A molecular dynamics study of the glass transition in CaAl2Si2O8: Thermodynamics and tracer diffusion

Molecular dynamics (MD) simulation provides a unique window into the microscopic processes controlling the properties of amorphous silicates of geochemical importance. Of special interest are changes in structure and dynamics around the glass transition temperature. Seventeen simulations for composition CaAl2Si2O8 in a microcanonical ensemble of 1300 particles (O + Si + Al + Ca) were conducted at temperatures from 1700 to 5000 K at approximately 1 GPa. A pair-wise potential allowing for Coulombic and Born-Mayer interaction was used. Simulation durations were in the range of 50 to 150 ps. Particle trajectories were collected and used to build a picture of the structure and dynamics of equilibrium liquid, supercooled liquid and glassy CaAl2Si2O8 as a function of temperature along the 1 GPa isobar. The computer glass transition was detected at Tg ~ 2800 K by study of thermodynamic properties, speciation equilibria and tracer diffusivity. Tg is observed as a change in slope of enthalpy (H) vs. temperature at T = Tg ~ 2800 K. The configurational isobaric heat capacity of supercooled melt relative to the glass is 53.3 J/(K·mol), within a factor of two of the experimental value. The “computer” isobaric heat capacity for equilibrium liquid at 3000 K is 457 ± 35 J/(K·mol) vs. the calorimetric value of 461 J/(K·mol). In equilibrium liquid, speciation defined by equilibria such as O + O = 2 O and TO4 + TO6 = 2 TO5 are temperature-dependant with ∆H and ∆S approximately equal to –39 kJ/mol and 19 J/mol K and –10 kJ/mol and 12 J/mol, respectively; these are in good agreement with laboratory values. The computer glass point at 2800 K is identified as the temperature at which speciation equilibria become “frozen”. The static structure factor for O-O confirms the conclusion, based on pair correlation statistics, that the glass transition is not associated with significant changes in the static structure. Dramatic differences in the mobility of all atoms monitored by tracer diffusion are noted as a function of temperature. Self-diffusivity orders at fixed temperature according to DCa > DO > DAl > DSi with DCa ~ 20% larger than DO and DO ~ 2 DSi. Activation energies for diffusion for all atoms lie in the range 170 to 190 kJ/mol. The small range in tracer diffusivity and activation energy (Ea) found for different atoms suggests cooperative motion is important. At Tg/T ≥ 1 for the nonequilibrium glass, Ea decreases by ~40% for all atoms compared to corresponding high-temperature (equilibrium melt) values. The crossover between continuous (hydrodynamic-like) motion and atomic hopping motion shows up clearly in the time-dependence of the mean square displacement as a function of temperature. The qualitative view is that a given particle and its neighbors remain trapped for a finite waiting time before undergoing cooperative thermally activated rearrangement. The waiting time distribution is strongly temperature-dependent and related to the rapid increase in structural relaxation time as temperature approaches Tg. Shear viscosity computed from relaxation of microscopic density fluctuations, the Eyring relation and the Green-Kubo formalism are 0.04 Pa·s, 0.09 Pa·s, and 0.02 Pa·s, respectively; these values are all somewhat higher than the extrapolated laboratory value of ~0.005 Pa·s. the 70 000-kilometer globe-encircling oceanic ridge system. Global geochemical interchange between hydrosphere, biosphere and lithosphere is influenced by reaction of natural glass with aqueous solutions of varying temperature, pressure and composition. Microbial biofilms develop along the glass-fluid interface and mediate geochemical exchange as well. Glass is also present on other planetary bodies. For example, thermal emission spectrometric data from the Mars Global Surveyor is interpreted to indicate that the Martian lowland in the northern hemisphere (basaltic andesite bulk composition) is dominated MORGAN AND SPERA: MD STUDY OF GLASS TRANSITION OF CaAl2Si2O8 916 by crystals of plagioclase feldspar and volcanic glass (Bandfield et al. 2000). Glass also forms during the ubiquitous process of shock compression during hypervelocity impact of planetary materials. In particular, study of the dynamics, properties and structure of amorphous CaAl2Si2O8, is relevant to shock amorphitization and generation of maskelynite. In crystalline form, plagioclase feldspar makes up a substantial part of the lunar crust as well as the oceanic and continental crust on Earth. Melting of subducted plagioclase-rich crust may have played an important role in the Archean and may contribute to island arc volcanism today. High pressure CaAl2Si2O8 in the hollandite structure may serve as a repository of elements normally considered incompatible. Additionally, CaAl2Si2O8 bears on the distribution of radiogenic heat sources within the Earth’s mantle and crystal-liquid partitioning of the large-ion lithophile elements in mantle-derived magma (Zhang et al. 1993; Fodor et al. 1994; Downs et al. 1995; Akaogi 2000). Understanding the nature of glasses, metastable (supercooled) liquids and equilibrium liquids is evidently important for a variety of environmental, geoscience and technological problems. Despite their ubiquity, some of the most basic questions surrounding amorphous materials remain unanswered. Foremost amongst these is the nature of the transition from equilibrium and supercooled (metastable) liquid to amorphous glassy solid. Glass is an unusual material because it retains the disorder present in normal liquid but shares many macroscopic properties with its corresponding crystalline form. Because many different kinds of materials–from metals to ionic solids to organic polymers to Lennard-Jones fluids–undergo glass transitions, it is clearly a fundamental phenomenon not restricted to a particular class of materials of specific bonding type or composition. The literature on glasses, structural relaxation, the glass transition and the connection between liquid and glass structure and properties is vast, reflecting its importance. The reviews of Zallen (1983), Hansen and McDonald (1986), Fredrickson (1988), Zarzycki (1991), Binder (1995), Kob (1995, 1999), Debenedetti (1996), Angell (1988, 1991), Angell et al. (1997), Bottinga (1994), Bottinga et al. (1995) and Ediger (1996) are especially informative. Many theories have been proposed to account for the dramatic increase in relaxation time and accompanying strong variation in transport properties such as shear viscosity and tracer diffusion at the glass transition. Since the 1980s, increased interest in the structure, dynamics, and properties of melts and glasses of geochemical importance has arisen. New experimental and computational techniques have been applied to supplement traditional phase equilibria and thermochemical investigations. The review volume edited by Stebbins et al. (1995) presents a comprehensive summary of the structure, dynamics and properties of silicate melts, supercooled liquids and glasses of special relevance to geochemical problems up until that time. In the present work, the structure and atomic mobility of amorphous CaAl2Si2O8 is studied as a function of temperature from 1700 K to 5000 K at a pressure of ~1 GPa using the Molecular Dynamics (MD) technique. In particular, we investigate how temperature affects the short range structure of the equilibrium liquid, how the mechanism of diffusion changes around Tg when speciation equilibria becomes “frozen,” and the role of cooperative motion in thermally activated self diffusion of Si, Al, O, and Ca. In another study (Morgan and Spera, unpublished manuscript, 2001), tagged particle dynamics are used to obtain a detailed microscopic picture of the glass transition and extant theories of viscosity and structural relaxation are discussed in light of insight gained from the MD simulations. CaAl2Si2O8: Previous work and melting relations AMD study of liquid anorthite at T = 4000 K, a temperature well above the “computer” glass transition of Tg ≈ 2800 K from 1 bar (10 GPa) to 76 GPa (~1800 km depth on Earth) has been presented by Nevins and Spera (1998). This work is briefly reviewed here because of its relevance to the present study. Profound changes in short-range structure and atom mobility were found to occur in molten (fully relaxed) anorthite (i.e., equilibrium liquid) as pressure increases at T = 4000 K. This temperature is greater than both the calorimetric and computer glass transition temperatures of 1160 K and ~2800 K, respectively. The abundance of TO4 and TO6 (T = Si, Al) polyhedra monotonically decrease and increase, respectively, as pressure increases. The concentration of pentahedrally coordinated T (T or TO5) attains a maximum at 5 GPa. Significantly, at ~5 GPa, the tracer diffusivity of both O and Si take on maximum values. Large changes in the O about O and O about T nearest neighbor coordination statistics occur as pressure increases. The former changes rapidly in the 0 to 10 GPa range whereas the latter exhibits a broad peak in O (that is, O with three nearest T neighbors as in the stishovite structure) around 40 GPa. The formation of significant numbers of TO5 polyhedra drastically alters intermediate range (0.5–1.5 nm) order by frustration of ring formation defined by linkage of corner-sharing TO4 tetrahedra. These results provide the microscopic basis for the pressure dependence of the macroscopic properties of molten CaAl2Si2O8. Because the MD simulations performed by Nevins and Spera (1998) were carried out isothermally, neither the temperature-dependence of short range structure, nor the glass transition were studied. By combining the results presented below with those obtained earlier, one obtains a first-order picture of the microscopic structure of molten CaAl2Si2O8 for pressures in the range 0 to 76 GPa and temperatures in the range 2800 to 5000 K. Amorphous CaAl2Si2O8 illustrates classic fragile-liquid behavior (Angell 1985, 1991, 1995). Although stoichiometrically a tetrahedral “2-4” fluid (O in nominal twofold coordination with T (T ≡ Si, Al) and T in fourfold coordination with O), CaAl2Si2O8 exhibits marked non-Arrhenian shear viscosity-temperature behavior at low pressure (Fig. 1a). The apparent activation energy for viscous flow at low-temperature is three to five times greater than the corresponding value at high temperatures in the equilibrium liquid. Unlike typical strong network melts, the change in isobaric heat capacity at the calorimetric glass transition, a measure of the configurational entropy of supercooled liquid relative to the crystal at Tg, is relatively large (Fig. 1b). For CaAl2Si2O8, Richet and Bottinga (1995) cite ∆CP ≈ 8 J/g atom K at the laboratory glass transition. In contrast, SiO2 and NaAlSi3O8, both “strong” fluids, MORGAN AND SPERA: MD STUDY OF GLASS TRANSITION OF CaAl2Si2O8 917 exhibit ∆CP values at Tg ≈ 1480 K and 1100 K, respectively, of about 2 J/g atom K. Morse (1980) summarizes the melting of crystalline anorthite. At ~1 GPa (the mean pressure of the MD simulations) anorthite melts congruently at 1841 K. Above ~1 GPa, anorthite melts incongruently to crystals of corundum (Al2O3) plus a liquid more rich in Ca and Si than stoichiometric anorthite. The incongruent melting reaction is CaAl2Si2O8 (an) → Al2O3 (cor) + CaSi2O5 (liq). Interestingly, Stebbins and Poe (1999) demonstrate that Si, along with octahedral Si is present in CaSi2O5 glass quenched from the liquid state at 2575 K and 10 GPa. This is consistent with the simulations of Nevins and Spera (1998) where both Si and Si were noted. Computer vs. laboratory glass transition The phenomenology of structural relaxation and the dependence of the glass transition temperature on cooling rate in laboratory studies has been discussed extensively in the literature (e.g., see Moynihan et al. 1976; Brawer 1985; Scherer 1986; Dingwell 1995). The calorimetric glass transition for CaAl2Si2O8 at 10 Pa is 1160 K at a cooling rate of γ = 10 K/ps [1 picosecond (ps) = 1·10 s] according to Richet and Bottinga (1995) and 1109 K at γ = 1.7·10 K/ps according to Moynihan (1995). These values, based on laboratory quench rates, may be compared to the “computer” glass transition found in this study (see below) of Tg ≈ 2800 ± 200 K at a quench rate of ~700 K/ps, thirteen orders of magnitude faster than the laboratory cooling rate. The computer Tg for anorthite composition is lower than that for silica of 3050 K and 3300 K for cooling rates of γ = 70K/ps and 700K/ps, respectively, using the BKS potential for silica (Vollmayr et al. 1996; van Beest et al. 1990). In contrast, Della Valle and Andersen (1992) estimate Tg ~ 2200 K for silica using the TTAM potential (Tsuneyuki et al. 1988; see also Rustad et al. 1990, 1991a, 1991b, 1991c, 1992). In comparison, the laboratory calorimetric glass transition for silica is ≈1480 K (Richet and Bottinga 1995) for γ = 10 K/ps. A useful heuristic is that the ratio of the computer to calorimetric glass transition temperature is ≈ 2.4 (see Bryce et al. 1999 for additional examples in the system NaAlO4-SiO2). Although the glass transition temperature varies with cooling rate, it is reasonable to believe that the underlying microscopic dynamics of the transition are similar regardless of quench rate. In broad terms, models for the glass transition can be broken into two classes: thermodynamic and nonthermo-dynamic. In the thermodynamic view, the observed glass transition is the kinetic manifestation of an underlying second order Ehrenfest phase transition at the Kauzmann temperature TK (TK < Tg) with discontinuous derivatives of thermodynamic state variables such as the isobaric expansivity, αp [≡V (∂V/∂T)p] and isobaric heat capacity CP [≡(∂H/∂T)p]. The vanishing of the entropy difference between metastable (supercooled) liquid and crystalline solid is a consequence of the change in isobaric heat capacity of the material at the Kauzmann temperature, TK, as new degrees of freedom come into play. In nonthermodynamic theories, structural arrest is viewed purely as a dynamical singularity associated with dramatic growth of relaxation time for decay of microscopic density fluctuations in the supercooled liquid. From this vantage, the glass point marks a transition from ergodic to nonergodic behavior in 6N+1-dimensional potential energyphase space. The hope is that although cooling rate clearly affects the numerical value of Tg, the underlying microscopic dynamics captured by carefully performed Molecular Dynamics simulation is relevant to thermal arrest, the drastic increase in relaxation time on approach to the glassy state and the configurational entropy of fully ergodic liquid. In the following sections, the structure, properties and dynamics of CaAl2Si2O8 at temperatures spanning the glass transition temperature are described. CaAl2Si2O8 is expected to exhibit somewhat more complicated behavior than a simple two-atom material such as silica. In CaAl2Si2O8, the TOn network (mainly TO4 at low pressure) is characterized by equal numbers of Si-O and Al-O bonds, not just Si-O bonds as in silica. There is therefore additional configurational entropy and steric distortion introduced by the mixing of Al and Si of different nominal size and charge. Moreover, the divalent alkaFIGURE 1. (a) Log viscosity (Pa·s) vs. Tg/T for CaAl2Si2O8 (Tg = 2800 K) from laboratory viscosity data fit to the VTF equation from data compiled by Hummel and Arndt (1985). Solid lines encompass temperature intervals over which laboratory data was fit; dotted lines are interpolated or extrapolated. (b) Laboratory isobaric heat capacity (J/mol·K) for CaAl2Si2O8 plotted vs. temperature from parameters provided by Richet and Bottinga (1984b). C p ( J/ m ol ⋅K ) lo g( η )( P a S ) MORGAN AND SPERA: MD STUDY OF GLASS TRANSITION OF CaAl2Si2O8 918 line earth metal Ca is present in CaAl2Si2O8 and missing in silica. The coordination, mobility and possible cooperative motion of O and Ca is a factor missing in compositionally simpler silicates. Finally, unlike the alkali aluminosilicates, where the ratio of Na to O diffusivity is of order 10–100, Ca has about the same tracer diffusivity as O (see below). The multicomponent nature of CaAl2Si2O8 complicates the analysis of structural relaxation. At the same time, at least this level of complexity is present in real geofluids and it appears worthwhile to study such systems further despite use of a rather simple form for the potential energy expression.