicknesses less than characteristic length there is a strong dependence of k⁎ on h⁎ as shown by the dashed-dotted line k⁎=πh⁎ in Fig. 2a . Fig. 2b also shows that the average Avrami exponent (�n) varies with h*, where �n is defined as the average of local Avrami exponent [24–26] over the trans- formed volume fraction from 10% to 90%. At very small film thick- ness, the average Avrami exponent starts from three as a characteristic of 2D growth for CNR and increases linearly to the point where film thickness equals the characteristic length (h⁎= 1). For thicknesses larger than the characteristic length, �n shows a smaller change with h⁎ until it reaches a plateau at h⁎=5 with a value close to four as given by 3D growth. SSN also exhibits similar behavior observable with sharp transition from almost 2D growth (�n ≈2) at small thickness to h⁎=1 and then a plateau toward 3D growth (�n≈3). Since all data in Fig. 2