Department of Mechanics: Seminar: Abstract Miller

Finite Temperature and Finite Deformation: New Tools for More Efficient and Accurate Atomistic Simulation

Two of the important challenges to using atomistic simulation to study material behaviour are the roles of finite temperature and finite deformation. We discuss a new method for finding activation energies for atomistic systems and an improved method for accurately controlling the true stress in molecular dynamics simulations.

When studying the deformation of solids at finite temperature, we are regularly confronted with the long time scales associated with activated processes. Diffusion, defect nucleation and dislocation motion are all processes that are too slow to study with direct atomistic simulations. Instead, we must appeal to models based on transition state theory, whereby we can predict process rates using accurately calculated parameters from key atomic configurations. Specifically, we require knowledge of three atomic configurations: two local minima on the potential energy surface (PES) of the atoms (the "reactant" and "product" states), and the saddle point ("transition" state) configuration between these minima. For complex processes involving many atoms, the challenge is to find these configurations both efficiently and with sufficient accuracy.

In this presentation, I will discuss a new method called TRREAT (the "Transition Rapidly-exploring Random Eigenvector Assisted Tree" method) that we have developed [1] for the purpose of searching the PES for transition and product states. The method is in the family including the nudged elastic band [2], dimer [3] and activation-relaxation [4] methods, but seems to have advantages for certain applications. Specifically, it can find numerous transition and product states from knowledge of only the reactant state and does so efficiently by a combination of Monte Carlo search, eigenvector biasing and a method to avoid previously searched regions of the PES. Further, because it works well with only approximate eigenvectors (due to its stochastic nature), it can be applied to large DFT calculations where accurate eigenvectors are prohibitively expensive to compute.

The second challenge we will address in this presentation, that of finite deformation, is important when studying soft materials or stress-induced phase transformations of crystals. When the stress state needs to be controlled in atomistic simulation of such systems, this is almost universally achieved using the Parrinello-Rahman technique [5] or some related variation (e.g. [6,7]). However, none of the these techniques are able to control the true (Cauchy) stress applied to the system. Instead, they apply an approximation to the second Piola-Kirchhoff stress (related to the "engineering" stress). The true Cauchy stress that results during such a simulation is dependent on the deformation of the simulation cell. Further, this true stress cannot be known a priori, and it can be significantly different than the apparent applied stress when the deformation is large.

In this presentation, I will discuss an alternative MD algorithm that controls the true Cauchy stress applied to the system. The "Cauchystat" is based on the constant stress ensemble presented by Tadmor and Miller [8], but with modified equations of motion that update the system boundary conditions in response to the resulting deformation of the simulation cell. As a clear example of the method's usefulness, we show that the correct stress control is important in the case of martensitic phase transformations, where the predicted martensitic start temperature and austenitic finish temperature are significantly altered as compared to the Parrinello-Rahman result.

References

[1] Carlos Campana and Ronald E. Miller. “Transiting the molecular potential energy surface along low energy pathways: The TRREAT algorithm.” Journal of Computational Chemistry, 2013, http://onlinelibrary.wiley.com/doi/10.1002/jcc.23408/abstract

[2] G. Mills and H. Jonsson. "Quantum and thermal effects in H2 dissociative adsorption – evaluation of free-energy barriers in multidimensional quantum-systems." Phys. Rev. Lett., 72(7):1124–1127, 1994.

[3] G. Henkelman and H. Jonsson. "A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives." J. Chem. Phys., 111(15):7010–7022, 1999.

[4] G. T. Barkema and N. Mousseau. "Event-based relaxation of continuous disordered systems." Phys. Rev. Lett., 77(21):4358–4361, 1996.

[5] M.Parrinello and A.Rahman. "Polymorphic transitions in single crystals: A new molecular dynamics method." J. Appl. Phys., 52(12):7182–7190, 1981.

[6] G.J. Martyna, D.J. Tobias and M.L. Klein. "Constant-pressure molecular-dynamics algorithms." J. Chem. Phys., 101(5) pp. 4177-4189, 1994.

[7] R. M. Wentzcovitch. "Invariant molecular-dynamics approach to structural phase transitions." Phys. Rev. B, 44(5):2358–2361, 1991.

[8] E. B. Tadmor and R. E. Miller. "Modeling Materials: Continuum, Atomistic and Multiscale Techniques." Cambridge: Cambridge University Press, 2011.

Ron Miller is currently a professor at Carleton University (Ottawa, Canada), and also a visiting professor at EPFL (Lausanne, Switzerland). He got his PhD in Mechanics at Brown University in 1997 and was a post-doc at Harvard, Division of Engineering and Applied Sciences, with John Hutchinson; a visiting professor at Brown, the Technion, INPG (France); a visiting researcher at CNRS (France); and an assistant professor at the University of Saskatchewan in Saskatoon, Saskatchewan.