Scripta Materialia 124 (2016) 164–168

Contents lists available at ScienceDirect

Scripta Materialia

j ourna l homepage: www.e lsev ie r .com/ locate /scr ip tamat

Thin film phase transformation kinetics: From theory to experiment

M.M. Moghadam ⁎, P.W. VoorheesDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA

⁎ Corresponding author.E-mail address: mahyar.moghadam@northwestern.ed

http://dx.doi.org/10.1016/j.scriptamat.2016.07.0101359-6462/© 2016 Acta Materialia Inc. Published by Elsev

a b s t r a c t

a r t i c l e i n f o

Article history:Received 23 May 2016Accepted 8 July 2016Available 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) descriptionof a nucleation and growth phase transformation that yields the average Avrami exponent and rate constant as afunction 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 allowsall the kinetic parameters, including the nucleation rate and interface growth velocity in films to be determinedfrom experiment.

© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords:Phase transformationKineticsThin filmsActivation analysisJMAK theory

Many first order transformations such as crystallization from amor-phous state, relaxation of polarized domain in ferroelectrics, spheruliticcrystallization in polymers, cellular precipitation and recrystallizationtake place through nucleation and interface-limited growth [1–11]. Ki-netics of these transformations is commonly addressed through theclassic theory of Johnson-Mehl- Avrami-Kolmogorov (JMAK) [12–16],due to its simplicity and effectiveness. However, the assumptions usedin this theory are often violated, which makes it necessary to considerthe 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 beenmany successful attempts removing the restrictive assumptions of theJMAK approach [17–20]. However, majority of these studies use sophis-ticated analytical methods and find complicated models as a result, incontrast 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 isotropicphase transformations in thin films obtained by level-set simulations.The virtue of level-set method is that it is possible to quantify the effectof film thickness on the coefficients of the JMAK theory and thuswe de-velop amethod to extract kinetic parameters such as the nucleation rateand interface velocity from experimental data.

JMAK theory is based on the geometrical evolution of an extendedvolume of transformations that occur via nucleation and growth. Nucle-ation mechanisms are often categorized based on how frequent nucleiappear in the system. If nucleation happens only at the beginning andgrowth takes place afterwards, nucleation mechanismwould be identi-fied as a Site-Saturate Nucleation (SSN). While if the system keeps

u (M.M. Moghadam).

ier Ltd. All rights reserved.

adding nuclei with the same rate over the entire transformation period,it would be defined as constant nucleation rate (CNR). The conventionalformat 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 Avramiexponent and k is the reaction rate constant. The Avrami exponent con-tains information about growth dimensionality (D). In particular, forconstant nucleation rate, the Avrami exponent is n=D+1, while forsite-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 andgrowth 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=4for 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 thetransformation process and defined as τ=λ/v. Clearly both λ and τare temperature dependent. In addition to fundamental informationthat is embedded in these two parameters, they can be used to scaleall 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 dimensionlessparameters, 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

Fig. 1.Avrami plot for transformation in different film thicknesses. (a) constant nucleationrate, (b) site-saturated nucleation. Solid lines demonstrate the slope associated with agiven n.

165M.M. Moghadam, P.W. Voorhees / Scripta Materialia 124 (2016) 164–168

nucleation rate (Eq. (2a)) and site-saturated nucleation (Eq. (2b)) asfollows [18–21]:

f V ¼ 1− exp −π3t�4

h ið2aÞ

f V ¼ 1− exp −4π3

t�3� �

ð2bÞ

Phase transformations via nucleation and interface-limited growthare an ideal case to be simulated by the level-set method. It can followgrain coalescence as well as account for the effects of contact with theexternal boundary. Evolution on experimental length and time scalesis another advantage of this method, which makes its results directlycomparablewith 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 withdifferent random initial configurations of the nuclei. Some of the dataappearing 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 quiteinformative and makes a comparison of the kinetic parameters mucheasier. As is evident from Eq. (3), in the Avrami plot the slope of eachline 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, theAvrami 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 theJMAK theory in the bulk system it is expected to be constant for allruns at the same temperature [25].

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, consistentwith Eq. (2a). Clearly, the value of the rate constant that is foundusing the classical JMAK 3D treatment does not hold in thin films, es-pecially when the film thickness is smaller than characteristic lengthof the system [24,25]. For very thin films (h⁎b1) where one assumesthat most of the growth occurs after surface impingement it yields arate constant of k⁎=πh⁎/3, shown by the dotted line in Fig. 2a [20]. Asimilar trend can be observed for site-saturated nucleation. As thefilm thickness becomes large, k⁎ approaches 4π/3 in agreementwith Eq. (2b). By contrast, for thicknesses less than characteristiclength there is a strong dependence of k⁎ on h⁎ as shown by thedashed-dotted line k⁎=πh⁎ in Fig. 2a [20]. Fig. 2b also shows thatthe average Avrami exponent (�n) varies with h*, where �n is definedas 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 acharacteristic of 2D growth for CNR and increases linearly to thepoint where film thickness equals the characteristic length (h⁎=1). For thicknesses larger than the characteristic length, �n shows asmaller change with h⁎ until it reaches a plateau at h⁎=5 with avalue close to four as given by 3D growth. SSN also exhibits similarbehavior observable with sharp transition from almost 2D growth (�n≈2) at small thickness to h⁎=1 and then a plateau toward 3Dgrowth (�n≈3). Since all data in Fig. 2 are dimensionless, one can

use them in another system with different characteristic parametersto predict the rate constant and effective Avrami exponent. To facil-itate using Fig. 2 plots, equations associated with the best fit of thesedata are provided in [27,28].

The significance of the results shown in Fig. 2 is more evident whilecalculating the activation energy from the change in the rate constantwith temperatures. Assuming that the dependence of the rate constanton temperature is:

k ¼ k0 exp −E=kBT½ � ð4Þ

where, E and k0 are activation energy and pre-factor of the phase trans-formation. Common practice is to plot ln[k] as a function of 1/kBT to ex-tract the slope and thus the activation energy of the transformation [29].This method seems reasonable for bulk systems where k0 and E are as-sumed to be temperature independent. However, an expansion of theactivation energy and pre-factor for CNR (Eq. (5a)) and SSN (Eq. (5b))reveals why the Eq. (4) does not hold in thin films.

k ¼ αN0vn−10

zfflfflfflfflfflffl}|fflfflfflfflfflffl{k0

exp − EN þ n−1ð ÞEvð Þzfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{E

=KBT

24

35 ð5aÞ

Fig. 2. (a) Variation of rate constant, k⁎with h⁎. (b) Variation of the average Avrami exponent, �nwith h⁎. CNR is constant nucleation rate, and SSN is site-saturated nucleation. The dashedlines denote the dependence of k* on h* for very thin films.

Fig. 3. Probability density of grain size distribution under (a) constant nucleation rate, (b)site-saturated nucleation. Same legend applied to both.

166 M.M. Moghadam, P.W. Voorhees / Scripta Materialia 124 (2016) 164–168

k ¼ αN0vn0zfflfflffl}|fflfflffl{k0

exp − EN þ nEvð Þzfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{E

=KBT

24

35 ð5bÞ

here α is a shape factor, N0, v0 are pre-factors and EN, Ev are the ac-tivation energies in Arrhenius equations for calculating nucleationrate (density) and growth velocity respectively [29]. Eqs. (5a) and(5b) indicate that the k0 and E are function of n. According to JMAKtheory in bulk systems, n is temperature independent and has thesame integer value at different temperatures, while as shown clearlyin Fig. 2 in thin film systems n is a function of h⁎, which is tempera-ture dependent through λ. In other words, h⁎ varies with tempera-ture even though the dimensional thickness of the film remainsconstant. That means both k0 and E become temperature dependentand the conventional method of calculating phase transformationactivation energy based on Eq. (4) can lead to spurious results inthin films. This analysis highlights the significance of determiningthe activation energy for nucleation and growth separately as an al-ternative way to study thin films phase transformations.

As mentioned earlier, there is a connection between the averagegrain size (�d

�) and the characteristic length of the system under trans-

formation. To investigate this connection,we take advantage of our sim-ulationmethod and construct the probability density distribution of thegrain size, P(d⁎). Fig. 3 reveals interesting details of grain size distribu-tion in CNR and SSN at different film thicknesses. Fig. 3a shows two re-gimes for constant nucleation rate transformations. For h⁎N1 alldistributions demonstrate similar almost unimodal shapes, covering abroad spectrum of grain size, but as film thickness decreases (h⁎b1),narrower peaks start to format particular values of d⁎. For site-saturatednucleation, Fig. 3b shows relatively narrow peaks, that are symmetricabout d⁎≈1.24 with peak probabilities 2–3 times greater than theCNR case. In the SSN case, the distribution also does not change signifi-cantly for all thicknesses investigated, which is consistent with the ho-mogeneous distribution of nuclei with the same nucleation density atthe beginning of the transformation, assumed in site-saturated nucle-ation [30,31].

Fig. 4. (a) Average grain size (�d�) as function of film thickness. (b) Final transformation time (tF⁎) as a function of film thickness. Same legend applied to both.

Table 1

Values of the interface velocity (v), nucleation rate ( _NV), average Avrami exponent (�n) andrate constant (k⁎) predicted by proposed method in comparison with experimental datafor crystallization of 0.2 μm, a-NiTi film at 768 k.

v[μms−1] _NV ½μm−3s−1� n k⁎

Model 1.54×10−2 9.53×10−2 3.14 0.32Experiment 2.05×10−2 9.81×10−2 3.12 0.34

167M.M. Moghadam, P.W. Voorhees / Scripta Materialia 124 (2016) 164–168

Using the results of level set simulations of nucleation and growthphase transformation in thin films we are now able to present a practi-cal framework for linking the experimental data to theoretical parame-ters in order to predict quantities such as the nucleation rate (density)and growth velocity. Experimental data for this purpose are commonlygathered using an in-situ characterization procedure such as X-ray scat-tering [1,31] or transmission electron microscopy [31,32]. Among alldata collected from these experiments, there are two key quantities

that most often overlooked. The first is the average grain size (�d�) of

the sample once the transformation is completed and the second isthe transformation time (tF⁎). In our analysis, the average grain size is

calculated based on a spherically equivalent volume (�d� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi6�V�

=π3q

)

where �V� is the average grain volume and the transformation time is de-fined as the time at which 99% of the transformation is complete. Thesetwo quantities can be used to determine the characteristic length (λ)and characteristic time (τ) of the system, essential information for fur-ther analysis of the kinetics. Fig. 4a, b show the relationship betweenthese two experimentally measurable quantities and the relative filmthickness. The equations of the lines shown in Fig. 4 are given in Eqs.(6a,b) and (7a,b) for constant nucleation rate and site-saturated nucle-ation respectively,

CNR d� ¼ −0:0564h�

−0:8409

þ 1:29t�F ¼ 0:3266h�

−0:8698

þ 1:43

(ð6a;bÞ

SSN d� ¼ 1:24

t�F ¼ 0:3463h�−0:9873 þ 0:99

�ð7a;bÞ

For a phase transformation that occurs under constant nucleationrate, Fig. 4a, the average grain size shows a strong dependence on thefilm thickness for h⁎b1, but a much weaker dependence for thickerfilms (h⁎N1), that asymptotes to 1.29, which is the average grain sizevalue for bulk transformations [20]. By contrast, the average grain sizein site-saturated nucleation does not show any dependence on filmthickness and has a constant value of 1.24, which agrees with a

theoretical calculation [31], although the grain size distribution doesslightly change, see Fig. 3b.

Based on the aforementioned analysis, it is possible to obtain kineticparameters directly from conventional experimental data. By followingthis methodology, the characteristic length and time of the system canbe determined at different temperatures, as well as nucleation rate(density) and interface growth velocity in order to calculate the activa-tion energies of the nucleation and growth processes. The step-by-stepprocedure is as follows. 1.Measure fV vs. t,�d, h and tF from experiment. 2.Make an Avrami plot to extract the average Avrami exponent and iden-tify nucleation mechanism (CNR or SSN). 3. Use �d and h to extract λusing the proper Eqs. (6a) or (7a), according to the nucleation mecha-nism. 4. Substitute h⁎ and tF in the appropriate Eqs. (6b) or (7b) to deter-mine τ. 5. From the knowledge of λ and τ and the equation τ=λ/v

determine interface growth velocity. 6. Use λ and v in the equationsλ ¼ð _NV=vÞ−1=4

orλ ¼ NV−1=3 to calculate nucleation rate or nucleation den-

sity respectively.We illustrate and test this approach using the data provided by Lee

et al. [32] from in-situ TEM experiment on a-TiNi thin film crystalliza-tion. Nucleation and growth parameters are directly measured duringthe transformation under constant nucleation rate conditions. Sincethe nucleation mechanism has already been identified, we skip firsttwo steps and start from step 3 to extract characteristic length and rel-ative thickness of this system by using the final average grain size in Eq.(6a). After determining λwe find h⁎ and thenmove forward to step 4 tocalculate characteristic time from Eq. (6b). Based on the average grainsize and crystallization time provided in [32], we predict that at 768K,the characteristic length is λ=0.63 (μm) and the characteristic time is

168 M.M. Moghadam, P.W. Voorhees / Scripta Materialia 124 (2016) 164–168

τ=45 (s). By proceeding to final steps we are able to calculate nucle-ation rate and interface growth velocity as well. We also utilize equa-tions in [27] to determine �n and k⁎ for a given h⁎.

Table 1 compares kinetics parameters predicted by proposed meth-odology with the values measured experimentally in [32]. Cleary themodel has been able to reproduce experimental data with reasonableaccuracy and these results can be seen as verification for the proposedprocedure.

Our approach to analyze thin film phase transformations via thelevel-set method enables us to quantify the effect of film thickness onkinetics parameters of JMAK equation including the Avrami exponentand rate constant. Furthermore the method presented can be used tomake a connection between theory and experiment in order to deter-mine the parameters governing the entire transformation process, andthus provide an accurate model that can be used to achieve the desiredmicrostructure and properties of the material.

The support of the NSF-MRSEC at Northwestern, DMR-1121262, isgratefully appreciated.

References

[1] D.E. Proffit, T. Philippe, J.D. Emery, Q. Ma, B.D. Buchholz, P.W. Voorhees, M.J. Bedzyk,R.P.H. Chang, T.O. Mason, J. Electroceram. 34 (2014) 167–174.

[2] Y. Kwon, D.-H. Kang, Scr. Mater. 78–79 (2014) 29–32.[3] C.S. Ganpule, A.L. Roytburd, V. Nagarajan, B.K. Hill, S.B. Ogale, E.D. Williams, R.

Ramesh, J.F. Scott, Phys. Rev. B 65 (2001) 014101.[4] V. Nagarajan, C.S. Ganpule, R. Ramesh, P.H. Ishiwara, P.M. Okuyama, D.Y. Arimoto

(Eds.), Ferroelectric Random Access Memories, Springer, Berlin Heidelberg 2004,pp. 47–70.

[5] L. Gránásy, T. Pusztai, G. Tegze, J.A. Warren, J.F. Douglas, Phys. Rev. E 72 (2005)011605.

[6] Z.J. Liu, J. Ouyang, W. Zhou, X.D. Wang, Comput. Mater. Sci. 97 (2015) 245–253.

[7] B. Amir Esgandari, H. Mehrjoo, B. Nami, S.M. Miresmaeili, Mater. Sci. Eng. A 528(2011) 5018–5024.

[8] L. Jiang, D. Zhang, Y. Dong, F. Guo, G. Hu, H. Xue, F. Pan, Mater. Sci. Technol. 31(2015) 1088–1095.

[9] A.D. Rollett, D.J. Srolovitz, R.D. Doherty, M.P. Anderson, Acta Metall. 37 (1989)627–639.

[10] Y. Lü, D.A. Molodov, G. Gottstein, Acta Mater. 59 (2011) 3229–3243.[11] J.E. Taylor, J.W. Cahn, C.A. Handwerker, Acta Metall. Mater. 40 (1992) 1443–1474.[12] A.N. Kolmogorov, Bull. Acad. Sci. USSR Math. Ser. 1 (1937) 355–359.[13] M. Avrami, J. Chem. Phys. 9 (1941) 177–184.[14] M. Avrami, J. Chem. Phys. 8 (1940) 212–224.[15] M. Avrami, J. Chem. Phys. 7 (1939) 1103–1112.[16] W.A. Johnson, R.F. Mehl, Trans. AIME 135 (1939) 396–415.[17] K. Sekimoto, Phys. A Stat. Mech. Appl. 135 (1986) 328–346.[18] L.E. Levine, K.L. Narayan, K.F. Kelton, J. Mater. Res. 12 (1997) 124–132.[19] V.I. Trofimov, I.V. Trofimov, J.-I. Kim, Thin Solid Films 495 (2006) 398–403.[20] J. Očenášek, P. Novák, S. Agbo, J. Appl. Phys. 115 (2014) 043505.[21] M.T. Todinov, Acta Mater. 48 (2000) 4217–4224.[22] S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer New

York, New York, NY, 2003.[23] J.A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in

Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science,Cambridge University Press, 1999.

[24] E.L. Pang, N.Q. Vo, T. Philippe, P.W. Voorhees, J. Appl. Phys. 117 (2015) 175304.[25] M.M. Moghadam, E.L. Pang, T. Philippe, P.W. Voorhees, Thin Solid Films 612 (2016)

437–444.[26] A. Calka, A.P. Radlinski, Mater. Sci. Eng. 97 (1988) 241–246.[27] Fitted functions for calculating rate constant and effective Avrami exponentwith re-

spect to film thickness based on data presented in Fig. 2-a,b for constant nucleationrate: 1/k⁎=0.4406 h⁎−1.362+0.9976 and 1=�n ¼ 0:0374 h�−0:6018 þ 0:244

[28] Fitted functions for calculating rate constant and effective Avrami exponentwith re-spect to film thickness based on data presented in Fig. 2-a,b for site-saturated nucle-ation: 1/k⁎=0.1628 h⁎−1.324+0.2416 and 1=�n ¼ 0:0602 h�−0:7371 þ 0:327

[29] F. Edelman, R. Weil, P. Werner, M. Reiche, W. Beyer, Phys. Status Solidi A 150 (1995)407–425.

[30] A.V. Teran, A. Bill, R.B. Bergmann, Phys. Rev. B 81 (2010) 075319.[31] S.P. Ahrenkiel, A.H. Mahan, D.S. Ginley, Y. Xu, Mater. Sci. Eng. B 176 (2011) 972–977.[32] H.-J. Lee, H. Ni, D.T. Wu, A.G. Ramirez, Appl. Phys. Lett. 87 (2005) 124102.