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Elementary mechanisms governing the dynamics of silica

Normand Mousseaua)

Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, and Computational Physics, Department of Applied Physics, TU Delft, Lorentzweg 1, 2628 CJ Delft, The Netherlands

G. T. Barkema

Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands Simon W. de Leeuw

Computational Physics, Department of Applied Physics, TU Delft, Lorentzweg 1, 2628 CJ Delft, The Netherlands

共Received 15 July 1999; accepted 15 October 1999兲

A full understanding of glasses requires an accurate atomistic picture of the complex activated processes that constitute the low-temperature dynamics of these materials. To this end, we generate over five thousand activated events in a model silica glass, using the activation–relaxation technique; these atomistic mechanisms are analyzed and classified according to their activation energies, their topological properties and their spatial extent. We find that these are collective processes, involving ten to hundreds of atoms with a continuous range of activation energies; that diffusion and relaxation occurs through the creation, annihilation and motion of single dangling bonds; and that silicon and oxygen have essentially the same diffusivity. © 2000 American Institute of Physics.关S0021-9606共00兲71402-4兴

I. INTRODUCTION

Glassiness is a dramatic slowing down of the kinetics of a liquid as the temperature decreases below some typical value. Experiments have yielded considerable information about the macroscopic character of this phenomenon, but very few techniques provide the local probe needed to un-derstand its microscopic origin.1–3 On the theoretical side, significant progress has been made recently in understanding the supercooled region, but little is known about the atomis-tic nature of the relaxation and diffusion dynamics taking place at temperatures below the glass transition.4 Using a new Monte Carlo technique, the activation–relaxation technique,5,6we map the activated processes of g-SiO2 tak-ing place at low temperatures.

At low temperatures, the dynamics is well characterized by a sequence of activated mechanisms that bring the con-figuration from one local energy minimum to another. The activation–relaxation technique 共ART兲 is a method that re-constructs such activated processes共events兲 in complex con-tinuous systems.5,6It does this without following the dynam-ics of the system: the probability for an event to occur in ART might differ greatly from true dynamics. Moves within ART are defined directly in the configurational energy land-scape and can reach any level of complexity required by the dynamics; they can involve hundreds of atoms crossing bar-riers as high as 25 eV.7In a two-step process, a configuration is first brought from a local minimum to an adjacent saddle point and then relaxed to a new minimum. Such an event is shown in Fig. 1.

The more traditional way of studying the activated dy-namics is by means of molecular dydy-namics at elevated

tem-peratures. This approach was followed by Litton and Garofalini8 as well as Horbach and Kob,9 using a semi-empirical potential, and by Sarnthein et al. using an ab initio method.10ART adds to the standard approach in a few ways: it generates events with a high efficiency, one at a time with-out entanglement, and thus provides for an easy identifica-tion and classificaidentifica-tion.

In this work, we have used ART to generate a large data base of activated events in glassy SiO2 described with the screened-Coulomb potential of Nakano et al.11,12The events are analyzed in terms of energetics, topological changes, etc.

II. GENERATION OF THE DATA BASE

We study two independent runs on 1200-atom cells of SiO2, modeled with the screened-Coulomb potential of Na-kano et al. which has been shown to give realistic structures and a good account of a number of dynamical properties.11,12 We prepare these runs starting from randomly packed unit cells; this procedure ensures the absence of correlation, both with the crystalline state as well as between runs. The samples are relaxed through a sequence of 5000 ART events. Each ART event consists of an activation and a relaxation step. In the activation, first a local displacement at a random location in the sample is generated, next a re-defined force Gជ is followed until a first-order saddle point in the 3N-dimensional configurational energy landscape is reached. This redefined force is given by

G⫽Fជ⫺共1⫹␣兲共F•rˆ兲rˆ, 共1兲

where Fis the 3N-dimensional force vector, rˆ the 3N-dimensional unit vector of the displacement from the lo-cal energy minimum and ␣ a parameter for which we chose

a兲Electronic mail: mousseau@helios.phy.ohiou.edu

960

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⫽0.15/(1⫹r). Once the saddle-point is passed, energy

minimization brings us to another local energy minimum, which is accepted or rejected following a standard Metropo-lis procedure with a fictitious temperature of 0.25 eV. A more detailed description of the precise ART implementation can be found in Refs. 5,6. After relaxation, a further 5000 ART iterations are performed on each cell. Slightly more than half of these iterations show a clean convergence to a saddle point, providing a database of 5645 events. An analy-sis of these events can give us a unique glimpse at the basic nature of activated mechanisms in this material.

We checked for systematic effects caused by the initial-ization procedure or by the potential used: a comparison with events from a shorter run, starting from an MD-prepared 576-atom sample, indicates that the nature of the events is independent of the preparation mode; a comparison with events from a shorter run in which the van Beest potential13 was used, with parameters as in Ref. 14, indicates that, un-less stated otherwise, the results presented here are at least qualitatively similar between these potentials. However, one limitation of these potentials is that they do not account for homopolar bonding; potentials that account for the covalent bonds appropriately, such as ab initio potentials, are

prefer-able but unfortunately computationally not feasible for the system sizes required. In spite of these limitations, we can still gain some knowledge about the nature of relaxation in this material, as shown below.

During the acquisition of events, the configurational en-ergy decreases by about 30 meV per atom. The density of coordination defects fluctuates but does not show a clear trend. With the Nakano potential, the samples have roughly the right density for g-SiO2but the defect density is too high: about one percent of the bonds between O and Si are miss-ing, compared to perfect coordination. The defects produced are almost uniquely dangling bonds; we do not create any homopolar bond nor, consequently, any superoxide radical or Frenkel pair.1,15

III. ENERGETICS AND SPATIAL EXTENT

We first look at properties averaged over the whole data base. A recurring issue in glasses is the spatial extent of activated processes, measured in the number of participating atoms, or total displacement. For each event, we determine the number of atoms displaced by more than a threshold distance rc; the number of atoms participating in an event

depends of course on the value of this threshold. We plot this number as a function of rc in Fig. 2, from the initial mini-mum to the saddle point and the final minimini-mum configura-tions.

Typically, an event is accompanied by a local volume contraction or expansion. In the elastic limit, the displace-ment of the surrounding atoms decreases quadratically away from the center of the event, and the number of atoms mov-ing more than a cut-off distance rcdecreases as rc⫺3/2; Fig. 2

shows that this scaling is obeyed over two orders of magni-tude, from 2 to 200 atoms. The departure from the elastic scaling above 200 atoms is explained by finite-size effects: the total number of atoms共1200兲 is approached. The extent

FIG. 1. An event takes place in two stages: the activation from the initial minimum共top兲 to a saddle point 共middle兲, and the relaxation from there to the final minimum 共bottom兲. For clarity, we have depicted only the 187 atoms that move more than 0.1 Å , plus their nearest neighbors. Large circles are Si atoms, small ones O. In the top and bottom figures, the few atoms that are actually involved in the change of topology, plus their neigh-bors, are identified by a different color; this particular event is the hopping of a dangling bond on an O to a near neighbor. In the middle figure, the different coloring separates the atoms involved from their immobile neigh-bors.

FIG. 2. Log–log plot of the averaged number of atoms having moved by more than a threshold distance for all the events created in g-SiO2. The

dashed and the dot–dashed lines are distributions for the displacement from the initial minimum to the barrier and to the new minimum, respectively. The solid line is given by the equation n⫽5.0r⫺3/2.

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of scaling at the other end is more surprising, it reaches almost all the way down to a single moving atom. This is due to a large number of different mechanisms that contribute to the average presented here. In a crystal the events are much more constrained by the symmetry, and this curve would show structure, a signature of these events, at threshold dis-tances corresponding to at least a dozen participating atoms. Although the exponent of the elastic deformation is uni-versal, the pre-factor is not. This quantity reflects the typical local volume change associated with the events; in this case we get 92 Å3, corresponding to a length scale of 4.5 Å .

In Fig. 3 we plot this distribution for a threshold of rc ⫽0.1 Å , the typical vibrational amplitude of silicon at room

temperature. This threshold emphasizes slightly the atoms with net displacement over the whole event, focusing on the collective nature of diffusion and relaxation in disordered systems. As can be seen from this figure, events typically involve the motion of hundreds of atoms with simultaneous diffusion of both species to varying degrees: diffusion there-fore should not be thought of in terms of elemental jumps but of complex rearrangements.

For each event, we also determine the height of the ac-tivation barrier and the energy asymmetry 共initial to final minimum energy difference兲. Figure 4 shows broad and con-tinuous distributions of these quantities. Both distributions are smooth, without any distinct feature. Moreover, the height of the barrier and the asymmetry of the well are only weakly correlated.

From the distribution of activation energies, it is not pos-sible to extract directly an estimate of the diffusion barrier although theoretical work by Limoge and Bocquet18suggests that the effective activation energy should be around the maximum of its distribution. In our case, this provides a

ball-park figure of about 5 eV. The exact value of the aver-age energy barrier depends on the sampling of the events and the potential used; any comparison with other theoretical and experimental results must therefore be made with some care. In terms of correlation, larger events are not found to require a higher activation energy: size and energetics are almost entirely uncorrelated. Moreover, no correlation is found between the distance by which Si or O move and the corresponding activation barriers; the difference in activation energies of Si and O should thus be small.

Based on their MD simulations of molten silica between 4800 and 7200 K, Litton and Garofalini also report that O and Si diffuse with similar activation energies.8Horbach and Kob9 report activation energies of 4.66 eV for oxygen and 5.18 eV for silicon, obtained at temperatures from 2800 to 3250 K. Experiments report activation energies of 6.0 eV for Si,19obtained in electrically fused quartz, and 4.7 eV for O, with a much smaller prefactor,20obtained in vapor-phase de-posited amorphous silica. The large difference in experimen-tal activation energies might be caused by the different sample preparation techniques, resulting in different types of impurities.8On the other hand, the small differences in the O and Si diffusion rate in simulations might be a short-coming of the potentials which do not feature full homopolar inter-actions and have a relatively high defect density.

IV. TOPOLOGY

More microscopic information on the nature of the events can be obtained by studying the topology of the net-work. For this purpose, we divide the events into three dis-tinct categories: perfect events where only perfectly coordi-nated atoms change neighbors, conserved events that involve

FIG. 3. Distribution of the number of atoms involved in events, i.e., .the number of atoms that have moved more than 0.1 Å , from the initial mini-mum to the saddle point共dashed curve兲 and to the final minimum 共solid curve兲. The number of displaced atoms increases as one moves from the saddle point to the new minimum. An event involves typically between 50 and 250 atoms, but a significant number of events involve up to 300 atoms and more. This shows that relaxation in glasses is a collective phenomenon, in agreement with experimental results by Je´roˆme and Commandeur共Ref. 16兲, but different from the related material a-Si, where only about 40 atoms are typically involved in events共Ref. 17兲. The size of these events puts a lower bound on the number of atoms necessary in numerical simulations to avoid large finite-size effects.

FIG. 4. Distributions of the energy barriers共dashed curve兲 and asymmetries

共solid curve兲, obtained from the data base of all events. Both distributions

are continuous; no gap is seen. The barrier distribution peaks around 5 eV and extends well beyond the physically relevant values, but also displays a significant weight below 5 eV. The asymmetry distribution is narrower, and peaks around 2.5 eV. A small fraction of the events show a negative asym-metry, i.e., lower the energy. One expects this fraction to be small, since the configurations are well relaxed. Because of the exponential nature of the activation process, the presence of high energy barriers and states is irrel-evant because they are not sampled on normal time scales; it is the low energy part of the spectrum which determines the dynamics.

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only diffusion of coordination defects共dangling and floating bonds兲 and events that create or anneal coordination defects. Amorphous Si and g-SiO2are thought to be conceptually similar, both described by Zachariasen’s continuous random networks. However, while perfect events play a central role for both relaxation and self-diffusion in a-Si,17,21 they are rare in g-SiO2, which strongly favors chemical ordering: per-fect events require atomic exchanges at the second neighbor level, inducing more strain or larger topological rearrange-ments than in a-Si. Perfect events comprise only about one percent of the total number produced. A third of these events involves local topological rearrangements, mostly two Si ex-changing a pair of neighboring O. Such moves can only hap-pen with a relatively low energy barrier if the local rigidity of the network is reduced by nearby undercoordinated atoms. Two thirds of the perfect events do not involve a topological modification but simply some slight local rearrangement, with displacements on the order of 0.1 Å and asymmetries of about 10⫺4 eV. Such events could be candidates for tun-neling states.

The relatively high density of coordination defects 共com-pared with the experimental values兲 could be the origin of this imbalance in the proportion of perfect events compared with that of conserved events. As mentioned above, simula-tions in a-Si,20,21where the density of coordination defects is many times that of our current samples of g-SiO2, show nev-ertheless more than 10% perfect events, most with topologi-cal changes to the network. The ratio of different events should also reflect, therefore, some topological constraints.

We find 906 conserved events, i.e., events describing the diffusion of defects. Such defects are almost exclusively dan-gling bonds, on both Si (E

centers兲 and O 共nonbridging oxygens兲, although a few highly energetic floating bonds on O are also present. We see no sign of point defects, which would show themselves by a strong spatial correlation be-tween dangling or floating bonds. Events describe over-whelmingly single-dangling-bond diffusion mechanisms. The simplest of these is a jump of a dangling bond from one atom to its neighbor, an example of which is given in Fig. 1. More complex events are also seen, involving jumps to the second or third neighbor, or local rearrangements along a loop. All these mechanisms have relatively well defined bar-riers and asymmetries. For instance, a comparison of near-neighbor dangling bond diffusion involving different topo-logical rearrangements shows that the average cost of creating a 3-fold ring in silica is 1.5⫾0.2 eV.

More than 80 percent of the events produced involve the creation or the annihilation of coordination defects, with a wide spectrum of energies and configurations. Events with a low barrier and asymmetry, the ones determining the dynam-ics, are often topologically simple, like the annihilation of one or two pairs of dangling bonds or their creation. In ef-fect, the creation 共or annihilation兲 of a pair of defect costs

共saves兲 much less energy than would be naively thought by

simply considering the breaking of a bond in a crystal or a molecule: the elastic energy stored in the network will often counter the bonding energy. Contrary to what is found in crystalline silica, the creation of a defect in the glass can have an activation energy and asymmetry that is comparable

to those associated with their diffusion. For example, creat-ing a pair of danglcreat-ing bonds in order to remove a 3-fold rcreat-ing costs only about 0.4 eV, much less than what would be ex-pected in an unstrained environment.

V. CONCLUSIONS

The above results provide the following picture regard-ing relaxation and diffusion in the Nakano model of g-SiO2. Mechanisms responsible for relaxation and diffusion in g-SiO2are the creation, diffusion and annihilation of coordina-tion defects, and can require the collective displacement of hundreds of atoms. The types of defects that dominate the dynamics are dangling bonds, either attached to a Si atom (E

centers兲, or to an O atom 共nonbridging oxygens兲; a pair of these defects can easily be created and annealed, with an activation energy that is often similar to what is required for the diffusion of these defects. Moreover, all these mecha-nisms involve O and Si with almost equal weight, indicating that the two species should diffuse with roughly the same activation barrier.

Further research is necessary to determine whether all these results also hold for real g-SiO2; in particular, the dominant role for coordination defects in the dynamics might be less prominent in real g-SiO2, where the defect density is lower by a few orders of magnitude. It is, however, clear that the elementary mechanisms in g-SiO2are fundamentally dif-ferent from those found in amorphous silicon, where, in spite of a higher density of coordination defects, 10% of the gen-erated events involve no defect; this underlines the rich di-versity in the microscopic dynamics of network glasses.

ACKNOWLEDGMENTS

Part of the calculations were carried out on the CRAY T3E of HPAC. This work is supported in part by the ‘‘Stich-ting voor Fundamenteel Onderzoek der Materie 共FOM兲,’’ which is financially supported by the ‘‘Nederlandse Organi-satie voor Wetenschappelijk Onderzoek共NWO兲,’’ and by the National Science Foundation under Grant No. DMR 9805848.

1D. L. Griscom, Mater. Res. Soc. Symp. Proc. 61, 213共1986兲.

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共Klu-wer Academic, 1990兲, p. 601.

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M. A. Lamkin, F. L. Riley, and R. J. Fordham, J. Eur. Ceram. Soc. 10, 347

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could be associated with radiation damage.

8D. A. Litton and S. H. Garofalini, J. Non-Cryst. Solids 217, 250共1997兲. 9J. Horbach and W. Kob, Phys. Rev. B 60, 3169共1999兲.

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J. Sarnthein, A. Pasquarello, and R. Car, Phys. Rev. B 52, 12690共1995兲.

11P. Vashishta, R. K. Kalia, J. P. Rino, and I. Ebbsjo¨, Phys. Rev. B 41,

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14K. Vollmayr, W. Kob, and K. Binder, Phys. Rev. B 54, 15808共1996兲. 15H. Hosono, H. Kawazoe, and N. Matsunami, Phys. Rev. Lett. 80, 317

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B. Je´roˆme and J. Commandeur, Nature共London兲 386, 589 共1997兲.

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19G. Bre´bec, R. Se´guin, C. Sella, J. Bevenot, and J. C. Martin, Acta Metall.

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