Studying Slow Fractures Using ART

 

1. Introduction

The dynamics of a complex system can be described as a walk on a high-dimensional free energy surface. A complete knowledge of the dynamics and equilibrium properties of the system can be obtained if one knows the distribution and properties of all the local minima of the energy landscape and the rates controlling the jumps between these local minima.

An event in a dynamic system can be considered as a configurational transition from one local minimum to another. In many materials and systems such as glassy and amorphous materials, the dynamics is slow in the sense that events occurs over time scales many orders magnitude larger than the associated natural atomistic time scale.

For most of the time, a system is confined in its deep local minimum. The rate for the system to jump over the surrounding barriers and move to a new minimum decays exponentially with increasing height of the barriers. For these reasons, simulation techniques of accelerated dynamics are crucial in studying long-term system dynamics. The ART (activation-relaxation technique) is one of these accelerated methods.

ART was introduced by Mousseau and his co-workers ~10 years ago. It is a generic method to explore the landscape of continuous energy functions through hops from a local minimum to another with a series of activated steps. It allows the system to evolve following well-defined paths between local energy minima.

We propose to apply ART to a fracture-growth problem in silicon crystals. An existing crack in the crystal is under minor stress and the tip proceeds by breaking bonds around it. Since the stress is very small, the growth rate of the fracture is tiny. To simulate the long-term behavior of the fracture growth, ART is an excellent method to use.

 
2. The Activation-Relaxation Technique
As stated in the original paper of Barkema & Moussear (1996), the underlying concept of ART is that it is the material itself, as defined by its total energy surface, that should determine events, not an external rule. ART is realized in two major phases: activation and relaxation.
(1) Activation
The activation phase can be further divided into two steps: escaping from the harmonic well (Fig. 1) and converging to the saddle point (Fig. 2). Starting from a local minimum (M0), the configuration is first pushed away from the local minimum in a random direction. The configuration is moved following modified force vectors obtained by inverting the component of the force parallel to the displacement form the current position to the local minimum. As each of these moves, the lowest eigenvalue of the local Hessian is evaluated using Lanczos method. A saddle point is found when the following two criteria is met: a zero total force and a negative eigenvalue.
Figure 1. Escaping the harmonic well.
Figure 2. Converging to a saddle point.
 
(2) Relaxation
To ensure that the system does not fall back to the previous local minimum, the configuration is pushed slightly over the saddle point before the relaxation. A modified conjugate-gradient algorithm is employed to relax the system from the saddle point to the new local minimum (Fig. 3). This modified CG method can be seen as a dual method that favor steepest-descent away from the minimum and CG nearby.
Figure 3. Relax to a new local minimum.
 
3. Test runs with amorphous silicon

The starting configuration of 1000-atom cell of a-Si is obtained from the website of Prof. Normand Mousseau at the Universit¨¦ de Montr¨¦al. A modified Stillinger-Weber potential fitted by Vink, Barkema, Mousseau and van der Weg is used. A low-energy event generated by ART is accepted or rejected based on a standard Metropolis algorithm. In the test runs, an effective threshold temperature of 0.5 eV is used. The starting configuration of the 1000 a-Si atoms is shown in Figure 4. From this starting local minimum, ART explores the energy landscape by traveling through saddle points and local minima. As shown in the movie of Fig. 5, the system goes through transition states by repeating the process of activation (from a local minimum to a saddle point) and relaxation (from a saddle to a new local minimum). In this test run, ART found the 25 local minima of the energy landscape (Fig. 6) for the specific starting a-Si configuration (Fig. 4). The energy differences between the local minima are minor suggesting that this starting configuration is already close to optimal (in a deep energy minimum) with its current potential and temperature condition. Test runs with other temperature thresholds such as 0.25 eV and 0.8 eV give similar results.

 
 
Figure 4. Starting configuration of the 1000-atom amorphous Silicon .
 
 
 
Figure 5. Amorphous silicon system going through the transition states. There are total 50 configurations: 25 saddle points and 25 local minima.
 
 
 
Figure 6. All 25 local minima.
 
 
4. Future work
Silicon is very brittle. It is well known that even at room temperature, static cracks in silicon can have tips that are atomically sharp. Crack in silicon crystals propagating by breaking bonds around the sharp tip. Instead of molecular dynamics simulation, we propose to use ART to accelerate the dynamics by exploring the energy landscape.
 
5. Acknowledgement
Many thanks to Ken-ichi for helping with the visualization.
 
References
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