Journal of Non-Crystalline Solids 357 (2011) 1564–1573

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Journal of Non-Crystalline Solids

j ourna l homepage: www.e lsev ie r.com/ locate / jnoncryso l

Silica molecular dynamic force fields—A practical assessment

Thomas F. Soules ⁎, George H. Gilmer, Manyalibo J. Matthews, James S. Stolken, Michael D. FeitNational Ignition Facility and Photon Sciences, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USA

⁎ Corresponding author.E-mail address: soules1@llnl.gov (T.F. Soules).

0022-3093/$ – see front matter. Published by Elsevierdoi:10.1016/j.jnoncrysol.2011.01.009

a b s t r a c t

a r t i c l e i n f o

Article history:Received 2 September 2010Received in revised form 2 January 2011Available online 4 February 2011

Keywords:Molecular dynamics;Silica glass structure;Silica glass properties;Silica force fields

The purpose of this paper is to compare simple and efficient pair-wise force fields for silica glass and assesstheir applicability for use in large scale molecular dynamic (MD) simulations of laser damage mitigation. Anumber of pair-wise force fields have been shown to give the random tetrahedral network of silica glass.Further, potentials obtained by fitting quantum mechanical results exhibit many of the properties of silicasuch as the low thermal expansion and densification. However with these potentials densification is observedat temperatures much higher than experiment. We also show that the thermodynamic melting point ofβ-crystobalite similarly occurs at temperatures much higher than observed experimentally. Softer empiricalpotentials can be constructed that do give liquid properties at experimental temperatures. However in allcases the activation energies for diffusion are lower than the experimental activation energies for viscosity.

B.V.

Published by Elsevier B.V.

1. Introduction

In the National Ignition Facility at Lawrence Livermore NationalLaboratory surface damage to silica optics caused by the high intensitiesof lasers used to induce fusion is currently being mitigated by pulsedCO2 laser treatments. The mitigating laser pulses can raise a small silicadamage site to temperatures of 2000–5000 K in 10−9 to 10−12 s. Thiscauses thedamaged site to “heal” or at least change its optical propertiesso that light traveling through does not cause further damage.Evaporation and ablation can also be induced depending on thepower and temporal shape of the pulses. The effects of these lasermitigation pulses are being modeled with finite-element tools [1].However fundamental properties of silica, such as, heat capacities,thermal conductivities, thermal expansion, etc. during the very rapidheating and cooling of silica to very high temperatures are frequentlyunknown and cannot be obtained independently experimentally. MDsimulations performed with simple potentials that faithfully reproduceexperimental data under less severe conditions could be a practical toolfor obtaining these properties and even simulating the mitigationexperiment [2]. Conversely the laser experimental results can probe thesilica glass and liquid under conditions not previously accessible.

Our initial objective was to simulate the laser mitigation using MDand the force fields developed by van Beest, Kramer and van Santen(BKS) [3] who fit self-consistent-field (SCF) Hartree–Fock calculationson small silica molecules. The BKS potential has impressively beenshown to reproduce the features of the complicated phase diagram ofSiO2 including qualitatively the regions of stability of the liquid,stishovite, coesite and β-quartz phases [4]. On the other hand a BKS

MD model of β-crystobalite homogenously melts at ~5000 K in theMD (NPT) heating versus the experimental thermodynamic meltingtemperature ~2000 K and boiling point~3000 K. The periodic bound-ary constraints of the MD simulation, its rapid heating rate and theabsence of any nucleating site are no doubt responsible for failure ofthe structure to melt at a lower temperature. However, Saiko-Voivodet al. [4] estimate a thermodynamic melting point for BKS β-quartz byfinding the point where the free energy is equal to that of the liquid atnear atmospheric temperature to be~3700 K. Further although theequilibrium BKS silica liquid shows a density maximum similar to thereal liquid silica [5] and other liquids having a random tetrahedralnetwork, this maximum occurs at ~4700 K versus the experimentalvitreous silica density increase at ~1820 K [6]. Hence it seems prudentto assess the behavior of vitreous silica simulations versus experimentwith different simple force fields before constructing large scale MDensembles that purport to model the behavior of real silica opticdamage sites under laser mitigation.

Because of its importance in many fields from geophysical scienceto photonics and its role as the archetypical strong glass former [7]there have been many MD simulations of silica with different forcefields. They fall roughly into three groups: The first simulations ofsilica by Woodcock, Cheeseman and Angell (WCA) [8] used a simpleBorn-Mayer (BM) potential with ionic charges of +4 for Si and−2 foroxygen. These authors showed that even with this crude ionicpotential the basic structure of silica glass as a random network ofcorner connected silica tetrahedrons is achieved in theMD simulation.The WCA potential was re-parameterized and the charges reduced byone of the authors [9] and Mitra et al. used the Pauling form for thetwo-body potential [10]. Later three-body terms were added andparameterized to reduce the spread in the OSiO bond angles andmoreimportantly to adjust the tetrahedral corner sharing SiOSi bond anglesso that these angles agree more closely with the experimental values

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deduced from the neutron scattering peaks [11–13]. The more recentgroup of pairwise force fields including BKS have been based on fits toquantum mechanical calculations. Carre et al. [14] re-parameterizedthe BKS potential (hereafter referred to as the CHIK potential) to fitthe structures they obtained using density functional Car-ParrinelloMD simulations. Still other potentials based on ab-initio calculationshave been proposed [15,16] including potentials that require the forcefield to be modified at each time step in the MD run [17]. The Tagneyand Scandolo (TS) [17] potential that evaluates the polarizability ofions at each time step based on the neighboring ion configuration hasbeen shown to give very good agreement with infrared spectra.Paramore et al. [18] attempted to map the TS force fields onto a newpairwise potential. These potentials have been compared [19].Demicralp et al. used a combination of a Morse potential andCoulombic interactions and allowed the Coulomb charges to varybased on a valence-averaged and electron affinity equation during therun [20]. A potential almost identical to that used by Demicalb wasused later by Takada et al. [21] with fixed charges to simulate silicaglass. Although effects requiring the force field to be updated at eachtime step must be important in the limit, for example, of vaporizationof SiO2 into SiO and O2 molecules, in keeping the limited aim of thisarticle here we focus on only simple pair-wise force fields that are notadjusted during the MD run. Our objective is limited to finding apractical and efficient perhaps empirical set of pair force field that isable to predict a specific set of properties over the temperature rangeand heating rate of the laser mitigation experiments.

This paper is organized as follows: After briefly describing themethods used, we graph four different simple two-body force fieldschosen both for their simplicity and efficiency in MD runs and forcovering a wide range of Si―O bond strengths. Then the results of NPTMD runs heating and cooling the canonical ensembles with each of thepotentials are presented. We estimate the thermodynamic meltingtemperature for simulated β-crystobalite ensembles by equilibrating theliquid and solid in a separate set of simulations. Fictive temperaturesduringcooling aredetermined fromthebreak in the slopeof theenthalpycurves and the glass transition and heat capacity behavior is discussed inthe final section. The density versus temperature and self diffusioncoefficients are also presented for each force field. The conclusion of thisstudy is that the potentials so far designed to fit ab-initio calculations,such as, BKS give melting and glass forming temperatures that are toohigh. A simple softer pairwise potential can fit the experimental meltingand cooling results for silica glass and liquid quite well. However, all thepotentials except those with extremely high melting temperatures andfictive temperatures give activation energies for self-diffusion that arelower than the experimental activation energy for viscosity.

2. Methods

2.1. MD methods

All the MD simulations reported here were carried out usingLAMMPS, a MD software code for parallel processing computersdeveloped at Scandia National Laboratory [22]. All runs were madeunder NPH conditions [23] with pressure maintained at ~1 atmosphere

Table 1Equations and parameters for the force fields compared in this work. The parameters in thlabeled Takada are used with Eq. (2). The C parameters in the Takada force field have unitsunits and scaled by 0.65 with the Si―Si short range repulsion set equal to 0.

qSi rcutoff Si―O

A (eV) B (Å−1) C (Å6 )

BKS3 2.4 11.0 18004 4.473 133.5CHIK14 1.910 11.0 27029 5.158 148.1Soules* 2.28 5.5 608.55 3.448 0Takada21 1.3 11.0 1.996 2.652 1.628

andwith heat added or subtracted using the Langevinmethod [24]withweak coupling to a random thermal bath simulating an NPT canonicalensemble. Inmost of the simulations, we startedwithMD ensembles of1000 Si atoms and 2000 O atoms initially at their positions in anidealized β-cristobalite configuration and used the Verlet algorithmwith a time step of 1 fs for the MD runs.

Arguably the most difficult problem in setting up these MDsimulations is how to treat the long-range Coulomb forces. LAMMPShas two built in options for long range Coulomb forces: a solution ofthe Ewald equation and a particle-particle particle mesh (PPPM)solver where the mesh is generated from a numerical solution ofPoisson's equation. However, both are computationally intensive anddifficult to apply to surfaces. After doingmany of our calculations withboth options, we adopted a screened potential proposed by one of theauthors [9] wherein the Coulomb force is replaced by the force field ofa charge at the center of a sphere containing uniform charge density ofopposite sign. The uniform charge density spheres will cancel in arandom system [9]. This can be checked by increasing the radius of thesphere. Carre et al. [25] recently compared two other methods forscreening the long range part of the BKS potential, the Wolfsummation and Yakawara equation, and showed that with a screeningdistance of 10.58 Å the dynamics and structure were essentially thesame as that obtained using the full Ewald potential. Using our simplescreening, we obtained the same structures and dynamical propertiesas with the Ewald and PPPM solutions in LAMMPSwhen using a cutoffof 11.0 Å. Even with this rather long range cutoff these calculationsran ~20 times faster than those using the Coulomb solvers included inLAMMPS, and with the cutoff used earlier [9] of 5.5 Å the calculationsran another factor of 20 times faster.

2.2. Force fields

The equations for the force fields investigated are shown inTable 1. They include three previously published force fields [3,14,21]and an empirical pairwise force field used by one of the authors [9].The empirical potential can be scaled without significantly disturbingthe calculated structure. We report results using a scaling factor of0.65 in order to investigate a second soft field. The BKS and CHIKequations include a 1/r7 term that must have another term added toprevent the force from diverging at small values of r and we used thescreening functions suggested [6,14].

Fr = AB exp −Brð Þ−6C = r7 + qiqje2

� �1−r2 = r2cutoff

� �ð1Þ

Fr = −2AB 1− exp B C−rð Þð Þ½ � exp B C−rð Þð Þ + qiqje2

� �1−r3 = r3cutoff

� �ð2Þ

The pairwise force field equations are graphed in Fig. 1. Themaximum attractive SiO forces span a wide range fromwhat might becalled the two strong attractive Si―O force fields (BKS and CHIK) totwo soft force fields (Takada and the scaled Soules' empirical forcefield). This is also reflected in the integral, the pair-wise potential

e rows labeled BKS, CHIK and Soules are used with Eq. (1). The parameters in the rowof Angstroms. Soules* is from Ref. [9], (Eq. (9)) page 51 of Ref. [9] converted to present

O―O Si―Si

A (eV) B (Å−1 ) C (Å6 ) A (eV) B (Å−1 ) C (Å6 )

1388. 2.76 175 0 0 0659.6 2.590 26.84 3150.1 2.852 626.8466.6 3.448 0 0 0 00.0233 1.373 3.791 0.0077 2.045 3.760

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Fig. 1. Graphs of the radial force fields being compared in this paper. The dashed lines indicate values at the nearest neighbor distances. The numbers labeled E* are activationenergies in eV/atom determined as difference in potential energies between the equilibrium positions and the point of inflection in the potential energy curves.

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energy curves. The graphs also show differences in the curvature ofthe Si―O force field going from broad to narrow depending onwhether an additional attractive term was added to the Coulombforce. On the figure are activation energies that are defined as thepotential energy differences between the pairwise SiO potentialenergy evaluated at the equilibrium distance in the simulations,~1.62 Å, and the potential energy at the inflection point or themaximum restoring force. When a SiO bond is stretched to beyondthis distance the SiOSi potential bifurcates and the oxygen atommidway between two silicon atoms will accelerate toward one or theother. We view this energy as a measure of the bond breakingactivation energy and it will be seen to roughly track properties, suchas, the activation energy for diffusion discussed later.

The O―O force field generally has an additional repulsive termadded to the Coulomb repulsion keeping the oxygen atoms wellseparated and maintaining the OSiO bond angle at the tetrahedralvalue, 109°. Several of the authors have also included an attractive OOforce field term to account for atomic polarization. However this isoverwhelmed by the repulsion.

The force field between two silicon ions is repulsive and in the BKSand modified Soules potential it is assumed to be entirely due to theCoulomb repulsion between the charges. Deleting additional attractiveand repulsive terms to the Si―Si force from other force fields has littleeffect on Si―Si force field especially at distances near the closest Si―Sidistance, ~ 3.15 Å.

3. Results

3.1. Melting

Fig. 2 compares the enthalpy versus temperature curves for each ofthe four force fields during heating of the MD canonical ensembles

starting with the idealized β-crytobalite structure for 3000 atoms. Theheating rates are ~4000 K/ns. The small steps or kinks in the curvesindicate a small heat of fusion in the enthalpy resulting fromhomogenous melting. The insert shows that, not surprisingly thetemperature at which homogenous melting occurs depends on theheating rate. Melting is delayed by the time required for nucleationand growth of liquid in the crystalline matrix so that these kinks occurat a higher temperature than the thermodynamic melting point.

BecauseMDheating rates are alwaysmuch faster than experimentalheating rates (except perhaps in some of our laser heating experiments)and because there are no nucleation sites in the MD simulation,thermodynamicmelting temperatureswere estimated using a differentapproach. An equilibrium between the crystal and liquid wasestablished without introducing a surface area driving force or a criticalsize seed by labeling the top half of the atoms in our MD cells “crystal”and those in bottom “glass.” The glass layer atoms are heated to abovethe homogenous melting temperature while the crystal layer atomswere kept at a low temperature using the Langevin coupling.While thiscannot be done in the laboratory it is easy to achieve in the MDsimulation. The only difference between atoms in the two regions ismagnitude of the Langevin interactions and the two regions remain inintimate contact. Because of the periodic boundary conditions in theMDsimulations this simulates a layered structure.NPT conditionswereusedwith the pressure set at 1 atmosphere in each direction. During thisinitial preparation of the system wherein the atoms in the glass grouparemelted very little exchange occurs between atoms in the crystal andglass layers.

The Langevin terms are then turned off allowing the temperatureof the glass and crystal atoms to equilibrate. By careful choice of initialtemperatures for both layers the atoms in the glass region remaindisordered while those in the crystal region while mostly orderedbecome disordered near the interface. A cross section of the

Fig. 2. Enthalpy versus temperature during heating of the β-crystobalite ensembles forthe four force fields. Kinks in the curves indicate the homogenous melting temperatureswhen the MD heating is carried out at ~4000K/ns. Insert shows effect of heating rate onthe melting temperature with BKS potential. Tm are estimated thermodynamic meltingtemperatures of the MD ensembles using the methods discussed in the text.

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configuration of such an ensemble before and after being allowed toequilibrate is shown in Fig. 3. Conversely the liquid region is not seento order in these simulations even at lower temperatures with longtime runs, 20 million time steps or 20 ns. This is not surprising since itis difficult to crystallize this strong glass former. The equilibriumtemperature was independent of the initial temperatures chosen solong as the system did not completely melt. Runs up to 10 ns(10 million time steps) were used.

Fig. 3. Cross section through a silica MD ensemble in which the atoms of the upper half are hheated above that temperature. Panel B shows a snapshot after allowing the two layers comregion indicating some melting of the crystal. Because the oxygen atoms show more disord

A clearer estimate of the melting temperature is achieved bytaking the split ensemble after melting the glass region rescaling thevelocities of all the atoms to a suitable temperature below the meltingpoint and then raising the temperature of the entire ensemble of glassand crystal atoms slowly using velocity rescaling. This procedure issimilar to the homogenous melting simulations but the crystal layersare in direct thermal contact with a liquid phase. Fig. 4 shows anexample using the CHIK force fields.

In our initial 3000 atom cubic box during the heating the largenumber of surface atoms relative to those in the bulk induced meltingat lower temperatures even though there was no surface areareduction driving force. The extent to which melting temperaturesare influenced by having very thin (b20 Å) alternate layers of liquidand crystal is not known. The phenomena observed may be morerelated to sintering which occurs at temperatures well belowmelting.A stable melting point independent of the simulation time and theMDbox size was achieved by increasing the simulation cell thickness thusdecreasing the fraction of atoms on the interfacial surface between theliquid and solid compared to the bulk. Taller prisms cells with adistance between layers of liquid and crystal in contact of 35.8 Å and71.6 Å containing 720 and1440 silicon atoms respectively with thecorresponding number of oxygen atoms melted at the sametemperatures.

Melting is observed when the MD β-crystobalite atoms near theinterface start to disorder and the interface between the crystal and theamorphous regions starts to move into the crystal. The CHIKβ-crystobalite ensemble is starting to melt while in equilibrium withthe amorphous phase between 3000 and 3100 K.

Another indicator of melting is a change in slope of the potentialenergy curve for the system. The potential energy of the system iscalculated by summing over the potential energy of all interactingpairs of atoms. Below themelting temperature, by the equipartition ofenergy the potential energy increases slowly and linearly withtemperature. As melting begins the slope of the potential energyincreases as surface atoms from the crystal group go into the liquidphase.

Still another indicator ofmelting is themean squared displacement(MSD) of atoms in the crystal group. The MSD is defined here as theaverage of the square of the displacement of each atom from itsposition at the beginning of the run. During the rapid linear ramp of

eld at a temperature below the homogenous melting temperature and the lower half ise to an equilibrium temperature. The liquid–crystal interface has moved into the crystaler we have displayed only the silicon atoms to identify crystalline order.

Fig. 4. Simulated MD melting of an ensemble of β-crystobalite (showing Si atoms as blue diamonds) in contact with a melted SiO2 glass (red diamonds) at the same temperatureusing the CHIK force field and a heating rate of 100 K/ns. The potential energy versus temperature of the system is the data with scatter. The smooth curve is the mean squareddisplacement (MSD) of the crystal atoms from the beginning of the run versus temperature. Melting is indicated by the change in slope in the MSD curve, movement of front ofdisorder in the lattice and a change in slope of the potential energy.

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Fig. 5. Enthalpy versus temperature during cooling of MD silica glass runs under NPHconditions at 1000 K/ns. Tf , the fictive temperatures, are the temperatures at which theliquid configuration is arrested on the time scale of the cooling indicated by theintersection of extrapolations of the solid and liquid curves.

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temperature the MSD of atoms in the crystal group increases due toincreasing vibration amplitude. When melting starts the slope of theMSD curve for the crystal group of atoms starts increasing as some ofthe atoms of the crystal group begin to diffuse with displacementscharacteristic of the liquid group and ultimately theMSD curve for thecrystal group of atoms starts to catch up to the MSD curve of atoms inthe liquid group (not shown the Figure). Assuming an Arrheniusbehavior of the diffusion coefficient with temperature, the theoreticalcurve for diffusion in the liquid during a linear ramp of thetemperature is a reasonablefit to the long time slope of thisMSD curve.

Also in Fig. 4, in order to increase the temperature as slow aspossible while maintaining a practical run time force fields weremodified by setting the long range cutoff for the Coulomb potential to5.5 Å. In comparison of the results of runs with cutoffs of 5.5 Å and11.0 Å the shorter cutoff gave the same simulated melting tempera-tures and had only a small effect on other dynamical properties, suchas, self-diffusion discussed later.

In the simulation shown the heating rate was 100 K/ns but slowerheating rates, as slow as 10 K/ns, did not change the temperature atwhichmelting was observed. These results suggest that we succeededin observing thermodynamic melting point of the CHIK β-crystobalite.The same approaches were used to determine the thermodynamicmelting temperatures for the different potentials indicated as Tms inFig. 2. The melting temperature determined in this way for BKS β-cristobalite at 1 atmosphere is 3600 K in good agreement with Saika-Voivod et al. [4] who determined the melting points of BKS quartz andcoesite at one atmosphere by calculating the temperature where thefree energy of the crystal and glass are equal. On the other hand, theMD melting point determined for the soft Takada force field isonly~10% higher than experiment and the melting point for thescaled Soules potential is actually less than the experimental meltingpoint, 1978 K.

3.2. Cooling and fictive temperatures

Fig. 5 shows the enthalpy versus temperature curves duringcooling at a rate of 1000 K/ns. The fictive temperatures indicated on

the figure are the temperatures at which a linear extrapolation of thesolid properties, in this case the enthalpy intersects an extrapolationof the liquid curve. In experiments and in simulations the fictivetemperatures decrease with a decrease in the cooling rate. The shift tolower temperatures of the fictive temperature with decreasing MD

Fig. 7. Sketch illustrating the ring structure of vitreous silica with corner connectedtetrahedra. The figure shows the low energy rocking modes of oxygen atomsperpendicular to the axis between silicon atoms.

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cooling rates in silica simulations was first shown by Soules [9] for asimple empirical potential and discussed in detail by Vollmayr et al.[6] for the case of the BKS silica potential and so will not be presentedhere.

The heat capacities corresponding to the enthalpy curves areshown in Fig. 6. Because of statistical noise in the enthalpy data heatcapacity curves were obtained by first fitting the low temperaturedata to a straight line. The difference between the line and theenthalpy data at higher temperatures was fit to a low orderpolynomial and the sum was differentiated to give the curves.

3.3. Density

A very interesting property of silica and some other randomtetrahedral network glasses that should be reproduced in a practicalMD model is the small thermal expansion of silica glass over much ofits solid glass temperature range and the density maximum in theliquid [26]. Both phenomena are easily understood qualitatively froma sketch (see Fig. 7) illustrating corner sharing randomly connectedsilica tetrahedra that form rings with predominantly six tetrahedraper ring. The rings are the ribs of an open fused cage-like structure.Vibrations of the oxygen atoms perpendicular to the Si―O―Si bondsconnecting tetrahedral are readily thermally excited and will rock thetetrahedra making up the cage ribs but will not increase the volume.When the vibrations are strong enough or when tetrahedra actuallybreak away from ribs in the liquid, the specific volume will collapseslightly to higher density amorphous structures.

Fig. 8 shows how the density varies as MD runs are cooled usingeach of the force fields investigated. The experimental density at roomtemperature is 2.2 g/cm3.

All force fields except the BKS give very close to the experimentaldensity at room temperature. Vollmayr et al. truncated and shifted theshort range terms in the BKS potential to correct the roomtemperature density [6]. Also while all the force fields result in asmall thermal expansion in the solid glass temperature range thevolume thermal expansion coefficients are still significantly largerthan the experimental value~1.6×10-6 K-1. Takada et al. argue thattheir potential also gives a very small 0.1% increase in density near

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Fig. 6. Heat capacity, Cp, curves obtained by taking the derivative of the enthalpy curvesin Fig. 4 as discussed in the text.

1700 K [21] but the thermal expansion of this force field is relativelylarge and the density maximum was not convincing in our simula-tions. On the other hand the other force fields show a distinctiveincrease in density, greater than what has been observed experimen-tally. We believe that this increase requires an attraction between thesilicon atoms and the next nearest neighbor oxygen atoms encour-aging the structure to collapse when vibrations disrupt the cornersharing tetrahedral rings. This next nearest neighbor distance asshown below is~4.0 Å. The Takada force field shows very little Si―Oattraction at this distance while the other force fields investigatedprobably have a longer range Si―O attraction at~4 Å that is toostrong.

3.4. Pair radial distribution functions (rdf's)

There are many similarities between the structures calculatedfrom all these force fields in spite of the very different strengths of thepairwise potentials (see Fig. 9). All of them give the same overallstructure for vitreous silica, namely, silica tetrahedra with a Si―Odistance of ~1.61 Å linked at corners and in rings of predominately 6members forming the ribs of an overall fused cage-like structure. This

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Fig. 8. Density during MD cooling for each of the force fields tested. The numbers in thelegend are volumetric thermal contraction coefficients per degree K in the solidtemperature range (300–1500 K).

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Fig. 9. The pair radial distribution functions (rdf's) at room temperature from MDcooling simulations. Also shown is the cumulative number of neighboring atoms as afunction of distance.

1570 T.F. Soules et al. / Journal of Non-Crystalline Solids 357 (2011) 1564–1573

shows that these features are mainly a function of relative atomicsizes.

There are some significant differences. For example the first Si―Opeak in our empirical potential is broader the first peak of the othermodels. The BKS model shows higher coordination numbers at nextnearest neighbor distances which is consistent with the higherdensity obtainedwith this potential. More detailed structural analysesof the MD rdf's are given by the respective authors. The calculatedrdf's are in good agreement with the x-ray diffraction [27] and theneutron diffraction correlation functions [28] although experimen-talists often notice differences [29] for example in bond angles, such asthe Si―O―Si bond angle that is deduced from experiment to be peakaround 140 degreeswhilemost of the simulations give a value of ~150degrees. Certainly effects, such as, repulsion by non-bonding electronpairs on the bridging oxygen atoms that would reduce this angle arereal and are not included in these simple radial pairwise force fields.

They can be included in three-body potentials but at the expense ofadding additional adjustable parameters or mapping out a force fieldin a many dimensional space.

3.5. Self-diffusion coefficients

The mean squared displacements (MSD's) of atoms have alreadybeen discussed in connection with melting. Diffusion constants areeasy to obtain from the slopes of the MSD curves starting at t=0 at aconstant temperatures after the initial displacements due to vibrations

(see Fig. 10a). Di⇒ 1= 6ð Þd →xi tð Þ−→xi 0ð Þ� �2

= dt. Fig. 10b shows that forall the potentials investigated the self-diffusion coefficients of siliconatoms track those of the oxygen atoms although in the case of ourempirical potential for which the Si―O bonding is much broader theoxygen atoms diffuse about twice as fast as the silicon atoms. The selfdiffusion coefficients fall in two groups one corresponding to the“strong” potentials and the “soft” potentials. Activation energies areroughly twice the activation energy estimates from Si―O potentials inFig. 1 suggesting that self diffusion of either Si or O atoms requires thebreaking of two bonds.

The diffusion coefficients for the BKS potential shown in Fig. 10b arein agreement with those determined by Shell et al. [30] and Saika-Voivod et al. [31]. Also like the results of Saiko-Voivod et al. our diffusionresults for all the potentials when plotted as lg (D) versus 1/T show acurvature at high temperatures, DN10−5, becoming Arrhenius at lowertemperatures,Db10−6 . Saiko-Voivod et al. account for this curvature inthe case of the BKS force fields by using the Adam-Gibbs expression forthe relaxation, D/T=μ0 exp(−A/TSC). They plotted lg(D/T)versus 1/TSCusing SC evaluated from their MD data (see Eq. (6) below).

Finally Bulk moduli for each force field was determined bycompressing the MD simulated silica glasses at 293 K. Only smallreductions in volume were made to avoid any coordination change.All force fields except the soft potential from Takada [21] give Bulkmoduli that are higher than experiment by ~30–40% (Table 2).

4. Discussion

4.1. Fictive temperatures

The high fictive temperatures shown in Fig. 5 are not surprising.Using the phenomenological non-linear Tool–Narayanaswamy [32]equations to calculate the fictive temperature with reasonable valuesfor the parameters (actually we used values selected by the authorsrather than those of Brunning et al. [33] that were obtained by fittingactual relaxation data but at much slower cooling rates), one predictsfictive temperatures of N2600 K at a cooling rate of 1000 K/ns [34].The equation for the fictive temperature is given by:

Tf = T−∫T

To

M T 0; Tf ; t

� �dT 0;

M = exp − t=τð Þβh i

;

τe τref exp½ H = RÞð−1= Tref +x= T+ 1−xð Þ= Tf Þ� �

H = R e73kK; x e0:7; β e1:0; τref = 120 s; Tref = 1400 K:

Ρ T0ð Þ−Ρ Tð Þ = αs T0−Tð Þ + αl−αsð Þ T0−Tfð Þ:

ð3Þ

Where τref is the structural relaxation time measured at areference temperature Tref. Relaxation curves have been measuredby monitoring the D2 Raman peak intensity in a furnace [35]. x is aparameter that weights the fictive temperature effect on theactivation energy and β is the Kohlrausch–Williams–Watts stretchedexponential decay parameter. P(T) is the value of a property thatdepends on the structural properties of the glass. The last equation

0

100

200

300

400

500

600

0 10 20 30 40 50 60 70 80 90 100

MSD

(r(

t)-r

(0))

2(Å

2 )

Time (ps)

6500 oK

5500 oK

3000 oK 3500 oK

4000 oK

4500 oK

(a)

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

x=1000 / T (oK)

D (

cm2 /s

) Soules Si D= 4.00e -31x

Soules O D= 3.91e -29x

BKSSi D= 0.33e -41x

BKSO D= 0.15e -37x

CHIKSi D= 0.31e -39x

CHIKO D= 0.16e -36x

Takada SiD= 0.04e -24x

Takada O D= 0.11e -27x

(b)

0.80.70.60.50.40.30.20.1

Fig. 10. The figure on the left is a sample of the mean square displacements of atoms at different temperatures, in this case Si atoms using the BKS potential. The MD ensemble wascooled to each temperature indicated in the curves and then run for 100 psec. The graphs on the right are diffusion constants obtained from the long time slopes of the mean squaredisplacements.

1571T.F. Soules et al. / Journal of Non-Crystalline Solids 357 (2011) 1564–1573

assumes that the derivatives of the property in the liquid l and solidstate s are constant over the temperature range of interest. If they arenot then this latter equation must be replaced by an integral over thetemperature range.

However, Vollmayr et al. [6] convincingly show that theseequations must be at least modified to describe the very hightemperatures and the very rapid cooling rates of MD simulations.Following Stillinger and Weber [36], steepest descent quenching ofMD runs with the BKS force fields show the properties of the inherentmechanically stable structures of the potential energy landscapewithout kinetic energy effects. The densities of the inherent structuresselected at different temperatures vary when the temperature fromwhich the quench is made is between 4840 K and 3200 K. However,the densities of inherent structures quenched from temperaturesN4840 K are the same as if they had been quenched from 4840 K. Itwas not possible to fall out of equilibrium with the liquid and into adifferent mechanically stable solid structure corresponding totemperatures greater than 4840 K due to the limited range ofmechanically stable energy minima. This would be equivalent in thephenomenological (Eq. (3)), above 4840 K, to making the relaxationfunction instantaneous or in another interpretation assuming that attemperatures above 4840 K the derivative of the property in theequilibrium liquid loses its configuration contributions. The latterinterpretation is supported by the shape of the heat capacity curvesshown in Fig. 6. Configuration contributions to the heat capacitydecline past the peak in heat capacity curve. Configuration contribu-tions to the heat capacity can be frozen in only from temperaturesnear the peak in the heat capacity curve down to lower temperatures.However this fictive temperature range is dependent on the choice ofthe potential. Useful potentials should be chosen so that differentinherent structures are selected in the range of say ~1300–2300 K, arange over which changes in density [37] and other fictivetemperature dependent properties, such as the Raman bands havebeen measured [38].

Table 2Bulk moduli, B=(1/V)(∂V/∂T), obtained by reducing slightly insteps the volume of the MD simulation box at room temperature.

Model Bulk modulus (GPa)

BKS 58.7CHIK 54.3Takada 43.5Soules 55.8Experiment ~39

4.2. Statistical mechanical model of liquid MD silica

The equilibrium liquid/glass graphs of energy and heat capacity(Figs. 5 and 6) can be understood assuming that the liquid samples a3 N dimensional potential energy hyper-surface where N is thenumber of atoms with many minima corresponding to mechanicallystable amorphous structures. In the classical limit,

H Tð Þ = Eharm Tð Þ + Eanh Tð Þ + eIS Tð Þ; ð4Þ

where Eharm(T)=3nkT is the classical mechanical contribution fromharmonic vibration about the inherent structure minima and Eanh(T)isthe anharmonic contribution to the vibration energy. Based on the factthat a the enthalpy follows the DuLong and Petit line very well to hightemperatures for our purposesweneglect the anharmonic contributions,whence

CP Tð Þ = ∂H Tð Þ=∂Tð Þ≈3nk + ∂eIS Tð Þ=∂Tð ÞP; ð5Þ

The configurational contribution to the entropy is

SC Tð Þ = SC T0ð Þ + ∫T

T0

1 = T 0� �∂eIS T 0� �

= ∂T 0� �PdT

0: ð6Þ

where T0 is a temperature at which SC(T) is a constant.In a previous publication [9] a statistical mechanical model was

suggested for the contributions to the energy and heat capacity frompopulating local amorphous minima. This model assumes that atopological lattice can be constructed and that populations of localdefect minima on this lattice are responsible for the inherentstructures. The configuration part of the partition function can thenbe written

ΖC Tð Þ = ∑igi exp −εi =kTð Þ

� �n

: ð7Þ

n is the number of sites thatmay contain defects. εi is theminimumpotential energy of the ith local defect and gi is its degeneracy. Thepartition function in this model is similar to the expression for thepartition function contributions for electronic states on a lattice,the difference being that the electronic excited states on each atom ormolecule are replaced by defect states on SiO2 lattice units in the glass.

1572 T.F. Soules et al. / Journal of Non-Crystalline Solids 357 (2011) 1564–1573

Using Eq. (7) yields a simple expression for the inherent structuralcontribution to the energy and heat capacity.

eIS Tð Þ≈kT2∂ ln ZC = ∂T = nkT∑xigi exp −xið Þ = ∑gi exp −xið Þ½ �;

CIS Tð Þ≈nk∑x2i gi exp −xið Þ

∑gi exp−xið Þ− ∑xigi exp −xið Þð Þ2

∑gi exp −xið Þð Þ2" #

; where xi = εi = kT

ð8Þ

Fig. 11 shows a graph illustrating the behavior of the heat capacityusing Eq. (8) with some representative values for local potentialenergy minima. The overall behavior is similar to the heat capacitycurves in Fig. 6. To fit the MD heat capacity curves would requireinherent structural minima at ~0.7–1 eV for the soft force fields whilethe strong force fields require minima at~1.3 eV . These are in therange of the energy required to break Si―O bonds shown in Fig. 1a. Asindicated in the previous publications [6,9], the defects correspondingto theseminima are three and five coordinated silicon atoms and non-bridging oxygen atoms. If the definition of a “fragile” glass is that thereis a significant increase in heat capacity of the liquid relative to thesolid glass then the MD glasses are fragile glasses at the highertemperatures. For the soft force fields there is an increase in heatcapacity relative to the solid of ~10% at 1800 K in agreement withexperiment [5]. Experimentally the glass transition occurs below theregion of increase in heat capacity and by this definition the glass isbehaving as a “strong” glass.

Assuming that topological changes in the ring structure areindependent, another partition function with the same form asEq. (8) might be constructed with low energy transitions representingdifferent topological ring structures in the glass and this partitionfunction would multiply the expression in Eq. (7). However, althoughtransitions involving the breaking and reforming of the ring structurehave a high activation energy giving rise to a high activation energyfor viscosity there is little net change in energy after the transition andhence these topological reconstructions responsible for glass flowcontribute little to the heat capacity.

0

1

2

3

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

7000

CIS

(nk

-gas

con

stan

t)

T(oK)

5 0.01 75 1.0

5 0.01 40 0.7 200 3.0

5 0.01 40 1.3

Fig. 11. Heat capacity curves generated using Eq. (8) and numbers in the legend thatindicating degeneracy and energy in eV. of levels. For example the green curve wasgenerated with two local defect energies: one with a degeneracy of 5 at 0.01 eV and thesecond a degeneracy of 75 at 1.0 eV.

Below the fictive temperature the term eIS(T) becomes a constanteIS(Tf) where Tf is the fictive temperature and as indicated in the heatcapacity of the MD simulations becomes ~3 n k.

4.3. Self-diffusion coefficients

The activation energies for self-diffusion for all four potentialsinvestigated here are much less the experimental activation energyfor 1/viscosity for silica, namely, ~73 kK (see Fig. 10 (b)). There issome experimental diffusion data for oxygen self-diffusion in vitreoussilica that supports a lower activation energy [39,40]. The activationenergies for diffusion are correlated with the melting temperatures sothat if a very strong potential force field is used which gives anactivation energy for self-diffusion in the range of the experimentalactivation energy for viscosity [41] the predicted melting tempera-tures and densification temperatures will become even much higherthan those found for the “strong” force fields. Efforts to decouple theactivation energy for diffusion frommelting temperatures by devisingpair-wise force fields for which the activation energy for bondbreaking shown in Fig. 1 was increased while maintaining areasonable cohesive enthalpy between 20 and 30 eV [42] were notsuccessful.

5. Concluding Remarks

While the strong potentials based on ab-initio calculations inparticular the BKS and CHIK potentials yield many of the properties ofthe liquid they do so at temperatures that are too high. The softer forcefields proposed by Takada et al. and Demicralp et al. fit theexperimental melting point of β-crytobalite and elastic constants ofsilica glass but fail to fit the low thermal expansion and densification.It is possible that a hybrid empirical force field could fit liquidexperimental properties at temperatures that are close to thosemeasured and fit the properties of the solid.

Our results also suggest that a strong glass former defined as onewith little change in heat capacity at the glass transition is one inwhich hopping from one minima to another results in no significantchange in potential energy. For example, topological changes in thesilica ring network by exchanging corner sharing tetrahedral connec-tions would have little effect on the enthalpy. On the other hand athigher temperatures even for these strong glasses high potentialenergy defect minima come into play and populating these defectstates does cause a change in potential and hence a structuralcontribution to the enthalpy and heat capacity.

Important practical results of this study include the fact thatrelatively short range empirical potentials that are very efficient inMD simulations can be used and appear to be just as accurate as thosehaving longer range Coulombic forces in reproducing certainexperimental results.

In a future publication we will refine the pairwise potentials inorder to find a single set that faithfully reproduces the experimentalproperties of liquid and solid vitreous silica and can be used to aidin our understanding of the properties silica under conditions oflaser mitigation. However, probably any pair-wise force field thatdoes reproduce these physical properties will likely yield self-diffusion coefficients with relatively low activation energies similarto those shown for the “soft” potentials in Fig. 10b. Newer forcefields that change with each time step as the type of bondingchanges, such as the ReaxxFF proposed for silica are being investigated[43].

Laser mitigation experiments will allow us to explore very fastheating and cooling rates and determine whether fragile liquidbehavior can be observed in real silica glass as well as otherinformation including the maximum fictive temperature possible.

1573T.F. Soules et al. / Journal of Non-Crystalline Solids 357 (2011) 1564–1573

Acknowledgments

This work was performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344. The authors also acknowledgeJeffrey Bude for his encouragement and support. The authors wouldalso like to thank Professor Bruce Berne for his potentials.

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