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Photostructural relaxation in As–Se–S glasses: Effect of network fragility
Journal of Non-Crystalline Solids 352 (2006) 2067–2072 www.elsevier.com/locate/jnoncrysol
Photostructural relaxation in As–Se–S glasses: Effect of network fragility Pierre Lucas a
a,*
, Ellyn A. King a, Adam D. Horner a, Bradley R. Johnson b, S.K. Sundaram b
Department of Materials Science and Engineering, University of Arizona, 4715 E. Fort Lowell Road, Tucson, AZ 85712, USA b Pacific Northwest National Laboratory, Richland, WA, USA Received 13 December 2005; received in revised form 3 March 2006 Available online 3 May 2006
Abstract The effect of photoinduced structural relaxation in As–S–Se glass is investigated during sub-bandgap irradiation. It is shown that the glass undergoes rapid optically induced structural relaxation upon photoexcitation of bonding electrons. Fragile systems exhibit larger relaxation as expected from their enthalpy profile. This suggests that the process is thermodynamically driven and that the kinetic impediment to relaxation at low temperature is lifted through photoinduced softening of the glass matrix. Activation energy for enthalpy relaxation measurement and an annealing study near Tg show that the photorelaxation effect is not a thermally activated process. The hri dependence of photostructural changes is addressed and explained using the energy landscape formalism. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.20.Gy; 61.82.Fk; 42.70.Gi Keywords: Chalcogenides; Laser–matter interactions; Photoinduced effects; Enthalpy relaxation; Fragility; Structural relaxation
1. Introduction Chalcogenide glasses exhibit a wide variety of photoinduced effects upon irradiation with sub-bandgap light [1,2]. This collection of effects can be divided into several categories depending on the characteristics of the inducing light source. Vectoral effects [3,4] are dependent on the polarization of the light and can introduce permanent anisotropy, such as dichroism [5] into the glass. Scalar effects on the other hand are independent of the polarization state and result in isotropic changes such as photodarkening and photoexpansion. These scalar effects can in turn be subdivided in two groups [6], the first type of effects are associated with a change in refractive index and include photodarkening and photorefraction [7]. These changes appear at relatively low irradiation intensity (1– *
Corresponding author. Tel.: +1 520 322 2311. E-mail address: [email protected] (P. Lucas).
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.03.004
10 W/cm2) and are used to produce Bragg gratings [8] and channel waveguides [9]. The second type of effect requires higher intensities (>100 W/cm2) and is associated with photoinduced fluidity [10]. This phenomenon involves a photoinduced softening of the glass matrix, allowing plastic deformation [11], mass transport [12], photomelting [13] and giant photoexpansion [14]. Photostructural changes are initiated by the photoexcitation of electron–hole pairs from localized tail states, followed by local structural rearrangements before recombination occurs. Several mechanisms for these structural rearrangements have been suggested including bond twisting motion [7,15], bond reconfiguration through charged defects intermediary states [16,17], or transient bond formation [18]. It is then suggested that for a sufficiently high photon flux, dynamic local structural rearrangements can lead to a macroscopic softening of the glass network resulting in photoinduced fluidity [19]. It is important to note that these phenomena are athermal [20] and actually show
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P. Lucas et al. / Journal of Non-Crystalline Solids 352 (2006) 2067–2072
gap light irradiation, which only induces surface effects. Bulk photorelaxation effects have also been observed in various chalcogenide systems under irradiation with highenergy c rays [21,22].
STRONG
FRAGILE
2. Experimental Thermally induced
Photoinduced
0 Troom
Tg
Temperature
Fig. 1. Schematic plot of the enthalpy variation with temperature for a strong chalcogenide glass (hri 2.4) and a fragile glass (hri 5 2.4). The arrows show the effect of enthalpy relaxation induced thermally near Tg and optically at room temperature.
an inverse temperature dependence, that is, the fluidity is higher a lower temperature [10]. Amorphous materials synthesized from the liquid state are intrinsically not in a state of thermodynamic equilibrium. Indeed, the glass transition occurs when a supercooled liquid becomes too viscous and cannot relax fast enough to reach thermal equilibrium. The liquid is then frozen into a rigid structure and remains in a high configurational enthalpy state as the temperature decreases. The system then falls out of equilibrium and departs further from the liquid equilibrium line (shown as a dashed line in Fig. 1). This departure from equilibrium is associated with a build up of the thermodynamic driving force for relaxation, however the relaxation time increases exponentially with decreasing temperature and quickly reaches values much larger than experimental timescale such that the relaxation process appears non-existent and is in many cases hardly detectable. It is predicted that the phenomenon of photoinduiced fluidity should allow for unusual effects in terms of enthalpy relaxation since it permits the introduction of structural degrees of freedom in the glass covalent network, without the need for annealing at high temperature. In other words the glass can be optically annealed at room temperature. One unique characteristic of this situation is that the relaxation process, which has a large thermodynamic driving force at room temperature but is kinetically prevented, can now be allowed to proceed. The kinetic impediment to relaxation can be optically lifted while the driving force for relaxation is still high. In this paper we show that the relaxation process in As– S–Se glass can be optically accelerated and observed at room temperature. The relaxation behavior for glasses of various network connectivity and fragility is investigated and compared with thermal relaxation kinetics. Throughout the study we use a sub-bandgap irradiation source and consequently the bulk of the glass samples are irradiated homogeneously. This is in contrast to the usual band-
2.1. Glass synthesis Glasses from the As–S–Se system were synthesized using high purity starting elements sealed in quartz ampoules under high vacuum. The samples were then heated and melted in a rocking furnace for up to 8 h in order to obtain a compositionally homogeneous liquid. The ampoules were then quenched in air and the resulting glasses were annealed at Tg for several hours. Three compositions corresponding to glasses with different network connectivity were prepared. Two chalcogen rich glasses and one stoichiomeric glass were synthesized: As18S41Se41, As24S38Se38 and As40S40Se20 corresponding to an average coordination number hri [23,24] of 2.18, 2.24, and 2.4 respectively. 2.2. Irradiation procedure Samples of bulk glasses were cut and polished into small cubes approximately 2 mm in size. This size was chosen in order to keep the sample smaller than the beam diameter of the laser source. This way every glass sample was homogeneously and reproducibly irradiated during each experiment. The samples were irradiated for various time intervals until the effect of irradiation saturated. The laser source was a 785 nm monomode diode laser from Toptica Photonics (Munchen, Germany). This laser wavelength is sub-bandgap and is still within the transparency domain for all glasses as shown in Fig. 2. This wavelength is associated with very low bandtail absorption, this way the bulk of the glass was irradiated and relaxation effects corresponding to the bulk glass, not only the surface, were observed. The laser intensity on the sample was 2.5 W/cm2.
6
As24S38Se38
Absorbance
H
5
As18S41Se41 As40S40Se20
4
785 nm
3 2 600
650
700
750
800
850
Wavelength (nm) Fig. 2. Absorption edge of the three glass compositions studied. The irradiation wavelength at 785 nm corresponds to low absorption in the Urbach tail for all glasses.
P. Lucas et al. / Journal of Non-Crystalline Solids 352 (2006) 2067–2072
Thermally relaxed
2
1.5
Irradiated 1 0.5
0 40
Reference
60
80
100
120
140
160
180
Temperature (oC) Fig. 3. MDSC trace of an As18S41Se41 glass relaxed thermally for 2.5 years and photorelaxed during sub-bandgap irradiation for time intervals varying from 10 to 120 min. The overshoot in heat capacity is indicative of the enthalpy regained by the glass during reheating and the difference in area between the reference curve and subsequent curves is a measure of enthalpy relaxation.
2.3. Thermal analysis Each glass sample was heated and cooled at 10 °C/min prior to irradiation in order to erase the thermal history and provide a reference point to evaluate the extent of relaxation. After irradiation the glass sample was reheated at 10 °C/min in a modulated differential scanning calorimeter (MDSC model Q-1000 from TA Instruments) and the resulting thermogram was integrated to quantify the enthalpy relaxation. Glass samples 10–16 mg in mass were sealed in hermetic pans and an empty pan was used as a reference. The overshoot in heat capacity shown in Fig. 3 corresponds to the regain in enthalpy of a relaxed glass relative to a fresh glass. The difference in surface area between the two thermograms is a measure of the enthalpy relaxation.
as well as a glass exposed to sub-bandgap light for various irradiation time intervals. The glass transition of As18S41Se41 (Tg = 97 °C) is unusually low for an inorganic glass in comparison to room temperature. This allows the glass structure to undergo sizable enthalpy relaxation within a few years at ambient temperature. The large Cp overshoot shown in Fig. 3 is the result of thermal annealing at room temperature for 2.5 years. The additional thermograms show the effect of irradiation for time intervals ranging from 10 to 120 min. The similar Cp overshoot reveals the presence of fast optically induced relaxation processes in As–S–Se glasses. The effect of photoinduced relaxation was quantified for several glasses and appeared to reach a saturation value for extended irradiation times. Fig. 4 displays the time dependence of photorelaxation for three glasses with different average coordination numbers. Fragile glasses with coordination numbers departing from hri = 2.4 are shown to undergo the most extensive relaxation. On the contrary, strong glass corresponding to the stoichiometric As40S20Se40 undergoes only barely detectable relaxation. The extent of photorelaxation is evaluated using two different
0
2 3 4
As24Se38S 38=2.24
5 6
As18Se41S41=2.18
7 8 0
50
100
Fig. 3 shows the DSC thermogram of an As18S41Se41 glass held at room temperature for a long period of time,
200
250
300
350
1
As40 Se 20 S40=2.4
0.95
3. Results
150
Time (min)
2.4. Error analysis
(b)
0.9 0.85
Tf/Tg
The temperature accuracy of the MDSC measurements is ±0.1 °C. The error reported on Tf in Fig. 5 is the standard deviation on a set of three measurements. The error reported for enthalpy relaxation in Figs. 5(a) and 6 correspond to the deviation over three series of data analysis. The error on normalized fictive temperature in Fig. 5(b) is the normalized product of the deviation on Tf and Tg measurements. The error on relaxation times is calculated based on a 3% error on activation energy. The activation energy determination is quite accurate as long as Tf is measured in a consistent and systematic manner as shown in Ref. [25].
(a)
As40Se20S 40=2.40
1
Enthalpy release (J/g)
Heat capacity (J/g C)
2.5
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As 24 Se 38 S38=2.24
0.8 0.75 0.7
As18 Se41 S41=2.18
0.65 0.6 0
50
100
150
200
250
300
350
Time (min) Fig. 4. Extent of relaxation during sub-bandgap irradiation for glasses with different network connectivity as defined by the average coordination number hri. Strong glasses (hri = 2.4) undergo very little relaxation. (a) Relaxation quantified by the release of enthalpy obtained using the method described in Fig. 3. (b) Relaxation quantified by the fictive temperature obtained using the construction shown in Fig. 5. All lines are guide to the eyes.
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measures of structural relaxation: the enthalpy release and the fictive temperature. Fig. 4(a) displays the enthalpy release, which is obtained by integrating the heat capacity curve of samples irradiated for various lengths of time. This integration quantifies the increasingly larger Cp overshoot that is observed for long irradiation time (Fig. 3). Ultimately, this effect saturates and the DSC trace remains unchanged with additional irradiation time. The same relaxation effect is evaluated using an alternative method for measuring the enthalpy state of the glass. The fictive temperature reported in Fig. 4(b) corresponds to the temperature at which the glass enthalpy line intersects the liquid enthalpy line (dashed line in Fig. 1). As the glass enthalpy line is lowered during relaxation the fictive temperature decreases [26]. This temperature can be obtained from the DSC trace using the construction shown in the inset of Fig. 5. For a detailed derivation of this method the reader is referred to Ref. [27]. The fictive temperatures in Fig. 4(b) are then normalized to the respective Tg in order to allow for a meaningful comparison of relaxation in the three glass compositions. In order to contrast the photorelaxation effect with thermally induced relaxation it is insightful to compare the kinetics of the two mechanisms. To that avail, the activation energy for enthalpy relaxation can be measured in order to assess the temperature dependence of the struc-
4
lnQ (K/min)
3.5 3 2.5 2 1.5 1 2.64
2.66
2.68
2.7
2.72
2.74
2.76
2.78
103/Tf (K-1) Fig. 5. Variation in fictive temperature with cooling rate for the As18S41Se41 glass. The slope obtained by linear regression corresponds to DH*/R where DH* is the activation energy for enthalpy relaxation. The inset shows the construction used to measure the fictive temperature Tf from the DSC trace. The line shown is obtained by linear regression of the data points.
tural relaxation time [28]. The activation energy DH* for As18S41Se41 is estimated using a method based on the cooling rate dependence of Tf [27,29]. The variation in fictive temperature as a function of cooling rate Q is measured, and the slope of the ln Q vs. 1/Tf plot provides an estimate of DH*/R according to Eq. (1) where R is the gas constant. d ln Q=dð1=T f Þ ¼ DH =R.
ð1Þ
The cooling rate dependence of Tf for As18S41Se41 is shown in Fig. 5. The slope yields an activation energy DH* = 152.8 kJ/mol which is within the expected range for this type of chalcogenide glass [30]. The temperature dependence of the enthalpy relaxation time can now be estimated using this value of DH* and the Arrhenius equation s = s0exp(DH*/RT). The value of s0 is calculated by assigning a value of s = 200 s at the regular glass transition temperature Tg. The values of relaxation times obtained this way for a series of annealing temperatures near Tg are reported in Table 1. Additionally the relaxation time at room temperature (T = 298 K) is estimated at 0.53 year. This suggests that the glass shown in Fig. 3, which was annealed at room temperature for 2.5 years, has almost reached equilibrium. The validity of the relaxation times estimated from DH* can be tested by measuring the extent of enthalpy relaxation during a 1 h anneal at the same temperature as the s values reported in Table 1. These s values should be consistent with the extent of relaxation predicted for a 1-h heat treatment. The results of this test are shown in Fig. 6. The glass is initially heated and cooled at 10 °C/min and then reheated at 10 °C/min up to the annealing temperature. After 1 h of isothermal annealing, the glass is quenched at the highest rate available in our DSC (40 °C/min) down to 0 °C. An upscan thermogram at 10 °C/min is then recorded in order to evaluate the enthalpy released during the annealing process. Fig. 6 shows that a maximum in relaxation occurs for a 1-h anneal at around 80 °C. This value is consistent with the relaxation times reported in Table 1, if we consider that the glass structure takes about 10 times the relaxation time to reach equilibrium. According to this approximation the glass annealed at 95 °C and 90 °C are in effect fully relaxed after 1 h but the glass annealed at 85 °C and 80 °C cannot fully reach equilibrium. Finally, the glass annealed at 75 °C and 70 °C cannot even relax half way to equilibrium, if we assume an exponential relaxation behavior. This relaxation behavior is described schematically in Fig. 1, where the arrows show the extent of thermally induced relaxation near Tg. At high temperature the relaxation is fast but ther-
Table 1 Relaxation times at various temperatures corresponding to the points in Fig. 6 T (°C)
70
75
80
85
90
95
s (min)
171 (68%)
79 (67%)
37 (66%)
18 (66%)
9 (65%)
4.2 (64%)
The enthalpy relaxation times for As18S41Se41 are estimated using the Arrhenius relation s = s0 exp(DH*/RT) where DH* is the activation energy for structural relaxation obtained from Fig. 5.
P. Lucas et al. / Journal of Non-Crystalline Solids 352 (2006) 2067–2072 -0.4
Enthalpy (J/g C)
-0.8
-1.2
-1.6
-2 70
75
80
85
90
95
Temperature (°C) Fig. 6. Enthalpy released by a As18S41Se41 glass during a 1 h thermal treatment at the temperature indicated. The enthalpy release is obtained using the method described in Fig. 3. DHQ correspond to the fraction of enthalpy decrease due to the finite quenching rate after thermal treatment.
modynamically limited and at lower temperature the thermodynamic drive is large but kinetically impeded. The optimum condition between these two extremes occurs around 80 °C as shown by the similarity in shape between the arrows in Fig. 1 and the data points in Fig. 6. 4. Discussion At this point it is important to emphasize the difference in mechanisms between optically and thermally induced relaxations. In particular, the possibility of thermal effects due to laser heating must be addressed. While the absorption is very low at 785 nm, the potential occurrence of photodarkening raises the concern that laser absorption could elevate the sample temperature and induce thermal relaxation. However a comparison of Figs. 6 and 4(a) clearly demonstrates that the photorelaxation effect cannot be solely due to laser heating. Indeed, Fig. 6 shows that the maximum relaxation attainable by annealing for 1 h at any temperature is 1.75 J/g, yet the extent of photorelaxation during the same 1 h is 2.5 times larger with 4.3 J/g (Fig. 4). This clearly indicates that the photorelaxation mechanism cannot entirely be of thermal origin and must involve some optically induced structural rearrangement, most likely associated with photoinduced fluidity. This is consistent with the findings that most known photostructural effects in chalcogenide are athermal [10,15,20]. We now consider the effect of network connectivity toward photostructural changes. Due to the covalent nature of the bond network in chalcogenide glasses, it is common to describe the matrix in terms of the average coordination number hri [31,32]. The quantity hri is defined as the average number of covalent bonds per atom and represents the bond density of the glassy matrix. It is observed that many physical properties reach an extremum for hri = 2.4 when the number of degrees of freedom is equal to the number of constraints to which the constituent par-
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ticles are subjected [33,34]. In particular, it is shown that the fragility of the glass-forming system follows that pattern and presents a minimum in fragility for hri = 2.4 [30,33,35]. The definition of fragility is based on the temperature-dependence of the viscosity of glassforming liquids [36]. Liquids that shows a strongly non-Arrhenius profile are considered fragile. In other words, liquids that undergo greater structural collapse for a given temperature increment are the most fragile. While this definition is based on the kinetic properties of the system, the fragility can equivalently be described in terms of thermodynamic properties as illustrated in Fig. 1. Following this approach, the fragile systems are the one loosing the most entropy or enthalpy for a given temperature step. This description is of greater interest for us since this implies that the fragile system will build up a greater thermodynamic drive toward relaxation after vitrification. As shown in Fig. 1, the enthalpy of the frozen glassy state departs from the equilibrium liquid line at a faster rate for fragile systems. The effect of hri (and therefore fragility) on the photorelaxation process is clearly seen in Fig. 4. The fragile glasses undergo much larger relaxation as predicted from the enthalpy curve in Fig. 1. On the other hand, the strong glass has very little ability to relax and shows almost no detectable change. Similar results in the Ge–Se system [35] indicate that it is a general feature of chalcogenide glasses. These results suggest that the relaxation process is indeed thermodynamically driven and only activated through laser exposure. The glass matrix is softened and allowed to rearrange during photoexcitation of bonding electrons. The structure can then relax to a lower free energy state within a short time scale. In effect, the kinetic impediment that prevents relaxation at room temperature is optically lifted. It should be realized however that the saturated photorelaxation state achieved after long exposure is not equivalent to a thermally relaxed glass structure. Indeed, the bond network is undergoing continuous photoexcitation and rearrangement during light exposure. The state achieved after light saturation was shown to correspond to a dynamic equilibrium between photoexcitation of the matrix and thermodynamically driven relaxation [37]. Alternatively, the effect of hri on photostructural changes can be explained by invoking the energy landscape formalism. This model describes the glass structure in terms of the density of configurational states or number of minima on the potential energy hypersurface [38]. Recent computer simulations on a Lennard–Jones system have lent credence to this model by presenting a probe of the energy landscape at different temperatures [39]. The model differentiates fragile and strong systems based on the density of minima on the energy landscape [40]. Fragile glasses have a large number of minima and can explore many configurational states (configurons), while strong glasses have fewer minima and more easily stay trapped in a potential well. This interpretation explains why the structure of a fragile system is less resilient to thermal
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degradation, showing a rapid increase in fluidity with temperature. It is then suggested that the same formalism should apply to structural changes induced by photoexcitation and could explain why fragile systems exhibit larger photostructural changes [37]. During irradiation, the glass matrix undergoes continuous rearrangement and is allowed to explore other configurational states. It is then reasonable to expect that the fragile system will undergo larger structural changes because of the availability of a greater number of different structural configurations to explore upon recombination of photoelectrons. 5. Conclusion Irradiation of various As–S–Se glasses with sub-bandgap light reveals that the glass matrix can undergo rapid, light activated relaxation. The relaxation behavior is consistent with the thermodynamic profile predicted for strong and fragile glasses according to the average coordination number hri. Fragile glasses undergo large relaxation and strong glasses show barely detectable changes. This behavior can be understood using the energy landscape formalism. It is shown that the photorelaxation process is too large to be a thermally induce process due to laser heating. It is important to consider these effects when choosing a glass composition for laser processing of photonic devices. References [1] K. Shimikawa, A. Kolobov, S.R. Elliott, Adv. Phys. 44 (1995) 475. [2] A.V. Kolobov, K. Tanaka, in: H.S. Nalwa (Ed.), Handbook of Advanced Electronic and Photonic Materials and Devices, vol. 5, Academic, New York, 2001, p. 47. [3] V.K. Tikhomirov, G.J. Adriaenssens, S.R. Elliot, Phys. Rev. B 55 (1997) R660. [4] P. Krecmer et al., Science 277 (1997) 1799. [5] V.M. Lyubin, V.K. Tikhomirov, J. Non-Cryst. Solids 114 (1989) 133. [6] K. Tanaka, Proc. SPIE 5061 (2003) 16. [7] K. Tanaka, J. Non-Cryst. Solids 59&60 (1983) 925. [8] K. Shiramine, H. Hisakuni, K. Tanaka, Appl. Phys. Lett. 64 (1994) 1771.
[9] S. Ramachandran, S.G. Bishop, Appl. Phys. Lett. 74 (1999) 13. [10] H. Hisakuni, K. Tanaka, Science 270 (1995) 974. [11] D.T. Kastrissios, G.N. Papatheodorou, S.N. Yannopoulos, Phys. Rev. B 65 (2002) 165211. [12] A. Saliminia, T.V. Galstian, A. Villeneuve, Phys. Rev. Lett. 19 (2000) 4112. [13] V.V. Poborchii, A.V. Kolobov, K. Tanaka, Appl. Phys. Lett 74 (1999) 215. [14] H. Hisakuni, K. Tanaka, Appl. Phys. Lett. 65 (1994) 2925. [15] K. Tanaka, J. Non-Cryst. Solids 266 (2000) 889. [16] R.A. Street, Solid-State Commun. 24 (1977) 363. [17] H. Fritzsche, Philos. Mag. B 68 (1993) 561. [18] A.V. Kolobov, H. Oyanagi, Ka. Tanaka, K. Tanaka, Phys. Rev. B 55 (1997) 726. [19] H. Fritzsche, Solid-State Commun. 99 (1996) 153. [20] C.R. Schardt, P. Lucas, A. Doraiswamy, P. Jivaganont, J.H. Simmons, J. Non-Cryst. Solids 351 (2005) 1653. [21] R. Calemczuk, E. Bonjour, J. Non-Cryst. Solids 43 (1981) 427. [22] R. Golovchak, O. Shpotyuk, A. Kozdras, Cz. Gorecki, M. Shpotyuk, Phys. Chem. Glass 46 (2005) 579. [23] J.C. Phillips, M.F. Thorpe, Solid-State Commun. 53 (1985) 699. [24] A.K. Varshneya, J. Non-Cryst. Solids 273 (2000) 1. [25] C.T. Moynihan, A.J. Easteal, J. Wilder, J. Phys. Chem. 78 (1974) 2673. [26] J.P. Larmagnac, J. Grenet, P. Michon, Philos. Mag. B 45 (1982) 627. [27] C.T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, J. Am. Ceram. Soc. 59 (1975) 12. [28] L. Boehm, M.D. Ingram, C.A. Angell, J. Non-Cryst. Solids 44 (1981) 305. [29] C.T. Moynihan, S.K. Lee, M. Tatsumisago, T. Minami, Thermochim. Acta 280 (1996) 153. [30] M. Tatsumisago, B.L. Halfpap, J.L. Green, S.M. Lindsay, C.A. Angell, Phys. Rev. Lett. 64 (1990) 1549. [31] J.C. Phillips, J. Non-Cryst. Solids 34 (1979) 153. [32] M.F. Thorpe, J. Non-Cryst. Solids 57 (1983) 355. [33] G.G. Naumis, Phys. Rev. B 61 (2000) R9205. [34] U. Senapati, A.K. Varshneya, J. Non-Cryst. Solids 185 (1995) 289. [35] P. Lucas, A. Doraiswamy, E.A. King, J. Non-Cryst. Solids 332 (2003) 35. [36] C.A. Angell, J. Non-Cryst. Solids 131–133 (1991) 13. [37] P. Lucas, E.A. King, A. Doraiswamy, P. Jivaganont, Phys. Rev. B 71 (2005) 104207. [38] C.A. Angell, Nature 393 (1998) 521. [39] S. Sastry, P.G. Debenedetti, F.H. Stillinger, Nature 393 (1998) 554. [40] C.A. Angell, J. Phys. Chem. Solids 49 (1988) 863.
Photostructural relaxation in As–Se–S glasses: Effect of network fragility Pierre Lucas a
a,*
, Ellyn A. King a, Adam D. Horner a, Bradley R. Johnson b, S.K. Sundaram b
Department of Materials Science and Engineering, University of Arizona, 4715 E. Fort Lowell Road, Tucson, AZ 85712, USA b Pacific Northwest National Laboratory, Richland, WA, USA Received 13 December 2005; received in revised form 3 March 2006 Available online 3 May 2006
Abstract The effect of photoinduced structural relaxation in As–S–Se glass is investigated during sub-bandgap irradiation. It is shown that the glass undergoes rapid optically induced structural relaxation upon photoexcitation of bonding electrons. Fragile systems exhibit larger relaxation as expected from their enthalpy profile. This suggests that the process is thermodynamically driven and that the kinetic impediment to relaxation at low temperature is lifted through photoinduced softening of the glass matrix. Activation energy for enthalpy relaxation measurement and an annealing study near Tg show that the photorelaxation effect is not a thermally activated process. The hri dependence of photostructural changes is addressed and explained using the energy landscape formalism. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.20.Gy; 61.82.Fk; 42.70.Gi Keywords: Chalcogenides; Laser–matter interactions; Photoinduced effects; Enthalpy relaxation; Fragility; Structural relaxation
1. Introduction Chalcogenide glasses exhibit a wide variety of photoinduced effects upon irradiation with sub-bandgap light [1,2]. This collection of effects can be divided into several categories depending on the characteristics of the inducing light source. Vectoral effects [3,4] are dependent on the polarization of the light and can introduce permanent anisotropy, such as dichroism [5] into the glass. Scalar effects on the other hand are independent of the polarization state and result in isotropic changes such as photodarkening and photoexpansion. These scalar effects can in turn be subdivided in two groups [6], the first type of effects are associated with a change in refractive index and include photodarkening and photorefraction [7]. These changes appear at relatively low irradiation intensity (1– *
Corresponding author. Tel.: +1 520 322 2311. E-mail address: [email protected] (P. Lucas).
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.03.004
10 W/cm2) and are used to produce Bragg gratings [8] and channel waveguides [9]. The second type of effect requires higher intensities (>100 W/cm2) and is associated with photoinduced fluidity [10]. This phenomenon involves a photoinduced softening of the glass matrix, allowing plastic deformation [11], mass transport [12], photomelting [13] and giant photoexpansion [14]. Photostructural changes are initiated by the photoexcitation of electron–hole pairs from localized tail states, followed by local structural rearrangements before recombination occurs. Several mechanisms for these structural rearrangements have been suggested including bond twisting motion [7,15], bond reconfiguration through charged defects intermediary states [16,17], or transient bond formation [18]. It is then suggested that for a sufficiently high photon flux, dynamic local structural rearrangements can lead to a macroscopic softening of the glass network resulting in photoinduced fluidity [19]. It is important to note that these phenomena are athermal [20] and actually show
2068
P. Lucas et al. / Journal of Non-Crystalline Solids 352 (2006) 2067–2072
gap light irradiation, which only induces surface effects. Bulk photorelaxation effects have also been observed in various chalcogenide systems under irradiation with highenergy c rays [21,22].
STRONG
FRAGILE
2. Experimental Thermally induced
Photoinduced
0 Troom
Tg
Temperature
Fig. 1. Schematic plot of the enthalpy variation with temperature for a strong chalcogenide glass (hri 2.4) and a fragile glass (hri 5 2.4). The arrows show the effect of enthalpy relaxation induced thermally near Tg and optically at room temperature.
an inverse temperature dependence, that is, the fluidity is higher a lower temperature [10]. Amorphous materials synthesized from the liquid state are intrinsically not in a state of thermodynamic equilibrium. Indeed, the glass transition occurs when a supercooled liquid becomes too viscous and cannot relax fast enough to reach thermal equilibrium. The liquid is then frozen into a rigid structure and remains in a high configurational enthalpy state as the temperature decreases. The system then falls out of equilibrium and departs further from the liquid equilibrium line (shown as a dashed line in Fig. 1). This departure from equilibrium is associated with a build up of the thermodynamic driving force for relaxation, however the relaxation time increases exponentially with decreasing temperature and quickly reaches values much larger than experimental timescale such that the relaxation process appears non-existent and is in many cases hardly detectable. It is predicted that the phenomenon of photoinduiced fluidity should allow for unusual effects in terms of enthalpy relaxation since it permits the introduction of structural degrees of freedom in the glass covalent network, without the need for annealing at high temperature. In other words the glass can be optically annealed at room temperature. One unique characteristic of this situation is that the relaxation process, which has a large thermodynamic driving force at room temperature but is kinetically prevented, can now be allowed to proceed. The kinetic impediment to relaxation can be optically lifted while the driving force for relaxation is still high. In this paper we show that the relaxation process in As– S–Se glass can be optically accelerated and observed at room temperature. The relaxation behavior for glasses of various network connectivity and fragility is investigated and compared with thermal relaxation kinetics. Throughout the study we use a sub-bandgap irradiation source and consequently the bulk of the glass samples are irradiated homogeneously. This is in contrast to the usual band-
2.1. Glass synthesis Glasses from the As–S–Se system were synthesized using high purity starting elements sealed in quartz ampoules under high vacuum. The samples were then heated and melted in a rocking furnace for up to 8 h in order to obtain a compositionally homogeneous liquid. The ampoules were then quenched in air and the resulting glasses were annealed at Tg for several hours. Three compositions corresponding to glasses with different network connectivity were prepared. Two chalcogen rich glasses and one stoichiomeric glass were synthesized: As18S41Se41, As24S38Se38 and As40S40Se20 corresponding to an average coordination number hri [23,24] of 2.18, 2.24, and 2.4 respectively. 2.2. Irradiation procedure Samples of bulk glasses were cut and polished into small cubes approximately 2 mm in size. This size was chosen in order to keep the sample smaller than the beam diameter of the laser source. This way every glass sample was homogeneously and reproducibly irradiated during each experiment. The samples were irradiated for various time intervals until the effect of irradiation saturated. The laser source was a 785 nm monomode diode laser from Toptica Photonics (Munchen, Germany). This laser wavelength is sub-bandgap and is still within the transparency domain for all glasses as shown in Fig. 2. This wavelength is associated with very low bandtail absorption, this way the bulk of the glass was irradiated and relaxation effects corresponding to the bulk glass, not only the surface, were observed. The laser intensity on the sample was 2.5 W/cm2.
6
As24S38Se38
Absorbance
H
5
As18S41Se41 As40S40Se20
4
785 nm
3 2 600
650
700
750
800
850
Wavelength (nm) Fig. 2. Absorption edge of the three glass compositions studied. The irradiation wavelength at 785 nm corresponds to low absorption in the Urbach tail for all glasses.
P. Lucas et al. / Journal of Non-Crystalline Solids 352 (2006) 2067–2072
Thermally relaxed
2
1.5
Irradiated 1 0.5
0 40
Reference
60
80
100
120
140
160
180
Temperature (oC) Fig. 3. MDSC trace of an As18S41Se41 glass relaxed thermally for 2.5 years and photorelaxed during sub-bandgap irradiation for time intervals varying from 10 to 120 min. The overshoot in heat capacity is indicative of the enthalpy regained by the glass during reheating and the difference in area between the reference curve and subsequent curves is a measure of enthalpy relaxation.
2.3. Thermal analysis Each glass sample was heated and cooled at 10 °C/min prior to irradiation in order to erase the thermal history and provide a reference point to evaluate the extent of relaxation. After irradiation the glass sample was reheated at 10 °C/min in a modulated differential scanning calorimeter (MDSC model Q-1000 from TA Instruments) and the resulting thermogram was integrated to quantify the enthalpy relaxation. Glass samples 10–16 mg in mass were sealed in hermetic pans and an empty pan was used as a reference. The overshoot in heat capacity shown in Fig. 3 corresponds to the regain in enthalpy of a relaxed glass relative to a fresh glass. The difference in surface area between the two thermograms is a measure of the enthalpy relaxation.
as well as a glass exposed to sub-bandgap light for various irradiation time intervals. The glass transition of As18S41Se41 (Tg = 97 °C) is unusually low for an inorganic glass in comparison to room temperature. This allows the glass structure to undergo sizable enthalpy relaxation within a few years at ambient temperature. The large Cp overshoot shown in Fig. 3 is the result of thermal annealing at room temperature for 2.5 years. The additional thermograms show the effect of irradiation for time intervals ranging from 10 to 120 min. The similar Cp overshoot reveals the presence of fast optically induced relaxation processes in As–S–Se glasses. The effect of photoinduced relaxation was quantified for several glasses and appeared to reach a saturation value for extended irradiation times. Fig. 4 displays the time dependence of photorelaxation for three glasses with different average coordination numbers. Fragile glasses with coordination numbers departing from hri = 2.4 are shown to undergo the most extensive relaxation. On the contrary, strong glass corresponding to the stoichiometric As40S20Se40 undergoes only barely detectable relaxation. The extent of photorelaxation is evaluated using two different
0
2 3 4
As24Se38S 38
5 6
As18Se41S41
7 8 0
50
100
Fig. 3 shows the DSC thermogram of an As18S41Se41 glass held at room temperature for a long period of time,
200
250
300
350
1
As40 Se 20 S40
0.95
3. Results
150
Time (min)
2.4. Error analysis
(b)
0.9 0.85
Tf/Tg
The temperature accuracy of the MDSC measurements is ±0.1 °C. The error reported on Tf in Fig. 5 is the standard deviation on a set of three measurements. The error reported for enthalpy relaxation in Figs. 5(a) and 6 correspond to the deviation over three series of data analysis. The error on normalized fictive temperature in Fig. 5(b) is the normalized product of the deviation on Tf and Tg measurements. The error on relaxation times is calculated based on a 3% error on activation energy. The activation energy determination is quite accurate as long as Tf is measured in a consistent and systematic manner as shown in Ref. [25].
(a)
As40Se20S 40
1
Enthalpy release (J/g)
Heat capacity (J/g C)
2.5
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As 24 Se 38 S38
0.8 0.75 0.7
As18 Se41 S41
0.65 0.6 0
50
100
150
200
250
300
350
Time (min) Fig. 4. Extent of relaxation during sub-bandgap irradiation for glasses with different network connectivity as defined by the average coordination number hri. Strong glasses (hri = 2.4) undergo very little relaxation. (a) Relaxation quantified by the release of enthalpy obtained using the method described in Fig. 3. (b) Relaxation quantified by the fictive temperature obtained using the construction shown in Fig. 5. All lines are guide to the eyes.
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measures of structural relaxation: the enthalpy release and the fictive temperature. Fig. 4(a) displays the enthalpy release, which is obtained by integrating the heat capacity curve of samples irradiated for various lengths of time. This integration quantifies the increasingly larger Cp overshoot that is observed for long irradiation time (Fig. 3). Ultimately, this effect saturates and the DSC trace remains unchanged with additional irradiation time. The same relaxation effect is evaluated using an alternative method for measuring the enthalpy state of the glass. The fictive temperature reported in Fig. 4(b) corresponds to the temperature at which the glass enthalpy line intersects the liquid enthalpy line (dashed line in Fig. 1). As the glass enthalpy line is lowered during relaxation the fictive temperature decreases [26]. This temperature can be obtained from the DSC trace using the construction shown in the inset of Fig. 5. For a detailed derivation of this method the reader is referred to Ref. [27]. The fictive temperatures in Fig. 4(b) are then normalized to the respective Tg in order to allow for a meaningful comparison of relaxation in the three glass compositions. In order to contrast the photorelaxation effect with thermally induced relaxation it is insightful to compare the kinetics of the two mechanisms. To that avail, the activation energy for enthalpy relaxation can be measured in order to assess the temperature dependence of the struc-
4
lnQ (K/min)
3.5 3 2.5 2 1.5 1 2.64
2.66
2.68
2.7
2.72
2.74
2.76
2.78
103/Tf (K-1) Fig. 5. Variation in fictive temperature with cooling rate for the As18S41Se41 glass. The slope obtained by linear regression corresponds to DH*/R where DH* is the activation energy for enthalpy relaxation. The inset shows the construction used to measure the fictive temperature Tf from the DSC trace. The line shown is obtained by linear regression of the data points.
tural relaxation time [28]. The activation energy DH* for As18S41Se41 is estimated using a method based on the cooling rate dependence of Tf [27,29]. The variation in fictive temperature as a function of cooling rate Q is measured, and the slope of the ln Q vs. 1/Tf plot provides an estimate of DH*/R according to Eq. (1) where R is the gas constant. d ln Q=dð1=T f Þ ¼ DH =R.
ð1Þ
The cooling rate dependence of Tf for As18S41Se41 is shown in Fig. 5. The slope yields an activation energy DH* = 152.8 kJ/mol which is within the expected range for this type of chalcogenide glass [30]. The temperature dependence of the enthalpy relaxation time can now be estimated using this value of DH* and the Arrhenius equation s = s0exp(DH*/RT). The value of s0 is calculated by assigning a value of s = 200 s at the regular glass transition temperature Tg. The values of relaxation times obtained this way for a series of annealing temperatures near Tg are reported in Table 1. Additionally the relaxation time at room temperature (T = 298 K) is estimated at 0.53 year. This suggests that the glass shown in Fig. 3, which was annealed at room temperature for 2.5 years, has almost reached equilibrium. The validity of the relaxation times estimated from DH* can be tested by measuring the extent of enthalpy relaxation during a 1 h anneal at the same temperature as the s values reported in Table 1. These s values should be consistent with the extent of relaxation predicted for a 1-h heat treatment. The results of this test are shown in Fig. 6. The glass is initially heated and cooled at 10 °C/min and then reheated at 10 °C/min up to the annealing temperature. After 1 h of isothermal annealing, the glass is quenched at the highest rate available in our DSC (40 °C/min) down to 0 °C. An upscan thermogram at 10 °C/min is then recorded in order to evaluate the enthalpy released during the annealing process. Fig. 6 shows that a maximum in relaxation occurs for a 1-h anneal at around 80 °C. This value is consistent with the relaxation times reported in Table 1, if we consider that the glass structure takes about 10 times the relaxation time to reach equilibrium. According to this approximation the glass annealed at 95 °C and 90 °C are in effect fully relaxed after 1 h but the glass annealed at 85 °C and 80 °C cannot fully reach equilibrium. Finally, the glass annealed at 75 °C and 70 °C cannot even relax half way to equilibrium, if we assume an exponential relaxation behavior. This relaxation behavior is described schematically in Fig. 1, where the arrows show the extent of thermally induced relaxation near Tg. At high temperature the relaxation is fast but ther-
Table 1 Relaxation times at various temperatures corresponding to the points in Fig. 6 T (°C)
70
75
80
85
90
95
s (min)
171 (68%)
79 (67%)
37 (66%)
18 (66%)
9 (65%)
4.2 (64%)
The enthalpy relaxation times for As18S41Se41 are estimated using the Arrhenius relation s = s0 exp(DH*/RT) where DH* is the activation energy for structural relaxation obtained from Fig. 5.
P. Lucas et al. / Journal of Non-Crystalline Solids 352 (2006) 2067–2072 -0.4
Enthalpy (J/g C)
-0.8
-1.2
-1.6
-2 70
75
80
85
90
95
Temperature (°C) Fig. 6. Enthalpy released by a As18S41Se41 glass during a 1 h thermal treatment at the temperature indicated. The enthalpy release is obtained using the method described in Fig. 3. DHQ correspond to the fraction of enthalpy decrease due to the finite quenching rate after thermal treatment.
modynamically limited and at lower temperature the thermodynamic drive is large but kinetically impeded. The optimum condition between these two extremes occurs around 80 °C as shown by the similarity in shape between the arrows in Fig. 1 and the data points in Fig. 6. 4. Discussion At this point it is important to emphasize the difference in mechanisms between optically and thermally induced relaxations. In particular, the possibility of thermal effects due to laser heating must be addressed. While the absorption is very low at 785 nm, the potential occurrence of photodarkening raises the concern that laser absorption could elevate the sample temperature and induce thermal relaxation. However a comparison of Figs. 6 and 4(a) clearly demonstrates that the photorelaxation effect cannot be solely due to laser heating. Indeed, Fig. 6 shows that the maximum relaxation attainable by annealing for 1 h at any temperature is 1.75 J/g, yet the extent of photorelaxation during the same 1 h is 2.5 times larger with 4.3 J/g (Fig. 4). This clearly indicates that the photorelaxation mechanism cannot entirely be of thermal origin and must involve some optically induced structural rearrangement, most likely associated with photoinduced fluidity. This is consistent with the findings that most known photostructural effects in chalcogenide are athermal [10,15,20]. We now consider the effect of network connectivity toward photostructural changes. Due to the covalent nature of the bond network in chalcogenide glasses, it is common to describe the matrix in terms of the average coordination number hri [31,32]. The quantity hri is defined as the average number of covalent bonds per atom and represents the bond density of the glassy matrix. It is observed that many physical properties reach an extremum for hri = 2.4 when the number of degrees of freedom is equal to the number of constraints to which the constituent par-
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ticles are subjected [33,34]. In particular, it is shown that the fragility of the glass-forming system follows that pattern and presents a minimum in fragility for hri = 2.4 [30,33,35]. The definition of fragility is based on the temperature-dependence of the viscosity of glassforming liquids [36]. Liquids that shows a strongly non-Arrhenius profile are considered fragile. In other words, liquids that undergo greater structural collapse for a given temperature increment are the most fragile. While this definition is based on the kinetic properties of the system, the fragility can equivalently be described in terms of thermodynamic properties as illustrated in Fig. 1. Following this approach, the fragile systems are the one loosing the most entropy or enthalpy for a given temperature step. This description is of greater interest for us since this implies that the fragile system will build up a greater thermodynamic drive toward relaxation after vitrification. As shown in Fig. 1, the enthalpy of the frozen glassy state departs from the equilibrium liquid line at a faster rate for fragile systems. The effect of hri (and therefore fragility) on the photorelaxation process is clearly seen in Fig. 4. The fragile glasses undergo much larger relaxation as predicted from the enthalpy curve in Fig. 1. On the other hand, the strong glass has very little ability to relax and shows almost no detectable change. Similar results in the Ge–Se system [35] indicate that it is a general feature of chalcogenide glasses. These results suggest that the relaxation process is indeed thermodynamically driven and only activated through laser exposure. The glass matrix is softened and allowed to rearrange during photoexcitation of bonding electrons. The structure can then relax to a lower free energy state within a short time scale. In effect, the kinetic impediment that prevents relaxation at room temperature is optically lifted. It should be realized however that the saturated photorelaxation state achieved after long exposure is not equivalent to a thermally relaxed glass structure. Indeed, the bond network is undergoing continuous photoexcitation and rearrangement during light exposure. The state achieved after light saturation was shown to correspond to a dynamic equilibrium between photoexcitation of the matrix and thermodynamically driven relaxation [37]. Alternatively, the effect of hri on photostructural changes can be explained by invoking the energy landscape formalism. This model describes the glass structure in terms of the density of configurational states or number of minima on the potential energy hypersurface [38]. Recent computer simulations on a Lennard–Jones system have lent credence to this model by presenting a probe of the energy landscape at different temperatures [39]. The model differentiates fragile and strong systems based on the density of minima on the energy landscape [40]. Fragile glasses have a large number of minima and can explore many configurational states (configurons), while strong glasses have fewer minima and more easily stay trapped in a potential well. This interpretation explains why the structure of a fragile system is less resilient to thermal
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degradation, showing a rapid increase in fluidity with temperature. It is then suggested that the same formalism should apply to structural changes induced by photoexcitation and could explain why fragile systems exhibit larger photostructural changes [37]. During irradiation, the glass matrix undergoes continuous rearrangement and is allowed to explore other configurational states. It is then reasonable to expect that the fragile system will undergo larger structural changes because of the availability of a greater number of different structural configurations to explore upon recombination of photoelectrons. 5. Conclusion Irradiation of various As–S–Se glasses with sub-bandgap light reveals that the glass matrix can undergo rapid, light activated relaxation. The relaxation behavior is consistent with the thermodynamic profile predicted for strong and fragile glasses according to the average coordination number hri. Fragile glasses undergo large relaxation and strong glasses show barely detectable changes. This behavior can be understood using the energy landscape formalism. It is shown that the photorelaxation process is too large to be a thermally induce process due to laser heating. It is important to consider these effects when choosing a glass composition for laser processing of photonic devices. References [1] K. Shimikawa, A. Kolobov, S.R. Elliott, Adv. Phys. 44 (1995) 475. [2] A.V. Kolobov, K. Tanaka, in: H.S. Nalwa (Ed.), Handbook of Advanced Electronic and Photonic Materials and Devices, vol. 5, Academic, New York, 2001, p. 47. [3] V.K. Tikhomirov, G.J. Adriaenssens, S.R. Elliot, Phys. Rev. B 55 (1997) R660. [4] P. Krecmer et al., Science 277 (1997) 1799. [5] V.M. Lyubin, V.K. Tikhomirov, J. Non-Cryst. Solids 114 (1989) 133. [6] K. Tanaka, Proc. SPIE 5061 (2003) 16. [7] K. Tanaka, J. Non-Cryst. Solids 59&60 (1983) 925. [8] K. Shiramine, H. Hisakuni, K. Tanaka, Appl. Phys. Lett. 64 (1994) 1771.
[9] S. Ramachandran, S.G. Bishop, Appl. Phys. Lett. 74 (1999) 13. [10] H. Hisakuni, K. Tanaka, Science 270 (1995) 974. [11] D.T. Kastrissios, G.N. Papatheodorou, S.N. Yannopoulos, Phys. Rev. B 65 (2002) 165211. [12] A. Saliminia, T.V. Galstian, A. Villeneuve, Phys. Rev. Lett. 19 (2000) 4112. [13] V.V. Poborchii, A.V. Kolobov, K. Tanaka, Appl. Phys. Lett 74 (1999) 215. [14] H. Hisakuni, K. Tanaka, Appl. Phys. Lett. 65 (1994) 2925. [15] K. Tanaka, J. Non-Cryst. Solids 266 (2000) 889. [16] R.A. Street, Solid-State Commun. 24 (1977) 363. [17] H. Fritzsche, Philos. Mag. B 68 (1993) 561. [18] A.V. Kolobov, H. Oyanagi, Ka. Tanaka, K. Tanaka, Phys. Rev. B 55 (1997) 726. [19] H. Fritzsche, Solid-State Commun. 99 (1996) 153. [20] C.R. Schardt, P. Lucas, A. Doraiswamy, P. Jivaganont, J.H. Simmons, J. Non-Cryst. Solids 351 (2005) 1653. [21] R. Calemczuk, E. Bonjour, J. Non-Cryst. Solids 43 (1981) 427. [22] R. Golovchak, O. Shpotyuk, A. Kozdras, Cz. Gorecki, M. Shpotyuk, Phys. Chem. Glass 46 (2005) 579. [23] J.C. Phillips, M.F. Thorpe, Solid-State Commun. 53 (1985) 699. [24] A.K. Varshneya, J. Non-Cryst. Solids 273 (2000) 1. [25] C.T. Moynihan, A.J. Easteal, J. Wilder, J. Phys. Chem. 78 (1974) 2673. [26] J.P. Larmagnac, J. Grenet, P. Michon, Philos. Mag. B 45 (1982) 627. [27] C.T. Moynihan, A.J. Easteal, M.A. DeBolt, J. Tucker, J. Am. Ceram. Soc. 59 (1975) 12. [28] L. Boehm, M.D. Ingram, C.A. Angell, J. Non-Cryst. Solids 44 (1981) 305. [29] C.T. Moynihan, S.K. Lee, M. Tatsumisago, T. Minami, Thermochim. Acta 280 (1996) 153. [30] M. Tatsumisago, B.L. Halfpap, J.L. Green, S.M. Lindsay, C.A. Angell, Phys. Rev. Lett. 64 (1990) 1549. [31] J.C. Phillips, J. Non-Cryst. Solids 34 (1979) 153. [32] M.F. Thorpe, J. Non-Cryst. Solids 57 (1983) 355. [33] G.G. Naumis, Phys. Rev. B 61 (2000) R9205. [34] U. Senapati, A.K. Varshneya, J. Non-Cryst. Solids 185 (1995) 289. [35] P. Lucas, A. Doraiswamy, E.A. King, J. Non-Cryst. Solids 332 (2003) 35. [36] C.A. Angell, J. Non-Cryst. Solids 131–133 (1991) 13. [37] P. Lucas, E.A. King, A. Doraiswamy, P. Jivaganont, Phys. Rev. B 71 (2005) 104207. [38] C.A. Angell, Nature 393 (1998) 521. [39] S. Sastry, P.G. Debenedetti, F.H. Stillinger, Nature 393 (1998) 554. [40] C.A. Angell, J. Phys. Chem. Solids 49 (1988) 863.