A Molecular Dynamics Approach to the Computer Glass Transition of CaAl2Si2O8
Neil Morgan
The capability of understanding structural and dynamic behavior of materials at high pressures and temperatures has been enhanced by the use of the Molecular Dynamics (MD) simulations in recent years. MD simulations are numerical solutions to classical equations of motion for a set of particles governed by some set of inter-atomic potentials. MD simulations allow one to investigate the dynamics of a system in microscopic detail and to record phenomena that occur at the femtosecond (10-15 seconds) time scale.
Twelve 50 picosecond MD simulations were carried out over a temperature range of 1707 to 4977 K and approximately 1 GPa to identify the computer glass transition, Tg, of CaAl2Si2O8. A total of 1300 particles were used in the NVE (microcanonical) simulations. A simple pairwise additive intermolecular potential containing only Coulomb attractions and repulsions and exponential repulsions of the Born-Mayer-Huggins form was used: Uij = qiqj/rij + Aij exp(-Bijrij). The full ionic charge between particle i, qi and particle j, qj is separated by the interparticle bond distance, rij. The Ewald method was used to compute the Coulomb interaction for each ion; the cutoff used in the Born-Mayer-Huggins summation was 0.8 nm. An initial momentum-free and thermally equilibrated system at a temperature of 10 000 K was quenched at a rate of 700 K/ps to a temperature of 3150 K before the 50 ps MD simulations were commenced. From this base temperature of 3150 K other desired temperatures were attained by a cooling or heating rate of 70 K/ps. In a separate study we set the initial quench rate at 70 K/ps and found that the quench rate had little effect on the structure of the glass.
The glass transition is a phenomenon observed when a system is cooled rapidly beyond its freezing point before crystallization occurs. It is not clear that the glass transition is a phase transformation, but it is known that macroscopic thermodynamic properties such as heat capacity, Cp, is sensitive to these system perturbations. In Figure 1, enthalpy (H, kJ/mol) is plotted against temperature (T, K) from which the slope of the line gives the heat capacity at constant pressure: ("H/"T)p = Cp. At high temperatures CaAl2Si2O8 is at an equilibrium state and the data fits to a straight line. Yet below 3000 K the data indicate a decrease in Cp by a change in the slope of the line through the higher temperature data. This implies that CaAl2Si2O8 is falling out of thermodynamic equilibrium as the time of observation becomes comparable to the relaxation time of the system which in turn suggests that Tg is a little below 3000 K at approximately 1 GPa.
The van Hove self-correlation function:
can be employed to verify whether the system is ergodic equilibrium or is in an arrested, glassy state. The probability function:
gives the likelihood of locating a particle at point r at time t when it was originally at r = 0 and t = 0. For an ergodic liquid, P(r,t) as a function of r will have a single maximum that migrates smoothly towards larger r as t increases. When the material becomes structurally arrested, as in a glass, one can expect P(r,t) to exhibit a double peaked structure with very little movement of the maximum with t. In Figures 2a through 2d, P(r,t) is plotted versus r for various values of t for the oxygen ion. Figure 2a exhibits an evolution of the van Hove that is characteristic of a completely ergodic liquid at a temperature of 3480 K. In Figure 2b, at a temperature of 2870 K, P(r,t) exhibits less peak migration as t increases and there is some evidence of a second maximum. Figure 2c suggests that CaAl2Si2O8 at 2540 K has undergone a transition from an ergodic liquid to an arrested state judging from the conspicuous second peaks and minimal migration of the main maximum in P(r,t). In Figure 2d, at 1700 K, these characteristics are exaggerated to the extent that there is so little movement in the system that almost all the peaks line up with each other. This implies that the system is in a perfectly glassy state with a frozen configuration lacking long range structure.
It has been proposed by Hiwatari et al., 1991 that another tool for distinguishing between materials with structurally arrested, glassy behavior and those with ergodic, liquid-like behavior is the non-Gaussian parameter:
that can be determined from higher order terms of the expanded form of the incoherent scattering function. The A(t) statistic decays towards 0 for materials that are relaxed during the simulation. Structurally arrested ions are characterized by A(t) functions approaching some constant non-zero value. In Figure 3, A(t) is plotted versus t for each of the MD simulations for the oxygen ion. For temperatures 3480 K and above, these clearly decay to zero and are ergodic liquids. For slightly lower temperatures, distinguishing between 2870 K and 3150 K is difficult, yet it seems that the 2540 K curve without a pronounced peak has transcended to a glassy state. Clearly, temperatures lower than 2540 K are structurally arrested and have wildly different asymptotic values.
One of the curiosities with the glass transition is how might the arrangement and packing structures of atoms change with a very fast cooling rate. In this study we looked at the effect of temperature on short range structure. Shown in Figure 4a is the temperature dependence of the coordination of O about T atoms in the CaAl2Si2O8 system, where T is either silicon or aluminum. The entire temperature range is dominated by [4]T and [5]T (approximately 80-90% of the total [n]T abundance). The remaining is taken up by [3]T and [6]T.
The coordination of oxygen about other O atoms is plotted versus temperature is shown in Figure 4b. Nine and tenfold coordination numbers are in relative abundance across the temperature range. Similar to Figure 4a, 4b indicates no, or very little structural dependence on temperature, even near the glass transition of CaAl2Si2O8. From these studies one can conclude that the glass transition is strictly a kinetic phenomena.
In this study we used MD simulations to locate the computer glass transition
of the CaAl2Si2O8 system and to determine if
there were any dynamic short range structural characteristics across the studied
temperature range. First looking at macroscopic thermodynamic properties, we
found that Cp begins decrease at approximately 3000 K, giving a gross
estimate of Tg. The van Hove and the non-Gaussian parameter provided
a microscopic look at the dynamics of the MD simulations and both suggested
that the CaAl2Si2O8 system approached the glass
transition in the approximate range of 2870 to 2540 K. Unfortunately, the van
Hove and A(t) could not put an exact number on Tg, but this can be
explained in part by the short range coordination statistics; the glass transition
is purely a kinetic phenomena because no structural changes occurred within
the studied temperature range. Additionally, Tg can fluctuate due
to the inherent statistical nature of the MD simulations, which simply means
that any property we calculate from the CaAl2Si2O8
system will have a range associated with it.
