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1 2 3 4 Kinetics of the β→α transformation of tin: Role of α–tin 5 6 nucleation 7 8 9 10 11 Guang Zeng1, Stuart D. McDonald1, Qinfen Gu2, Syo Matsumura3 and Kazuhiro Nogita1* 12 13 14 15 1 Nihon Superior Centre for the Manufacture of Electronic Materials (NS CMEM), School of 16 17 Mechanical and Mining Engineering, The University of Queensland, St Lucia, QLD 4072, 18 19 Australia; 20 21 22 2 Powder Diffraction Beamline, The Australian Synchrotron, Clayton, VIC 3168, Australia; 23 24 3 Department of Nuclear Engineering and Quantum Physics, Kyushu University, Fukuoka 819- 25 26 0395, Japan 27 28 29 30 31 ABSTRACT The role of α-Sn nucleation on the kinetics of the β→α phase transformation is investigated in this 32 study, using in situ variable temperature synchrotron powder x-ray diffraction (PXRD). In the entire thermal history 33 34 of α→β→α transformation, the population of α-Sn at the onset of the β→α transformation was varied by controlling 35 the α→β transformation under isothermal conditions. The degree of α-Sn nucleation is found to have a strong 36 37 influence on the kinetics of the β→α phase transformation. By introducing an impingement factor c=2, the 38 transformation curves yield good fits to the modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) model. Inceases 39 40 in annealing time at 45 ºC resulted in insufficient nucleation at the commencement of the reaction at the kinetically 41 optimal temperature of -45 ºC and additional nucleation is required. 42 43 44 45 46 1. Introduction 47 48 The crystallography, mechanical properties and physical properties of tin have been the subjects 49 1-5 50 of considerable interest from experimental and calculation/modeling studies . The β→α 51 transformation in tin has been of great theoretical and practical interest for over 100 years 2, 3, 5-14. 52 53 It is the only known solid-state transformation in which a metallic material (β-Sn) transforms 54 12-15 55 into a non-metallic semi-conductive solid (α-Sn) . The significant density change (27.0% at - 56 5, 14 57 100°C to 26.1% at 50°C) that is induced by this transformation can lead to the blistering of 58 59 60 ACS Paragon Plus Environment 1 Crystal Growth & Design Page 2 of 16

1 2 3 the free surface of the tin, cracking and even complete disintegration. This phenomenon has been 4 5 traditionally known as “tin pest” 5 because the blisters that form on the surface of the tin have an 6 7 appearance similar to those which occur on the skin of animals and humans afflicted by some 8 14 9 diseases . This blistering and cracking is the result of the combination of the large increase in 10 14, 16 11 volume as the β-Sn transforms to α-Sn and the very low ductility of the α-Sn . With the 12 widespread adoption of lead-free solders by the electronics industry there has been renewed 13 14 interest in tin pest because of concern that these Sn-rich alloys might be susceptible to this 15 14 16 phenomenon . It has been concluded that commercial lead-free solder made with standard purity 17 5, 17 18 Sn have less potential for "tin pest" than high purity Sn due to impurity effects . However, it is 19 still difficult to predict "tin pest" and the deep understanding required to exercise any practical 20 21 control of the β→α transformation is lacking5. 22 23 Previous studies have confirmed that the β→α transformation is a nucleation and growth process 24 25 2, 5, 18. The equilibrium transformation temperature is 13.2°C but in practice the reaction occurs 26 27 only with significant undercooling and after a considerable incubation period2, 3, 6, 10, 13. The 28 13 29 nucleation of α-Sn is particularly difficult so the majority of the time for the β→α 30 31 transformation is taken up by the incubation period with the subsequent growth being relatively 32 rapid 5. It has been reported that the kinetics of the transformation can be greatly accelerated by 33 34 the introduction of heterogeneous nuclei, known as seeding 13. Seeding can be achieved by 35 36 pressing into the β-Sn, powdered crystals of α-Sn or other crystals that are isomorphic with α-Sn 37 6, 13 38 with similar lattice parameters such as CdTe, InSb and Ge . Once nucleated the growth 39 kinetics of the β→α transformation are determined mainly by the impurities and alloying 40 41 additions in the tin, the shape and size of the sample, the tin grain size, the temperature and the 42 43 thermo mechanical history prior to transformation 5, 9-11, 19-23. The kinetics of the β→α 44 5 24 45 transformation have been intensively investigated by methods such as XRD , DSC , 46 2 9 8-12, 19-26 47 dilatometry , electrical resistance measurement and microstructural observation . 48 Despite these investigations over the past 60 years the kinetics of the transformation are still not 49 50 well understood, and this may be due to the difficulty in identifying the quantity of any α-Sn 51 52 particles within the β-Sn as potential nucleation sites for the transformation. 53 54 X-ray powder diffraction is one of most powerful methods of obtaining quantitative phase 55 27 56 information from multicomponent mixtures . The variable temperature in situ synchrotron 57 58 PXRD that is now available provides a method of studying the transformation more precisely 59 60 ACS Paragon Plus Environment 2 Page 3 of 16 Crystal Growth & Design

1 2 3 than has been possible in the past. In our recent study of the transformation using this method, 4 5 data sufficient to plot a temperature, time, transformation (TTT) diagram for the β→α 6 7 transformation were obtained 5. This TTT diagram has a typical “C” shape with a “nose” at - 8 9 50°C and is reasonably consistent with the results of the 1957 kinetic study by Burgers and 10 2 11 Groen who used dilatometry. In that study the samples were of similar purity (99.99%) and had 12 a thermal history of previous β→α transformations. It should be noted that in our study where 13 14 samples were annealed for 10 minutes at 50°C prior to cooling, the β→α transformation kinetics 15 28-30 16 were a good fit with the Johnson-Mehl-Avrami-Komolgorov (JMAK) model with an Avrami 17 18 exponent of 3, which implies that the transformation occurred by three-dimensional growth from 19 pre-existing nuclei5. However, as the annealing time at 50°C was increased to 20 minutes, the 20 21 kinetics failed to fit the JMAK equation and additional nucleation was required for the β→α 22 17 23 transformation to commence . In our recent study, additions of 1 wt.% Pb, Cu, Ge, or Si, in the 24 17 25 form of powder mixtures, all inhibited the β⟶α transformation . The method used minimised 26 27 the effects of element solubility and residual stresses and the primary effect of the additive was 28 assumed to be that of intimate physical contact. Of the addition elements studied, Pb and Cu 29 30 show the strongest ability to suppress the β⟶α transition. Ge and Si also inhibited the β→α 31 17 32 transformation despite having a similar crystal structure to α-Sn . The kinetics of the β→α 33 4-6, 17 34 transformation is largely determined by the nucleation of α-Sn . The presence of pre-existing 35 α-Sn has been proven to greatly accelerate the formation and growth of the α-Sn phase from β- 36 37 Sn. However, it has not been quantitatively or accurately investigated in previous studies. In this 38 39 study, high purity α-Sn samples were in situ annealed during powder diffraction at 45°C for 40 41 various durations, aiming to control the amount of α-Sn remaining after α⟶β transformations of 42 43 varying degrees of completeness. Afterwards, the influences of α-Sn nucleation on the β⟶α 44 transformation in tin were examined. 45 46 47 2. Experimental Section 48 49 To prepare α-Sn samples for powder XRD experiments, high purity 99.99% tin cylinders (20mm 50 51 diameter and 10mm height) were cast and then kept for 180 days at -45ºC. During this time, the 52 cylinders converted to α-Sn powder. The samples used in this study are identical to those used in 53 54 previous studies5, 31. 55 56 57 58 59 60 ACS Paragon Plus Environment 3 Crystal Growth & Design Page 4 of 16

1 2 3 Samples of ~0.15g of the powder mixtures were loaded into quartz capillaries (0.3 mm in 4 5 diameter) and stored at -20 ºC in preparation for experiments at the the Powder Diffraction 6 7 beamline at the Australian Synchrotron. During powder x-ray diffraction (PXRD), the 8 9 temperature of the samples was controlled using a Cryostream (Oxford Cryosystems, Cryostream 10 11 Plus) with temperature accuracy within +/- 0.5 ºC. High-resolution PXRD data were collected in 12 the 2 theta range of 10 to 80 degrees using a Mythen-II detector with a 18 keV beam. A LaB6 13 14 standard (NISTLaB6 660b) was used at room temperature for wavelength calibration. The 15 16 calibrated wavelength was 0.6887 Å. Differential Scanning Calorimetry (DSC) study at a heating 17 18 rate of 6ºC/min indicated that 46.7 % of α-Sn was transformed to β-Sn at the point of 45ºC 19 during continuous linear heating, as shown in Figure 1a 17. This estimate was based on the 20 21 assumption that the degree of transformation is equal to the fraction of heat absorbed or released 22 32 23 . Therefore, it is expected that the amount of α-Sn is able to be controlled by varying the 24 25 annealing time at 45ºC. As illustrated in Figure 1b, for the β⟶α transformation, α-Sn powder 26 27 was first heated up to 45 ºC at the rate of 6 ºC/min and held for 5, 10, 15, 20, 30 and 60 minutes 28 to control the population of α-Sn remaining. 29 30 31 Isothermal PXRD measurement was continually conducted every 5 minutes during annealing at 32 45ºC. This was followed by cooling to -45ºC at a rate of 6ºC/min, and subsequently performing 5 33 34 minutes of continuous data collection for each of up to 60 measurements at -45ºC. The Rietveld 35 36 refinements of the PXRD pattern were conducted using TOPAS 4.2 software (Bruker-AXS, 37 38 Germany). The fundamental parameters (FP) approach was employed for the peak refinements in 39 TOPAS to minimize the reliability factor “Rwp”, which follows directly from the square root of 40 41 the quantity minimized, scaled by the weighted intensities 27. The structural data reported for α- 42 43 Sn by Tsukeva et al. 33 and β-Sn by Pietzka et al. 34 were used as starting parameters for 44 45 quantitative phase analysis in TOPAS. 46 47 3. Results and Discussions 48 49 The in situ isothermal PXRD results at -45°C are shown in Figure 2. It is clearly seen that peaks 50 51 indexed as tetragonal β-Sn occur initially, with α-Sn peaks emerging and increasing in intensity 52 53 over the period of measurement. In most cases, the peaks of β-Sn remained in the final stage 54 55 indicating that the phase transformations were not completed within the range of time and 56 temperatures used in this study. Generally it is shown in Figure 2 that the transformation rate 57 58 59 60 ACS Paragon Plus Environment 4 Page 5 of 16 Crystal Growth & Design

1 2 3 exhibits a sigmoidal shape. The initially slow rate of transformation accelerating to a maximum 4 5 a maximum rate at some intermediate time before the transformation slows near completion. It 6 7 can be seen that the duration of annealing at 45°C haa an influence on the kinetics of the β→α 8 9 transformation. 10 11 Quantitative analysis is requried for obtaining the accurate transformation curves of β→α at - 12 13 45°C. The sample was fully rotated with 360° in the around the axis of the capillary during the 14 35 15 measurement, thus texture effects of samples was somewhat reduced . The whole pattern 16 Rietveld method of structure refinement was conduted to obtain the weight fraction of α-Sn and 17 18 β-Sn phase from each pattern. As shown in the Rietveld plot of Figure 3, a suitable range of 10° 19 20 to 80° 2θ was chosen to include the most major intensity lines and stepped at 0.002° increments. 21 22 The data plotted are from the 60.95% α-Sn and 39.05 % β-Sn. The fraction transformed over the 23 time of the experiments is plotted in Figure 4. As shown in Figure 4, there is a decreasing trend 24 25 in the transformation rate as the prior annealing time at 45°C increases. The only exception is 26 27 that the sample with the 10 minute annealing transforming faster than that with the 5 minute 28 29 annealing, which may arise due to experimental errors of controling the nucleation population or 30 31 there are factors other than annealing time also at play. Therefore, it can be concluded that the 32 kinetics of the β→α phase transformation is sensitive to the amount of α-Sn although the 33 34 differences in the amount of pre-existing α-Sn that give rise to different kinetics are not possible 35 36 to measure within the resolution of synchrotron PXRD. 37 38 The pre-existing α-Sn greatly influenced the kinetics of the β→α phase transformation. 39 40 Therefore, it is worth re-examining the process of α→β transformation and its contribution to the 41 42 whole thermal history of the α→β→α transformation. As shown in our previous study (replotted 43 in Figure 5a)5, in the case of isothermal conditions, high temperautres favor a faster α→β 44 45 transformation and at each temperature the slope of the JMAK plot decreased from n~4 to n~1. It 46 47 can be infered that the kinetics of the α→β phase transformation are significantly influenced by 48 49 nucleation at relatively low temperatures and also by growth rates at high temperautre. As shown 50 51 in Figure 5b, the driving force for the α→β transformation increased as the temperauture raised, 52 as calculated by Ravelo et al.36. In this study, after annealing at 45°C, the α→β transformation is 53 54 almost completed and the amount of α-Sn available acting as nucleation sites decreases as 55 56 annealing progresses. It is impossibile to distinguish the amount of α-tin prior to the β→α 57 58 59 60 ACS Paragon Plus Environment 5 Crystal Growth & Design Page 6 of 16

1 2 3 transformation from Rietveld analysis, but it is confirmed that α-Sn nucleation has a significant 4 5 impact on the kinetics of the β→α transformation and this has been quantitafied in this study. 6 7 Various kind of models have been proposed for describing isothermal transformation kinetics, 8 9 such as the JMAK 28-30 equation, the Austin-Rickett (AR)37 equation, and the time law for 10 11 normal grain growth 38, 39. All these kinetic models consist of two factors:(i) the rate-time factor 12 28-30 13 and (ii) the impingement related to the fraction untransformed. The JMAK model is a 14 15 common model used to illustrate the kinetics of phase transformation with considerations of 16 nucleation and growth. If the kinetics are well described by a JMAK model, plotting of the 17 18 transformation data in the form of Eq.(1) (c=0) yields a straight line of gradient n and intercept 19 20 lnk. However, it is applicable to all kinetics processes. The consideration of impingment using 21 28-30, 38 22 the JMAK approach is only valid when nucleation is random and growth is linear . This 23 issue of impingement in the JMAK equation has been considered intensively in previous work 37- 24 25 40. The impingement factor is commonly used to correct some effects such as depletion of solute 26 27 content in an untransformed matrix caused by competitive growth of reaction products, a direct 28 38 29 collision of two advancing reaction products, or an exhaustion of nucleation sites . A modified 30 38, 39 31 JMAK/AR type kinetic model was developed by Lee and Kim et al. , who included a 32 generalised impingement exponent c, to describe the isothermal kinetics of phase transformation 33 34 35  (1− x ) −c 36 ln forc ≠ 0 c 37 Gc()≡  38  1 39 ln ln forc = 0 (JMAK) 40  (1− x ) 41 (Eq. 1) 42 43 =⋅ntnkln +⋅ ln 44 45 where x = the fraction transformed, k=k0exp(-Q/RT), k0= the pre-exponential factor , Q = the 46 47 activation energy, t = time, R = the gas constant, T = the absolute temperaute. The time 48 40 49 exponent, n, is a parameter depending on the nucleation mechanism and the growth process . 50 51 As shown in Figure 6(c), the plots of the G(c=2) exhibit a much better fit than the JMAK plots, 52 and nearly yield a straight line except for the transformation involving less than 10 minutes of 53 54 annealing at 45°C. It should be note that the variation is shown in the slope ‘n’ among these 55 56 linear fits, from n=3.9 to n=7.6, indicating that the kinetics have been greatly altered by the 57 58 change in the number of nucleation sites. However, the intercept lnk shows less variation for 59 60 ACS Paragon Plus Environment 6 Page 7 of 16 Crystal Growth & Design

1 2 3 different anneanling times at 45°C (different number of α-Sn phase nucleation sites). It can be 4 5 infered that the nuclei population has minimal effect on the acitivation energy of the phase 6 7 transformation, which is considered to be temperaure dependant (constant for the temperature of 8 9 -45ºC in this study). It is suggested that, inceases in the annealing time at 45ºC resulted in 10 11 insufficient nucleation at the commencement of the reaction at the temperature of -45ºC and 12 additional nucleation is required which is reflected in the fitting parameters of the JMAK plots. 13 14 15 4. Conclusions 16 17 The dependence of kinetics of the β→α phase transformation on α-Sn nucleation were semi- 18 19 quantatively investigated in this study. It has been proven that the population of α-Sn during the 20 onset of transformation decreased with an increase in annealing time, and delayed the kinetics of 21 22 the β→α phase transformation. With introducing an impingement factor c=2, the transformation 23 24 curves yield good fits to the modified JMAK model. Inceases in annealing time at 45ºC resulted 25 26 in insufficient nucleation at the commencement of the reaction at the temperature of -45ºC and 27 28 additional nucleation is required. 29 30 Acknowledgments 31 32 We gratefully acknowledge financial support from the University of Queensland-Nihon Superior 33 34 collaborative research program, and Australian Research Council Linkage project 35 36 (LP140100485). PXRD experiments were performed at the Australian Synchrotron Powder 37 Diffraction Beamline (AS123/PD/6371). The author thanks Dr. J. Khan of the Queensland Node 38 39 of the Australian National Fabrication Facility (ANFF-Q) at UQ for use of the DSC facility. G. 40 41 Zeng is financially supported by a University of Queensland International (UQI) Scholarship, 42 43 and Graduate School International Travel Award (GSITA) of UQ, as well as a China Scholarship 44 45 Council (CSC) Scholarship. 46 47 References 48 49 (1) Musgrave, M. J. P., On the Relation Between Grey and White Tin (α-Sn and β-Sn). 50 51 Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 52 1963, 272, (1351), 503-528. 53 (2) Burgers, W. G.; Groen, L. J., Mechanism and kinetics of the allotropic transformation of 54 tin. Discuss. Faraday Soc. 1957, 23, (0), 183-195. 55 (3) Raynor, G.; Smith, R., The transition temperature of the transition between grey and 56 57 white tin. Proceedings of the Royal Society of London. Series A. Mathematical and Physical 58 Sciences 1958, 244, (1236), 101-109. 59 60 ACS Paragon Plus Environment 7 Crystal Growth & Design Page 8 of 16

1 2 3 (4) Gallerneault, W.; Vnuk, F.; Smith, R., Siliconstabilized grey tin. J. Appl. Phys. 1983, 4 5 54, 4200. 6 (5) K. Nogita; C. M. Gourlay; S. D. McDonald; S. Suenaga; J. Read; G. Zeng ; F.Gu, Q., 7 XRD study of the kinetics of β↔α transformations in tin. Philos. Mag. 2013, 93, (27), 3627- 8 3647. 9 (6) G. Mitchell, D.; Donnelly, S., A transmission electron microscopy study of the β→ α- 10 11 phase transformation of tin. Philos. Mag. A 1991, 63, (4), 747-755. 12 (7) Kariya, Y.; Williams, N.; Gagg, C.; Plumbridge, W., Tin pest in Sn-0.5 wt.% Cu lead- 13 free solder. JOM 2001, 53, (6), 39-41. 14 (8) Semenova, O.; Flandorfer, H.; Ipser, H., On the non-occurrence of tin pest in tin–silver– 15 indium solders. Scr. Mater. 2005, 52, (2), 89-92. 16 (9) Plumbridge, W. J., Tin pest issues in lead-free electronic solders. J. Mater. Sci.: Mater. 17 18 Electron. 2006, 18, (1-3), 307-318. 19 (10) Plumbridge, W., Tin pest in electronics? Circuit World 2007, 33, (1), 9-14. 20 (11) Plumbridge, W. J., Recent Observations on Tin Pest Formation in Solder Alloys. J. 21 Electron. Mater. 2008, 37, (2), 218-223. 22 (12) Di Maio, D.; Hunt, C., Time-lapse photography of the β-Sn/α-Sn allotropic 23 24 transformation. J. Mater. Sci.: Mater. Electron. 2009, 20, (4), 386-391. 25 (13) Di Maio, D.; Hunt, C. P., Monitoring the Growth of the α Phase in Tin Alloys by 26 Electrical Resistance Measurements. J. Electron. Mater. 2009, 38, (9), 1874-1880. 27 (14) Starink, M., On the meaning of the impingement parameter in kinetic equations for 28 nucleation and growth reactions. Journal of Materials Science 2001, 36, (18), 4433-4441. 29 (15) Vnuk, F., Preparation of compact α-tin specimens. J. Cryst. Growth 1980, 48, (3), 486- 30 31 488. 32 (16) Vnuk, F.; DeMonte, A.; Smith, R. W., The composition dependence of the grey tin → 33 white tin transition in dilute tin-germanium alloys. Mater. Lett. 1983, 2, (1), 67-70. 34 (17) G. Zeng; S. D. McDonald; Q.F.Gu; K. Sweatman; Nogita, K., Elemental effects on the 35 β→α transformation in tin. Philos. Mag. Lett. 2013, 94, (2), 53-62. 36 37 (18) Christian, J. W., The theory of transformations in metals and alloys. 3rd ed.; Pergamon 38 Press: Oxford ; New York :, 2002. 39 (19) Peng, W., An investigation of Sn pest in pure Sn and Sn-based solders. Microelectronics 40 Reliability 2009, 49, (1), 86-91. 41 (20) Leodolter-Dworak, M.; Steffan, I.; Plumbridge, W.; Ipser, H., Tin Pest in Sn-0.5Cu Lead- 42 Free Solder Alloys: A Chemical Analysis of Trace Elements. J. Electron. Mater. 2010, 39, (1), 43 44 105-108. 45 (21) Plumbridge, W., Tin pest in lead-containing solders. Soldering & Surface Mount 46 Technology 2010, 22, (1), 56-57. 47 (22) Plumbridge, W. J., Further Observations on Tin Pest Formation in Solder Alloys. J. 48 Electron. Mater. 2010, 39, (4), 433-440. 49 50 (23) Di Maio, D.; Hunt, C., On the absence of the β to α Sn allotropic transformation in solder 51 joints made from paste and metal powder. Microelectron. Eng. 2011, 88, (1), 117-120. 52 (24) Gialanella, S.; Deflorian, F.; Girardi, F.; Lonardelli, I.; Rossi, S., Kinetics and 53 microstructural aspects of the allotropic transition in tin. J. Alloys Compd. 2009, 474, (1–2), 134- 54 139. 55 (25) Joo, Y. J.; Takemoto, T., Transformation of Sn–Cu alloy from white tin to gray tin. 56 57 Mater. Lett. 2002, 56, (5), 793-796. 58 59 60 ACS Paragon Plus Environment 8 Page 9 of 16 Crystal Growth & Design

1 2 3 (26) Skwarek, A.; Sroda, M.; Pluska, M.; Czerwinski, A.; Ratajczak, J.; Witek, K., 4 5 Occurrence of tin pest on the surface of tin-rich lead-free alloys. Soldering & Surface Mount 6 Technology 2011, 23, (3), 184-190. 7 (27) Young, R. A.; Prince, E.; Sparks, R. A., Suggested guidelines for the publication of 8 Rietveld analyses and pattern decomposition studies. J. Appl. Crystallogr. 1982, 15, (3), 357- 9 359. 10 11 (28) Avrami, M., Kinetics of Phase Change. I General Theory. The Journal of Chemical 12 Physics 1939, 7, (12), 1103. 13 (29) Avrami, M., Kinetics of Phase Change. II Transformation-Time Relations for Random 14 Distribution of Nuclei. The Journal of Chemical Physics 1940, 8, (2), 212. 15 (30) Avrami, M., Granulation, Phase Change, and Microstructure Kinetics of Phase Change. 16 III. The Journal of Chemical Physics 1941, 9, (2), 177. 17 18 (31) Zeng, G.; McDonald, S. D.; Gu, Q.; Suenaga, S.; Zhang, Y.; Chen, J.; Nogita, K., Phase 19 stability and thermal expansion behavior of Cu6Sn5 intermetallics doped with Zn, Au and In. 20 Intermetallics 2013, 43, (0), 85-98. 21 (32) Malinov, S.; Guo, Z.; Sha, W.; Wilson, A., Differential scanning calorimetry study and 22 computer modeling of β  α phase transformation in a Ti-6Al-4V alloy. Metallurgical and 23 24 Materials Transactions A 2001, 32, (4), 879-887. 25 (33) Tsukeva, E. A.; Tsakin, E.; Peneva, S. K.; Djuneva, K. D., On the possibility to grow thin 26 tin films with unusual tin structures from the vapour. Zeitschrift für Kristallographie - 27 Crystalline Materials 1989, 187, (1-2), 63-70. 28 (34) Pietzka, M. A.; Schuster, J. C., Phase equilibria of the quaternary system Ti-Al-Sn-N at 29 30 900 °C. J. Alloys Compd. 1997, 247, (1–2), 198-201. 31 (35) McCusker, L.; Von Dreele, R.; Cox, D.; Louer, D.; Scardi, P., Rietveld refinement 32 guidelines. J. Appl. Crystallogr. 1999, 32, (1), 36-50. 33 (36) Ravelo, R.; Baskes, M., Equilibrium and thermodynamic properties of grey, white, and 34 liquid tin. Phys. Rev. Lett. 1997, 79, (13), 2482. 35 (37) Austin, J.; Rickett, R., Kinetics of the decomposition of austenite at constant temperature. 36 37 Trans. AIME 1939, 135, (8), 396-415. 38 (38) Lee, E.-S.; Kim, Y. G., A transformation kinetic model and its application to Cu Zn 39 40 Al shape memory alloys—I. Isothermal conditions. Acta Metall. Mater. 1990, 38, (9), 1669- 41 1676. 42 (39) Lee, E.-S.; Kim, Y. G., A transformation kinetic model and its application to Cu Zn 43 44 Al shape memory alloys—II. Non-isothermal conditions. Acta Metall. Mater. 1990, 38, (9), 45 1677-1686. 46 (40) Starink, M., Kinetic equations for diffusion-controlled precipitation reactions. J. Mater. 47 Sci. 1997, 32, (15), 4061-4070. 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 9 Crystal Growth & Design Page 10 of 16

1 2 3 Figure Captions 4 5 Figure 1 (a) DSC curve of α-tin sample heated up from 0°C to 100°C at the rate of 6°C/min; and (b) thermal history 6 7 of synchrotron PXRD measurements. 8 9 Figure 2 Synchrotron PXRD of the isothermal β→α transformation. Each spectrum corresponds to 5 min of holding 10 at -45°C. Pure Sn with (a) 5 min., (b) 10 min., (c) 15min., (d) 20min., (e) 30 min. and (f) 60 min. period of 11 12 annealing prior to being cooled down to -45°C at the rate of 6°C/min. 13 14 Figure 3 Rietveld plots of observed and final calculated intensities versus 2θ degree. Observed intensities are 15 represented by the blue line, the calculated intensities by the red line, the difference plot by the grey line and the 16 17 reflection markers are also shown in blue ; (a) start; (b) in progress and (c) end of transformation. 18 19 Figure 4 Isothermal transformation curves of β⟶α at -45°C, previously subjected to various annealing times at 20 45°C, obtained from the Rietveld analysis of XRD patterns in Figure 3. 21 22 Figure 5 (a) Isothermal α→β transformation after heating from 0°C to each of 30, 35, 40, 45 and 50 °C, at a rate of 23 24 6 °C/min [5]; (b) ) Free-energy differences between α and β-tin as a function of temperature calculated by first 25 principles calculations in Ref.[36]. 26 27 Figure 6 Modified JMAK plots of β→α transformation from data displayed in Figure 4a. (a) JMAK plot (c=0); (b) 28 29 c=1 and (c) c=2 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 10 Page 11 of 16 Crystal Growth & Design

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Figure 1 (a) DSC ce f -tin sample heated up from 0°C to 100°C at the rate of 6°C/min; and (b) thermal history of 33 34 synchrotron PXRD measurements. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Figure 2 Snchn PXRD f he ihemal anfmain. Each ecm cend 5 min f hlding a -45°C. 49 Pure Sn with (a) 5 min., (b) 10 min., (c) 15min., (d) 20min., (e) 30 min. and (f) 60 min. period of annealing prior to being cooled 50 51 down to -45°C at the rate of 6°C/min. 52 53 54 55 56 57 58 59 60

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35 Figure 3 Rietveld plots f beed and final calclaed ineniie e 2 degee. Obeed ineniie ae eeened b he 36 37 blue line, the calculated intensities by the red line, the difference plot by the grey line and the reflection markers are also shown in 38 blue ; (a) start; (b) in progress and (c) end of transformation. 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Figure 4 Ihemal anfmain ce f ⟶ a -45°C, previously subjected to various annealing times at 45°C, obtained 21 from the Rietveld analysis of XRD patterns in Figure 3. 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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32 33 34 Figure 5 (a) Isothemal anfmain afe heating from 0°C to each of 30, 35, 40, 45 and 50 °C, at a rate of 6 °C/min [5]; 35 36 (b) ) Free-energy differences between and -tin as a function of temperature calculated by first principles calculations in 37 Ref.[36]. 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Figure 6 Modified JMAK plots of anfmain fm daa dilaed in Fige 4a. (a) JMAK l ( c=0); (b) c=1 and (c) 46 c=2. 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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