American Mineralogist. Volume 80, pages 417-431, 1995

Molecular dynamics simulations of liquids and glasses in the systemNaAISi04-Si02: Methodology and melt structures

DANIEL J. STEIN, FRANK J. SPERADepartment of Geological Sciences and Institute for Crustal Studies, University of California,

Santa Barbara, California 93106-9630, U.S.A.

ABSTRACT

This is the first of a two-part molecular dynamics (MD) study that examines the effectsof temperature, pressure, and composition on the structure and properties of ten compo-sitions in the system NaAISiO.-Si02. Results were obtained for collections of at least 1300atoms at temperatures between 2500 and 4500 K, pressures of 2-5 GPa, and simulationdurations on the order of 0.1 ns. Durations and numbers of particles are both about threetimes larger than for previous MD simulations on molten aluminosilicates. This studyaddresses experimental matters, including aspects of simulation methodology that affectthe accuracy of atom trajectories (hence computed properties). These include Nt (totalnumber of atoms in the MD box) and spatial resolution achieved in the interparticle forcecalculation. Static melt structures and their systematic variation with temperature andcomposition are also explored. In the second part (Stein and Spera, in preparation), themechanism of diffusion is studied in detail, and MD-computed results for a variety ofthermodynamic and transport properties are reported and related explicitly to melt struc-ture.

The present study employs a pairwise-additive form of the interatomic potential energyfunction, using the parameters of Dempsey and Kawamura (1984) with electrostatic in-teractions computed by the Ewald sum technique. Greater than 90% of Si and AI arefourfold-coordinated by 0 and >90% of 0 is in twofold coordination by the tetrahedral(T) cations, showing little change with composition and temperature. T -T coordination isfound to have a more distinct dependence on composition than does T -0 coordination.Structures and properties determined with this set of parameters differ from those com-puted using the parameters published by Angell et al. (1987), which give a less stronglyordered tetrahedral network and rates of diffusion for network-forming ions that are largerby about an order of magnitude.

Results indicate that transport properties computed from time-correlation functions(e.g., diffusivity and ionic conductivity) apparently become asymptotic for system sizes(Nt) greater than about 1000 particles. Relative to these values, properties computed withNt less than about 400-600 particles are generally overestimated by 10-100% and showincreased variance. Truncation of the Ewald sum (limiting the number of reciprocal spacevectors) introduces additional variance in computed properties. Melt structure (e.g., near-est-neighbor coordination statistics, static pair-correlation functions, and intertetrahedralbridging angle distributions) are less dependent upon system size than transport properties.The particular form and parameterization ofinterionic potential influence computed prop-erties much more than does system size or spatial resolution in the force calculation.

Substitution of (Na + AI) for Si is accompanied by a decrease in the mean intertetra-hedral bridging (T -0- T) angle related to increasing numbers of AI-O-AI bridges and pro-duces broader and less sharply peaked angle distributions. Radial density functions com-puted from the MD configurations reveal muted structure beyond the T -T 1 coordinationshell in comparison with published XRD analyses, although there is evidence of structuredue to T-02 and T-T2 correlation near 5 A, especially in the simulation of molten NaAlSiO..Structures produced by the MD model reflect the effects of using simplistic potential energyfunctions, as well as characteristically high experiment temperatures and rapid quenchrates required by the practical limitations of the MD method.

0003-004X/95/0506-0417$02.00 417

-

-_-0--

_ .~ _~___

---------..-- --,-- ------

INTRODUCI10N

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES418

The ability to extrapolate knowledge of the physicalproperties of silicate melts and magmas to extreme con-ditions gives insight into a variety of geophysical andgeochemical problems. Of particular concern to geo-chemists are the properties of glassy or molten silicatesrelevant to the generation, segregation, ascent, eruption,and crystallization of magma. These phenomena are im-portant in characterizing the origin and evolution of oce-anic and continental crust, as well as the parent bodies ofthe differentiated meteorites, and in understanding theorigin and fate of magma oceans, which probably devel-oped during accretion of the Earth and other terrestrialplanets (Wetherill, 1990). Solutions to problems in geo-dynamic transport typically involve analysis of appropri-ate forms of macroscopic conservation equations. Theseare unlikely to be meaningful without accurate knowledgeof the physical properties of silicate materials. Further-more, elucidating the relationship between the properties(both static and dynamic) of melts and glasses and theirstructure at the atomic level facilitates comprehension ofa wide variety of complex problems regarding advancedmaterials of technological importance.

Because of the increasing capability of computers, mo-lecular dynamics (MD) is becoming a cost-effective meansof investigating the thermodynamic and transport prop-erties of Earth materials at physical conditions that aredifficult or expensive to achieve in the laboratory. Anadvantage of MD simulations over conventional experi-ments is the ability to perform many experiments rapidlyso that a variety of interaction potentials or a range ofparameters for a given form of the potential may be con-sidered. At the same time, the limitations of the MDmethod should be noted. Specification of the true poten-tial for even simple melts has not been accomplished(Gibbs, 1982; Navrotsky et aI., 1985). However, empir-ical pairwise-additive potentials, which provide for elec-trostatic (Coulomb) forces and short-range repulsion, andwhich sometimes include dipole-dipole (van der Waals)interaction, appear to recover the properties of simplemolten alkali halides reasonably well (Woodcock, 1975;Sangster and Dixon, 1976). These types of simple central-force potentials similarly enable one to recover the short-range structure and other properties of molten silicatesunder a range of temperature and pressure conditions. Aprimary goal of our studies (Stein, 1993; Stein and Spera,in preparation) is to examine the practical capabilities ofMD in recovering material properties by making com-parisons between simulation and laboratory results. An-other goal is to use MD simulations to gain insight intothe mechanisms of diffusion, viscous flow, and ionic con-ductivity. Where appropriate, the MD results are dis-cussed in relation to the intrinsic limitations imposed byuse of a simple pairwise-additive potential and by theexperimental time scales currently practical for the MDmethod.

The first applications of MD to silicate compositions

reproduced the short-range structure of liquid Si02 andbinary metal oxide-silica systems (Woodcock et at, 1976;Soules, 1979; Mitra, 1982; Matsui et aI., 1982; Matsuiand Kawamum, 1984; Erikson and Hostetler, 1987). Sincethen, the scope of application to problems in physicalgeochemistry has expanded to include examination of theinfluence of pressure, temperature, and composition onstructure and transport properties (principally ionic self-diffusivity; see, for example, Soules and Busbey, 1981;Angell et aI., 1983, 1987; Kubicki and Lasaga, 1988, 1990,1991; Scamehorn and Angell, 1991). Subsequent effortshave documented the importance of system size (numberof particles, Nt) in the determination of high-pressure ion-ic self-diffusion coefficients (Rustad et aI., 1990) and thesensitivity of diffusivity and other properties to varia-tions in the interatomic potentials used in the simulations(Rustad et aI., 1991a). As the MD method has been de-veloped, its application in condensed matter physics hasemployed sophisticated techniques for determining tmns-port properties such as shear viscosity (Ogawa et aI., 1990;Wasserman et aI., 1993a, 1993b) and thermal conductiv-ity (Lindan and Gillan, 1991).

At tempemtures just above the liquidus, substantial struc-tural (volume) reorganization occurs in silicate liquidsover time scales on the order of fractions of a microsec-ond. Measurements of silicate diffusion at 1400-1500 °C(Lesher and Walker, 1986) suggest that network-formingcations may undergo displacements of approximately 1-2 nm over time scales on the order of 10-7 s. This averagemotion is an order of magnitude larger than the averagenetwork (T-0) bond length (-0.16 nm). Structural relax-ation mechanisms in near-liquidus silicates are found tooccur with frequencies at or just above the MHz range,as indicated by NMR studies of liquid alkali alumino-silicates (Liu et aI., 1987, 1988) and also by ultrasonictechniques (Kress et aI., 1988, 1989). These relaxationtimes are greater by a factor of 106 than the time scalesimposed by MD experiments. The high tempemtures usedin MD experiments are necessary to make relaxation timessmall in comparison with the duration of a typical MDexperiment. Diffusivities calculated from simulations re-sult in estimated viscosities on the order of 10-1 or 10-2Pa.s at a temperature of 3000 K, using the Stokes-Ein-stein or Eyring relations. Given unrelaxed shear modulifor typical silicate melts (-25 GPa, see Dingwell andWebb, 1989), a relaxation time of about 10-13 s (0.1 ps)characterizes the dynamics at simulation temperatures(-3500 K). Most of the simulations that this study re-ports were carried out for durations of 50-100 ps.

Because ofthe formidable computational requirementsof a l500-particle MD simulation, it is possible to studythe computer melt for only a few tenths of a nanosecondof real time. This necessitates simulated cooling rates onthe order of 10' °C or more per nanosecond (about abillion times greater than the most rapid labomtoryquench rates). Because viscosity increases extremely rap-idly near the glass transition temperature (Tg), the relax-ation time for velocity and stress autocorrelation rises

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

dramatically and becomes :» 10-9 s. T. is known to de-pend on cooling rate and increases with the rate ofquenching such that, in the glass transition region, mostproperties become history- or frequency-dependent (e.g.,thermal expansivity, cf. Scherer, 1986; Zarzycki, 1991).Investigations of silicates spanning the liquid-to-glasstransition in temperature have, by altering the coolinghistory, detected changes in the temperature at whichthermodynamic equilibrium is lost, arresting the materialin a glassy state (e.g., Moynihan et aI., 1976). Examina-tion of the phase-space trajectories ofMD systems makesit possible to evaluate the point and degree of departureof a simulation from ergodic behavior owing to the ex-treme quench rates typical of MD simulations (see, forexample, Barrat et aI., 1990; Barrat and Klein, 1991).This limitation should be kept in mind when determiningthe range of validity of MD-generated properties. Fornetwork silicate systems at high temperature, deviationsfrom Arrhenian behavior remain small, and the activa-tion energy observed for melt viscosity and ionic self-diffusion is nearly temperature independent. This facili-tates extrapolation of high-temperature MD results tolower temperature conditions for comparison with labo-ratory data. One purpose of this study is to examine indetail the breakdown of ergodic behavior with respect tothe mobility of Na, AI, Si, and 0 atoms in simulatedsodium aluminosilicate melts (Stein and Spera, in prep-aration).

Laboratory data are essential to test the ability of thesimulations to make valid predictions. The systemNaAlSi04-Si02 has been particularly well studied exper-imentally, especially regarding the relationship of struc-ture to static and dynamic properties. Studies includemeasurements of physical properties such as density(Riebling, 1966), viscosity (Riebling, 1966; Stein andSpera, 1993, and references therein), glass and crystallinephase spectroscopy (Sharma et aI., 1978; Taylor andBrown, 1979a, 1979b; McMillan et aI., 1982; Seifert etaI., 1982; Mysen et aI., 1983), phase equilibria (Greig andBarth, 1938; Schairer and Bowen, 1956), and thermo-chemical properties (Navrotsky et aI., 1982; Richet andBottinga, 1984). Additional studies include experimentalmeasurements of the viscosity of albite and jadeite meltsat high temperatures and pressures (Kushiro, 1976, 1978)and diffusivity of tetrahedral cations and 0 in jadeitemelt (Kushiro, 1983; Shimizu and Kushiro, 1984).

Several previous MD studies exist for sodium alumi-nosilicate compositions. Dempsey and Kawamura (1984)discussed short- and medium-range melt structure forsystems containing about 500 atoms and compared theirresults to existing X-ray diffraction data. Angell et al.(1983) and Scamehorn and Angell (1991) related struc-ture to patterns of diffusive behavior, particularly at ex-treme pressures (20-40 GPa). The studies of Angell andcoworkers focus on illustrating the concepts of strengthand fragility with respect to relaxation mechanisms inliquids (Angell, 1988, 1991); comparison between simu-lation and laboratory results is not emphasized. These

419

studies were performed with 400 particles (Nt) for dura-tions on the order of 10-11 s (10 ps) and do not reportthe sensitivity of computed properties to system size orthe choice of parameters for computing potential energy.A study of sodium aluminosilicates by Zirl and Garofal-ini (1990) made use of three-body forces for O-(Si,Al)-Otriplets, with Nt <400 atoms per simulation. The presentstudy shows that for properties that require the deter-mination of time-correlation functions (such as self-dif-fusivity), the roles of the spatial resolution (number ofreciprocal lattice vectors used in the Ewald sum calcula-tion) and the size of the MD cell (or total particle number,Nt) must be carefully evaluated.

There has been little work on the sensitivity of MD-computed properties to variations in the parameteriza-tion and form of the interionic potential. For this reason,our molecular dynamics study presents results on thestructure, thermodynamic, and transport properties inNaAISi04-Si02, using parameters for the interaction po-tentials different from those employed by Angell and co-workers and by Rustad et al. (1990, 1991a, 1991b) forSi02. It reports results on ten sodium alumino silicatecompositions at simulation temperatures between 2500and 4000 K and pressures in the range of 2-5 GPa. Prop-erties discussed include short-range structural parameters(radial distribution function or RDF, distribution of in-tra- and intertetrahedral angles, and coordination statis-tics for the network cations). Heat capacity, isothermalcompressibility, coefficients of self-diffusion ofNa, AI, Si,and 0 ions, and ionic conductivity are addressed in thesecond part of the study (Stein and Spera, in preparation).The second part also examines the mechanism of thetransition from liquid-like to glassy behavior observed inthe kinetics of network cations and O. At least forNaAISi04-rich compositions in this system, the range oftemperature studied here spans the critical temperatureat which this transition from ergodic to nonergodic be-havior occurs.

EXPERIMENTAL MATTERS

Molecular dynamics technique

We performed equilibrium molecular dynamics simu-lations often compositions in the system NaAISi04-Si02in the microcanonical (NVE) ensemble. In contrast tolaboratory experiments, which are inevitably slightly off-composition, the common molar ratio Na/ Al = 1 is spec-ified exactly for each simulation. In the molten state at3-4 GPa pressure, all simulated compositions have struc-tures (see below) in which virtually all 0 ions occur asbridges between pairs of silicate or aluminate tetrahedra(i.e., 0 is twofold-coordinated by Si or AI). Densities sim-ilar to those measured for the respective glasses understandard conditions of temperature and pressure (in therange 2250-2350 kglm3) resulted in pressures around 3.5:t 1 GPa at the simulation temperatures (2500, 3000,3500, 4000, and 4500 K). Several simulations in the NPTensemble were also made in order to compute isothermalcompressibilities for comparison with laboratory results.

~----

TABLE 1. Born-Mayer parameters

A,(J) B, (m-1 x 10-9)

Pair DK84 SA91 DK84 SA91

Si-O 1.3402 x 10 15 3.1295 X 10-16 46.512 34.483

0-0 9.6024 x 10-17 1.7014 X 10-1. 27.778 34.483

AI-O 1.1251 x 10-15 3.0914 x 10-1. 45.455 34.483

Na-O 6.2917 x 10-1. 1.6683 x 10-1. 40.000 34.483

AI-Si 9.7093 x 10-1. 3.7906 x 10-1. 133.333 34.483

Na-Si 3.7202 x 10-12 2.2794 x 10". 95.238 34.483

Si-Si 4.7705 x 10-9 3.6839 x 10-1. 142.857 34.483

Na-AI 1.7710 x 10-12 2.3094 x 10.-1. 90.909 34.483

AI-AI 2.4213 x 10-10 3.8832 x 10-1. 125.000 34.483

Na-Na 1.1310 x 10-13 1.3353 X 10-1. 71.429 34.483

420 STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

Note: sources for parameters are as follows: DK84 = Dempsey andKawamura(1984); SA91 ~ ScamehornandAngell(1991).Forinteractionsinvolving Si, entries for SA91 correct an error in reporting "s.

by Scamehornand Angell (1991).

Three ofthe selected compositions are those of the in-terior invariant points in the low-pressure phase diagramof the system (cf. Greig and Barth, 1938; Schairer andBowen, 1956) and have NaAlSi04-Si02 ratios near 1:4,2:3, and 3:2 (labeled N20, N41, and N60, respectively,to indicate mole percent of NaAISi04 component). Oth-ers include compositions of the mineral phases albite,jadeite, and nepheline (labeled N33, N50, and NIOO, re-spectively). These six compositions were simulated po-lythermally and were the same as those studied in therheometric laboratory experiments of Stein and Spera(1993). Simulations were also obtained for compositionswith NaAlSi04-Si02 ratios of 8:7, 4:3, 5:3, and 2:1 (la-beled N53, N57, N63, and N66, respectively) at 4000 Konly. Most experiments were 40 ps in duration, althoughsome were extended to 70 or 80 ps, and systems con-tained 1300 ::!:2 particles, depending on stoichiometry.Limited results for simulations ofSi02 are included whereappropriate as a point of reference with respect to thenumerous MD studies available for that composition.

The computer program for the simulation was basedon FORTRAN routines referred to in Allen and Tildesley(1987) and modified and enhanced by Rustad et at. (1990,199Ia). Additional codes to perform correlation-ensem-ble statistical mechanics and visualization were devel-oped as part of the present study (see also Stein and Spera,in preparation). Simulations were performed on CRA YC90 and CRAY Y-MP systems and on IBM RS/6000-350 workstations. On a CRA Y C90, the version of thecomputer program used in this study required 0.3 CPU-second per time step for a 1300-particle simulation; onthe workstations, each time step required about 10 CPUseconds.

A pairwise-additive interparticle potential energy wascomputed according to

with full ionic charges qi and q}, separated by distance rij.Repulsion energy representing overlap of electronic chargedistribution at small particle separations is parameterized

according to the Born-Mayer form: Ai} and Bij are givenunique values for each ion pair in the simulation on thebasis of empirical ionic size and hardness parameters,which effectively determine the curvature of the potentialenergy function. Parameters Ai}and Bi}for the exponentialrepulsion term were taken from Dempsey and Kawamura(1984) and are reproduced in Table I, along with valuesused in previous MD studies of sodium aluminosilicates(Angell et aI., 1983, 1987; Scamehorn and Angell, 1991,denoted in the text, tables, and figures as SA91; the pa-rameterization of Dempsey and Kawamura, 1984, is de-noted as DK84). Note that in Table 1, calculation ofSA91pair interactions involving Si incorporate correction of atypographical error in reporting the Si size parameter <TSi

by Scamehorn and Angell (1991). Interactions for eachof the cations coordinated with 0 in appropriate sym-metry are compared graphically in Figure la-Ic for theparameterizations DK84 and SA91. Differences betweenthe two parameter sets are displayed in the sharpness ofthe repulsive barrier at short-range, as well as the locationof and curvature near the potential energy minimumwhere electrostatic attraction is significant.

Ewald's summation method was used to compute thelong-range Coulomb contribution to force and energy, withK = 5/L (the convergenceparameter describingthe widthof the Gaussian canceling distribution surrounding eachion) for cubic cell dimension L. The reciprocal space por-tion of the sum was limited to reciprocal lattice vectorsk = 27rn/D such that In 12.::S81. In addition to the trun-cation of the Ewald sum in k-space, a cutoff of 0.8 nmwas employed in the computation of short-range repul-sions. The velocity version of the Verlet algorithm (Allenand Tildesley, 1987) was used to recompute particle po-sitions and velocities after each time step, and periodicboundary conditions were implemented such that the pri-mary MD cell was surrounded by replicate images in simplecubic symmetry to minimize effects owing to the presenceof surfaces. Each simulation contained approximately1300 particles, with density near 2300 kg/m3, requiringa cell-edge dimension of about 2.6 nm. For comparison,in a simulation with Nt = 195 particles, L

""1.4 nm since

Nt"" L3.

Sensitivity to parameterization of interionic potential

It is instructive to consider the effects of different val-ues of the Born-Mayer repulsion parameters on the meltstructures and properties obtained in the simulations. Forthe purposes of this comparison, coordination statisticshave been obtained for NaAISi30s at 4000 K by neighborcounting within the distance of the first minimum of thepair-correlation function (see below). Computationscomparing the two parameter sets were performed with1300 particles and used over 900 reciprocal lattice vec-tors in the Ewald sum calculation. Considerable differencesbetween the two potentials appear in the coordinationvolumes of 0 about Si, 0 about AI, and of tetrahedralcations about O.

Simulations using SA91 give significantly larger near-neighbor coordination shells with higher average COOf-

50.0

0.0---:>.. '\'-" -50.0

" ~--- --100.0 (b)

400.0

350.0

--- 300.0:>~250.0

~200.0

150.0

100.01.0

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES 421

dination of Si and Al by 0 and of 0 by (Si + AI). TheDK84 parameters lead to a more nearly tetrahedral fluidwith about 94% of Si in fourfold-coordination by 0 and6% in fivefold-coordination. In contrast, the SA91 pa-rameters lead to about 70% [4JSiand the remaining 30%[5JSi(Fig. 2a). Consistent variations in Al and 0 coordi-

!lation are evident (Fig. 2b and 2c). Angell et al. (1983)and Scamehorn and Angell (1991) presented MD resultsfor three of the compositions investigated here (N33, N50,and N 100) using the SA91 parameters for the interionicpair potentials (see Table 1). SA91 incorporates a T-Opotential with softer short-range repulsions and a shal-lower attractive well than DK84 (Fig. la and Ib), gen-erating simulation pressures (at similar density) that wereslightly lower than those of the present study.

Coordination statistics presented by Angell and co-workers were obtained from constant-pressure simula-tions at high temperature (see Fig. 4 in Scamehorn andAngell, 1991). These simulations display greater 0 co-ordination for Al relative to Si at elevated temperaturesand prominent temperature variation in Al coordination.In the simulation of NaAlSi04 performed by Scamehornand Angell, significant numbers of [6JAIwere observed ina 300 K quenched configuration with a reported fictivetemperature of 4000 K (Fig. 5 of that study). Differencesbetween Si and Al coordination are not so pronouncedin our constant- volume simulations using SA91. Spectro-scopic experiments (Stebbins and Sykes, 1990) have de-tected only very limited concentrations of Si and Al withcoordination greater than four in high-pressure alumi-nosilicate melts (i.e., in glasses quenched from these melts).The populations of anomalously coordinated 0 generatedfrom the SA91 parameterization suggest differences inthermal expansivity between the two parameterizationswithin the low- to moderate-pressure conditions of thesimulations.

The differences between the respective structural re-sults of the two parameter sets are matched by significantdifferences in particle mobility. Although we have notmade an extensive study of this matter, results are givenfor the motion of 0 in molten NaAlSi,Os at 4000 K (Fig.2d). 0 mean square displacement (MSD) is approxi-mately 10 times greater with SA91 than in results ob-tained from DK84. These variations in structural anddynamic properties are the direct outcome of differencesamong various features of the potential energy surface(Fig. la and Ib). The potential well of the DK84 param-eterization is deeper for silicate and aluminate clusterscompared with that for the SA91 values. Additionally, ata given Si-O separation within the distance of the mini-mum, DK84 is considerably steeper and hence morestrongly repulsive. Both of these aspects lead to a morecaged behavior of network ions at a given temperatureand pressure, resulting in decreased network mobility.

Effects of system size and spatial resolution ofEwald sum calculation

A brief study was made of the variation of computedDroperties with system size (L) or total number of parti-

50.0

0.0 (a) 1-- - D K841

- SA91---:>~ -50.0

~-100.0

-150.01.0 1.5 2.0 2.5 3.0

Si-Q Distance (A)3.5 4.0

1---DK841-SA91

AIO -S4

-150.01.0 2.0 2.5 3.0

AI-Q Distance (A)3.5 4.01.5

1---DK841- SA91

(c)NaO.ll

/I

1.5 2.0 2.5 3~0 3.5 4.0Na-Q Distance (A)

Fig. 1. (a) Potential energy, V, vs. Si-O distance forSiO:-(tetrahedral symmetry). Vertical axis is energy in electronvolts; horizontal axis is Si-O distance in Angstroms. Parametersobtained from sources denoted in legend: DK84 = Dempsey and

Kawamura (1984); SA91 = Scamehom and Angell (1991). (b)Same as a for AI-O interaction. (c) Same as a for Na-O (octa-hedral symmetry).

cles (Nt"" D) and the number of reciprocal lattice vectorsutilized in the indirect portion of the Ewald sum calcu-lation. Self-diffusivities for Na, AI, Si, and 0 and a typicalstatic (thermodynamic) property, the isochoric heat ca-pacity, C" were considered. One result is that underre-solution in space (i.e., too small N" L, or In 12) leads tocalculated C, values that are systematically too high by10-30%. The effect of poor spatial resolution on self-diffusivities is similar: underresolved simulations tend to

-- ..----

422

1.0

0.8I:

.E 0.6....<:.ICIS

"'" 0.4r-

0.2

0.0

. DK84~SA91

1 2 3 4 5 6Coordination Number

-8-- DK84 - G- -SA9!15 1.5

~~~....10 1."=- QC)

-< (d)~!:I:) c:IQ /

Q!:I:) 5"JY'

Network Mobility 0.5 ~!:I:)...- NaAISi308 :;

0 .-if4000 K 0

0 00 0.01 0.02 0.03 0.04

Time (ns)

-N = 1300 - - -N = 390 --N=390H__uN = 195

'"'I s 81 1ft'I S 81

'" 'I S 36 ~'IS 81

50

40

~30Q

'"::1! 200

10

00

1.050

(c)0.8 ~40

I: O-T Q30

'".E 0.6 NaAISi308 ::1!....<:.I

4000 K::;;: 20CIS

"'" 0.4r- 10~DK84

0.2 SA91 00

0.01 2 3 4 5 6

Coordination Number

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

Si -°NaAISi308

4000 K

(a)

Fig. 2. (a) Comparison of Si-O coordination for NaAlSi30.at 4000 K from simulations using DK84 and SA91 parameters.(b) Same as a but showing AI-O coordination. (c) Same as a butshowing coordination of 0 by nearest-neighbor tetrahedral (T)cations (Si and AI). (d) Effects of choice of potential parameterson 0 mean square displacement (MSD) for NaAlSi30. at 4000K. Parameters from sources indicated: DK84 = Dempsey andKawamura (1984), plotted on right axis; SA91 = Scamehorn andAngell (1991), plotted on left axis. Displacements show differ-

NaAISi308

4000 K

SA91 Parameters

---

(e)

0.01 0.02 0.03 0.04 0.05 0.06 0.07Time (os)

-N = 1300 - - -N = 3901ft 'I

,; 81 !It'I

,; 81- -N = 390 HuN = 195

Ift'IS36 ",'1,;8160

-.'-'./;- --

,-,.-

,-SA91 Parameters . -. ~

. . .~;.t..~ n. /.....

..-;-:';-"'-

t<'"'

~>""""';:---

./.._r.=-

0.01 0.02 0.03 0.04 0.05 0.06 0.07Time (os)

enees of about a factor of IO. Vertical scales are MSD in squared/lngstroms; horizontal scale is time in nanoseconds. (e) Plot ofo mean square displacement for simulations of NaAISi30. usingvarious system sizes (N) and reciprocal space cutoff (I n 12) forEwald summation; parameters of SA91 at 4000 K. Reductionof system size or spatial resolution results in overestimation ofdiffusivity and increased uncertainty in values of D (see text). (f)Same as e for Al ions. Note that variances are larger owing tosmaller proportions of Al ions relative to O.

generate estimates of D, that are too large by 10-100%.In sodium aluminosilicate melts the combination of lowmobility and abundance of Al ions results in relatively

large variations of Al MSD as a function of both Nt (orL) and the reciprocal space cutoff. By way of example,Figure 2e and 2f show MSD for 0 and AI in molten

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES 423

NaAlSi30s at 4000 K using the SA91 potential parame-ters (see Table I) in systems with 195, 390, and 1300particles and variations in the reciprocal space cutoff forthe Ewald sum.

Since D varies directly with MSD, one may note thatthe effect of using too few particles is that estimates ofMD-generated self-diffusivity may vary by as much as100%. For example, Al diffusivity for Nt = 390 and re-ciprocal space cutoff In 12 :S 36 or 81 varies from about1 x 10-5 to 1.5 X 10-5 cm2/s with the less well-resolvedcase exhibiting greater Al mobility. For more mobile ionsor those present in larger numbers, this effect appearssmaller but still significant.

Another way to study the effect of system size (L or Nt)on computed transport properties is by analysis of themagnitude of temperature and pressure fluctuations dur-ing the production portion of the MD experiment. Themagnitude of these fluctuations depends on Nt such thatlarger fluctuations occur in MD systems of smaller size.Given typical values of the activation energy (Ea) andactivation volume (V.) for self-diffusion, it is possible torecast the temperature and pressure fluctuations duringthe MD experiment into statistical uncertainties for com-puted D values. This analysis suggests that (]'D values foro self-diffusion in molten NaAlSi30s at 4000 K and 3GPa are 3-5 times larger in 195-particle simulations, incomparison with the Nt = 1300 experiments employed inthe current study.

The fundamental information from any MD simula-tion is the evolution of location (r;) and velocity (u;) ofeach particle in a specified time interval. Although short-range structure results are relatively insensitive to systemsize (see below), without adequate phase-space trajecto-ries, dynamic properties will not be accurately deter-mined given, of course, the limitations in the effective-ness of the potential energy functions Vi}' The MD studyof Rustad et ai. (1990) notes that for systems containingsmall numbers of particles, the particle MSD curves donot converge to a well-defined slope at increasingly longtimes. Thus the self-diffusion coefficient, which is basedon the limiting (infinite time) slope of the MSD curve,cannot be uniquely determined. In effect, the MSD curvesare one-dimensional representations of the trajectoriescomputed in the simulations and show that the trajectorycomputations in small systems are subject to increasinguncertainty with decreasing system size. What is mostevident in these comparisons is the increase in size offluctuations in the rates of ionic diffusivities as spatialresolution (L or reciprocal space cutoff) decreases. Thesefluctuations necessarily result in greater uncertainty incomputed diffusivities, especially when consideration isgiven to the larger fluctuations of experiment pressureand temperature that occur in systems with smaller tV..In summary, for a given potential, both system size (L ortV.) and the details of the Ewald summation (specifically,the total number of reciprocal lattice vectors utilized) havea demonstrable effect on the computed material proper-ties. MD primary cells of dimension 25 A corresponding

to Nt ~ 1300 with In 12 :S 81 recover properties that areat least independent of further increases in spatial reso-lution. Discrepancies in computed static and transportproperties of up to 100% may be incurred in simulationsthat lack sufficient spatial resolution.

Experimental cooling and equilibration schedule

Random particle positions and a Gaussian velocity dis-tribution appropriate to a temperature of 10000 K wereused to begin the computations. The initial steps of thesimulation were computed using the repulsion parame-ters Ai} and Hi}of SA91 and a time step of 0.1 fs (1 fs =10-15 s); when cell energies became negative, these werereplaced by the DK84 parameters and a time step of 1 fswas used throughout the remainder of the simulation.Velocity-scaling methods (Berendsen et aI., 1984) wereused to bring the system to 10000 K and to change thestate of the computer glasses when establishing the finalexperiment temperatures. Equilibration in the NVE en-semble for lOps at 10000 K was followed by velocity-scaled cooling to 3000 K at a rate of 200-300 Kips. After20 ps at constant volume and total (potential and kinetic)energy at 3000 K, velocity scaling was again employed toalter the resulting configurations to temperatures near2500, 3500, or 4000 K where additional NVE trajectorieswere computed. Total length ofNVE simulations for eachof the four temperatures was 40 ps, although some ex-periments were extended to 80 ps in order to developbetter statistics. Density was held constant for each com-position throughout the simulations; energies remainedconstant to within one part in 4000; net momenta in theNVE cells were found to be zero to within one part in10000. For the additional simulations performed at 4000K (N53, N57, N63, and N66), the NVE experiment du-rations were 70 ps. For the purposes of obtaining self-diffusion coefficients and structural information, particlepositions were saved every 0.05 ps once the target exper-iment temperature had been attained and stabilized.

REsULTS: SIMULATED MELT STRUCTURES

Once phase-space trajectories (the arrays of positionsand velocities saved every 0.05 ps) have been deter-mined, one has the opportunity to determine many cor-relation functions and hence to recover several static(thermodynamic) and transport properties. For example,application of the Voronoi tessellation (Bernal, 1959,1960; Bernal and Mason, 1960) and the fluctuation-dis-sipation theorem (Kubo, 1966) on MD-generated trajec-tories for the microcanonical (constant NVE) ensemblefor silica have been used to correlate high-pressure meltstructure with spectral and transport properties (Rustadet aI., 1991a, 1991b, 1991c, 1992).

The short-range structural details ofthe simulation re-sults may be summarized in terms of nearest-neighborcoordination statistics. The potential energy parametersdeveloped by Dempsey and Kawamura (1984) were con-strained by their adequacy in reproducing crystal struc-

m ___

424

---

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

15 15N33Si - 0

3000 KI-DK841

-- -- - SA91I

I 1010 I

I'I', ,

I',', ,

,, <55

- - -

oI

I o

o 1 2 3r (A)

4 5

15

10

N33AI - 03000 K

15

10

o o

5

o 1 83 4 5r (A)

6 72

Fig. 3. (a) Pair-correlation function, g(r), and running coor-dination number, n(r), for the ion pair Si-O in NaAISi30. at3000 K, comparing results from DK84 and SA91 parameteri-zations. The function n(r) was obtained by integration of g(r).Horizontal axis in Angstroms; vertical axis for g(r) indicatesprobability of finding 0 at various distances from Si relative tooverall distribution of 0 [as rbecomes large, function g(r) attainslimiting value of unity]; vertical scale for n(r) indicates runningaverage number of 0 at a specified distance from Si ions. (b)Same as a for pair AI-O, results using DK84 parameters.

tures of sodium aluminosilicate minerals. Soules (1979)and others have noted that structural features reproducedby the MD calculation demonstrate that it is not imper-ative to employ a model with directional or covalentbonding in order to recover the first-order feature of net-work silicate fluids-their essentially tetrahedral near-neighbor geometry.

Description of short- and intermediate-range structurein condensed matter simulations generally proceeds fromanalysis of the pair-correlation function g(r) computedaccording to standard procedures (see, for example, Allenand Tildesley, 1987). In the present study, features ofnear-neighbor coordination have been determined by nu-merical integration of g(r) to obtain n(r) or by countingneighbors within the distance of the first minimum ing(r). Figure 3a shows the relation between g(r) and n(r)for Si-O pairs in the composition NaAISi30. (N33) at3000 K and presents results from simulations using both

sets of interaction parameters (DK84 and SA91). Thefirst peak in g(r) is very sharp and, along with the obviouscorresponding well-defined plateau of n(r) at just over 4,indicates the high degree of order in the first 0 coordi-nation sphere around Si. Most 0 around Si is found with-in a restricted range of distances, and the first maximumdenotes that the most-probable Si-O separation is 0.163nm. DK84 gives a narrower Si-O distribution than doesSA91, generating distinctly smaller average coordinationnumbers; these evident differences are further addressedbelow. The AI-O coordination environment is less sharp-ly defined than that for Si-O. Figure 3b presents the g(r)and n(r) functions for the AI-O pair from simulationsusing DK84; the peak is broader than that for Si-O, andthe most probable AI-O separation is 0.171 nm. In thecoordination region, values for n(r) are slightly smallerfor Al-O than for Si-O. Note that g(r) shows little struc-ture beyond the range of 0.5 to 0.8 nm, which is a func-tion of the high-temperature state and simple form of thepotential energy function chosen for the simulations.

Coordination variations with temperatureand composition

The difference between the Si-O and AI-O coordinationenvironments is more apparent than are the variation ofeach respective cation environment with temperature andcomposition. This distinction between the Si-O and AI-Opairs represents a significant feature of the particle inter-action model (DK84) used for these pairs in the simula-tions, i.e., the Al-O bond is longer and weaker than theSi-O bond. It is this feature that establishes the compo-sitional patterns in various properties computed in thesimulations.

For the tetrahedral sites, little variation of n(r) withcomposition is seen at constant temperature, and evenless is seen with temperature at constant composition,although variations are slightly greater for the AI-O pairthan for Si-O. Figure 4a illustrates variation of n(r) forthe Si-O pair for six compositions at 4000 K. Figure 4bcompares n(r) for Si-O in NaAISiO. at all temperatures.Figure 4c illustrates the differences in n(r) for Si-O andAl-O pairs in NaAlSiO. at 2500 and 4000 K; note thegreater temperature variation of n(r) for AI-O.

Coordination statistics for the six compositions of Fig-ure 4a were also determined by counting the variousnearest-neighbor pairs in a sequence of 800 spatial con-figurations, each separated in time by 0.05 ps. Table 2summarizes variations with composition at 4000 K ofnearest-neighbor coordination statistics for Si-O, AI-O,and O-T. Spatial variation of average coordination num-ber (Fig. 4a-4c) is obtained from n(r) by integration ofg(r) to the distance of its first minimum. 0 is mostly infourfold coordination around Si, with the remainder(about 10%) in fivefold coordination, accounting for n(r)values slightly >4 in the 2-3 A range (Fig. 4a and 4b).o coordination is more variable for AI, with a slightlylower proportion in fourfold coordination than for Si-O,about 5% in fivefold, and the remainder in threefold coor-

4.5 TABLE2. Compositionalvariationof coordinationnumberSi - 0 CN N20 N33 N41 N50 N60 N100

4000 KDK84 Si-O

3 0.0 0.0 0.0 0.0 0.022 0.0

""'"4 0.936 0.929 0.934 0.933 0.923 0.908..

4.0 .... -N20 5 0.064 0.070 0.066 0.067 0.055 0.092'-';:: .......u N33 6 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001

- - - - - N41 AI-O- -

. N50 1 0.0 0.0 0.0 0.0 0.005 0.0(a) -N60 2 <0.001 0.0 0.0 0.0 0.005 0.0-----NI00 3 0.068 0.036 0.042 0.040 0.066 0.0244 0.888 0.912 0.908 0.917 0.887 0.918

3.5 5 0.044 0.051 0.050 0.043 0.035 0.0571.5 2.0 2.5 3.0 3.5 6 <0.001 <0.001 <0.001 <0.001 <0.001 <0.001

r (..\.) O-T1 0.011 0.015 0.021 0.018 0.052 0.0402 0.953 0.941 0.934 0.941 0.892 0.889

4.5 3 0.035 0.044 0.045 0.041 0.051 0.070

NaAISiO4

1/'4 0.0 <0.001 <0.001 <0.001 <0.001 <0.001

J' Note: coordination number (CN) = fraction of neighboring sites by pair,Si - 0 (.;. .

DK84 .--:.0"'-determined by counting to distance of first minimum in g(r) for all com-

~.~-_. positions simulated polythermally. O-T refers to coordination of 0 by com-:....J?-.~~~ (b) bined 5i + AI.Results collected at 4000 K.,..-.,

..d"~4.0 .:;}

;:: j .....2500K

:} ........-3000 K

j - -- -- 3500 Kj - 4000 K

3.51.5 2.0 2.5 3.0

r (..\.)

4.5NaAISiO 4 ..

'T-O .,1;DK84 ~..~.~..:

............ -- - .....-- (c),..-.,

",-:;;-.. 4.0'-' /1;:: - Si -0, 2500 K

, ........- Si -0, 4000 K

,:/ -----AI-0,25OOK- - AI- 0, 4000K

3.51.5 2.0 2.5 3.0 3.5

r (..\.)

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

Fig. 4. (a) Plot of running coordination number n(r) for thepair Si-O for selected compositions at 4000 K; note the verymuted compositional variation. Horizontal axis in Angstroms.The function n(r) was computed by integration of g(r). Simula-tions used DK84 parameterization. (b) Same as a but showingvariations of n(r) with temperature at constant composition(NaAISiO.). Very little temperature dependence is evident inplateau region. (c) Detail of n(r) for the pairs Si-O and AI-O inNaAISiO., comparing results at extremes of temperature. Notethat total range on horizontal axis is 2 A. At a given distancein the coordination shell, Al has lower average coordinationthan Si.

--.--

425

3.5

dination; no obvious trend in composition can be detected.The average 0 coordination number of AI is actuallyslightly lower than that for Si, resulting in systematicallyhigher average coordination of 0 by tetrahedral cationswith increasing abundance of Ai. Coordination of 0 byAl + Si is dominantly twofold, and the proportion oftwofold 0 decreases slightly (but systematically) with in-creasing Al/(Si + AI). As Al/(Si + AI) increases, the slightincrease in concentration of more highly-coordinated 0that results from mixing of units of NaAI02 with Si02varies consistently with increasing 0 mobility (Brawer,1985), decreasing isothermal melt viscosity (Stein andSpera, 1993), and consequent increasing melt fragility(Angell, 1988, 1991). In the constant volume simulations,there is little change in coordination with temperature inthe range 2500 K ~ T ~ 4000 K. A consequence of theuse of the DK84 parameters is a significantly lower con-centration of[3]0 compared to the SA91 simulations (Fig.2c). The fraction of 0 with only one tetrahedral cation(Si or AI) within the cutoff distance (non-bridging 0) nec-essarily increases with the increase in [3]0. 0 mobility(Fig. 2d) is a sensitive function of the concentrations ofanomalously coordinated 0 generated depending onchoice of parameter scheme.

Considerable order is seen in the T- T coordination en-vironment. Figure 5a shows g(r) for all T -nearest- T pairsfor NaAISi206 at 3000 K, including Si-Si, Si-A1, and Al-Ai. Compositional ordering of n(r) is similar to that forT-O pairs (Fig. 5b). Figure 5c and 5d show that AI co-ordination by T is distinctly greater than that of Si by Tacross the compositional range. This difference betweenAl and Si coordination is more apparent than in the caseof T-O pairs and shows how weakening of the networkordering is amplified in passing from short to interme-diate distance scales. This contributes to the explanationof compositional patterns in properties determined by the

426

6.0

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

5.0N50

3000 K

4.0 - - -Si - Si

-AI-Si

-- -- - AI - AI

,-,~ 3.0

2.0

1.0 ,,0.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0

r (A)

4.50

4.25

Si - T3000 K

-:- 4.00'-' ---+- N20

- & - N33--'Y--N41

x--- NSO--B--N60---0--- NI00

3.75(c)

3.503.25 3.50 3.75

r (A)

4.00 4.25

5.0

T-T3000 K

4.5

4.0---+- N20- & - N33- - 'Y- - N41---k-- NSO--B--N60---0--- NI00

3.53.25 3.50 3.75

r (A)

4.00 4.25

5.00

4.75

---+- N20- & - N33- - 'Y- - N41---x--- NSO--G--N60---0--- NI00,-,

6 4.50

4.25

AI - T3000 K

(d)

4.003.25 3.50 3.75

r (A)4.00 4.25

Fig. 5. (a) Pair-correlation function g(r) for Si-Si, AI-Si, and AI-AI pairs in composition NaAISi,O. (N50) at 3000 K; someordering ofT-T pairs persists out to 8 A. (b) The function n(r) for T-T pairs for all compositions at 3000 K. (c) Same as b for T-Tpairs centered on Si. Systematic compositional variation is more evident than for T-O pairs. (d) Same as c for T-T pairs centeredon AI. Note that T-T coordination of AI is less ordered than that for Si.

simulations, as well as in laboratory experiments. Indi-cations of systematic compositional variation in non-Ar-rhenian viscosity behavior in this system have previouslybeen noted (Richet and Bottinga, 1984; Stein and Spera,1993). Discussion of the concepts of strength and fragilityin viscous liquids (Angell, 1988, 1991) relates changes inrelaxation mechanisms to patterns in the density ofmin-ima on the potential energy surface. Figure 5c and 5d, inillustrating the significant differences of Si-T and AI- Tordering, suggest that increases in the density of minimaare contributed by AI sites in the network.

Conspicuously absent from the present series of sim-ulations is the double peak in g(r) for AI-AI that wasnoted by Dempsey and Kawamura (1984), as well as byScamehorn and Angell (1991), and attributed to irregularAI polyhedra sharing edges with AI-centered 0 tetrahe-dra. We are currently unable to account for this discrep-ancy, but it is not the result of using small numbers ofparticles, nor is it simply explained in terms of details ofthe Ewald sum calculation, because, the double peak doesnot appear in our results using Nt = 390 or in simulations

using lower reciprocal space resolution, regardless of pa-rameterization of repulsions.

Bridging angle distributions

Tetrahedral structural members may be treated moreor less as units, and changes in silicate structure and prop-erties are commonly expressed in terms ofSi-O-Si (moregenerally, of intertetrahedral or T -0- T) angle variationsor of the fraction of non bridging O. Ofadditional interestare properties of the nontetrahedral polyhedra, well doc-umented in crystalline silicates (e.g., Hazen and Finger,1982) and more recently in glasses as well (Richet et aI.,1993). The conventional definition ofa bridging 0 as theo connecting two neighboring (Si,AI)-centered tetrahedrais used. Features of the distribution of intertetrahedrala.ngles are a direct result of the packing geometry of thevarious (spherical) ions, modified by ongoing diffusiveevents. However, as has been noted in numerous otherstudies, the failure to include covalency or directionalityin the bonding model causes the average angle to be about8° too large in comparison with results from spectroscopy

40000

35000

30000

.., 25000'"=... 20000=cr...

15000...

"" 10000

5000

075

5000

4000

N 3000

2000

1000

0

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES 427

(Taylor and Brown, I 979b; Okuno and Marumo, 1982).Any melt property that can be related to T -0 bond

strength will be affected by changes in the intertetrahedralangle (and vice versa). Increasing T -0 bond length is as-sociated with decreasing T -0- T angle (Brown et aI., 1969).This relationship is particularly well illustrated in the re-sults of the calculation of potential energy surfaces as afunction of bond angle and bond distance (Gibbs et aI.,1981; Chakoumakos et aI., 1981). These studies showthat the minimum in potential energy follows a broadtrend in which increasing bond angle is correlated withdecreasing bond distance. For example, activation ener-gies of diffusion, conductivity, and viscous flow are cor-related with the strength ofT-O bonds (e.g., de long andBrown, 1980a, 1980b). Thus, a lowering of the intertetra-hedral angle is associated with enhanced network mobil-ity (increases in diffusivity and ionic conductivity anddecreases in viscosity). In crystalline materials of fixeddegree of polymerization, as are the compositions in thepresent study in which all 0 is nominally bridging, changesin T -0 bond length and T -0- T bond angle are interre-lated (Gibbs et aI., 1972; Brown et aI., 1969). As is shownby our results, this aspect of silicate melt structure is wellreproduced in the simulations.

The intratetrahedral 0- T -0 angle distributions for N33,N50, and N100 at 300 K are given in Figure 6a, and thedistributions of intertetrahedral bridging (T-0- T) anglesat 3000 K obtained for these same compositions are pre-sented in Figure 6b. At low temperature, the 0- T -0 angledistributions show little variation with composition. Theaverage O-T-O angle is 108.9° for NaAlSi,O. and Na-AISi206 and 109.0° for NaAlSiO. (the normal interiortetrahedral angle is 109.5°). The distributions generallyhave a standard deviation ofabout 8 or 9 ° and are slight-ly positive-skew. Statistics were collected on 400 config-urations, each separated by 0.05 ps. Some of the widththe T -0- T distributions should be attributed to the factthat ionic mobility precludes computation of angles onthe basis of average particle positions, as has been donein simulations of glass structure at lower temperatures.In the case of the 0- T -0 distribution, it is evident thatrapid quenching in MD simulations preserves the char-acteristics of higher temperature configurations.

The principal features of the bridging (T-0- T) anglestatistics revealed by the simulations consist of a generalbroadening of the peak of the skew-symmetric distribu-tions (increased standard deviation) with increasing All(Si + AI), a trend that is accompanied by a gradual in-crease in area of the low-angle tail of the distribution. Ata representative temperature of 3000 K, the average angleof the distributions decreases by over 5° as compositionchanges from N20 to N100, as summarized in Table 3.The relation of these patterns to the increasing content ofAI in the network cation positions will be discussed in asubsequent section. In Table 3, the statistics of T-O-Tangles are further broken down in terms of Si-O-Si, Si-O-Al, and AI-O-AI angles, and the standard deviationalso shows systematic compositional behavior, which is

--

(a) -N33N50- - -NI00

T = 300 K

90 105 120 135 150O-T-O Angle (deg)

(b)

T = 3000 K

nom',_ N33

-N50. . . . .NI00

50 70 90 110 130 150 170

T-O-T Angle (deg)

Fig. 6. (a) Distribution of intratetrahedral (0- T -0) angles forNaAlSi30. (N33), NaAlSi206 (N50), and NaAlSiO. (NIOO) at300 K. Average is 108.9° and standard deviation is 8°; geometricintratetrahedral angle is 109.5°. (b) Distribution ofintertetrahed-ral bridging (T-O-T) angles for composition NaAlSi30. (N33),NaAlSi206 (N50), and NaAlSiO. (NIOO) at 3000 K. Averagesand standard deviations of distributions are given in Table 3.

linked to the compositional variation in T -T neighbordistances (Fig. 5b). Dempsey and Kawamura (1984) not-ed from examination of the population of AI-O-Al anglesin their MD simulations that the AI-avoidance principleis not obeyed in these simulations. We confirm that thedistribution of AI-AI neighbors in the simulations is sta-tistically random.

In the simulations performed for the present study, thelow T -0- T angle population is correlated with larger T-Oand T -T distances, which in turn are correlated with Sior Al sites having larger average coordination numbers.Scamehorn and Angell (1991) presented results showingthat the small-angle portion of the distributions con-tained a much larger proportion of the angles; this is con-sistent with the differences in their results for averagecoordination number. However, in examining the g(r)results ofScamehorn and Angell (1991) and Angell et aI.(1983), it is not possible to determine if the correlationextends to T-O distance. It may be speculated that thesedifferences are related to the broader, shallower T-O po-tential well of SA91, permitting significantly greaterbreakdown of T -0 and, hence, T -T order than does the

---------

TABLE 3. Intertetrahedral bridging angles (3000 K)

Angle N Av. s.d.

SiO, Total 208 776 153.163 13.442N20 Total 211221 150.62 15.996

AI-O-AI 4076 128.02 24.653Si-O-AI 64113 144.97 17.6605i-O-5i 143 032 153.80 13.474

N33 Total 213404 149.35 16.470AI-O-AI 11842 129.22 23.115Si-O-AI 86 031 146.59 16.387Si-O-Si 115 531 153.46 13.500

N41 Total 214431 148.98 16.826AI-O-AI 15072 129.15 22.2495i-O-AI 95 852 146.33 16.364Si-O-Si 103 507 154.32 13.195

N50 Total 214085 148.39 17.038AI-O-AI 21 464 135.39 21.473Si-O-AI 104 872 146.34 16.794Si-O-SI 87749 154.02 13.402

N60 Total 209 711 146.81 17.439AI-O-AI 29 328 133.07 21.008Si-O-AI 104 206 146.29 16.280Si-O-Si 76 177 152.79 13.942

N100 Total 217 001 145.14 18.330AI-O-AI 50172 133.31 20.5875i-0-AI 120618 146.71 16.274Si-O-Si 46211 153.92 13.799

Note: angles computed on all T-O-T triplets for T-O distances <2.5 A;based on about 300 configurations each separated in time by 0.05 ps.

428 STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

potential of Dempsey and Kawamura (1984) (see Fig. 1).Comparison with laboratory data awaits the results frommore definitive spectroscopic studies on these composi-tions.

DISCUSSION

Recent comments in the literature (Wright, 1993) havenoted the inability of the simple potentials used in MDsimulations to reproduce the detailed structure of labo-ratory glasses much beyond the range of the first coor-dination shell. Furthermore, as noted above, the lack ofcovalence in the bonding model used in these simulationsgenerated average Si-O-Si angles somewhat larger thanthose found in actual glasses. Figure 7a-7c compares theradial distribution functions for NaAISi30s (N33), Na-AlSi206 (N50), and NaA1SiO. (N100) obtained from theMD results with a radial distribution analysis of Na-A1Si30s and NaA1Si206 glasses quenched from high pres-sure (Rochella and Brown, 1985) and NaAlSiO. glassquenched at low pressure (Aoki et aI., 1986). The relativepositions ofthe peaks may be compared, but the intensityscale should be regarded as arbitrary (simulated RDFshave been scaled to match the intensities of the T-O peakand are computed without reference to an actual electrondensity corresponding to the distribution of nuclei re-sulting from the MD simulation). The low intensityshoulder at 2.6 Awas interpreted by Rochella and Brownto be due to nearest-neighbor 0 atoms (those sharing asingle T site); this is the location of the first peak of g(r)for 0-0 in the simulations. In their experiments, consid-erable reduction in intensity of this peak at higher pres-sures was attributed to a more diverse set of 0-0 coor-dination environments. This peak is relatively prominent

in the simulation results and indicates that the potentialenergy functions preserve significant 0-0 correlation ata single T site. Little change with moderate pressure in-creases is expected in the 0- T-0 angle distribution (Fig.6a) in simulations using the DK84 parameters in com-parison with SA91. The absence of the prominent RDFpeaks beyond about 3.5 A indicates that such structureis not well defined in the simulations. A peak at 5.0 Awas attributed by Taylor and Brown (l979a, 1979b) totetrahedral cation separations on opposite sides of ringsconsisting of six tetrahedra but is not evident in the sim-ulation results. The high-temperature structural statecontributes to the damping of intermediate range order.

The simple form of ionic pair potential energy func-tions used in these simulations adequately reproduces theaverage nearest-neighbor coordination tetrahedral struc-ture of these fully polymerized melts. Coordination sta-tistics also show a significant population o[[S]Si (5-10%)as well as 13]0(4-7%) and 2-5% nonbriding 0 sites (Fig.

2a-2c). Spectroscopic studies have detected similar con-centrations of anomalously coordinated species in glassesquenched from melts at pressures up to 10 GPa, but onlyin significantly depolymerized alkali silicate composi-tions. Spectroscopic results on NaAISi30s (Stebbins andSykes, 1990) have measured barely detectable quantitiesof these anomalously coordinated species. The reason forthe discrepancy as well as the generally high degree ofmedium range disorder (e.g., in the angle distributions)is that our coordination statistics were collected in a dy-namic system capable of recording diffusive events (orsimilar vibrational states that do not relax by diffusion),whereas the high-pressure glasses were examined spectro-scopically at low temperatures in a static (i.e., frozen)state.

The diffusion rates of network forming species in thesesimulated melts are controlled by the motions of severalions around a defect site (cf. Brawer, 1985; Kubicki andLasaga, 1988) and are thus sensitive to intermediate-rangestructure. Aspects of these diffusive events are discussedin detail in the second part of this study. Insofar as thebonding model is pairwise-additive, the intent of the sim-ulation is not to mimic faithfully the energetics of siteexchange in diffusive events that involve the cooperativemotions of several particles. The details of such processesmust be left to future quantum mechanical studies ofclusters of atoms and MD simulations that incorporatemulti-body forces. In any event, there is a difference inthe ionicity of the AI-O and Si-O bonds, and changes inthe bonding character of the tetrahedral silicate structurewith temperature and pressure, which have been dis-cussed in other studies, may find compositional analogiesin this system. MD simulations of sodium aluminate cur-rently in progress may provide additional insight into thisproblem.

CONCLUSIONS

Results obtained in the present study demonstrate thatdynamic properties such as self-diffusivity determined by

MD simulation are strongly influenced by choice of pa- 6000rameterization and, to a lesser extent, by system size and

4000spatial resolution incorporated in details of the Ewaldsum calculation. Order of magnitude differences in trans- 2000port behavior result from different parameterization of~the interionic repulsions, while differences of 10-100% ~0in material properties (static and transport) may resultfrom changes in system size and spatial resolution. Use -2000of increasingly larger numbers of particles appears to lead

-4000to stabilization of transport values at a lower bound.Structural details are less sensitive to system size than 0 1 2 3 4 5 6 7properties determined from time-correlation functions. r (A)Progressive changes in structure are observed by substi-tuting Na + Al for Si in a continuous aluminosilicatenetwork, and the choice of parameterization of the inter-particle potential affects both structural features and par-ticle mobilities in an internally consistent manner. 4000

I

==~\\DI

In the results reported here, patterns in coordinationstatistics reveal weak compositional variation of the tet- 2000rahedral (near-neighbor) structures. Stronger patterns areseen in the second neighbor (e.g., T-T coordination, Fig. ~05, and intra- and intertetrahedral angle distributions, Fig.~6).T -T coordination number systematically increases in -2000

the compositional series from N20 to N100. In the same N50 (b)compositional series, the distributions ofintertetrahedral -4000

(T -0- T) angles broaden and become less sharply peaked,0 1 2 3 4 5 6 7with growth in the low-angle tail. The average angle de- r (A)

creases and the standard deviation increases systemati-cally. This is a function of a less ordered coordinationenvironment around the Al sites (related to the densityof local minima on the potential energy surface), which 4000increase in concentration as NaAI02 is added to molten

I

==~\\DI

silica. 2000A significant detail of the bonding model used in the

present study is the longer and weaker AI-O bond relative0to Si-O. As Na + Al replaces Si in the silicate network,

~average coordination of°

by tetrahedral cations and co-~NIOO

ordination ofSi and Al by their nearest tetrahedral cation -2000neighbors increase systematically. A similar composi- (c)tional trend holds true for the intertetrahedral bridging

-4000(T -0- T) angle distributions, which become broader and0 1 2 3 4 5 6 7

skewed toward smaller angles, related to the increasing r (A)population of AI-O-AI bridges. Structural results have alsobeen compared to available X-ray diffraction measure-ments of radial distribution functions obtained forNaAISi3Os, NaAISi206, and NaAISi04 compositions.Structural information obtained in this study also reflectsthe effects of the high simulation temperatures and therapid quench rates of the simulations. These conditionsare imposed by limitations in computational resources inorder to observe dynamic properties (e.g., self-diffusivity)sensitive to melt structural relaxation times. In a com-panion study (Stein and Spera, in preparation), these resultsare related to compositional and temperature variationsin thermodynamic and transport properties obtained inthe simulations. The range of composition, temperature,and simulation duration spanned in the present study issufficient to encompass the melt-to-glass transition withrespect to ion mobilities.

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES 429

Fig. 7. (a) Comparison of radial density functions computedfrom simulation results for NaA1Si,O. (N33) at 3000 K withthose measured in X-ray difITaction experiments of Rochellaand Brown (1985). Radial density is defined by D;/r) = 4...r2 [per)

- Po],where per) is the number density of species j with respectto species i evaluated at any r, and Po is the average numberdensity of species j with respect to species i. MD indicates sim-ulation result; XRD indicates result from X-ray difITaction mea-surements. D(r) from simulations computed from sums of in-dividual pair correlations and normalized to match T-O peak(near 1.6 A) intensity measured in X-ray diffraction (XRD) stud-ies. Peak near 3.2 A is due to T-T correlations; peak between 4and 4.5 A is attributed to T-second 0 and that around 5 A isattributed to T-second-nearest T neighbors (see Fig. 5a). (b) Same

as a for NaA1Si206 (N50). XRD results from Rochella and Brown(1985). (c) Same as a for NaA1Si04 (NlOO). XRD results from

Aoki et aI. (1986).

- ---, -.-

430

ACKNOWLEDGMENTS

STEIN AND SPERA: MOLECULAR DYNAMICS OF LIQUIDS AND GLASSES

We express appreciation to J.R. Rustad and D.A. Yuen for assistanceregarding implementation of the molecular dynamics program during theearly stages of this study. We are indebted to the Minnesota Supercom-puter Institute, NASA Ames Research Center, the San Diego Supercom-puter Center, the UCLA Office of Academic Computing, and the NationalEnergy Research Supercomputer Center at Lawrence Livermore NationalLaboratory for the use of computing facilities. This research was under-taken through financial support from the U.S. Department of Energy (DE-FG03-9IERI4211) and the National Science Foundation [EAR-9303906(Geochemistry)]. The authors thank J.D. Kubicki and an anonymous re-viewer for valuable assistance during the editing process. Valuable dis-cussions with C.A. Angell and P.F. McMillan greatly improved the qualityof the manuscript. This is contribution 0190-34CM of the Institute forCrustal Studies, University of California, Santa Barbara.

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