For temperatures below the glass temperature Tg we have envisioned a model that describes the dynamics of the atomistic system as a series of transitions, following first-order kinetics and Poisson statistics, between basins surrounding the inherent structures. Our validation of this model is based on its ability to reproduce the mean square atomic displacement resulting from a full-blown atomistic molecular dynamics (MD) trajectory. The projection of MD trajectories onto the Poisson process model has been made possible thanks to the use of hazard plot analysis, which allowed the evaluation of the rate constants for individual transitions between the basins. Our results for a model two-component Lennard-Jones mixture [3-4] clearly show that our Poisson process approach is able to reproduce the inherent structure dynamics obtained directly from MD over a wide range of sub-Tg temperatures, from 9K to 38 K. At higher temperatures, the long-time diffusive motion of atoms in the system can be captured simply by tracking the mean square displacement between inherent structures.
We utilize and extend this methodology in order fully to reconstruct the system dynamics in the form of the mean square atomic displacement as a function of time at finite temperatures based on the succession of transitions in a network of basins or "states," the contribution of vibrational motion around each inherent structure being evaluated through short MD runs artificially trapped within each one of the states. To quantify the vibrational contribution we propose a procedure that traps the system within each of the basins by applying reflective conditions, that is, reversing the atomic momenta in Newton's equations of motion every time the minimization procedure shows that a switch to the basin of a neighboring inherent structure has occurred. The inversion of the atomic momenta is followed by a randomization of the momentum of the thermostat that smoothly leads to a new trajectory. This procedure has been inspired by the novel methods developed by Voter [5] for analyzing the dynamics of infrequent events, where reflective conditions were introduced in the context of biasing the local search for transitions, by increasing the temperature.
We provide [2] the mathematical formulation for lifting the coarse-grained Poisson process model of transitions between states back to the atomistic level and thereby reproducing the full dynamics of the atomistic system within the Poisson approximation. Our results for the mean square atomic displacement as a function of time show excellent agreement with the full MD for temperatures around and below the glass temperature of our model Lennard-Jones two-component mixtures. Clearly, our approach is able to reproduce the full dynamics of the atomistic system at low temperatures, where the Poisson approximation is valid.
The proposed methodology creates a network analogous to a network of first-order, reversible chemical reactions, with basins or states around the inherent structures playing the role of chemical species. When the system remains trapped in a small number of inherent structures, as happens at temperatures lower than the glass temperature, it is possible to derive an analytical solution for the evolution of the probability distribution of the system among states, given the transition rate constants. The ability to identify new states to be accessed and calculate the transition rate constants on the fly can be invoked in order to build an ever-expanding network of states and use it to evaluate analytically averages over all possible dynamic outcomes of the Poisson process up to a given observation time. This has been achieved by Boulougouris and Theodorou [6] via the development of Dynamic Integration over a Markovian Web (DIMW). This method relies upon a distinction between "explored" states and "boundary" states and invokes absorbing boundary conditions to evaluate analytically the mean first passage time to reach any of the boundary states. The augmentation of the network (by appending a boundary state to the set of explored states) is accomplished by a stochastic selection scheme based on the flux to each of the boundary states.
Our ultimate aim is to develop a methodology that enables mapping the dynamics around and below Tg to a coarse-grained first-order kinetic model based on the Poisson process approximation. To do so, we are combining our MD-based approach with the DIMW method [6]. This will allow us not only to observe the time evolution of the system, but also to sample over a time range far beyond the accessible times of classical MD. Therefore, it will allow us to create computer glasses at cooling rates closer to those used experimentally.
References
[1] Tsalikis D. G., Lempesis N., Boulougouris G. C. and Theodorou D. N., submitted.
[2] Tsalikis D. G., Lempesis N., Boulougouris G. C. and Theodorou D. N., submitted.
[3] Kob, W.; Andersen, H. C. Phys. Rev. Lett. 1994, 73, 1376.
[4] Kob, W. Phys. Rev. E. 1995, 51, 4626.
[5] Voter, A. F. Phys. Rev. Lett. 1997, 78, 3908; Voter, A. F.; Doll, J. D. J. Chem. Phys. 1985, 82, 80.
[6] Boulougouris, G. C.; Theodorou, D. N. J. Chem. Phys. 2007, 127, 084903.