Research Article 
Corresponding author: Jamie P. Wooding ( jwooding3@gatech.edu ) Academic editor: J. Ruud Van Ommen
© 2023 Jamie P. Wooding, Shawn A. Gregory, Amalie Atassi, Guillaume Freychet, Kyriaki Kalaitzidou, Mark D. Losego.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Wooding JP, Gregory SA, Atassi A, Freychet G, Kalaitzidou K, Losego MD (2023) Transformation kinetics for low temperature postdeposition crystallization of TiO _{2} thin films prepared via atomic layer deposition (ALD) from tetrakis(dimethylamino)titanium(IV) (TDMAT) and water. Atomic Layer Deposition 1: 118. https://doi.org/10.3897/aldj.1.101276

Background: We report on the fundamental crystallization kinetics of atomic layer deposited (ALD) TiO2 thin films undergoing a postdeposition anneal (PDA) at low temperatures to probe differences in the asdeposited film microstructure.
Methods: The system of study is ALD TiO2 thin films prepared from tetrakis(dimethylamino)titanium(IV) (TDMAT) and water at 120 °C, 140 °C and 160 °C followed by ex situ low temperature annealing at temperatures ranging from 140 °C to 220 °C. All asdeposited TiO2 thin films are amorphous by Xray diffraction (XRD). Postdeposition annealing (PDA) produces large grain anatase crystals, confirmed by XRD and topview scanning electron microscopy (SEM). A detailed SEM study is performed to quantify the nucleation and growth kinetics by fitting microstructural data to the JohnsonMehlAvramiKolmogorov (JMAK) equation. Finally, a timetemperaturetransformation (TTT) diagram is constructed to summarize the differences in crystallization behavior at different ALD deposition temperatures.
Results and conclusions: Fitting microstructural data to the JMAK equation reveals an Avrami exponent close to 3 with continuous nucleation, suggesting twodimensional, platelike crystal growth. Applying an Arrhenius relationship to the phase transformation data, the combined activation energy for nucleation and growth is found to be 1.40–1.58 eV atom1. Nucleation rates are determined, and an Arrhenius relationship is used to calculate the critical Gibbs free energy for nucleation (~1.3–1.4 eV atom1). As such, nucleation is the ratelimiting step for the amorphous to anatase phase transformation. ALD growth temperature is found to dictate film microstructure with lower deposition temperatures reducing the nucleation rate and leading to larger grain sizes irrespective of PDA conditions. The nucleation rate preexponential frequency factor increases with increasing deposition temperature, thereby increasing the likelihood for nucleation. Interestingly, it is this difference in the vibrational modes of the amorphous structure, as indicated by the variation in the nucleation rate preexponential frequency factor, that alters the phase transformation rates and not a change in the activation energies for the transformation.
Nucleation and Growth, Microstructure, TiO2, crystal growth, anatasexray diffraction (XRD), JohnsonMehlAvramiKolmogorov (JMAK) equation, Thin Film growth, kinetics, phase transformation
Titanium dioxide (TiO_{2}) is a widebandgap semiconductor with phasedependent high refractive index and high dielectric constant [
TiO_{2}ALD has been studied extensively using TiCl_{4} and H_{2}O as coreactants [
Crystalline asdeposited ALD films often show columnar grains through the film thickness. The grain size is typically less than the film thickness given a high density of nuclei during the initial stage of film growth [
Plot of the crystalsizetofilmthickness ratio as a function of the crystallization temperature compiled from various TiO2 ALD literature reports. All reports use thermalALD with water as the coreactant. Varying titanium precursor and substrate chemistries are noted by data point shape as indicated in the legend. Conditions reported include: TiCl4/H2O on Si as deposited (black square), TiCl4/H2O on Al2O3 as deposited (green square), TiCl4/H2O on Al2O3 with PDA (circled green square), TDMAT/H2O on Si as deposited (red circle), and TDMAT/H2O on Si with PDA, which are the results from this study (circled red star). Films that undergo PDA are all circled in blue.
Other reports have probed resultant ALDTiO_{2} film structure with postdeposition annealing (PDA) [
Here, we describe the first detailed study of anatase formation in TiO2 thin films deposited on a silicon wafer substrate from TDMAT/H2O thermal ALD and postdeposition annealing (PDA) in air. This study provides new information on TiO2 crystal growth during postdeposition annealing and challenges the present understanding of the temperatures required for crystallization to occur. The temperaturedependent crystallization kinetics presented are determined from an array of deposition and annealing temperatures. Using the JohnsonMehlAvramiKolmogorov (JMAK) equation, the Avrami exponent is calculated and the lowtemperature anatase crystal growth mode is identified. Further, the activation energy for anatase nucleation and growth is calculated, nucleation rates are determined, and the critical Gibbs free energy for nucleation is deconvoluted from the activation energy for crystal growth. This study advances understanding of the amorphous to crystalline phase transformation in ALD TiO2 thin films and identifies how the asdeposited amorphous structure of a film can influence crystallization kinetics and final film microstructure.
ALD was conducted in a homebuilt, hotwalled, flowtube reactor with custom LabVIEW control software [
Tetrakis(dimethylamino)titanium(IV) (TDMAT, 99% purity) from Strem Chemicals, Inc. (Newburyport, MA, USA) and deionized water were used as the precursor and coreactant for ALD of TiO2 films. TDMAT was dosed in a bubbler configuration with delivery lines to the reactor heated to 82 °C. Each ALD cycle used a deposition sequence of 1.0 s TDMAT / 5 s N2 purge / 0.4 s H2O / 85 s N2 purge. For all conditions, 1120 ALD cycles were deposited to yield 50 nm to 58 nm TiO2 films. ALD growth per cycle (GPC) values varied from 0.44 Å to 0.54 Å and are presented in Suppl. material
Spectroscopic ellipsometry (alphaSE, J.A. Woollam) was used to measure film thickness and index of refraction at 550 nm. A CodyLor model was used to describe the TiO_{2} layer on top of the native oxide layer on a Si substrate. Before ALD, the measured SiO_{2} native oxide layer on Si was approximately 1.7 nm, determined by modeling a native oxide layer on Si in J.A. Woollam’s CompleteEASE software. Grazing incidence Xray diffraction (GIXRD) was used to identify the crystalline phase of the deposited films. GIXRD was conducted on a PANalytical Empyrean system using CuKα radiation, a BBHD source optic, and a PIXcel 2dimensional detector. 2θω scans were taken. Additionally, grazing incidence wideangle Xray scattering (GIWAXS) measurements were performed on a TiO_{2} film set for asdeposited, short PDA, and long PDA conditions. GIWAXS was used to verify that the initial crystal growth viewed in SEM is indeed the TiO_{2} anatase crystalline phase that is measurable by GIXRD upon further transformation. Measurements were performed at the Soft Matter Interfaces Beamline at the National Synchrotron Light Source II with an incident energy of 12 keV and an incident angle of 0.15 degrees. Scattering patterns were recorded on a Pilatus 900 K−W detector, consisting of 0.172 mm square pixels, mounted at a fixed distance of 0.279 m from the sample position. The range of scattering angles was acquired by moving the detector horizontally on a fixed arc and postprocessing the images using Xi CAM software (Pandolfi et al. 2018). Both the samples holder and the detector were enclosed in a vacuum chamber.
A Hitachi SU8230 field emission SEM (FESEM) operated at 1 kV accelerating voltage and 15–20 µA emission current was used to image microstructure and crystal growth from the secondary electron (SE) signal. In the images, amorphous regions appear dark while crystalline regions tend to appear bright. For each condition, at least two TiO_{2} films were prepared and imaged, and images were taken top down at three different lateral locations on the film surface. Image analysis was performed with ImageJ to quantify the crystalline fraction transformed and to determine crystalline area density and size.
Figure
a) SEM micrographs for ALD films grown at 120 °C, 140 °C, and 160 °C: asdeposited (left) vs. postdeposition annealed at 200 °C in air (right). Anneal times to achieve apparent full crystallinity are listed on micrographs. Note the scale bar is different in the 120 °C PDA 200 °C micrograph to capture the large grain size. b) GIXRD scans for asdeposited and annealed (PDA at 200 °C in air) films deposited at 120 °C, 140 °C, and 160 °C. The three primary TiO2anatase peaks are labeled: 2θ = 25.3°, 37.8°, 48.0°.
Regardless, such large grain growth (>1 μm grain size in ~50 nm thick films) has not been observed before in ALD TiO_{2} films using the TDMAT/H_{2}O chemistry. Further, it is surprising that the resultant microstructure varies so significantly depending on deposition temperature, with grain size decreasing from 4.6 µm to 540 nm with a 40 °C increase in temperature. To provide further insights into this crystallization process and to determine why resultant, annealed microstructure is determined by ALD temperature, we undertake a comprehensive timedependent study of the crystalline transformation kinetics to understand the underlying activation energies.
Crystallization kinetics are studied over a range of ALD temperatures and postdeposition annealing (PDA) temperatures. Specifically, TiO_{2} films deposited at substrate temperatures of 140 °C and 160 °C are annealed in air at 160 °C, 180 °C, and 200 °C, while TiO_{2} films deposited at 120 °C are annealed in air at 180 °C, 200 °C, and 220 °C. Figure
SEM micrographs for TiO2 films deposited at a) 140 °C and postdeposition annealed (PDA) at 200 °C for i) 0 h, ii) 3.9 h, and iii) 10 h, at 180 °C for iv) 0 h, v) 12 h, and vi) 47 h, and at 160 °C for vii) 0 h, viii) 110 h, and ix) 327 h. SEM micrographs for TiO2 films deposited at b) 160 °C and postdeposition annealed (PDA) at 200 °C for i) 0 h, ii) 3.9 h, and iii) 5.9 h, at 180 °C for iv) 0 h, v) 12 h, and vi) 25 h, and at 160 °C for vii) 0 h, viii) 67 h, and ix) 109 h. Annealing time in hours is for the specified time, t, duration.
$X\left(t\right)=1\mathrm{exp}\left(k{t}^{n}\right)$ Equation 1
Here, k is the reaction rate for combined anatase nucleation and growth and n is the Avrami exponent, which often characterizes the mechanism and dimensionality of the phase transformation. The JMAK equation as presented in Equation 1 is linearized to the following form:
$\mathrm{ln}\left[\mathrm{ln}\frac{1}{1X\left(t\right)}\right]=n\mathrm{ln}k+n\mathrm{ln}t$ Equation 2
In plotting $\mathrm{ln}\left[\mathrm{ln}\frac{1}{1X\left(t\right)}\right]$ vs. ln t, the slope of this line is equal to the Avrami exponent n and the yintercept is equal to n ln t. Applying this linearization to both deposition temperatures at each of the three PDA temperatures (Figure
Calculated Avrami exponent n and reaction rate k for each ALD and PDA condition.
Dep T (°C)  PDA T (°C)  Avrami exponent, n  Reaction rate, k 

160  200  2.51 ± 0.04  9.6 × 10^{5} ± 0.05 × 10^{5} 
160  180  3.39 ± 0.08  2.2 × 10^{5} ± 0.007 × 10^{5} 
160  160  2.85 ± 0.08  4.0 × 10^{6} ± 0.04 × 10^{6} 
140  200  2.80 ± 0.02  4.6 × 10^{5} ± 0.004 × 10^{5} 
140  180  2.82 ± 0.09  8.8 × 10^{6} ± 0.02 × 10^{6} 
140  160  3.25 ± 0.04  1.3 × 10^{6} ± 0.005 × 10^{6} 
120  220  2.34 ± 0.02  6.8 × 10^{6} ± 0.04 × 10^{6} 
120  200  2.31 ± 0.02  2.2 × 10^{6} ± 0.009 × 10^{6} 
120  180  2.49 ± 0.04  4.1 × 10^{7} ± 0.05 × 10^{7} 
a) Amorphous TiO2 to anatase phase transformation curves for films deposited at (a) 120 °C (solid square marker), (b) 140 °C (solid triangle marker) and (c) 160 °C (solid circle marker) and postdeposition annealed at 160 °C (goldcolored), 180 °C (bluecolored), 200 °C (redcolored), and 220 °C (blackcolored). The solid lines are the X(t) model fit for the calculated reaction rate k and Avrami exponent n values. Linearized form of the crystalline fraction transformation curves to extract calculated reaction rate k and Avrami exponent n values for films deposited at (d) 120 °C, (e) 140 °C, and (f) 160 °C.
For the complete experimental set, the values for the Avrami exponent n vary from 2.31 to 3.39, with an average value of n = 2.75. The experimental variation in n may be attributed to the nature of these ex situ SEM measurements: the film is removed for imaging, which causes small deviations in time and temperature associated with the cooling and reheating of the film. The Avrami exponent should remain constant despite changes in temperature given no apparent changes in the crystal nucleation and growth mode. Given the similarity in crystalline microstructures, the crystal nucleation and growth mode at these deposition temperatures appears to be constant. The value for the Avrami exponent is dependent on the dimensionality of the product phase, the time dependence of the nucleation, and the time dependence of the rate limiting growth step. The ratelimiting growth step for an amorphoustocrystalline transformation is the interfacial reaction. Here, nucleation appears to occur continuously with film annealing, rather than all at once. Given these conditions, an Avrami exponent close to 3 is characteristic of a twodimensional growth mode, indicative of platelike microstructure observed here where there are micronsized grains of only 50 nm thickness [
As shown in Table
To calculate the combined activation energy for nucleation and growth, an Arrhenius relationship [
$\mathrm{k}={k}_{0}*\mathrm{exp}\left(\frac{{E}_{nucgrowth}}{{k}_{B}{T}_{PDA}}\right)$ Equation 3
In Equation 5, E_{nucgrowth} is the activation energy for crystal nucleation and growth, k_{B} is the Boltzmann constant, T_{PDA} is the annealing temperature, and k_{0} is a materialdependent frequency factor. The linearized form of Equation 3 is:
$\mathrm{ln}\left(k\right)=\left(\frac{{E}_{nucgrowth}}{{k}_{B}}\right)\frac{1}{{T}_{PDA}}+\mathrm{ln}\left({k}_{0}\right)$ Equation 4
Figure
To better understand how anatase nucleation and growth rates vary as a function of ALD and PDA temperature, we further consider how the reaction rate k from the JMAK equation is defined (Equation 1). Equation 1 presents the generalized form of the JMAK equation. However, in considering a twodimensional transforming area with continuous nucleation and growth, as is the case in this experimental work, the equation can be given as:
$X\left(t\right)=1\mathrm{exp}\left(\frac{\pi}{3}\dot{N}{\dot{v}}^{2}{t}^{3}\right)$ Equation 5
Here, Ṅ is defined as the nucleation rate and v˙ as the crystal growth rate. This derivation is presented in detail in Suppl. material
$k=\frac{\pi}{3}\dot{N}{\dot{v}}^{2}$ Equation 6
Recall that the reaction rate k is determined experimentally from the slope equal to n ln k in the linearized JMAK equation (Equation 2). By deconvoluting the reaction rate into nucleation rate and crystal growth rate, we can separate nucleation kinetics and crystal growth kinetics.
To accomplish this deconvolution, crystal nucleation rate Ṅ is independently determined from short PDA times, prior to impingement of growing crystals. The number of crystals per area in the SEM images are counted at these short PDA times to give a nucleation rate of number of crystals m^{2} s^{1}. The instantaneous nucleation rate Ṅ is plotted on a log scale against PDA temperature as a function of deposition temperature in Figure
a) Loglinear plot of nucleation rates as a function of postdeposition annealing (PDA) temperature for ALDTiO2 thin films deposited at 120 °C, 140 °C, and 160 °C. b) Linearized nucleation rate equation for films deposited at 120 °C, 140 °C, and 160 °C with linear regression line of best fit.
$\dot{N}={N}_{j}\mathrm{exp}\left(\frac{\Delta {G}_{nuc}^{*}}{{k}_{B}{T}_{PDA}}\right)$ Equation 7
In Equation 7, ∆G^{*}_{nuc} is the critical Gibbs free energy for nucleation, k_{B} is the Boltzmann constant, T is the PDA temperature, and N_{j} is the nucleation exponential precursor. Equation 7 is linearized to:
$\mathrm{ln}\left(\dot{N}\right)=\left(\frac{\Delta {G}_{nuc}^{*}}{{k}_{B}}\right)\frac{1}{{T}_{PDA}}+\mathrm{ln}\left({N}_{j}\right)$ Equation 8
Here, N_{j} is equal to f_{nucleation} * C_{het} where f_{nucleation} is a frequency factor for the phase transformation and C_{het} is the concentration of heterogeneous nucleation sites per areal density. The nucleation frequency factor includes the vibrational frequency of the atoms, the area of the critical nucleus, and an activation energy for atomic migration [
As seen in Figure
Experimentally determined values for reaction rate k and nucleation rate Ṅ are used to calculate the crystal growth rate for each deposition and annealing condition. For twodimensional, platelike growth, Equation 6 can be solved for crystal growth rate v˙ as follows:
$\dot{v}=\sqrt{\frac{3k}{\pi \dot{N}}}$ Equation 9
Here, the reaction rate and the average nucleation rate at a given deposition temperature and annealing temperature are used to calculate each crystal growth rate. These instantaneous crystal growth rates are plotted against deposition temperature in Figure
a) Crystal growth rates plotted as a function of deposition temperature for ALDTiO2 films deposited at 120 °C, 140 °C, and 160 °C. b) Arrhenius plot for crystal growth rate for TiO2 films deposited at 120 °C, 140 °C, and 160 °C with linear regression lines of best fit and 95% confidence intervals. The Rsquare values for 120 °C, 140 °C, and 160 °C lines respectively are: 0.001, 0.123, and 0.056.
Next, we apply the nucleation rate analysis structure to crystal growth rate. Here, the Arrhenius relationship for crystal growth rate is given as:
$\dot{v}=B\mathrm{exp}\left(\frac{{E}_{\text{crystal growth}}}{{k}_{B}{T}_{PDA}}\right)$ Equation 10
With the linearized form:
$\mathrm{ln}\left(\dot{v}\right)=\left(\frac{{E}_{\text{crystal growth}}}{{k}_{B}}\right)\frac{1}{{T}_{PDA}}+\mathrm{ln}\left(A\right)$ Equation 11
In Equation 10 and Equation 11, k_{B} is the Boltzmann constant, T is the PDA temperature, and A is the exponential prefactor. PDA temperature is used in Equation 10 and Equation 11 because that is the temperature at which the isothermal grain growth occurs. Figure
Calculated values for the critical Gibbs free energy for nucleation, ∆G^{*}_{nuc}, and the combined activation energy for nucleation and growth, E_{nucgrowth}, are reported in Table
Calculated fundamental activation energies: critical Gibbs free energy for nucleation and combined nucleationgrain growth for each ALD temperature, and reaction rate frequency factor.
Deposition Temperature (°C)  ∆G^{*}_{nuc} (eV atom^{1})  E_{nucgrowth} (eV atom^{1})  Reaction rate frequency factor, k_{0} (s^{1}) 

120  1.41 ± 0.08  1.46 ± 0.06  1.1 × 10^{10} ± 0.04 × 10^{10} 
140  1.35 ± 0.15  1.58 ± 0.03  2.5 × 10^{11} ± 0.2 × 10^{11} 
160  1.32 ± 0.10  1.40 ± 0.03  6.2 × 10^{11} ± 0.3 × 10^{11} 
The above analysis deconvolutes the amorphous to anatase phase transformation reaction rate into separate nucleation and growth rates. We can apply this understanding of these fundamental kinetics to understand why the crystalline microstructures after PDA vary with ALD deposition temperature.
Figure
To discern the contribution of the crystal growth rate to final grain size, we compare the crystal size of films deposited at 160 °C and annealed at 180 °C (ALD160 °C / PDA180 °C) to those deposited at 140 °C and annealed at 200 °C (ALD140 °C / PDA200 °C). Referencing Figure
With the presented phase transformation study for TiO_{2} thin films, isothermal timetemperaturetransformation (TTT) diagrams can be developed to describe the TiO_{2} amorphous to anatase solidstate phase transformation via postdeposition annealing (PDA). TTT diagrams can offer useful information for metal oxide thin films where the crystallization is heatingrate dependent and occurs in the solid state [
Timetemperaturetransformation (TTT) diagrams for TiO2 amorphous to anatase phase transformation via isothermal postdeposition anneals for 1120 cycle ALD films deposited at temperatures: (a) 120 °C, (b) 140 °C, (c) 160 °C. Transformation curves are labeled for 5%, 50%, and 95% transformation to the TiO2anatase phase. The abscissa axis is in base10 log time.
We report an atomic layer deposition (ALD) process with a PDA step to grow large grain TiO_{2} anatase on a Si substrate from TDMAT/H_{2}O chemistry. Largegrained anatase minimizes grain boundary volume and enables a greater proportion of the film to contribute its functional performance. The TiO_{2}ALD is deposited from 120 °C to 160 °C with ex situ annealing in air from 140–220 °C. A detailed SEM study of the TiO_{2}amorphous to TiO_{2}anatase phase transformation is presented to extract fundamental nucleation and growth kinetics values and to highlight structural differences in the asdeposited film structure as a function of deposition temperature. The JMAK equation is used to determine an average Avrami exponent close to 3 for the transforming volume, consistent with a twodimensional, platelike anatase growth mode with continuous nucleation. The critical Gibbs free energy for anatase nucleation is 1.32–1.41 eV atom^{1} and the combined activation energy for nucleation and growth is 1.40–1.58 eV atom^{1}, while the activation energy for crystal growth is near zero. Thus, nucleation appears to be the ratelimiting step. Interestingly, in studying the nucleation rate at different PDA temperatures as a function of ALD temperature, we identify the nucleation frequency factor, not the activation energy, as the differentiating quantity between the asdeposited TiO_{2} films. Amorphous TiO_{2} films deposited at higher temperatures have an increased nucleation frequency factor compared to those deposited at lower temperatures, potentially indicating that asdeposited higher temperature films have increased vibrational modes to attempt nuclei formation. As a result, ALD deposition temperature has a significant effect on the resultant grain size, with 20 °C increases in temperature decreasing the grain size by almost an order of magnitude. Finally, a timetemperaturetransformation (TTT) diagram architecture has been applied to highlight the differences in crystallization behavior between ALD temperatures.
In this work, we present a method to study differences in an asdeposited, amorphous TiO_{2} film as a function of ALD deposition temperature and introduce the nucleation rate frequency factor f_{nucleation} as an indicator of resultant nucleation and microstructure. Further, we separate the nucleation and growth kinetics for the amorphous to anatase phase transformation, demonstrating that nucleation is the ratelimiting step for the phase transformation under standard ALD process conditions. Finally, we demonstrate a practical lowtemperature route to grow largegrained anatase films which expand their use in applications requiring temperaturesensitive substrates and devices.
J.P.W. acknowledges this material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE2039655. A significant portion of this work was performed at the Georgia Tech Institute for Electronics and Nanotechnology, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (ECCS2025462). Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors recognize Proposal GU309504 for scattering measurements at Beamline 12ID at National Synchrotron Light SourceII at Brookhaven National Lab. A.A. appreciates the support of the National Science Foundation Graduate Research Fellowship under Grant No. DGE1650044. S.A.G. acknowledges financial support from a Link Foundation Energy Fellowship. J.P.W. thanks D. Tavakoli, T. Walters, and R. Monikandan for assistance with XRD and SEM.
Additional information
Data type: figures and tables (PDF file)
Explanation note: fig. S1. a) ALD growth per cycle (GPC) from 1120 TDMAT/H2O cycles for asdeposited TiO2 thin films at ALD temperatures: 120 °C, 140 °C, and 160 °C. b) Index of refraction, n, measured at 550 nm for asdeposited and postdeposition anneal TiO2 thin films at ALD temperatures: 120 °C, 140 °C, and 160 °C; fig. S2. a) GIWAXS scans for films deposited at 140 °C: asdeposited, PDA at 200 °C for 2 h, and PDA at 200 °C for greater than 95% X(t) transformed. b) GIWAXS scans for films deposited at 160 °C: asdeposited, PDA at 200 °C for 1 h, and PDA at 200 °C for greater than 95% X(t) transformed; fig. S3. SEM image grid for TiO2 films deposited at 120 °C and annealed at 180 °C, 200 °C, and 220 °C for the specified time duration; fig. S4. a) Original SEM image and b) image after processing: auto contrast/brightness, despeckle, threshold, area count. In c) dark regions are crystalline, white regions are amorphous. Note 1: Derivation for JMAK equation to describe a twodimensional transforming area with continuous nucleation and growth. Note 2: Statistical analysis for Arrhenius relationships presented in this report. tables S1, S2. For each dataset, reported Pearson’s r value, critical values, and whether the correlation is statistically significant.