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Relations between flash onset-, Debye-, and glass transition temperature in flash sintering of oxide nanoparticles
Scripta Materialia 169 (2019) 6–8
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Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Relations between flash onset-, Debye-, and glass transition temperature in flash sintering of oxide nanoparticles Rachman Chaim Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history: Received 8 April 2019 Received in revised form 30 April 2019 Accepted 30 April 2019 Available online 9 May 2019 Keywords: Flash-sintering Liquid-film Capillarity Melt-fragility Oxides
a b s t r a c t The liquid-film capillary model describes the ultrafast densification kinetics of oxide nanoparticles during flash sintering. The glass transition temperature of the supercooled liquid film dictates the lower bound of the flash sintering temperature in oxides with fragile liquids. This lower bound tends to Debye temperature of the oxide in its crystalline form. The tendency of the two bounds to coincide increases with the decrease in liquid fragility. Spontaneous crystallization of the fragile liquid leads to temperature difference between the lower bound flash onset and Debye temperatures. Fragility of the liquid affects the kinetics and thermodynamics of the flash sintering process. © 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Flash sintering of ceramic powders is a revolutionary technique for ultrafast densification of ceramic bodies within a few seconds [1,2]. However, despite the vast number of investigations of different oxide systems, the densification mechanisms responsible for the rapid kinetics are still under debate. The atomistic mechanisms suggested for flash sintering of oxide powders include solid-state [3–5] as well as liquid-film assisted sintering [6–8]. Whereas the solid-state sintering approach assumes the formation of point defect avalanches at the flash onset, the liquid-film assisted sintering assumes local surface softening/melting at the particle contacts. Here we will analyze and discuss recent literature data that supports our approach about the involvement of the liquid-film during ultrafast densification of ceramic nanopowders subjected to flash sintering. Recently, Yadav and Raj [9,10] used the flash temperature data of 3YSZ and 8YSZ (3 mol% and 8 mol% yttria-stabilized zirconia) using single- and polycrystals for correlation with their Debye temperatures, θD. Normalization of the data at high electric fields led to asymptotic behavior of the flash temperature lower bound to the Debye temperature range. They hypothesized that “the significance of Joule heating in Stage I is related to nonlinear lattice vibrations” [10]. However, as they explained in other words, the Debye temperature represents the upper limit of the lattice vibrations (hottest phonons) in the crystalline solid, hence the upper bound temperature for the elastic behavior. We emphasize that formation of local melt at the particle contacts and its preservation as a supercooled liquid-film depends on the oxide melt properties and its glass transition temperature, Tg. Below we will
E-mail address: [email protected].
https://doi.org/10.1016/j.scriptamat.2019.04.040 1359-6462/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
show that Yadav & Raj [10] findings about the Debye temperature, as a lower bound for flash temperature, are in agreement with the liquid-film assisted densification mechanism. Assuming the formation of a liquid film at the particle contacts at the flash temperature [8], the transient liquid can survive as a supercooled liquid for the particle rearrangement, provided sufficient power is supplied to the contact. The lower bound for the existence of the liquid film as viscous melt, if formed, is its glass transition temperature. The glass transition temperature, Tg, is defined as the temperature at which the melt viscosity reaches the value between 1011 and 1012 Pa·s [11]; below glass transition temperature the glass becomes almost an isotropic elastic continuum solid. Approaching from the low temperatures, Debye temperature represents the upper bound temperature below which the crystalline solid counterpart behaves elastically, yet with maximum lattice vibrations. Consequently, at certain conditions, these two temperatures resemble the similar physics of the atomic vibrations, either in the amorphous or in the crystalline state. Angel [12] has thoroughly discussed this aspect. The main difference between the crystalline oxide and its amorphous form (glass) is the long-range crystalline order in the former; the same interionic short-range order and ionic/covalent bonds persist in the two solids. Therefore, the main thermodynamic difference between the crystalline oxide and its amorphous counterpart is the fusion entropy. Consequently, the Debye temperature in terms of the hottest phonons in the crystal, manifested by the glass transition temperature in the glassy state. The glass above Tg experiences a softening behavior due to the hottest phonons in the otherwise crystalline form. Therefore, one expects the glass transition temperature of the glass to approach from above to the Debye temperature of its crystalline counterpart
R. Chaim / Scripta Materialia 169 (2019) 6–8
from below. The fact that heat capacity of a crystalline oxide at the Debye temperature reaches its maximum value (which depends on the atomic mass) irrespective of its crystal-type supports this expectation. Strong correlation exists between the melting point, Tm, and the Debye temperature, θD, of crystals via the Lindemann's criterion [13]. In addition, Turnbull [14] discussed the various material aspects important to glass formation, and proposed the Tg/Tm ratio of 0.66 for most glasses. Wang et al. [15] used the above considerations together with accumulated experimental data to show that the ratio Tg/θD falls in the range between 1.23 and 3.17 in metallic glasses. Unfortunately, the majority of cations in oxides are six-coordinated or higher, and are not glass formers and hence cannot retain stable glass from their melts. Thus, there is lack of data on the Tg of these oxides. Nevertheless, we will discuss the Tg - θD relation in terms of the melt properties and especially its ‘fragility’. The ‘fragility’ of the melt is defined by the slope of the viscosity change versus temperature as it approaches the glass transition temperature from above [16]. It is a measure of the dynamic relaxation of the molecular species comprising the melt. In this respect, and for the present treatment, one can classify the oxides according to the Zachariasen's rules for glass formation [11] into two classes: First, the glass formers, the cations of which are four coordinated or less, with partial covalent bonding. These include SiO2, GeO2, B2O3, and P2O5. These oxides are characterized by their relatively open crystal structures, and strong tendency to retain their amorphous character, even during slow cooling rate of the melt. This class of glasses are termed ‘strong’ glasses, and their viscosity follows the Arrhenius-type temperature dependence [12]. Most oxides belong to the second class, and do not form stable glass as mentioned above. The fusion entropies of the later oxides are relatively higher than that of the glass-forming oxides. As a result, there is a significant drop in the melt viscosity recorded in their supercooled melts with the temperature decrease, and hence they are termed ‘fragile’ melts. Thus, due to the same physics of the thermal ionic vibrations, one expects the Debye temperature of the glass-former oxide to be very close to the Tg of its amorphous counterpart. This proximity between the two temperatures is well expressed in GeO2 that exhibits pure Arrhenian behavior [17] with θD of 1000 K [18], and Tg of 853 K [19] and 980 K [20]. For comparison, Sapphire exhibits a fragile melt [21]. The values of 1042 K as θD for Sapphire [22], and 1300 K [21] and 1395 K [23] as Tg for supercooled alumina melt were reported. As the fragility of the oxide melt increases, its Tg increases due to the drastic non-Arrhenian increase in viscosity. Accordingly, the theoretical viscosity and activation energy for viscous flow are much smaller for the fragile oxide melts than for stable glass (i.e. silica based) melts [24]. The low viscosity and the narrow range of the glass transition temperature in oxides with fragile melts may explain the transient and ultrafast effect of the liquid during the particle rearrangement
Table 1 Flash sintering conditions in air and Debye temperatures for different oxides. Oxide
Experimental data Particle size nm
Al2O3 (0.25 wt% MgO) CeO2 (Sm-doped) MgTiO3 SrTiO3 (cubic) TiO2 (rutile) ThO2 Y2O3 8YSZ (cubic) ZnO
100–300 100 66 150 20 265 20 20 16
Field Flash V·cm−1 temp. Tf °C 500 120 500 500 500 533 500 500 160
1320 750 1241 900 700 970 1133 572 625
T [K] = T [°C] + 298. a Debye temperature values are for pure oxides.
Ref. Debye Tf, LB °C (K) Extrapolated temp. −1 for 3 kV·cm θD K
Ref.
950a
35
a
36 37 35 35 38 39 40 37
774 (1072) 173 (471) 524 (822) 563 (861) 506 (804) 508 (806) 778 (1076) 379 (677) 491 (789)
3 26 27 28 29 30 31 32 33
475 900 690 452 468 491 590 440
7
Fig. 1. Experimental flash temperature - electric field relations in different oxides: MgOdoped Al2O3 [3], Sm-doped CeO2 [26], MgTiO3 [27], SrTiO3 [28], TiO2 [29], ThO2 [30], Y2O3 [31], 8YZS-cubic [32], and ZnO [33].
immediately after the flash event. Homologous nucleation of the ‘fragile’ melt on its solid counterpart, immediately after the particle rearrangement, is mandatory, due to zero activation energy for the heterogeneous nucleation of the exothermic solidification [25]. In order to support the present analysis, we used the literature data of different oxides flash sintered in air at different temperature - electric field conditions [3,26–33] (with known Debye temperatures) as summarized in Table 1 and shown in Fig. 1. These experiments often differ in their powder particle size and the heating rates (Table 1). As was mentioned [34] the effect of the heating rate on the flash temperature is negligible in comparison to the particle size. Nevertheless, first we fitted all these data to curves (solid and dashed lines in Fig. 1) using power-law or exponential fittings the equations of which were summarized in Table 2. Extrapolation of these curves to a very high electric field of 3 kV·cm−1 yielded asymptotic behaviour from which the lower bounds for the flash temperatures were determined as Tf, LB in Table 1. Reliable Debye temperatures for these oxides were acquired from different sources in the literature [35–40] (Table 1). We plotted the lower bound flash temperature, Tf, LB versus Debye temperature of the oxides in Fig. 2. The line in Fig. 2 is the geometric location where these two temperatures of interest are equal. As is evident from Fig. 2, while some oxides, such as CeO2, 8YSZ, Al2O3 and MgTiO3, exhibit similarity between Tf, LB and θD, other oxides such as ZnO, TiO2, ThO2 and Y2O3 exhibit significant differences between the two temperatures of interest, as high as 600 K. Considering the nanoparticle size of ZnO, TiO2, and Y2O3 in Fig. 2 (and Table 1) higher flash temperatures are expected for larger particle size, hence far less compatibility with the Debye temperatures. Therefore, the tendency of the lower bound flash temperature to Debye temperature is not universal for all oxides. Nevertheless, as mentioned above, the glass transition temperature of the oxide melt in its
Table 2 Fitting curve equations used to calculate Tf, LB with extrapolation to 3 kV·cm−1. Oxide
Fitting equationa
Al2O3 (0.25 wt% MgO) CeO2 (Sm-doped) MgTiO3 SrTiO3 (cubic) TiO2 (rutile) ThO2 Y2O3 8YSZ (cubic) ZnO
Y = 5603.4 · x^-0.24725 Y = 3451.9 · x^-0.37354 Y = 1488.6 · exp (-0.000348 · x) Y = 4350.6 · x^-0.25539 Y = 2185.7 · x^-0.18268 Y = 28,381 · x^-0.50243 Y = 4079.5 · x^-0.20698 Y = 2511.4 · x^-0.23614 Y = 969.68 · x^-0.084919
a
Y = flash temperature [°C]; x = electric field [V·cm−1].
Fitting R value
Ref.
0.99856 0.99929 0.96345 0.99931 0.99270 0.93825 0.99737 0.99824 0.99222
3 26 27 28 29 30 31 32 33
8
R. Chaim / Scripta Materialia 169 (2019) 6–8
Acknowledgments I am grateful to Dr. Mark Rispler for his critical reading of the manuscript. References
Fig. 2. Lower bound flash temperature calculated at the electric field of 3 kV·cm−1 versus Debye temperature of the oxides used in Fig. 1. The lower bounds consistently are higher than the corresponding Debye temperature of the oxide.
supercooled liquid state depends on the liquid fragility and is always higher (by the factor of 1.2 to 3.17 in metallic systems) compared to the Debye temperature. As the liquid fragility is directly related to the fusion entropy, the lower bound flash onset temperature is expected to depart from Debye temperature with increase in the liquid fragility. Full confirmation of this hypothesis will be possible in the future when Tg data of non-glass forming supercooled oxide liquids will be available. Flash sintering investigations by Jain [41,42] and Sglavo [43–46] groups on silica based (strong glass-former) systems at different conditions revealed important results about the nature of the flash sintering in these systems. First, enhanced electric field-induced softening (EFIS) effect by which the applied electric field decreases the glass softening point in alkali silicate glasses was detected [41]. Biesuz et al. [43] observed a similar effect by adding a magnesia-silica glass to Al2O3 to facilitate its densification via liquid-phase sintering at temperatures lower than needed for conventional liquid-phase sintering. Overall, the flash sintering was mainly related to the dielectric breakdown caused by excitation of the alkali ions [40,43,44]. Biesuz et al. [46] also showed that enhanced densification of the porcelain stoneware by vitrification during flash sintering takes place beyond a critical current density. They related the enhanced liquid-phase densification to changes in the rheological behavior of the glass [46]. Therefore, it is evident that the electric field and current act as additional sources of energy, similar to temperature and pressure, in affecting the glass thermodynamic properties and especially its viscosity and glass transition temperature. In this respect, Mel'cuk et al. [47] investigated the existence of long-lived clusters within the fragile glass-forming liquids using molecular dynamics simulations. They deduced that the glass transition in the supercooled liquid represents the early stage of the glass thermodynamic instability. Therefore, one may expect that the viscosity, hence the glass transition temperature of the glass-forming oxides depends on the applied electric field, hence is not constant. This may be the cause for the successful flash sintering of glass-based systems below their conventional glass transition temperatures [41–46]. Finally, application of the present model for glass-forming systems should consider the decrease in the glass transition temperature caused by the applied electric field.
[1] M. Yu, S. Grasso, R. Mckinnon, T. Saunders, M.J. Reece, Adv. Appl. Ceram. 116 (2017) 24–60. [2] M. Biesuz, V.M. Sglavo, J. Eur. Ceram. Soc. 39 (2019) 115–143. [3] M. Cologna, J.S.C. Francis, R. Raj, J. Eur. Ceram. Soc. 31 (2011) 2827–2837. [4] H. Yoshida, A. Uehashi, T. Tokunaga, K. Sasaki, T. Yamamoto, J. Ceram. Soc. Japan 124 (2016) 388–392. [5] D. Kok, D. Yadav, E. Sortino, S.J. McCormack, K.P. Tseng, W.M. Kriven, R. Raj, M.L. Mecartney, J. Am. Ceram. Soc. 102 (2019) 644–653. [6] J. Narayan, Scr. Mater. 68 (2013) 785–788. [7] R. Chaim, Materials 9 (2016) 280. [8] R. Chaim, Scr. Mater. 158 (2019) 88–90. [9] D. Yadav, R. Raj, Scr. Mater. 134 (2017) 123–127. [10] D. Yadav, R. Raj, J. Am. Ceram. Soc. 100 (2017) 5374–5378. [11] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, John Wiley & Sons, New York, 1976. [12] C.A. Angell, J. Phys. Chem. Solids 49 (1988) 863–871. [13] P. Hofmann, Solid State Physics: An Introduction, Wiley-VCH Verlag Gmb H & Co, Weinheim, 2008. [14] D. Turnbull, Contemp. Phys. 10 (1969) 473–488. [15] W.H. Wang, P. Wen, D.Q. Zhao, M.X. Pan, R.J. Wang, J. Mater. Res. 18 (2003) 2747–2751. [16] J.C. Mauro, Y. Yue, A.J. Allison, P.K. Gupta, D.C. Allan, PNAS 106 (2009) 19780–19784. [17] L.M. Martinez, C.A. Angell, Nature 410 (2001) 663–667. [18] P. Hermet, A. Lignie, G. Fraysse, P. Armand, Ph. Papet, Phys. Chem. Chem. Phys. 15 (2013) 15943–15948. [19] D.B. Dingwell, R. Knoche, S.L. Webb, Phys. Chem. Minerals 19 (1993) 445–453. [20] P. Richet, Phys. Chem. Minerals 17 (1990) 79–88. [21] P.C. Nordine, J.K.R. Weber, J.G. Abadie, Pure Appl. Chem. 72 (2000) 2127–2136. [22] R. Viswanathan, J. Appl. Phys. 46 (1975) 4086–4087. [23] C. Shi, O.L.G. Alderman, D. Berman, J. Du, J. Neuefeind, A. Tamalonis, J.K. Richard Weber, J. You, C. Benmore, Frontiers in Mater. 6 (2019), 38. [24] G. Wu, E. Yazhenskikh, K. Hack, E. Wosch, M. Muller, Fuel Process. Tech. 137 (2015) 93–103. [25] R. Chaim, C. Estournès, J. Mater. Sci. 53 (2018) 6378–6389. [26] J. Li, L. Guan, W. Zhang, M. Luo, J. Song, X. Song, S. An, Ceram. Int. 44 (2018) 2470–2477. [27] X. Su, G. Bai, J. Zhang, J. Zhou, Y. Jia, Appl. Surf. Sci. 442 (2018) 12–19. [28] A. Karakuscu, M. Cologna, D. Yarotski, J. Won, J.S.C. Francis, R. Raj, B.P. Uberuaga, J. Am. Ceram. Soc. 95 (2012) 2531–2536. [29] S.K. Jha, R. Raj, J. Am. Ceram. Soc. 97 (2014) 527–534. [30] W. Straka, S. Amoah, J. Schwartz, MRS Comm. 7 (2017) 677–682. [31] H. Yoshida, Y. Sakka, T. Yamamoto, J.M. Lebrun, R. Raj, J. Eur. Ceram. Soc. 34 (2014) 991–1000. [32] J.A. Downs, V.M. Sglavo, J. Am. Ceram. Soc. 96 (2013) 1342–1344. [33] C. Schmerbauch, J. Gonzalez-Julian, R. Röder, C. Ronning, O. Guillon, J. Am. Ceram. Soc. 97 (2014) 1728–1735. [34] R. Chaim, C. Estournès, Scr. Mater. 163 (2019) 130–132. [35] E. Langenberg, E. Ferreiro-Vila, V. Leborán, A.O. Fumega, V. Pardo, F. Rivadulla, APL Mater. 4 (2016) 104815. [36] K. Suzuki, M. Kato, T. Sunaoshi, H. Uno, U. Carvajal-Nunez, A.T. Nelson, K.J. McClellan, J. Am. Ceram. Soc. 102 (2019) 1994–2008. [37] R.A. Robie, H.T. Haselton Jr., B.S. Hemingway, J. Chem. Thermodynamics 21 (1989) 743–749. [38] T.D. Kelly, J.C. Petrosky, J.W. McClory, T. Zens, D. Turner, J.M. Mann, J.W. Kolis, J.A. Colόn Santana, P.A. Dowben, Mater. Res. Soc. Symp. Proc. 1567 (2013). [39] W.R. Manning, O. Hunter Jr., B.R. Powell Jr., J. Am. Ceram. Soc. 52 (1969) 436–442. [40] C. Degueldre, P. Tissot, H. Lartigue, M. Puochon, Thermochimia Acta 403 (2003) 267–273. [41] C. McLaren, W. Heffner, R. Tessarollo, R. Raj, H. Jain, Appl. Phys. Lett. 107 (2015), 184101. [42] C. McLaren, B. Roling, R. Raj, H. Jain, J. Non-Cryst. Solids 471 (2017) 384–395. [43] M. Biesuz, V.M. Sglavo, J. Eur. Ceram. Soc. 37 (2017) 705–713. [44] L. Pinter, M. Biesuz, V.M. Sglavo, T. Saunders, J. Binner, M. Reece, S. Grasso, Scr. Mater. 151 (2018) 14–18. [45] M.O. Prado, M. Biesuz, M. Frasnelli, F.E. Benedetto, V.M. Sglavo, J. Non-Crystal. Solids 476 (2017) 60–66. [46] M. Biesuz, W.D. Abate, V.M. Sglavo, J. Am. Ceram. Soc. 101 (2018) 71–81. [47] A.I. Mal'cuk, R.A. Ramos, H. Gould, W. Klein, R.D. Mountain, Phys. Rev. Lett. 75 (1995) 2522–2525.
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Relations between flash onset-, Debye-, and glass transition temperature in flash sintering of oxide nanoparticles Rachman Chaim Department of Materials Science and Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e
i n f o
Article history: Received 8 April 2019 Received in revised form 30 April 2019 Accepted 30 April 2019 Available online 9 May 2019 Keywords: Flash-sintering Liquid-film Capillarity Melt-fragility Oxides
a b s t r a c t The liquid-film capillary model describes the ultrafast densification kinetics of oxide nanoparticles during flash sintering. The glass transition temperature of the supercooled liquid film dictates the lower bound of the flash sintering temperature in oxides with fragile liquids. This lower bound tends to Debye temperature of the oxide in its crystalline form. The tendency of the two bounds to coincide increases with the decrease in liquid fragility. Spontaneous crystallization of the fragile liquid leads to temperature difference between the lower bound flash onset and Debye temperatures. Fragility of the liquid affects the kinetics and thermodynamics of the flash sintering process. © 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Flash sintering of ceramic powders is a revolutionary technique for ultrafast densification of ceramic bodies within a few seconds [1,2]. However, despite the vast number of investigations of different oxide systems, the densification mechanisms responsible for the rapid kinetics are still under debate. The atomistic mechanisms suggested for flash sintering of oxide powders include solid-state [3–5] as well as liquid-film assisted sintering [6–8]. Whereas the solid-state sintering approach assumes the formation of point defect avalanches at the flash onset, the liquid-film assisted sintering assumes local surface softening/melting at the particle contacts. Here we will analyze and discuss recent literature data that supports our approach about the involvement of the liquid-film during ultrafast densification of ceramic nanopowders subjected to flash sintering. Recently, Yadav and Raj [9,10] used the flash temperature data of 3YSZ and 8YSZ (3 mol% and 8 mol% yttria-stabilized zirconia) using single- and polycrystals for correlation with their Debye temperatures, θD. Normalization of the data at high electric fields led to asymptotic behavior of the flash temperature lower bound to the Debye temperature range. They hypothesized that “the significance of Joule heating in Stage I is related to nonlinear lattice vibrations” [10]. However, as they explained in other words, the Debye temperature represents the upper limit of the lattice vibrations (hottest phonons) in the crystalline solid, hence the upper bound temperature for the elastic behavior. We emphasize that formation of local melt at the particle contacts and its preservation as a supercooled liquid-film depends on the oxide melt properties and its glass transition temperature, Tg. Below we will
E-mail address: [email protected].
https://doi.org/10.1016/j.scriptamat.2019.04.040 1359-6462/© 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
show that Yadav & Raj [10] findings about the Debye temperature, as a lower bound for flash temperature, are in agreement with the liquid-film assisted densification mechanism. Assuming the formation of a liquid film at the particle contacts at the flash temperature [8], the transient liquid can survive as a supercooled liquid for the particle rearrangement, provided sufficient power is supplied to the contact. The lower bound for the existence of the liquid film as viscous melt, if formed, is its glass transition temperature. The glass transition temperature, Tg, is defined as the temperature at which the melt viscosity reaches the value between 1011 and 1012 Pa·s [11]; below glass transition temperature the glass becomes almost an isotropic elastic continuum solid. Approaching from the low temperatures, Debye temperature represents the upper bound temperature below which the crystalline solid counterpart behaves elastically, yet with maximum lattice vibrations. Consequently, at certain conditions, these two temperatures resemble the similar physics of the atomic vibrations, either in the amorphous or in the crystalline state. Angel [12] has thoroughly discussed this aspect. The main difference between the crystalline oxide and its amorphous form (glass) is the long-range crystalline order in the former; the same interionic short-range order and ionic/covalent bonds persist in the two solids. Therefore, the main thermodynamic difference between the crystalline oxide and its amorphous counterpart is the fusion entropy. Consequently, the Debye temperature in terms of the hottest phonons in the crystal, manifested by the glass transition temperature in the glassy state. The glass above Tg experiences a softening behavior due to the hottest phonons in the otherwise crystalline form. Therefore, one expects the glass transition temperature of the glass to approach from above to the Debye temperature of its crystalline counterpart
R. Chaim / Scripta Materialia 169 (2019) 6–8
from below. The fact that heat capacity of a crystalline oxide at the Debye temperature reaches its maximum value (which depends on the atomic mass) irrespective of its crystal-type supports this expectation. Strong correlation exists between the melting point, Tm, and the Debye temperature, θD, of crystals via the Lindemann's criterion [13]. In addition, Turnbull [14] discussed the various material aspects important to glass formation, and proposed the Tg/Tm ratio of 0.66 for most glasses. Wang et al. [15] used the above considerations together with accumulated experimental data to show that the ratio Tg/θD falls in the range between 1.23 and 3.17 in metallic glasses. Unfortunately, the majority of cations in oxides are six-coordinated or higher, and are not glass formers and hence cannot retain stable glass from their melts. Thus, there is lack of data on the Tg of these oxides. Nevertheless, we will discuss the Tg - θD relation in terms of the melt properties and especially its ‘fragility’. The ‘fragility’ of the melt is defined by the slope of the viscosity change versus temperature as it approaches the glass transition temperature from above [16]. It is a measure of the dynamic relaxation of the molecular species comprising the melt. In this respect, and for the present treatment, one can classify the oxides according to the Zachariasen's rules for glass formation [11] into two classes: First, the glass formers, the cations of which are four coordinated or less, with partial covalent bonding. These include SiO2, GeO2, B2O3, and P2O5. These oxides are characterized by their relatively open crystal structures, and strong tendency to retain their amorphous character, even during slow cooling rate of the melt. This class of glasses are termed ‘strong’ glasses, and their viscosity follows the Arrhenius-type temperature dependence [12]. Most oxides belong to the second class, and do not form stable glass as mentioned above. The fusion entropies of the later oxides are relatively higher than that of the glass-forming oxides. As a result, there is a significant drop in the melt viscosity recorded in their supercooled melts with the temperature decrease, and hence they are termed ‘fragile’ melts. Thus, due to the same physics of the thermal ionic vibrations, one expects the Debye temperature of the glass-former oxide to be very close to the Tg of its amorphous counterpart. This proximity between the two temperatures is well expressed in GeO2 that exhibits pure Arrhenian behavior [17] with θD of 1000 K [18], and Tg of 853 K [19] and 980 K [20]. For comparison, Sapphire exhibits a fragile melt [21]. The values of 1042 K as θD for Sapphire [22], and 1300 K [21] and 1395 K [23] as Tg for supercooled alumina melt were reported. As the fragility of the oxide melt increases, its Tg increases due to the drastic non-Arrhenian increase in viscosity. Accordingly, the theoretical viscosity and activation energy for viscous flow are much smaller for the fragile oxide melts than for stable glass (i.e. silica based) melts [24]. The low viscosity and the narrow range of the glass transition temperature in oxides with fragile melts may explain the transient and ultrafast effect of the liquid during the particle rearrangement
Table 1 Flash sintering conditions in air and Debye temperatures for different oxides. Oxide
Experimental data Particle size nm
Al2O3 (0.25 wt% MgO) CeO2 (Sm-doped) MgTiO3 SrTiO3 (cubic) TiO2 (rutile) ThO2 Y2O3 8YSZ (cubic) ZnO
100–300 100 66 150 20 265 20 20 16
Field Flash V·cm−1 temp. Tf °C 500 120 500 500 500 533 500 500 160
1320 750 1241 900 700 970 1133 572 625
T [K] = T [°C] + 298. a Debye temperature values are for pure oxides.
Ref. Debye Tf, LB °C (K) Extrapolated temp. −1 for 3 kV·cm θD K
Ref.
950a
35
a
36 37 35 35 38 39 40 37
774 (1072) 173 (471) 524 (822) 563 (861) 506 (804) 508 (806) 778 (1076) 379 (677) 491 (789)
3 26 27 28 29 30 31 32 33
475 900 690 452 468 491 590 440
7
Fig. 1. Experimental flash temperature - electric field relations in different oxides: MgOdoped Al2O3 [3], Sm-doped CeO2 [26], MgTiO3 [27], SrTiO3 [28], TiO2 [29], ThO2 [30], Y2O3 [31], 8YZS-cubic [32], and ZnO [33].
immediately after the flash event. Homologous nucleation of the ‘fragile’ melt on its solid counterpart, immediately after the particle rearrangement, is mandatory, due to zero activation energy for the heterogeneous nucleation of the exothermic solidification [25]. In order to support the present analysis, we used the literature data of different oxides flash sintered in air at different temperature - electric field conditions [3,26–33] (with known Debye temperatures) as summarized in Table 1 and shown in Fig. 1. These experiments often differ in their powder particle size and the heating rates (Table 1). As was mentioned [34] the effect of the heating rate on the flash temperature is negligible in comparison to the particle size. Nevertheless, first we fitted all these data to curves (solid and dashed lines in Fig. 1) using power-law or exponential fittings the equations of which were summarized in Table 2. Extrapolation of these curves to a very high electric field of 3 kV·cm−1 yielded asymptotic behaviour from which the lower bounds for the flash temperatures were determined as Tf, LB in Table 1. Reliable Debye temperatures for these oxides were acquired from different sources in the literature [35–40] (Table 1). We plotted the lower bound flash temperature, Tf, LB versus Debye temperature of the oxides in Fig. 2. The line in Fig. 2 is the geometric location where these two temperatures of interest are equal. As is evident from Fig. 2, while some oxides, such as CeO2, 8YSZ, Al2O3 and MgTiO3, exhibit similarity between Tf, LB and θD, other oxides such as ZnO, TiO2, ThO2 and Y2O3 exhibit significant differences between the two temperatures of interest, as high as 600 K. Considering the nanoparticle size of ZnO, TiO2, and Y2O3 in Fig. 2 (and Table 1) higher flash temperatures are expected for larger particle size, hence far less compatibility with the Debye temperatures. Therefore, the tendency of the lower bound flash temperature to Debye temperature is not universal for all oxides. Nevertheless, as mentioned above, the glass transition temperature of the oxide melt in its
Table 2 Fitting curve equations used to calculate Tf, LB with extrapolation to 3 kV·cm−1. Oxide
Fitting equationa
Al2O3 (0.25 wt% MgO) CeO2 (Sm-doped) MgTiO3 SrTiO3 (cubic) TiO2 (rutile) ThO2 Y2O3 8YSZ (cubic) ZnO
Y = 5603.4 · x^-0.24725 Y = 3451.9 · x^-0.37354 Y = 1488.6 · exp (-0.000348 · x) Y = 4350.6 · x^-0.25539 Y = 2185.7 · x^-0.18268 Y = 28,381 · x^-0.50243 Y = 4079.5 · x^-0.20698 Y = 2511.4 · x^-0.23614 Y = 969.68 · x^-0.084919
a
Y = flash temperature [°C]; x = electric field [V·cm−1].
Fitting R value
Ref.
0.99856 0.99929 0.96345 0.99931 0.99270 0.93825 0.99737 0.99824 0.99222
3 26 27 28 29 30 31 32 33
8
R. Chaim / Scripta Materialia 169 (2019) 6–8
Acknowledgments I am grateful to Dr. Mark Rispler for his critical reading of the manuscript. References
Fig. 2. Lower bound flash temperature calculated at the electric field of 3 kV·cm−1 versus Debye temperature of the oxides used in Fig. 1. The lower bounds consistently are higher than the corresponding Debye temperature of the oxide.
supercooled liquid state depends on the liquid fragility and is always higher (by the factor of 1.2 to 3.17 in metallic systems) compared to the Debye temperature. As the liquid fragility is directly related to the fusion entropy, the lower bound flash onset temperature is expected to depart from Debye temperature with increase in the liquid fragility. Full confirmation of this hypothesis will be possible in the future when Tg data of non-glass forming supercooled oxide liquids will be available. Flash sintering investigations by Jain [41,42] and Sglavo [43–46] groups on silica based (strong glass-former) systems at different conditions revealed important results about the nature of the flash sintering in these systems. First, enhanced electric field-induced softening (EFIS) effect by which the applied electric field decreases the glass softening point in alkali silicate glasses was detected [41]. Biesuz et al. [43] observed a similar effect by adding a magnesia-silica glass to Al2O3 to facilitate its densification via liquid-phase sintering at temperatures lower than needed for conventional liquid-phase sintering. Overall, the flash sintering was mainly related to the dielectric breakdown caused by excitation of the alkali ions [40,43,44]. Biesuz et al. [46] also showed that enhanced densification of the porcelain stoneware by vitrification during flash sintering takes place beyond a critical current density. They related the enhanced liquid-phase densification to changes in the rheological behavior of the glass [46]. Therefore, it is evident that the electric field and current act as additional sources of energy, similar to temperature and pressure, in affecting the glass thermodynamic properties and especially its viscosity and glass transition temperature. In this respect, Mel'cuk et al. [47] investigated the existence of long-lived clusters within the fragile glass-forming liquids using molecular dynamics simulations. They deduced that the glass transition in the supercooled liquid represents the early stage of the glass thermodynamic instability. Therefore, one may expect that the viscosity, hence the glass transition temperature of the glass-forming oxides depends on the applied electric field, hence is not constant. This may be the cause for the successful flash sintering of glass-based systems below their conventional glass transition temperatures [41–46]. Finally, application of the present model for glass-forming systems should consider the decrease in the glass transition temperature caused by the applied electric field.
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