b. b. karki, d. bhattarai and l. stixrude

8/8/2019 B. B. Karki, D. Bhattarai And L. Stixrude

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gate its structural and dynamical properties. The organizationof the paper is as follows: Section II presents the computa-tional details. Section III presents the calculated results withanalysis of the equation of state, thermodynamic properties,geometric structure, and dynamical properties. Section IVdiscusses implications of our results and draws some conclu-sions.

II. METHODOLOGY

The computations have been performed using the rst-principles molecular dynamics methodVASP parallelcode47,48 as in our recent studies of liquid MgO andMgSiO3.14,49,50 The interatomic forces are computed at eachtime step from a fully self-consistent solution of the elec-tronic structure to the Born-Oppenheimer surface, within thenite temperature formulation of density functional theory.51The local density approximationLDA is used52 as this hasbeen found to yield superior agreement with the equation of state, structure, and elasticity of silicates and oxides,53 al-though we also perform a limited number of computations

with the generalized gradient approximationGGA54 forcomparison. Ultrasoft Si0.82 and 3s23 p4 and O 0.82 and 2s22 p4 pseudopotentials47 are used with a plane wavecutoff of 400 eV and gamma point sampling. The Pulaycorrection,55 which varies with compression from2.8 to 6.4 GPa over the volume range considered in thisstudy, is added to the calculated total pressure. To correct forthe well-known overbinding tendency of LDA, we followprevious work14,56,57 by adding a uniform correction to thepressure of 1.5 GPa, evaluated on the basis of comparisonbetween the LDA equation of state of quartz to experimentaldata.

Our FPMD simulations are based on the canonical NVT

ensemble in which the number of atoms in the periodicallyrepeated unit cell N , the volumeV , and the temperatureT are xed.58 The simulation box contains 24 SiO2 units72 atoms. A series of FPMD simulations of the liquid state

have been performed covering seven volumes:V / V X =1.0,0.9, 0.8, 0.7, 0.6, 0.5, and 0.4 along four isotherms, 3000,4000, 5000, and 6000 K. Here,V X =45.80 3 /SiO2 is thereference volume, similar to the experimental volume of theliquid at the ambient melting point.21 The correspondingpressure-temperature conditions span the freezing curve asestimated on the basis of shock-wave compression: 2000 Kat zero pressure to approximately 4500 K at 150 GPaRef.59 and so include the liquid stability eld and also the high-pressure supercooled regime. The simulation schedule is asfollows: The crystalline structure at each volume is rstmelted and equilibrated at 10 000 K for a period of 3 ps. Atime step of 1 fs is used. We then quench the system to adesired lower temperature. At this temperature, we run thesimulation long enough to reach the diffusional regime inwhich the mean-square displacementMSD of atoms varieslinearly with timesee Sec. III C. We conrm that the sys-tem at eachV -T condition simulated is in the liquid state byanalysis of the radial distribution function as well as themean-square displacement. For instance, atV / V X =1.0 thereference volume, simulation time periods of 3, 7, 20, and

58 ps, respectively, are required at 6000, 5000, 4000, a3000 K. Our simulation time periods far exceed previoFPMD simulation durations of 8 ps at 3500 K from Ref.43,2 ps at 3500 K from Ref.45, and 12.5 ps at 3500 K and22.5 ps at 3000 K from Ref.46. In our simulations, the meandisplacement of the atoms from initial to nal conguratiis 2 at 3000 K and more than 5 at higher temperatuUncertainties in the energy and pressure are computed usthe proper statistics via the blocking method.60 The conver-gence of the time-averaged properties and systematic energydrift are tested as in our previous study of liquid MgO.49 Wend that equilibrium propertiese.g., structure, pressure,energy converge much more quicklyi.e., within a fewpicoseconds at all temperaturesas in previous FPMDstudies4346 than dynamical propertiese.g., diffusion coef-cient. Previous MD studies have not predicted any signcant effects of nite size on equilibrium properties or strture of silica liquid. However, this is not true in the caseits dynamical properties, and we make a critical assessmof the potential uncertainties in our calculationssee the Sec.III C.

III. RESULTS AND ANALYSIS

A. Structural properties

The radial distribution functionRDF, g r , is computedto examine the structural properties of the simulated liqsystem. Figure1 shows the partialboth like and unlikeradial distribution functionsgSi-O r , gSi-Si r , andgO-O r atfour different compression and temperature conditiowhich are denoted as V1T1V =1.0V X and T =3000 K,V2T2 V =0.8V X and T =4000 K, V3T3 V =0.6V X and T =5000 K, and V4T4V =0.4V X andT =6000 K. TheseV -T points are chosen to lie uniformly 1000 K above the sil

freezing curve over the compression range considered hThe size of the uctuations after the rst peak in each cdecreases rapidly with distance and the RDF approacunity at larger distances, indicating the short-range order long-range disorder characteristic of the liquid state. The sitions of the peaks at low compression agree well with perimental data on silica melt and glass as does the asymmtry in the second Si-O peak23,61 see TableI . For all RDFs,with increasing temperature and compression, both the and second peaks decrease in amplitude and become broaFig.1 . The positions of both peaks tend to shift to smal

distances except the second peak in the Si-Si RDF, wheventually disappears at high compression and instead a nsecond peak appears at a larger distanceat condition V4T4.After the rst peak, the likeSi-Si and O-Oradial distribu-tion functions are very similar to each other at V1T1 aV2T2. This behavior has been seen in previous molecudynamics simulations38 and is remarkable because it is typical of symmetrical ionic liquids such as the alkali halidHowever, at higher compressionV3T3 and V4T4, the likedistributions begin to differ signicantly.

The structure of the liquid demonstrates that it compresvia mechanisms that are entirely different from those opetive in the crystalline polymorphs. We calculate the averaSi-O, O-O, and Si-Si distances asFig.2

KARKI, BHATTARAI, AND STIXRUDE PHYSICAL REVIEW B76 , 104205 2007

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r = 0

r minrg r dr

0

r ming r dr

, 1

wherer min is the position of the rst minimum in the corre-spondingg r . Remarkably, all three average distancesgradually increase in the liquid up to 30% compression. Thisis in contrast to the behavior of tetrahedrally coordinated

crystalline phases of silica, which are stable over a simrange of volumes, and which compress almost entirelyshortening Si-Si distances. AtV / V X =0.7, O-O distances be-gin to decrease rapidly, while Si-Si and Si-O distances ctinue to increase, before nally beginning to decreaseV / V X =0.6 andV / V X =0.5, respectively.

We calculate the average coordination numbersFig. 3via

C = 4 x 0

r minr 2g r dr , 2

where is the number density andx is the concentration N / N of species . The calculated Si-O, Si-Si, and O-Ocoordination numbers at V1T1 are, respectively, 4.02, 4.and 7.54, compared to the corresponding experimental vues of 3.9, 3.8, and 5.7 for silica meltTable I and corre-sponding values of 4, 4, and 6 for the ambient pressure crtalline polymorph. The Si-O coordination number increaon compression, as expected by analogy with the crystallpolymorphs, but much more slowly initially than in MgS3

0

2

4

6

8

10

12

R D F

V = V XT = 3000 K

Si-O

O-O Si-Si

4

6

8

R D F

V = 0.8V XT = 4000 K

0

2

0

2

4

6

R D F

V = 0.6V XT = 5000 K

0

2

4

0 2 4 6 8 10

R D F

r()

V = 0.4V XT = 6000 K

FIG. 1. Color online Partial radial distribution functionsRDFs for Si-O black, Si-Si blue , and O-Ored atomic pairs at

condition V1T1, V2T2, V3T3, and V4T4.

TABLE I. The calculated positions of the rst peak, the minimum after the rst peak, and the second peak for Si-O, O-O, and and the corresponding coordination numbers at fourV -T conditions. The experimental results for vitreousRef.23 and liquidRef.61 silicaat ambient pressure are shown.

Si-O O-O Si-Si

RP1 RM RP2 CN RP1 RM RP2 CN RP1 RM RP2 CN

V1T1 1.625 2.225 4.025 4.03 2.625 3.425 4.925 7.57 3.025 3.575 5.225 4V2T2 1.625 2.375 3.925 4.21 2.625 3.425 4.875 10.59 3.025 3.575 4.775 5V3T3 1.625 2.425 3.775 5.06 2.475 3.425 4.625 13.47 2.925 3.925 5.175 7V4T4 1.625 2.375 3.675 6.52 2.175 2.925 4.175 12.17 2.825 3.725 5.175 13Ref.23 1.62 4.15 2.6 4.95 3.08 5.18Ref.61 1.62 4.2 3.9 2.65 5 5.7 3.12 5 3.8

2.4

2.6

2.8

3.0

3.2

e ( )

Si-Si

O-O 3000 K4000 K

1.6

1.8

2.0

2.2

.

0.4 0.5 0.6 0.7 0.8 0.9 1

D i s t a n

V/V X

5000 K6000 KAveraged

Si-O

FIG. 2. Color onlineAverage Si-O, Si-Si, and O-O distanceas a function of compression and temperature. Lines representtemperature-averaged distances.

FIRST-PRINCIPLES SIMULATIONS OF LIQUID PHYSICAL REVIEW B76 , 104205 2007

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liquid Fig.3, top . In contrast to the silicate liquid, in whichthe coordination number depends nearly linearly on com-pression, in silica the coordination number remains verynearly 4 untilV / V X =0.8 where it begins to increase morerapidly. This behavior is consistent with the behavior of sodium-silicate glasses for which it is found that thepressure-induced Si-O coordination increase occurs morereadily at a lower pressurein less siliceous compositions.63The interpretation of these results is also consistent with ourndings, namely, that the Si-O coordination number in-creases initially at the expense of nonbridging oxygennomi-nally absent in the pure silica compositions of our study.62,64A similar behaviorinitial slow increasewith compressionof O-Si coordination in silica liquid is also predicted by ourcalculationsFig. 3, top . On the other hand, the O-OandSi-Si coordination number increases rapidly with compres-sion and is close to 12 at the highest compression of ourstudy, i.e., that of close-packed structuresFig. 3, bottom.

The Si-Si coordination number, however, continues to crease with compression even at the highest pressure of study, indicating that it is not accurate to characterize furtcompression of the liquid simply in terms of close-pack

structures.At each volume-temperature condition, a variety of loSi-O coordination environments existFig. 4 . We calculatethe relative proportions of different Si-O coordinatiowhich vary from threefold to eightfold. At low compressifourfold coordination is dominant with noticeable contritions from three- and vefold coordination. The proportiof these odd coordination environments decrease with creasing temperature in the simulationsFig.4, top , consis-tent with the experimental observation that they are esstially absent in the nearly perfectly tetrahedral rootemperature glass.23,25 With increasing compression, contributions from ve- and sixfold coordination increase, wherthose from three- and fourfold coordination decrease. Tpreponderance of vefold Si-O coordination environmentmidcompressionV / V X =0.6 is signicant for two reasons.First, it demonstrates that liquid structure is not merelydisordered version of the structure of the crystalline pomorphs, vefold coordination being extremely rare in crtalline silicates. Second, vefold coordination has been sgested as a particularly unstable transition state that enhandiffusion.41 At high compression, sixfold coordination dominant with its contribution increasing with decreastemperature at the expenses of ve- and sevenfold coordition Fig.4, bottom. Our results are consistent with the pre

4

5

6

7

n N u m

b e r

3000 K4000 K5000 K6000 KAveragedSolid phaseSilicate liquid

High P phases

Si-O

1

2

3

0.4 0.5 0.6 0.7 0.8 0.9 1.0

C o o r d

i n a t

i o

V/V X

Low P Phases

O-Si

9

11

13

15

n N u m

b e r

3000 K4000 K5000 K6000 KAveraged

3

5

7

0.4 0.5 0.6 0.7 0.8 0.9 1.0

C o o r d

i n a t

i o

V/V X

O-OSi-Si

(a)

(b)

FIG. 3. Color onlineThe calculated Si-O, O-Si, Si-Si, and O-Ocoordination numbers as a function of compression at four differenttemperatures. Solid lines represent the temperature-averaged values.Long dashed line represents crystalline silica phases. Short dashedline represents MgSiO3 liquid Ref.14 .

V 3000 K 4000 K 5000 K 6000 K Avg P at 3000 K1 4.03 4.04 4.02 3.95 4.01 0

0.9 4.05 4.19 4.06 4.11 4.1043 1.10.8 4.06 4.17 4.23 4.23 4.1724 70.7 4.28 4.43 4.54 4.54 4.4467 140.6 4.7 4.88 4.85 5.29 4.9298 240.5 5.67 5.7 5.66 5.91 5.7354 500.4 6.16 6.35 6.44 6.52 6.3695 150

1 0.9 0.8 0.7 0.6 0.5 0.43000 k

3 0 .0 31 77 8 0 .0 01 33 3 0 .0 09 08 3 0 .0 12 76 4 0 .0 02 40 3 0 0

4 0.926847 0.946583 0.926694 0.717847 0.414903 0.054181 0.0005835 0.040764 0.050889 0.063819 0.243944 0.4645 0.247861 0.0423196 0.000611 0.001194 0.000403 0.025347 0.117667 0.668056 0.7699867 0 0 0 0 0.000528 0.029583 0.1676538 0 0 0 0 0 0.000319 0.019042

4000k0.081793 0.00859 0.040708 0.020444 0.006861 0.000264 0

0.78548 0.85 0.753847 0.582069 0.315389 0.056056 0.0019277 7 7

0.0

0.2

0.4

0.6

0.8

1.0

D

i f f e r e n t S i - O C o o r d

i n a t

i o n

3 45 67 8

1.03 4 5

. . . . . . .0.003207 0.001106 0.004056 0.049569 0.186958 0.61 0.628672

0 0 0 0 0.009931 0.040361 0.2853650 0 0 0 0 0.001347 0.051172

5000 k0.140361 0.089167 0.078014 0.038042 0.015861 0.000625 00.678778 0.634236 0.620736 0.467375 0.3115 0.060875 0.0014030.163639 0.237361 0.273278 0.410681 0.48075 0.345222 0.0568610.007375 0.030486 0.023222 0.080417 0.181931 0.47 0.519139

0 0.000625 0.000236 0.003306 0.008806 0.106875 0.3474440 0 0 0 0.00025 0.008681 0.071264

6000 K0.220639 0.174681 0.127125 0.067083 0.013878 0.001125 00.539986 0.520278 0.5205 0.428736 0.167788 0.040042 0.0030420.191222 0.245319 0.300514 0.390361 0.414279 0.259375 0.0670420.019194 0.039653 0.038625 0.103208 0.327115 0.48 0.443375

0.0005 0.001 0.001889 0.005472 0.072228 0.188694 0.3862780 0 0 .0 00 01 4 0 0 .0 04 23 1 0 .0 30 41 7 0 .0 94 31 9

0.0

0.2

0.4

0.6

0.8

0.4 0.5 0.6 0.7 0.8 0.9 1.0

D i f f e r e n t S i - O C o o r d

i n a t

i o n

V/V X

6 7 8

FIG. 4. Color onlineThe calculated Si-O coordination numbers of different typesfrom three- to eightfold coordinationas afunction of compression at four different temperaturesthin solidlines. Thick solid lines and dashed lines represent the temperataveraged values of different contributions. Arrows indicate therections of increasing temperature.


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vious FPMD study that considered pressures up to 27 GPa.45For instance, atV / V X =0.8 and 3500 K, we nd proportionsof four- together with threefold, ve-, and sixfold coordi-nation of 87%, 13%, and 0%, respectively, compared to thecorresponding values from Ref.45 of 85%, 11%, and 0%.

Visualization of coordination environments65 reveals themechanism by which vefold coordination states are formedat low compressionFig.5 . The mechanism is a momentaryclosing of the Si-O-Si angle between two tetrahedra, whichbrings a fth oxygen within the coordination shell of one of the Si atoms. The product is a shared edge between a pen-trahedron and a tetrahedron and one three-coordinated oxy-gen. The Si-Si coordination number is unchanged by this

process as is the network connectivity: no new connectionsare formed between coordination environments. At highercompression, not only higher coordination numbers but alsoedge sharing is much more common. At V3T3, a wide vari-ety of coordination environments and shared elements arepresent, while a signicant amount of free volume remainsbetween the coordination polyhedra. Finally, at V4T4, thestructure begins to resemble a close-packing arrangement,with little free volume, and common face sharing as well asedge sharing and little corner sharing.

We now examine the distributions of the O-Si-O and Si-O-Si bond angles in liquid silicaFig.6 . The average anglesof O-Si-O distributions at V1T1 and V2T2 are 108.4 and107.6, respectively, in agreement with the previous FPMDstudy43 and close to the ideal tetrahedral angle109.47found with small deviations in the tetrahedral crystallinepolymorphs and silica glass at ambient conditions. With in-creasing temperature, the distribution becomes broader withits average value slightly shifting to smaller anglesFig. 6,top . On compression beyondV / V X =0.7, the distribution be-comes bimodal; at V4T4, the two modes are centered at 87and 137 and become narrower with decreasing temperature.The rst peak represents the characteristic octahedral angleof 90, whereas the second peak represents the angle madeby two opposite O atoms with the center Si atom. The dis-

tribution of the Si-O-Si bond angle shows a single bropeak at all conditionsFig.6, bottom. With increasing tem-perature, the distribution broadens, and the small-angle extends to smaller angles. On compression, the distributbecomes sharper and more asymmetric with the peak asmall-angle tail shifts to smaller angles. The average valof the distribution at V1T1 and V2T2 are 133.5 and 126compared to an angle of 144 for the glass at 300 K.23 Thedistribution extending below 120 down to angles as smal70 contains contributions from edge-sharing polyhedra.

conditions V3T3 and V4T4, the average values of the disbution are 114.9 and 110.8, as expected of edge- and fasharing polyhedra.

The initial compression mechanisms in silica liquid frV / V X =1.0 to 0.8 cannot be understood on the basis of tlocal structural features explored so far. The coordinatincrease is slight over this range, and nearest neighbor bodistances increase with compression. It has been suggesthat changes in intermediate range order in the form of rstatistics can account for the initial compression of silliquid based on simulations using a semiempirical foeld.66 In particular, it was argued that a decrease in thnumber of small rings and an increase in the characterisring size are of the primary compression mechanisms.order to test this idea, we compute the ring statistics in simulations. Our results show an initial decrease in the pportions of 3-rings with compressionFig.7, top . Moreover,the characteristic ring size calculated by using the approof Ref.67 increases on compressionFig. 7, bottom. Thenumber of small rings increases upon further compressV / V X 0.7 due to the increase in Si-O coordination num

ber, a feature absent from the previous simulations,66 whichused a semiempirical potential model that forced fourfcoordination to be maintained. We also nd a small proption of 2-rings, which are not included in the analysis of

FIG. 5. Color Visualization of the Si-O coordination environ-ment using the color-coded polyhedral representation at conditionV1T1 top left, V2T2 top right, V3T3 bottom left, and V4T4bottom right. The black, red, yellow, green, cyan, blue, magenta,

and white colors represent the local coordination numbers of 3, 4,5, 6, 7, 8, 9, and 10, respectively. The bond length is dened bythe rst minimum in the Si-O radial distribution function.

O - S

i - O A n g l e D i s t r i b u t i o n

V1T1

V2T2

V3T3

V4T4

40 60 80 100 120 140 160 180

S i - O

- S i A n g l e D i s t r i b u t i o n

Angle (degrees)

FIG. 6. Color onlineO-Si-O top and Si-O-Sibottom bond-angle distributions in liquid silica at conditions V1T1, V2T2, Vand V4T4.


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characteristic ring size. These are primarily associated withthe formation of vefold coordinated Si, by the mechanismdiscussed above, and so do not inuence the network con-nectivity. Our results are at odds with those of a previousFPMD study45 that found the number of 3-rings increasing

on compression. The origin of this discrepancy is unclear,but may be due to longer run duration of our simulations.

B. Equations of state

The calculated pressure-volume-temperature results aredescribed with the Mie-Grneisen equation of state,

P V ,T = P V ,T 0 + P th V ,T , 3whereP V ,T 0 is the pressure on the reference isotherm,T =T 0, and P th V ,T is the thermal pressure. As shown in theinset of Fig.8, the thermal pressure increases nearly linearlywith temperature at all volumes. The increase of thermalpressure on compression is initially gradual and then be-

comes more rapid atV / V X 0.7. The thermal pressure isrepresented by

P th V ,T = B V T T 0 , 4where the thermal pressure coefcientB may be representedover the range of our simulations by

B V = 10.4 9.5

1 + exp 25.2V /1.86, 5

in units of MPa K1 Fig. 8 . The reference isotherm, takento be T 0=3000 K, is described by the fth order Birch-

Murnaghan equation of state with zero pressure, volumbulk modulus, and rst and second pressure derivatiof the bulk modulus:V 0=45.8 0.2 3 /SiO2, K 0=5.2 1 GPa, K 0=22.5 3 , and K 0K 0=452.3 24 .The necessity of using such a high order t further illustrathe differences in compression mechanisms already dcussed between silica liquid and MgSiO3 liquid, which re-

quires only a third order t. The calculatedP -V isothermsinitially remain close and nearly parallel to each other compression, and then begin to diverge on further comprsion beyondV / V X =0.7 Fig. 9 . For comparison, we alsoperform simulations of the crystalline phases quartz, stisvite, and seifertitealpha-PbO2 structure at 3000 K.6872 Bycomparing the slope of the equation of state in pressuvolume space, we see that the liquid phase is more comprible than any of the solid phases when compared at the savolume. The density of the liquid exceeds that of quartza pressure of about 4 GPa, consistent with previoexperimental73 and theoretical analyses.45 At 3000 K, thedensity of the liquid also exceeds that of stishovite at ab90 GPa and seifertite at approximately 120 GPa. Thliquid-crystal density inversions are consistent with a recanalysis of the high-pressure melting curve of silica, whnds vanishing dT M / dP melting slopes at similarpressures.59 A complete analysis of the melting curve is byond the scope of this study.

Our results agree well with experimental determinatioof the equation of state of liquid SiO2. At low pressure, vari-ous experimental analyses1822 have yielded signicantly dif-ferent results, which span our predicted equation of stinset of Fig.9 . Our results are in excellent agreement wit

high-pressure high-temperature shock-wave data.59 Our cal-

0.0

0.5

1.0

1.5

2.0

2.5

N

u m

b e r o

f 3

- R i n g

3000 K

4000 K

6 02.366 1.447 1.532 0.756

1.878 1.975 2.013 1.816

2.793 2.075 3.627 4.303

3.493 4.521 4.841 7.713

5.712 6.883 7.895 11.134

4.0

4.5

5.0

5.5

6.0

0.6 0.7 0.8 0.9 1.0

C h a r a c t e r i s t

i c R i n g

S i z e

V/V X

FIG. 7. Ring statistics of liquid silica: The number of 3-ringtop and characteristic ring sizebottom at 3000 and 4000 K.

49.09784 0.90002459

48.5482 0.90003318

47.99856 0.90004474

47.44892 0.90006024

46.89928 0.9000811

46.34964 0.90010911

45.8 0.90014678

45.25036 0.90019739

44.70072 0.90026534

44.15108 0.90035671

43.60144 0.9004795

43.0518 0.90064448

42.50216 0.90086615

41.95252 0.901164

41.40288 0.9015642440.85324 0.90210205

40.3036 0.9028247

4

6

8

10

B ( M P a K - 1 )

-5

20

45

70

3000 4000 5000 6000

P ( G P a )

T (K)39.75396 0.9037956

39.20432 0.90509993

38.65468 0.90685219

38.10504 0.90920591

37.5554 0.91236693

37.00576 0.91661143

36.45612 0.92230928

35.90648 0.92995524

35.35684 0.94021058

34. 072 0.9539567

0

2

15 20 25 30 35 40 45 50

V(3)

FIG. 8. Volume dependence of thermal pressure coefcienBdened by Eq.5 in the text. The symbols represent the calculattemperature derivatives of the pressure at different volumes tained from linear t to the temperature variation of the total psure shown in the inset. The symbols in the inset represent

calculated results atV / V X =1.0 circles, 0.9 diamonds, 0.8 tri-angles, 0.7 squares, 0.6 asterisks, 0.5 crosses, and 0.4 pluses.Errors are within the size of the symbols. Lines represent the lits. For the sake of convenience, pressures atV / V X =0.5 and 0.4 areshifted by 15 and 90 GPai.e., downward, respectively.


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culated pressure for density of 4.71 g/cm3 at 8000 K is106.5 GPa, compared with the value from a previous FPMD

simulation of 110 GPa.74

Our GGA simulations agree wellwith previous GGA results45 inset of Fig.9 .We calculate the thermal Grneisen parameter, dened as

= V / C V P th / T , whereC V = E / T is the heat capacityat constant volume. As in the case of the thermal pressure, alinear equation is tted to the calculated energy-temperatureresults at each volumeinset of Fig.10 . Our results showthat C V and tend to decrease and increase, respectively,with compression in a nonlinear wayFig.10 . Two distinctregions appear: a low-compression region characterized bylittle variation with compression and large and small valuesof C V and , respectively, and a high-compression regioncharacterized by more rapid variations with volume and bysmallC

V and large . The heat capacity of the silica liquid

substantially exceeds the Dulong-Petit value. Our results forC V are larger than previous MD results,11 and both sets of theoretical results are larger than the experimental values.75The reason for the difference between theory and experimentin this case is not clear. It should be noted that our previousrst-principles molecular dynamics simulations of MgSiO3liquid are in excellent agreement with experimental heat ca-pacity data.14 It is possible that the heat capacity dependsstrongly on temperature at temperatures below the range of our study, although this seems unlikely based on the linearityof the energy-temperature relation.

C. Dynamical properties

We compute the self-diffusion coefcient as

D = limt

r t 2

6t , 6

where

r t 2 =1

N i=1

N

r i t r i 0 2 7

is the MSD andr i t is the position of theith atom at timet Figs. 11 and 12 . The partial MSD is then calculated b

60

100

140

180

P ( G P a )

3000 K4000 K5000 K6000 KQtStSeShock data

-5

5

15

25

26 30 34 38 42 46

P ( G P a )

V 3

LDAGGAPrev: GGAExpt 1Expt 2Expt 3eos-fit

-20

20

15 20 25 30 35 40 45 50V(3)

FIG. 9. Color onlineEquation of state of silica liquid: Linesare the Mie-Grneisen ts to the calculated pressure-volume resultsshown by open circles3000 K, squares 4000 K, triangles

5000 K, and diamonds6000 K. Errors are within the size of thesymbols. The dashed lines show the equation of state for the solidphases along 3000 K isothermQt, quartz; St, stishovite; Se, seif-ertite. The shock-compression datared asterisks at 4900 and5320 K and black asterisks at 5760 and 6500 Kfor silica melt arefrom Ref.59. Inset: Comparison of our results at 3500 K with pre-vious GGA results at 3500 KRef. 45 and with the experimentalresults Expt 1 from Ref.18, Expt 2 from Refs.18 and20, and Expt3 from Ref.22 at 1623 K. The uncertainties for liquid at 3500 Kare based on the averages taken between results at 3000 and4000 K.

0.4

0.6

0.8

1.0

1.2

-26

-24

-22

-20

3000 4000 5000 6000

E n e r g y ( e V )

T (K)

0.0

0.2

0.4 0.5 0.6 0.7 0.8 0.9 1.0V/V X

4.5

5.0

5.5

R

This study

Expt

Prev Calc

3.0

3.5

4.0

0.4 0.5 0.6 0.7 0.8 0.9 1.0

C V

/ n R

V/V X

(a)

(b)

FIG. 10. Calculated temperature-averaged Grneisen paramand heat capacity as a function of compression. Previous MDsults averaged between 3000 and 6000 K Ref. 11 and experi-mental dataRef. 59 for C V are shown. The inset shows the temperature variations of the total energy at different volumes:V / V X =1.0 circles, 0.9 diamonds, 0.8 triangles, 0.7 squares, 0.6asterisks, 0.5 crosses, and 0.4 pluses. Errors are within the size

of the symbols. Lines represent the linear ts. For the sake of venience, energies atV / V X =1.0, 0.9, 0.8, 0.7, 0.6, 0.5, and 0.4 arshifted by 1.5, 1.0, 0.5, 0.0, 0.5, 1.0, and 0.0 eV, respective


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averaging over atoms of a given species. We account for thetranslational symmetry of the system so that the MSD valueis not restricted by the size of the supercell. At high tempera-tures of 5000 and 6000 K, each MSD clearly shows twotemporal regimes. The rst is the ballistic regimefor shorttimes in which the atoms move without interacting stronglywith their neighbors and MSD is proportional tot 2. The sec-ond is the diffusion regimefor long timesin which MSD isproportional tot . However, at 3000 and 4000 K, an interme-diate regime appears where MSD increases slowly due to theso-called cage effect. The atoms are temporarily trapped inthe cages made by their neighbors. The duration of the cageregime increases rapidly with decreasing temperature span-ning 0.11 ps at 4000 K and 0.220 psat 3000 K. This isconsistent with a previous FPMD study.46 At 3000 K, thecurves show additional features; immediately after the ballis-tic regime, the curves show a small shoulder at around0.03 ps and a peak at around 0.2 ps. As suggestedpreviously,40 the rst feature can be associated with the com-plex local motion of the atoms in the open tetrahedral net-work and the second with the so-called boson peak related toa low-frequency vibrational mode.

The calculated diffusion coefcient at the reference vume V =V X as a function of the inverse temperature followthe Arrhenius relationFig.13 . Our analysis is made at con-stant volumealongV / V X =1.0 isochore, and the correctionsto constant pressure are within uncertainties. The calculaactivation energies are 3.438 and 3.2712 eV for siliconand oxygen, respectively. These numbers are relativsmall, compared to the experimental values for vitreosilica at lower temperatures of 6.0 and 4.7 eV, respectivefor Si Ref. 31 and O Ref. 32 , and a more recent experi-mental value of 4.74 eV for Si diffusion.33 The differencebetween our results and experiment may be explained bcrossover from strong Arrhenian behavior at the low teperatures of the experiments to fragile, non-Arrhenian behior within the temperature range of our simulations.40 Previ-ous MD studies based on semiempirical interatomic fomodels have produced a wide range of activation energiethe predicted diffusion constants differ by up to twdecades.40,41,77,78 Our results are slightly smaller than thos

1

10

100

D ( 2 )

6000 K5000 K

4000 K

3000 K

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1 10 100

M S

Time (ps)

Si atoms

1

10

100

2 )

6000 K5000 K

4000 K

3000 K

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1 10 100

M S D (

Time (ps)

O atoms

(a)

(b)

FIG. 11. Color onlineMean-square displacementMSD ver-sus time relationships for Si and O atoms atV / V X =1.0 for fourdifferent temperatures: 3000, 4000, 5000, and 6000 Kfrom lowerto upper curves.

1

10

100

( 2 )

V4T4

V3T3V1T1

V2T2

0.0001

0.001

0.01

.

0.001 0.01 0.1 1 10 100

M

S

Time (ps)

Si atoms

0.4V X, 3000K

1

10

100

2 )

V4T4

V3T3V1T1

V2T2

0.0001

0.001

0.01

.

0.001 0.01 0.1 1 10 100

M S D (

Time (ps)

O atoms

0.4V X, 3000K

(a)

(b)

FIG. 12. Color onlineMean-square displacementMSD ver-sus time relationships for Si and O atoms at four differcompression-temperature conditions: V1T1, V2T2, V3T3, V4T4. The lowermost curve in each plot is MSD at 3000 K forsmallest volumeV / V X =0.4 .


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rst-principles methods. Predictions of these anomalies havebeen based on semiempirical interatomic force models,including van BeestKramervan SantenBKS76 andWoodcock-Angell-CheesemanWAC.77 The prediction of aTMD has led several authors to draw analogies between thebehavior of silica and water.3 More recent studies haveshown that the dependence of TMD on volume and the mag-nitude of negative thermal expansivity are sensitive to the

assumed form of the atomic interactions.9,10

Our results show no evidence for a TMD. In particular,the pressure increases monotonically and nearly linearly withincreasing temperature along all isochores investigatedFig.8 . This is in contrast to the strongly concave dependence of pressure with temperature along isochores found in previousresults based on semiempirical potentials.9 For example, at adensity of 2.2 g/cm3, similar to ourV / V X =1.0, Ref.9 ndsthat the pressure decreases with increasing temperature up toa temperature of 9000 K, where the pressure is 10 GPa lessthan that at 3000 K according to the WAC model. The mag-nitude of this local pressure minimum lies well outside theuncertainties of our simulations. The BKS model producesmore subtle effectspressure at 5000 K is 4 GPa less thanthat at 2500 K, which can nevertheless also be excluded byour simulations.

We nd no evidence of spinodal instabilitiesvanishingbulk modulusover the range of our rst-principles simula-tions. The pressure monotonically increases with decreasingvolume along all isothermsFig. 9 . Spinodals have beenfound in previous simulations of liquid silica, behavior thathas been connected to the possibility of liquid-liquid phasetransformations.911 The WAC potential predicts two spin-odals on the 3000 K isotherm at 8 and 10 GPa, whereas theBKS potential predicts spinodals at positive pressure forT

6500 K andT 2000 K,9 outside the range of our results.We cannot rule out the presence of spinodal instabilities out-

side the range of our results. Indeed, extrapolation of ourequation of state t at 3000 K shows a spinodal at a pressureof 0.12 GPa andV / V X =1.05. Our results show no signs of spinodal instability at temperatures higher than the range in-vestigated here on the basis of the nearly linear dependenceof pressure on temperature.

The disagreement between pair potentials and rst-principles simulations calls into question the accuracy of thepair potentials. Even if two quite different pair potentialsyield qualitatively similar results, it does not necessarily fol-low that the qualitative trends are realistic. This may be be-cause of the limitations in the functional form of the pairpotentials: the well-known Born-Mayer form81 may not cap-ture the essential physics of bonding.42 Indeed, by includingpolarizability, Ref.82nds much better agreement with rst-principles simulations of liquid silica.

We nd that silica liquid is characterized by two distinctcompressional regimes: a low-pressure regimeV / V X 0.7in which the Si-O coordination number is nearly constantand a high-pressure regimeV / V X 0.7 in which the Si-Ocoordination number increases relatively rapidly. These tworegimes are characterized by distinct thermodynamic proper-ties. In the low-pressure regime, the liquid is much morecompressible, the heat capacity is higher, and the Grneisenparameter is smaller than in the high-pressure regime. The

behavior of silica liquid is thus quite distinct from MgS3liquid in which the Si-O coordination number and the Gneisen parameter depend essentially linearly on compress

Structural compression mechanisms in these two regimare fundamentally different. In the high-pressure regimcompression is primarily associated with an increase in Si-O coordination number. The low-pressure regime displa unique compression mechanism that is not seen in a

crystalline silicate. Whereas crystalline silicates compressther by decreasing nearest neighbor bond lengthse.g.,stishovite or the distancesangles between coordinationpolyhedrae.g., quartz or coesite, neither of these two com-pression mechanisms occur in the liquid. Instead, the liqcompresses by altering itsmedium range structure as mani-fested in the ring statistics.66 In particular, the number ofsmall rings decreases with compression, and the ring sincreases. These results are understood in terms of the retive pruning efciency of different sized rings as they formed in a Bethe lattice, and rationalized by compariswith a suite of crystalline silicate structures, which showwide variety of ring statistics and a range of framework dsities over a factor of 3, with the least dense framewoassociated with the largest number of small rings.67 The to-pological compression mechanism found in silica liquidpotentially very important for understanding the behaviopolymerized silicate liquids at elevated pressure. The reais that the topological mechanism is operative in the pressregime where the liquid is most compressible, and therefis responsible for a large fraction of the compression overpressure regime of Earths mantle.

An alternative interpretation of compression in the lopressure regime was offered by Traveet al. 45 They arguedthat increased network connectivity due to the presence osmall fraction of vefold coordinated Si is primarily respsible for compression in this regime. While this may also

a contributing factor, it is not clear how this mechanism clead to substantial amounts of compression. Unlike the rigrowth mechanism discussed above, it is not possible to crystalline models to quantify the relationship between compression mechanism and density. Moreover, we nd tformation of vefold coordination Si does not increase nwork connectivity, at least at low pressure.

The dynamical properties of silica liquid are also foundshow two distinct regimes, a low-compression anomaloregime showing a temperature-dependent diffusivity mamum and a high-compression regime in which the diffuscoefcient decreases with increasing pressure. We nd tthe diffusivity maximum is closely associated with the prence of vefold coordinated Si, in agreement with previstudies.41 The pentahedral coordination environments can considered as transition states for atomic diffusion. In pticular, the pressure at which vefold coordinated Si is mabundant coincides with the pressure of the diffusivity mamum at 4000 K. We conrm from rst principles the difsivity maximum in silica liquid, which was previously pdicted by MD simulations based on interatomic formodels.13,38,41,78,79 A clear pattern in our results has not beewidely discussed: the vanishing of the diffusivity maximwith increasing temperature. Indeed, the strong temperatdependence is at odds with a previous explanation of


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diffusivity maximum as being associated with the crystallinefourfold to sixfold coordination change.79 This model wouldpredict that the diffusivity maximum should either be inde-pendent of temperature or move to higher pressures withincreasing temperature, contrary to our ndings. Our resultsare consistent with a picture in which silica liquid behaveslike a strong liquid at low pressure and low temperature anda fragile liquid at high pressure and high temperature. Thisexplains the diffusivity maximum, its temperature depen-dence, and the difference between the high-temperature acti-

vation energy obtained from our simulations and the larglower-temperature activation energy found in experiment

ACKNOWLEDGMENTS

This work was supported by the NSF Grants No. EA0409074 B.B.K. and No. EAR-048182L.S. . Computingfacilities were provided by CCT at Louisiana State Univsity.

1C. A. Angell and H. Kanno, Science193 , 1121 1976.2F. X. Prielmeier, E. W. Lang, R. J. Speedy, and H.-D. Ldemann,

Phys. Rev. Lett.59 , 1128 1987.3P. H. Poole, M. Hemmati, and C. A. Angell, Phys. Rev. Lett.79 ,

2281 1997.4O. Mishima and H. E. Stanley, NatureLondon 396 , 329 1998.5M. Hemmati, C. T. Moynihan, and C. A. Angell, J. Chem. Phys.

115 , 6663 2001.6J. C. Phillips, J. Non-Cryst. Solids34 , 153 1979.7W. A. Crichton, M. Mezouar, T. Grande, S. Stolen, and A.

Grzechnik, NatureLondon 414 , 622 2001.8C. A. Angell, R. D. Bressel, M. Hemmati, E. J. Sare, and J. C.

Tuker, Phys. Chem. Chem. Phys.2, 1559 2000.9I. Saika-Voivod, F. Sciortino, and P. H. Poole, Phys. Rev. E63 ,

011202 2000.10I. Saika-Voivod, P. H. Poole, and F. Sciortino, NatureLondon

412 , 514 2001.11S. R. Ren, I. W. Hamley, P. I. C. Teixeira, and P. D. Olmsted,

Phys. Rev. E63 , 041503 2001.12S. Stolen, T. Grande, and H. B. Johnsen, Phys. Chem. Chem.

Phys. 4, 3396 2002.13R. Sharma, A. Mudi, and C. Chakravarty, J. Chem. Phys.125 ,044705 2006.

14L. Stixrude and B. B. Karki, Science310 , 297 2005.15Q. Williams and J. Garnero, Science273 , 1528 1996.16E. Stolper, D. Walker, B. H. Hager, and J. F. Hays, J. Geophys.

Res. 86 , 6261 1981.17G. H. Miller, E. M. Stolper, and T. J. Ahrens, J. Geophys. Res.

96 , 11849 1991.18G. A. Gaetani, P. D. Asimow, and E. M. Stolper, Geochim. Cos-

mochim. Acta62 , 2499 1998.19J. Zhang, R. C. Liebermann, T. Gasparik, and C. T. Herzberg, J.

Geophys. Res.98 , 19785 1993.20R. A. Lange and I. S. E. Carmichael, Geochim. Cosmochim. Acta

51 , 2931 1987.21V. C. Kress and I. S. E. Carmichael, Contrib. Mineral. Petrol.

108 , 82 1991.22P. Hudon, I. H. Jung, and D. R. Baker, Phys. Earth Planet. Inter.

130 , 159 2002.23R. L. Mozzi and B. E. Warren, J. Appl. Crystallogr.2, 164

1969.24R. J. Hemley, H. K. Mao, P. M. Bell, and B. O. Mysen, Phys.

Rev. Lett.57 , 747 1986.25D. L. Price and J. M. Carpenter, J. Non-Cryst. Solids92 , 153

1987.

26Q. Williams and R. Jeanloz, Science239 , 902 1988.27P. A. V. Johnson, A. C. Wright, and R. N. Sinclair, J. Non-Cry

Solids 58 , 109 1990.28X. Xue, J. F. Stebbins, M. Kanzaki, P. F. McMillan, and B. P

Am. Mineral.76 , 8 1991.29S. Susman, K. J. Volin, D. L. Price, M. Grimsditch, J. P. Rino,

K. Kalia, P. Vashishta, G. Gwanmesia, Y. Wang, and R. Liebermann, Phys. Rev. B43 , 1194 1991.

30C. Meade, R. J. Hemley, and H. K. Mao, Phys. Rev. Lett.69 ,1387 1992.

31G. Brebec, R. Sefuin, C. Sella, J. Bevenot, and J. C. Martin, AMetall. 28 , 327 1980.

32J. C. Mikkelsen, Appl. Phys. Lett.45 , 1187 1984.33D. Tsoukalas, C. Tsamis, and P. Normand, J. Appl. Phys.89 ,

7809 2006.34J. R. Rustad, D. A. Yuen, and F. J. Spera, Phys. Rev. B44 , 2108

1991; Chem. Geol.96 , 421 1992.35L. Stixrude and M. S. T. Bukowinski, Phys. Rev. B44 , 2523

1991.36J. P. Rino, I. Ebbsjo, R. K. Kalia, A. Nakano, and P. Vashish

Phys. Rev. B47 , 3053 1993.37W. Jin, R. K. Kalia, P. Vashishta, and J. P. Rino, Phys. Rev. L71 , 3146 1993.

38J. R. Rustad, D. A. Yuen, and F. J. Spera, Phys. Rev. A42 , 20811990.

39C. S. Marians and L. W. Hobbs, J. Non-Cryst. Solids124 , 2421990.

40J. Horbach and W. Kob, Phys. Rev. B60 , 3169 1999.41C. A. Angell, P. A. Cheeseman, and S. Tamaddon, Science218 ,

885 1982.42L. Stixrude, inStructure and Imperfections in Amorphous and

Crystalline Silicon Dioxide , edited by R. A. B. Devine, J.-P.Duraud, and E. DooryheWiley, New York, 2000, pp. 69103.

43J. Sarnthein, A. Pasquarello, and R. Car, Phys. Rev. Lett.74 ,4682 1995.

44J. Sarnthein, A. Pasquarello, and R. Car, Phys. Rev. B52 , 126901995.

45A. Trave, P. Tangney, S. Scandolo, A. Pasquarello, and R. CPhys. Rev. Lett.89 , 245504 2002.

46M. Pohlmann, M. Benoit, and W. Kob, Phys. Rev. B70 , 1842092004.

47G. Kresse, J. Hafner, and R. J. Needs, J. Phys.: Condens. Ma4, 7451 1992.

48G. Kresse and J. Furthmuller, Comput. Mater. Sci.6, 15 1996.49B. B. Karki, D. Bhattarai, and L. Stixrude, Phys. Rev. B73 ,


104205-11


12/12

b. b. karki, d. bhattarai and l. stixrude

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