PHYSICAL REVIEW B 74,

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1 Crystaization kinetics and gass-forming abiity of buk metaic gasses Pd 40 Cu 30 Ni 10 P 20 and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 from cassica theory Donghua Xu 1,2, * and Wiiam L. Johnson 1 1 Division of Engineering and Appied Science, Mai Code , Caifornia Institute of Technoogy, Pasadena, Caifornia 91125, USA 2 Department of Nucear Engineering, University of Caifornia, Berkeey, Caifornia , USA Received 19 March 2006; revised manuscript received 13 Apri 2006; pubished 27 Juy 2006 Due to their scientific significance and potentia engineering appications, buk metaic gasses are among the most intensivey studied advanced materias. Understanding the gass-forming abiity GFA of these metaic aoys is a ong-standing subject. Whie a arge number of empirica factors have been proposed to correate with GFA of the aoys, a fu understanding of GFA remains a goa to achieve. Since gass formation is a competing process against crystaization, we have performed a systematic anaysis on the crystaization kinetics of two known best metaic gass-formers Pd 40 Cu 30 Ni 10 P 20 in at. % and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 based on cassica nuceation and growth theory. Our resuts show that there is a dramatic difference between the two aoys in their nuceation behavior athough they possess comparabe GFA. Particuary, an extremey sharp nuceation peak /m 3 s is found for Pd 40 Cu 30 Ni 10 P 20 around 632 K with a very sma haf maximum width of 42 K, impying that this aoy is an exceent candidate for nanocrystaization studies. Moreover, we have aso found that the GFA of these aoys can be cacuated to a high accuracy and precision based on the cassica theory, suggesting that the cassica theory may be sufficient to account for gass formation mechanism in these metaic aoys. DOI: /PhysRevB PACS number s : Dq, Kf, Bx, Fb I. INTRODUCTION II. CRYSTAL NUCLEATION AND GROWTH KINETICS Because of their unique structure and properties, buk metaic gasses BMGs, a new category of advanced materias, are receiving tremendous attention from both fundamenta and practica research discipines. Gass-forming abiity GFA is one of the most important subjects in the study of BMGs, because it determines, to a arge degree, the potentia for this new category of materias to be utiized in various appications. 1 8 Upon continuous cooing from its moten state, an aoy can eventuay form a gass if, and ony if, the cooing rate exceeds a critica vaue. This critica cooing rate R c is a direct and universa measurement of the GFA of any substance incuding the conventiona oxide gasses. Therefore, a compete understanding of GFA shoud utimatey be supported by an accurate and precise cacuation of the R c. Whie a arge number of empirica factors e.g., Refs have been proposed over the past decades to correate with the GFA of various metaic gasses, they either were not verified against R c or coud not reproduce the experimentay measured R c to a satisfactory accuracy and precision. The accurate and precise cacuation of R c, and accordingy, the fu understanding of metaic gass formation mechanism remain unaccompished. Considering that gass formation is a competing process against crystaization, we perform a systematic study on the crysta nuceation and growth kinetics of two known best metaic gass formers Pd 40 Cu 30 Ni 10 P 20 in at. % and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 based on cassica theory. We aso cacuate the R c vaues for the two aoys by virtue of a continuous integra method first introduced by Weinberg et a. 15 for oxide gasses in the framework of the cassica theory, and anayze the accuracy and precision of the R c cacuation. A. Pd 40 Cu 30 Ni 10 P 20 Pd 40 Cu 30 Ni 10 P 20 and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 were seected for this study primariy because of the avaiabiity of their thermodynamic and kinetic data as we as reiabe R c vaues determined directy from continuous cooing experiments In Ref. 16, the isotherma temperature-timetransformation TTT diagram of Pd 40 Cu 30 Ni 10 P 20 was measured and successfuy fitted to the cassica nuceation and growth theory using the foowing equations: and u = I = A v exp k B T G 2, k BT exp n G k B T, 1 2 t = 3 3f 1/4 c, 3 I u where I is the steady-state nuceation rate, u is the diffusioncontroed crysta growth rate, A v is a fitting parameter, is the temperature-dependent viscosity, k B is the Botzmann constant, and G are the interfacia energy per area and the Gibbs free energy difference per voume between the iquid and the crystaine phases, respectivey, is the average atomic diameter, n is the average atomic voume, and f c is the critica crystaized voume fraction detectabe by instruments f c is usuay assigned to a somewhat arbitrary vaue for the fitting of a TTT diagram, and A v and are the ony two fitting parameters. The choice of f c vaue, however, does not affect the fitting Eq. 3 or the cacuation of R c /2006/74 2 / The American Physica Society

2 DONGHUA XU AND WILLIAM L. JOHNSON TABLE I. Thermodynamic and kinetic data for Pd 40 Cu 30 Ni 10 P 20 taken from Ref Pa s D T 0 H F J/m 3 T J/m 2 A v Pa/m 3 n m 3 T g m f c T n t n s a a The vaue of J/m 2 for given in Ref. 16 is not accurate. Eq. 8 beow because the intrinsic fitting parameter besides is actuay the ratio A v / f c rather than A v. It does, on the other hand, affect the absoute vaue of I Eq. 1 through A v. In the fitting, a Voge-Fucher-Tamman VFT form was used to represent the temperature dependence of viscosity T = 0 exp DT 0 T T 0, where 0, D, and T 0 are three constants. In addition, a first order approximation was used for G, i.e., G= H F 1 T/T, where H F is the enthapy of fusion per unit voume, and T is the iquidus temperature. Tabe I summarizes a the reevant thermodynamic and kinetic data for Pd 40 Cu 30 Ni 10 P 20 taken from Ref. 16. Using these data, one can cacuate and pot the nuceation rate I and the growth rate u as functions of temperature. The resuts are shown in Fig. 1. Upon cooing from the iquidus temperature 823 K, the crysta growth rate increases immediatey and quicky reaches its maximum vaue of m/s at 794 K, and then drops rapidy afterwards. By contrast, the nuceation rate remains very ow over a reativey arge temperature interva of 123 K beow the iquidus temperature. Starting from 700 K, however, the nuceation rate rises steepy as the temperature further decreases, and rapidy assumes its maximum vaue of /m 3 s at 632 K, foowed by a steep decrease afterwards. 4 Both the nuceation rate and the growth rate are jointy controed by thermodynamics and kinetics. In fact, one can separate the thermodynamic and kinetic factors by dividing each of the rates into two parts, with representing kinetics and I or u representing thermodynamics. Figure 2 shows the pots of the two thermodynamic factors, for the nuceation rate and the growth rate of Pd 40 Cu 30 Ni 10 P 20, respectivey. It can be seen that within the temperature range between the iquidus temperature and the gass transition temperature T g, 582 K for Pd 40 Cu 30 Ni 10 P 20, refer to Tabe I, the two thermodynamic factors both increase monotonicay as temperature decreases, which means that thermodynamics consistenty enhances crysta nuceation and growth. On the other hand, kinetics consistenty restrains crysta nuceation and growth, as represented by monotonicay increasing pot not shown during continuous cooing process. It is the competition between thermodynamics and kinetics that has resuted in the peaks of I and u that are shown in Fig. 1. As we wi discuss in next section, when compared with Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5, the nuceation behavior of Pd 40 Cu 30 Ni 10 P 20 appears very unique and has important impications for future nanocrystaization studies. B. Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 In Ref. 17, the kinetic time scaes for viscous fow, atomic transport, and crystaization isotherma in the iquid and FIG. 1. Nuceation rate I and crysta growth rate u as functions of temperature for Pd 40 Cu 30 Ni 10 P 20. FIG. 2. Thermodynamic factors, I and u, for the nuceation rate I and the crysta growth rate u, respectivey, as functions of temperature for Pd 40 Cu 30 Ni 10 P

3 CRYSTALLIZATION KINETICS AND GLASS-FORMING¼ TABLE II. Thermodynamic and kinetic data for Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 taken from Ref. 17 uness otherwise noted. 0 Pa s c 1 c 2 T 0 D 0 m 2 /s Q eff ev H F J/m 3 T J/m 2 A v m 5 n m 3 T g m f c T n t n s a a a Taken from Ref. 19. supercooed iquid states of Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 Vitreoy-1 were reported. It was found that the measured viscosity of Vitreoy-1 cannot be fitted we by a VFT equation in the supercooed iquid range. Rather, the Cohen-Grest CG free voume mode 20 provides a better description for the temperature dependence of viscosity, i.e. = 0 exp c 1 / T T 0 + T T0 2 + c 2 T. 5 The measured isotherma TTT diagram was fitted we using Eq. 3 and the foowing expressions for the nuceation rate I and the crysta growth rate u: 2 I = A v D eff exp k B T G and u = D eff 1 exp n G, 7 k B T where D eff is a piecewise-defined effective diffusivity: above 850 K, the effective diffusivity obeys a Stokes-Einstein reationship with viscosity, i.e., D eff =k B T/ 3 ; beow 850 K, however, the effective diffusivity is better described by an Arrhenius aw, i.e., D eff =D 0 exp Q eff /k B T with an activation energy Q eff =1.2 ev. Tabe II provides a ist of the reevant data for Vitreoy-1 taken from Refs. 17 and 19. The nuceation rate I and the crysta growth rate u are potted versus temperature in Fig. 3. The genera profies of I and u for Vitreoy-1 are very simiar to those for Pd 40 Cu 30 Ni 10 P 20, both increasing initiay as temperature decreases, reaching a peak vaue and then decreasing subsequenty. The peak positions of I and u for Vitreoy-1 are 793 K and 989 K, respectivey. More carefu comparison of Vitreoy-1 with Pd 40 Cu 30 Ni 10 P 20, however, reveas that whie the maximum crysta growth rates for the two aoys are different by ony two orders of magnitude, the maximum nuceation rates differ by amost fourteen orders of magnitude Note that there is a factor of 50 difference between the f c vaues used for the two aoys, which may affect I by the same factor through A v. However, this cannot at a account for the 14 orders of magnitude difference in the two maximum nuceation rates.. Nevertheess, the two aoys show very simiar shortest crystaization time, i.e., nose time t n, on their measured isotherma TTT diagrams, being 50 s and 60 s for Pd 40 Cu 30 Ni 10 P 20 and for Vitreoy-1, respectivey. 16,17 This has resuted from the different degree of overapping and the different sharpness with respect to temperature of I and u for the two aoys. Particuary worth noting is the distinct sharpness of the I peak for Pd 40 Cu 30 Ni 10 P 20, being characterized by its very sma haf maximum width of 42 K and extremey high magnitude of /m 3 s, as compared to 131 K and /m 3 s, respectivey, for Vitreoy-1 refer to Fig. 1, Fig. 3, and aso Tabe III. From these resuts, one can expect that nanocrystaization can be obtained more easiy in Pd 40 Cu 30 Ni 10 P 20 than in Vitreoy-1. An extremey arge number of nucei wi form when Pd 40 Cu 30 Ni 10 P 20 is maintained at around 632 K, and yet they cannot easiy grow into arge grains due to the very ow growth rate corresponding to that temperature. The characteristic data isted in Tabe III together with the information contained in Figs. 1 3 wi provide important guideines for future nanocrystaization experiments. III. CALCULATION OF CRITICAL COOLING RATE A. Continuous integra method In Ref. 15, Weinberg et a. first utiized a continuous integra method to evauate the reative effects of different factors such as iquid-crysta interfacia energy on the cacuated critica cooing rate R c for certain oxide gasses, on the basis of the Johnson-Meh-Avrami JMA phase transformation mode 21,22 and the cassica theory for crysta nuceation and growth. In the method, assuming a spherica mode of crysta growth, the crystaized voume fraction f of a gass-forming iquid upon continuous cooing, if f 1, depends on the cooing rate R as foows: f = 4 T g T g 3R 4 I u T dt 3. T T dt T Therefore, the critica cooing rate required for gass formation is determined as TABLE III. Characteristics of the nuceation and growth peaks for Pd 40 Cu 30 Ni 10 P 20 and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be Notation: T p peak temperature, HMW haf maximum width. Aoy Composition at. % u max m/s T p u HMW u I v,max m 3 s 1 T p I v HMW I v Pd 40 Cu 30 Ni 10 P Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be

4 DONGHUA XU AND WILLIAM L. JOHNSON FIG. 3. Nuceation rate I and crysta growth rate u as functions of temperature for Zr = 41.2 Ti 13.8 Cu 12.5 Ni 10 Be R 4 T g T g c I u T dt 3 1/4. 8 3f c T T dt T By substituting Eqs. 1, 2, and 4 into Eq. 8 and using the data isted in Tabe I, the critica cooing rate R c for Pd 40 Cu 30 Ni 10 P 20 is cacuated to be 0.46 K/s. This cacuated vaue agrees remarkaby we with the experimentay measured 0.33 K/s, 16 particuary considering the arge variation, usuay by severa orders of magnitude, in the critica cooing rates of different metaic aoys. Simiary, the critica cooing rate for Vitreoy-1 can be cacuated using Eqs. 5 8, the piecewise-defined effective diffusivity D eff, and the thermodynamic and kinetic data isted in Tabe II. The cacuation yieds R c =0.91 K/s. As for the case of Pd 40 Cu 30 Ni 10 P 20, this cacuated vaue for Vitreoy-1 agrees strikingy we with the experimentay measured 1.4 K/s. 18 FIG. 4. Measured and cacuated isotherma TTT diagrams of Pd 40 Cu 30 Ni 10 P 20 : a experimenta data Ref. 16 ; b cacuated using A v = Pa/m 3 and =0.061 J/m 2 ; c cacuated using A v = Pa/m 3 and = J/m 2 ; d cacuated using A v = Pa/m 3 and = J/m 2. B. Accuracy and precision of the cacuations It was warned in the origina paper by Weinberg et a. 15 that sma error in the interfacia energy may cause arge uncertainty in the cacuated R c. For exampe, the authors demonstrated see Fig. 3 in Ref. 15 when proportiona to a Turnbu s constant = N A V 2 m 1/3 / H F Ref. 9 increases by 15%, the cacuated R c for SiO 2 drops by one order of magnitude. As a matter of fact, the use of the absoute vaue of the cacuated R c was not recommended by the authors. However, the error or, sensitivity anaysis in Ref. 15 was performed in a way that errors from different factors were isoated and were not aowed to interfere with each other. Nevertheess, this is not the case when a measured isotherma TTT diagram or part of it, particuary the part around its nose is avaiabe, as in the present study. In the present cacuation of R c, the measured TTT diagrams pace constraints on the errors in and A v. Take Pd 40 Cu 30 Ni 10 P 20 as an exampe. As an iustration, Fig. 4 dispays the experimenta data together with three cacuated TTT diagrams for Pd 40 Cu 30 Ni 10 P 20. In Fig. 4, curve b is the best fit, with A v = Pa/m 3 and =0.061 J/m 2, curve c is cacuated by fixing A v = Pa/m 3 and introducing 10% error to and thus making = J/m 2, and curve d is cacuated using A v = Pa/m 3 and = J/m 2 such that the nose time t n is restored to the measured 50 s whie keeping the 10% error in. Apparenty, when introduced isoatedy, the 10% error can change the position of the cacuated TTT diagram dramaticay. In order to keep reasonabe fit to the measured data, accordingy, the 10% error in has to be accommodated with a three orders of magnitude change in A v. The mutuay constrained errors in and A v resut in a fairy stabiized precise R c, e.g., R c =0.42 K/s for A v = Pa/m 3 and = J/m 2,in comparison with R c =0.46 K/s obtained using the best fit of and A v. A further extension to the above anaysis is to aow ±10 s uncertainties in the nose time t n whie keeping the 10% error in i.e., fixing = J/m 2. The cacuated TTT diagrams are shown in Fig. 5. Correspondingy, the critica cooing rate is evauated to be: R c =0.34 K/s for A v = Pa/m 3, and R c =0.60 K/s for A v = Pa/m 3. Simiar anaysis has aso been performed on Vitreoy-1. It is concuded that the error bars to the cacuated R c =0.46 K/s for Pd 40 Cu 30 Ni 10 P 20 and R c =0.91 K/s for Vitreoy-1, caused by the uncertainties in their measured TTT diagrams, are both within ±0.2 K/s. Therefore, the cacuation is very precise. On the other hand, comparing the cacuated vaues with the measured 0.33 K/s and 1.4 K/s for Pd 40 Cu 30 Ni 10 P 20 and Vitreoy-1, respectivey, indicates the high accuracy of the cacuation. Certain empirica methods, such as the Turnbu s T rg rue 9 and the one based on the parameter, 10 have been used frequenty to obtain a quick and rough estimation of GFA since they do not require a the data input as needed for the

5 CRYSTALLIZATION KINETICS AND GLASS-FORMING¼ another quick and rough criterion that wi amost certainy encounter many exceptions. IV. CONCLUSIONS FIG. 5. Measured and cacuated with fixed = J/m 2 isotherma TTT diagrams of Pd 40 Cu 30 Ni 10 P 20 : a experimenta data Ref. 16 ; b cacuated using A v = Pa/m 3 ; c cacuated using A v = Pa/m 3 ; d cacuated using A v = Pa/m 3. integra method and accordingy much simper to appy. Whie many aoy systems were reported to support the predictive power of the empirica methods, there were aso a comparabe number of exceptions with distinct deviations. It shoud be noted that the main concern here in this present paper is to obtain a more compete and accurate fundamenta understanding of why a metaic gass can form from thermodynamic and kinetic point of view, rather than to estabish Within the framework of cassica nuceation and growth theory, we have performed a systematic study on the crystaization kinetics of two known best metaic gass formers Pd 40 Cu 30 Ni 10 P 20 and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5, and cacuated the critica cooing rates R c for the two aoys using a continuous integra method. We have found the existence of an extremey sharp nuceation peak /m 3 s around 632 K for Pd 40 Cu 30 Ni 10 P 20 with a very sma haf maximum width of 42 K, which impies that this aoy is an exceent candidate for nanocrystaization studies. The cacuated R c vaues agree remarkaby we with experimenta measurements, with ony very sma deviation of 0.13 K/ s and 0.5 K/s for Pd 40 Cu 30 Ni 10 P 20 and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5, respectivey. The error bars for the cacuations, caused by the uncertainties in the measured TTT diagrams, are evauated to be within ±0.2 K/s for both aoys. The high accuracy and precision of the cacuations suggest that the cassica theory may be sufficient to account for gass formation mechanism in these metaic aoys. ACKNOWLEDGMENTS D. H. Xu woud ike to thank B. D. Wirth at the University of Caifornia at Berkeey for his financia support and vauabe discussions. J. F. Loffer at ETH Zurich and J. Schroers at Yae University are acknowedged for providing some of the origina data. E. Ma at Johns Hopkins University is appreciated for proofreading the manuscript. *Corresponding author. Emai address: xudh@berkeey.edu 1 W. L. Johnson, MRS Bu. 24, D. Xu, G. Duan, and W. L. Johnson, Phys. Rev. Lett. 92, Y. He, S. J. Poon, and G. J. Shifet, Science 241, V. Ponnambaam, S. J. Poon, and G. J. Shifet, J. Mater. Res. 19, B. Zhang, R. J. Wang, D. Q. Zhao, M. X. Pan, and W. H. Wang, Phys. Rev. B 70, J. Das, M. B. Tang, K. B. Kim, R. Theissmann, F. Baier, W. H. Wang, and J. Eckert, Phys. Rev. Lett. 94, H. W. Sheng, W. K. Luo, F. M. Aamgir, J. M. Bai, and E. Ma, Nature London 439, D. Xu, B. Lohwongwatana, G. Duan, W. L. Johnson, and C. Garand, Acta Mater. 52, D. Turnbu, Contemp. Phys. 10, Z. P. Lu and C. T. Liu, Acta Mater. 50, A. Inoue and A. Takeuchi, Mater. Trans., JIM 43, D. N. Perera, J. Phys.: Condens. Matter 11, Y. M. Wang et a., Scr. Mater. 50, D. B. Mirace, W. S. Sanders, and O. N. Senkov, Phios. Mag. 83, M. C. Weinberg, B. J. Zeinski, and D. R. Uhmann, J. Non-Cryst. Soids 123, J. F. Loffer, J. Schroers, and W. L. Johnson, App. Phys. Lett. 77, A. Masuhr, T. A. Waniuk, R. Busch, and W. L. Johnson, Phys. Rev. Lett. 82, T. A. Waniuk, J. Schroers, and W. L. Johnson, App. Phys. Lett. 78, R. Busch, Y. J. Kim, and W. L. Johnson, J. App. Phys. 77, G. S. Grest and M. H. Cohen, in Advances in Chemica Physics, edited by I. Prigogine and S. A. Rice Wiey, New York, 1981, p W. A. Johnson and R. F. Meh, Trans. AIME 135, M. Avrami, J. Chem. Phys. 7,