Abstract
We seek to model the shock wave induced structural changes in silicate glass at the atomic scale. We use both direct shock propagation with non-equilibrium molecular dynamics (NEMD) and bulk simulations in the Hugoniot ensemble to characterize the structure and topology of the shocked glass. Despite the lack of long-range interactions in our model, the close agreement between our structures and those obtained by experimental and simulation studies alike, underlines the importance of the role played by first neighbor interactions on the structure of silicate glass. The results obtained from this study show that, in agreement with experimental work, the structure and topology of the shock-induced densified phase is unique in its structure as can be revealed by medium-range order measurements. The modifications include a reduction of the average tetrahedra size and an increase in the proportion of 3-4 and 8-10 membered Si-rings. Application of a Hugoniostat method based on constraint dynamics shows near-perfect agreement with the NEMD results. Besides validating the former method, this opens the prospect of studying shock-induced effects at a fraction of the cost required to run large scale shock simulations while using much more complicated potentials and setups.
Introduction
High power lasers are used to achieve extreme temperature and pressure conditions. During such experiments, deterioration of the optics due to the passage of the laser has been observed in the form of small craters at the back of the lenses [1, 2]. Is is believed that these craters originate from the absorption of the laser by defects which create a dense absorbing plasma. The induced temperature rise in turn triggers the propagation of a mechanical shock wave creating regions of permanently densified glass [3].
Experimental work on shock compression shows the creation of a densified phase upon propagation of shock wave through silica glass [4]. Kubota et al. [5, 6] have used molecular dynamics to propagate shock waves with up to 2.0 Km/s piston velocity through slabs of up to 240×103 atoms. The authors show drastic shock induced modifications in the structure and topology of the SiO2 network. It has also been shown that the structure of the resulting densified phase is unique as can be revealed by medium-range order measurements such as the ring size distribution [7–9].
Despite the major increase in the available computer power which allows simulation of several hundred million atoms, non-equilibrium molecular dynamics (NEMD) simulations of shocked materials still represent a computationally major undertaking. This is mainly due to the large system sizes needed to achieve a steady shock front; additionally a non-negligible part of the total simulation time is spent calculating the trajectories of particles in the unshocked region. Another NEMD method has been used in [10] where the simulation box consists of a moving window which follows the shock front. The method achieves accurate description of the shock induced effects provided steady shock waves are obtained. Alternatively, to address the system size problem, several groups have attempted to construct an alternativemethod where equilibrium simulations are performed on bulk systems in the Hugoniot ensemble. Such a method allows usage of much smaller systems to achieve results equivalent to those obtained with NEMD. The first such method (GNVHug) was proposed by Soulard [11] and uses the Gauss principle of least constraints to integrate the equations of motions and constrain the system on the Hugoniot. This method presents the double advantage of reaching equilibrium very rapidly (O(10−13)s) and guaranteeing that the system obeys the Hugoniot relations at all times. The method was applied to the simulation of shocked liquid argon and nitromethane [11, 12] and has shown good agreement with NEMD simulations. An alternative approach was proposed by Maillet et al. [13] which uses a Nos´e-Hoover like method to couple the system, instantaneously, uniaxially and homogeneously compressed to the shock density, with a thermostat to achieve the final Hugoniot state. Applied to the shock deformation of Lennard-Jones crystals along the h100i direction, the method has shown to reproduce both the Hugoniot curve and the shock induced defect structure. This method was later extended to ab initio molecular-dynamics to obtain the shock Hugoniot of tin up to 200 GPa [14]. Answering the criticism that the instantaneous compression of the system can lead to unrealistically large temperature and stresses, Ravelo et al. [15] improved the uniaxial Hugoniostat by adding an strain-rate dynamical variable which acts as a piston. The method allows gradual compression of the system to the shock density and therefore a more natural evolution of both the temperature and stress. Applied to the calculations of the Hugoniot of Lennard-Jones crystals equivalent to those performed in [13], the method improves the agreement with NEMD, especially for large compressions.
In this paper, we seek to model the shock induced structural changes in silicon dioxide at the atomic scale. To this end, after validating that our model achieves an accurate description of the glassy state of silicon dioxide, we use NEMD simulations to characterize precisely the structure induced by a mechanical shock wave. We then apply the GNVHug method and show this gives very good agreement with the NEMD results. The remainder of this paper is organized as follows. In Section II we present the potential used to describe the inter-atomic interactions in silicon dioxide. In Section III, we describe the methods and results obtained to create and analyze a bulk system of silicon dioxide in the glass phase in order to check our potential leads to an appropriate structure before shocking it. In Section IV, we turn on to the study of the structural changes induced by the propagation of a mechanical shock wave using NEMD simulations, while Section V concentrates on obtaining similar results with the Hugoniostat. Section VI concludes this work with a discussion of the results drawn from this work and proposes some directions for future developments.
Conclusions
In this paper we have used atomistic simulations to model the shock-induced structural changes in silicate glass. Our model is a modified version of the BKS potential where the long-range part of the interaction is cut and shifted, based on the assumption that the glass structure is mainly dictated through interactions within the first neighbor shell. We show that the glass produced with this potential is structurally equivalent to those generated using the BKS potential with long range interaction and the three-body VKRE potential. In addition, our structural parameters are consistent with those found in experimental studies. This therefore suggests the validity of the assumption made here. The short range interactions are responsible for the average tetrahedra size and shape as well as their relative positions. The glass being a continuous random network of connected tetrahedra, the long range order is controlled by the cumulative effects of setting the individual tetrahedra structure and their relative positions with respect to their immediate neighbors.
We have subsequently used non-equilibrium molecular dynamics to propagate mechanical shock waves through our glassy system. Comparison of the structures measured in the shocked and unshocked regions reveal that, above the plastic regime limit, the shock wave induces profound and irreversible structural changes. These are manifested by the reduction in the average tetrahedra size associated with the material densification. The relative positions of the tetrahedra is distributed with equal probabilities among two configurations where the angle made by connected tetrahedra equals 60◦ or 103◦. Analysis of the ring size distributions shows that this is connected with an increase in the proportion of both 3-4 and 7-10 membered rings. Thus, in addition to the material densification manifested by smaller tetrahedra and narrow inter-tetrahedra angles, the shock wave also stabilizes the formation of large rings in the network created by wide inter-tetrahedra angles.
We have then applied the GNVHug Hugoniostat method to perform equilibrium bulk simulations of the shockinduced structural changes. The method is shown to reproduce well the D(u) and P1(ρ1) curves in the plastic regime. Analysis of the structures shows very good agreement with those obtained using NEMD simulations; the correspondence improving with increased shock wave velocities. As the Hugoniostat method allows use of much smaller systems, these results open up the prospect of achieving modeling of the shock wave induced effects using complicated potentials at a fraction of the cost required to run large scale shock simulations.
This will form the basis of future work where the Hugoniostat method will be used to explore more complicated potentials including long-range forces and dynamics charges but also more complicated setups in an attempt to obtain a more realistic model of the mechanisms underlying damage of the optics by high-power lasers. Additionally, the results obtained from the bulk simulation will be completed by a more refined study to quantify the influence of long range forces on the structural properties of the glass comparing the glass structure obtained with and without treatment of the long range forces.
References
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