We report density-functional based molecular dynamics simulations, that show
that, with increasing pressure, liquid carbon undergoes a gradual
transformation from a liquid with local three-fold coordination to a
'diamond-like' liquid. We demonstrate that this unusual structural change is
well reproduced by an empirical bond order potential with isotropic long range
interactions, supplemented by torsional terms. In contrast, state-of-the-art
short-range bond-order potentials do not reproduce this diamond structure. This
suggests that a correct description of long-range interactions is crucial for a
unified description of the solid and liquid phases of carbon.Comment: 4 pages, 5 figure

A. Fasolino

A. Ferraz

A.D. Becke

C.J. Wu

D.W. Brenner

Daan Frenkel

Evert Jan Meijer

F.P. Bundy

G. Franzese

J.H. Los

J.N. Glosli

J.P. Perdew

J.R. Morris

Jan H. Los

K. Kobayashi

K. Raghavachari

Luca M. Ghiringhelli

M. Togaya

M. van Thiel

M.P. Grumbach

M.P. Tosi

N. Marks

N.A. Marks

P.E. Blöchl

P.G. Dacosta

R. Car

W.G. Hoover

Y. Katayama

English

arXiv

We report density-functional based molecular-dynamics simulations, which show that, with increasing pressure, liquid carbon undergoes a gradual transformation from a liquid with local threefold coordination to a "diamondlike" liquid. We demonstrate that this unusual structural change is well reproduced by an empirical bond-order potential with isotropic long-range interactions, supplemented by torsional terms. In contrast, state-of-the-art short-range bond-order potentials do not reproduce this diamond structure. This suggests that a correct description of long-range interactions is crucial for a unified description of the solid and liquid phases of carbon

Ghiringhelli, L.M.

Los, J.H.

Meijer, E.J.

Fasolino, A.

Frenkel, D.

NARCIS 

High-pressure diamondlike liquid carbon

International Migration, Integration and Social Cohesion online publications

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)High pressure diamond-like liquid carbonGhiringhelli, L.M.; Los, J.H.; Meijer, E.J.; Fasolino, A.; Frenkel, D.Publication date2004Published inPhysical Review BLink to publicationCitation for published version (APA):Ghiringhelli, L. M., Los, J. H., Meijer, E. J., Fasolino, A., & Frenkel, D. (2004). High pressurediamond-like liquid carbon. Physical Review B, 69(10), 100101/1-100101/4.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:31 Aug 2023RAPID COMMUNICATIONSPHYSICAL REVIEW B 69, 100101~R! ~2004!High-pressure diamondlike liquid carbonLuca M. Ghiringhelli,1 Jan H. Los,2 Evert Jan Meijer,1 A. Fasolino,1,2 and Daan Frenkel1,31van ’t Hoff Institute for Molecular Sciences, Universiteit van Amsterdam, Nieuwe Achtergracht 166,1018 WV Amsterdam, The Netherlands2Theoretical Physics, NSRIM, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands3FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ, Amsterdam, The Netherlands~Received 31 October 2003; published 12 March 2004!We report density-functional based molecular-dynamics simulations, which show that, with increasing pres-sure, liquid carbon undergoes a gradual transformation from a liquid with local threefold coordination to a‘‘diamondlike’’ liquid. We demonstrate that this unusual structural change is well reproduced by an empiricalbond-order potential with isotropic long-range interactions, supplemented by torsional terms. In contrast,state-of-the-art short-range bond-order potentials do not reproduce this diamond structure. This suggests that acorrect description of long-range interactions is crucial for a unified description of the solid and liquid phasesof carbon.DOI: 10.1103/PhysRevB.69.100101 PACS number~s!: 61.20.Ja, 34.20.Cf, 71.15.PdtherorrdnhntutooeleneenauerintlindnTtod-u--ithh-ledh-stthere-tethethetheitsWhen a simple fluid, such as argon, is cooled belowcritical temperature, it can form a liquid phase. Howevmore complex fluids, such as molten phosphorus, can ftwo distinct liquid phases that are separated by a first-oliquid-liquid phase transition~LLPT!.1 There is increasingevidence that a variety of network-forming liquids can udergo an LLPT~see, e.g., Franzeseet al.2!. In particular, ithas long been suspected that elemental carbon may exan LLPT.3,4As direct experiments are difficult in the relevatemperature regime~4000–6000 K!, much of the evidencefor such a transition comes from simulations. Classical simlations based on the Brenner bond-order potential withsional terms5,6 predicted a coexistence line between a twfold and a fourfold coordinated liquids starting from thmelting line of graphite end ending in a critical point~at;8800 K)7 as shown in Fig. 1. Recent first-principinvestigations8 have ruled out the occurrence of such a trasition in the range of specific volumes 6.59–15.6 Å3/atom at6000 K. More importantly, theseab initio results found amajority of three-fold coordination over the studied rangWu et al.8 pointed out that the presence ofp bonds in theliquid phase makes it difficult to represent the torsion pottial adequately. They concluded that empirical potentialsunable to capture these subtle electronic effects.In this Rapid Communication, we reexamine these issby extending the range of studied densities while compadensity-functional theory based molecular dynamics~DF-MD! simulations to Monte Carlo~MC! results based on twostate-of-the-art empirical potentials. The first is the recenproposed long-range carbon bond-order potential~LCBOP!~Ref. 9! extended with torsional interactions; the secondthe reactive empirical bond-order~REBO! ~Ref. 10! potentialwhich improves the earlier version due to Brenner.5 LCBOPintroduces long-range terms to account for interplanar biing of graphite which is not described by the REBO potetial.On the basis of DF-MD we find that, while no true LLPoccurs even for higher densities than in Wuet al.,8 a welldefined switching of the dominant coordination from threefour is found at;5.8 Å3/atom, just beyond the range stu0163-1829/2004/69~10!/100101~4!/$22.50 69 1001e,mer-ibit-r---.-resgys--ied in Wu et al.8 This ‘‘diamondlike’’ fourfold coordinatedliquid is highly structured, with a strongly anisotropic anglar distribution of the first neighbors. Interestingly, the empirical potential LCBOP gives an impressive agreement wthe DF-MD results for the structural properties of this higdensity liquid whereas the short-ranged REBO potentia10yields a graphitelike liquid in this range. It should be notthat both the low-density threefold coordinated and higdensity diamondlike liquid are very unusual. In fact mocovalent semiconductors become highly coordinated inliquid phase.14 Highly coordinated phases have been pdicted by Grumbach and Martin15 also for carbon albeit apressures of TPa.FIG. 1. ~Color online! Equation of state of liquid carbon atT56000 K. The inset shows theP-T phase diagram, with diamond~D!, graphite~G!, and liquid ~L! domains: the dotted curve is thsampled isotherm, the solid curves are experimental data—diamond-graphite coexistence curve from Bundy Ref. 11,graphite melting curve from Togaya~Ref. 12!. The dot-dashed lineis the estimated diamond melting line from Glosli and Ree~R f.13!, obtained with Brenner potential, the dashed curves areliquid-liquid coexistence curve ending in a critical point andgraphite melting curve from Glosli and Ree~Ref. 7!. Thel in theinset indicates the switching in coordination shown in Fig. 2.©2004 The American Physical Society01-1ngoathcfolypfoopoxhemmrlusinardstaiaicathhacethuenglth-mnsehdnoheelaythemeofereedDntairnialinldngdn-derht-tOPureofb-ghri-nsi-entdin-easRAPID COMMUNICATIONSGHIRINGHELLI, LOS, MEIJER, FASOLINO, AND FRENKEL PHYSICAL REVIEW B69, 100101~R! ~2004!We performed constant volume DF-MD simulations usithe Car-Parrinello method16 as implemented inCPMDpackage.17 The system consisted of 128 atoms in a cubic bwith periodic boundary conditions at nine densities andtemperature T56000 K, imposed by means ofNosé-Hoover18 thermostat. We used the Becke19 exchangeand Perdew20 correlation gradient corrected functional~BP!with a plane-wave basis set cut off at 35 Ry and sampledBrillouin zone only in the gamma point. BP gives a corredescription of bulk diamond. Each state point was studied5 ps, starting from a sample equilibrated via LCBOP; onminor structural changes occurred in the first tenths ofSince liquid carbon is metallic, we imposed a thermostatthe electronic degrees of freedom in order to ensure a primplementation of the Car-Parrinello scheme.21 We per-formed the MC simulations of 128 particles in a cubic bwith periodic boundary conditions with the LCBOP and tREBO potential. We sampled at 6000 K the constant voluensemble for specific volumes larger than 8 Å3/atom and theconstant pressure ensemble for smaller specific voluwhere the increase of pressure is steeper, with an ovebetween the two regions to check for consistency. MC simlations at selected volumes with 512 and 4096 particles uthe LCBOP showed negligible differences with the 128 pticle results for the local structure and equation of state.In order to reproduce theab initio results, we needed torefine the LCBOP.9 The detailed procedure will be describeelsewhere.22 Here we give a brief summary of the changeFollowing the strategy of the REBO potential10 we have in-troduced a correction to the angular function in order to sbilize small clusters in shapes other than chainlike. It isfact known ~see, e.g., Raghavachari and Binkley23! that,while odd numbered clusters prefer to arrange in a cheven numbered ones can also have metastable ring orstructures. We focused attention on a planarC4 cluster withC2v symmetry~a rhombus with two;60° angles23! and acubic C8.24 It turned out that the angular dependence ofbond order had to be weakened for coordination lower tthree. More important is the addition of torsional interations, in such a way as to describe the conjugation depdence of the torsional energy barrier in agreement withab initio calculations for double and conjugated bonds of Wet al.8 This dependence is not captured by the REBO pottial. There, the barrier as a function of the dihedral anv i jkl is always described by sin2(vijkl) and differs only by ascale factor between these two bonding situations. Inoriginal LCBOP, a variableNi jcon j was already defined, assuming values 1 or 0, respectively, in these two extrecases, to account for the presence or absence of ap bond.We fit the results of Wuet al.8 for Ni jcon j50,1 by two poly-nomials in cos2(vijkl) and interpolate between them by meaof a function decaying rather quickly away fromNi jcon j51.The torsional energy is activated only for a bond betwethreefold coordinated atoms, as for the REBO potential. Tsmaller torsional energy of a bond between fourfold coornated atoms is already taken into account by the long-rainteractions between the third neighbors.In Fig. 1 we compare the pressure-volume isothermsour DF-MD simulations with our MC results based on t10010xaetrs.rereesap-g-.-nn,geen-n-e-eeenei-gefempirical REBO and LCBOP potentials, and with thDF-MD data of Wuet al.,8 all at a temperature of 6000 K. Inview of the relatively low cutoff~35 Ry!, we had to correctthe pressures for the spurious contribution due to Puforces.25 In the density range where we can compare withresults of Wuet al.,8 the pressures that we compute are so15% lower than those reported by Wuet al. This relativelysmall difference is probably due to the different choicedensity functionals. We have checked that our samples windeed liquid. Over the whole isotherm we have observdiffusive behavior in both the MC-LCBOP and the DF-Msimulations, the latter indicating a self-diffusion coefficieat least of order 1025 cm2/s.Up to 60 GPa both empirical potentials seem to be in fagreement withab initio data. However, the coordinatiofractions shown in Fig. 2 indicate that the REBO potentfails to reproduce the correct liquid structure. It predicts,the range between 6 and 7.5 Å3/atom, mainly twofold coor-dination, whereas the present DF-MD results~and those ofWu et al.8!, as well as the LCBOP, show dominant threefocoordination and low twofold coordination.At higher pressures, DF-MD predicts a marked switchiof dominant coordination from three to four aroun5.8 Å3/atom. Judging from the isotherm of Fig. 1, the trasition seems to be continuous with no sign of a vanWaals loop. These results are consistent with the tigbinding MD simulations of Morriset al.26 In contrast, be-tween 5.6 and 6.0 Å3/atom, where the switch of dominancoordination takes place, the MC results based on LCBdisplay large fluctuations in density at the imposed pressof 100 GPa, resulting in a slight bending of the isothermFig.1. Again, qualitatively the coordination fractions otained by LCBOP agree with the DF-MD results even thouwith a more pronounced switch to coordination four. Cuously, the REBO potential predicts a first-order phase tration in the same density range but to a completely differstructure, a threefold graphitelike liquid with well orderesliding sheets which eventually get stuck upon furtherFIG. 2. ~Color online! Coordination fractions vs specific volumof liquid carbon. Note that atV;6 Å3/atom DF-MD and LCBOPpredict a switch from threefold to fourfold coordination, whereREBO evolves from a twofold to a graphitelike liquid.1-2iveo-iotinl-firfbu-nd–5rabodyentheaToid,gh-rstborthetri-d antheheq-o-m-OPtedre-akeon-ce,forc-de-isn-r--tdforfcc6.RAPID COMMUNICATIONSHIGH-PRESSURE DIAMONDLIKE LIQUID CARBON PHYSICAL REVIEW B69, 100101~R! ~2004!creasing of the pressure. The REBO potential never grise to fourfold coordination.If the threefold to fourfold coordination change extraplates to lower temperatures, it might provide an explanatfor the sudden change of the slope of the graphite melline.12 The origin of the latter is still an open question, athough it has been suggested that it is associated with aorder phase transition.4,12 Our results provide no evidence oa first-order transition but rather indicate a pronouncedcontinuous change of dominant coordination.Both the DF-MD and LCBOP simulations of the highdensity fourfold coordinated liquid show a strong diamolike positional and orientational order, as shown in Figs. 3In Fig. 3, we compare the pair correlationsg(r ) of this high-density liquid with that of a~meta!stable bulk diamond neathe estimated coexistence point, at 5000 K and 150 GPa~alsothe LCBOP and DF-MD liquid samples were equilibratedthat temperature!. One can see that up to the second neighshell the liquid has a structure almost as pronounced asFIG. 3. ~Color online! Pair-correlation functions at 5000 K. Diamond~dotted curve! is at 150 GPa; the LCBOP liquid~solid curve!is at 150 GPa and 5.14 Å3/atom; DF-MD liquid~dashed curve! is at5.14 Å3/atom.FIG. 4. ~Color online! Angular correlation functions of firsneighbors with state parameters as in Fig. 3.10010sngst-t.tria-mond. Theg(r )’s obtained by LCBOP globally agree fairlwell with those obtained by DF-MD,22 except around 2 Å,where the LCBOP minimum is too deep. In Fig. 4 we presthe calculated angular correlationg(3)(u) for first neighbors,i.e., those atoms that fall within the short-range cutoff of tLCBOP. Again, the first shell of neighbors in the liquid hasstrong tetrahedral ordering, comparable to bulk diamond.test how far the local diamond structure persists in the liquwe define the angular correlation function for second neibors~i.e., all those particles that are first neighbors of the fineighbors, excluding the central atom and the first-neighshell!. Figure 5 shows that, in the second-neighbor shell,diamond structure is completely lost. Yet, the angular disbution is not structureless: we find a peak around 60° anshoulder at;35°; in a diamond lattice, the latter feature cabe attributed to cross correlations between the first- andsecond-neighbor shells.It is rather surprising that the LCBOP reproduces ttransformation to a predominantly fourfold coordinated liuid, while the REBO potential does not. Apparently, the istropic long-ranged interactions play a crucial role. To deonstrate this, we verified that the short-ranged CBreproduces the behavior of the REBO potential.9 This behav-ior is rather puzzling as long-range interactions are expecto play a negligible role at these high densities~see, e.g.,Glosli and Ree13!. Moreover, long-ranged interactions weintroduced in Los and Fasolino9 to describe threefold coordinated graphitic phases, and no attempt was made to mthe long-range interactions dependent on the local envirment. Torsional interactions appear to be important, sinwithout them, the calculated pressures would be too highhigh densities and too low at low densities.We conjecture that the combination of torsional interations, and long-range forces is required to give the bestscription of the liquid. It would be interesting to check thconjecture by studying the liquid with other empirical potetials, in particular with Marks’ environment-dependent inteaction potential~EDIP!.27 The study of the liquid with thisFIG. 5. ~Color online! Angular correlation functions for seconneighbors~see text! with state parameters as in Fig. 3. The peaksthe diamond are spread around the theoretical values for anlattice of21, 20.5, 0, 0.5, with weights 6/66, 24/66, 12/66, 24/61-3aWreordlavnsnangheheasoldatiesseh-ni-ci-Owl-cil-n-rRAPID COMMUNICATIONSGHIRINGHELLI, LOS, MEIJER, FASOLINO, AND FRENKEL PHYSICAL REVIEW B69, 100101~R! ~2004!potential presented in Marks28 is limited to volumes of6.2 Å3/atom where it gives coordination fractions comprable to those obtained here by DF and by LCBOP and inet al.8In summary, on the basis of DF-MD simulations, we pdict that liquid carbon should exhibit a continuous transfmation from a threefold to a mostly fourfold coordinateliquid at high pressure. This liquid has a strong local anguordering, reminiscent of the diamond structure. We hacompared this finding with the results of MC simulatiobased on two empirical bond-order potentials with torsioterms, the short-range REBO potential and the long-raLCBOP. We find that the latter reproduces quite well tstructure of both low-density and high-density liquid. TREBO simulations instead display a marked first-order phtransition between a mostly twofold and a mostly threef-.Ein10010-u--releegraphite fluid in this range. It is tempting to speculate ththe existence of a fourfold coordinate liquid at high densitwill greatly facilitate the nucleation of diamonds from denliquid carbon.This work is part of the research program of the ‘‘Sticting voor Fundamenteel Onderzoek der Materie~FOM!,’’which is financially supported by the ‘‘Nederlandse Orgasatie voor Wetenschappelijk Onderzoek~NWO!.’’ E.J.M. ac-knowledges the Royal Netherlands Academy of Art and Sences for financial support. J.L. and A.F. acknowledge NWProject No. 015.000.031 for financial support. We acknoedge support from the Stichting Nationale Computerfaiteiten ~NCF! and the Nederlandse Organisatie voor Weteschappelijk Onderzoek~NWO! for the use of supercomputefacilities..ury,ev.1Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata, and K. Funakoshi, Nature~London! 403, 170 ~2000!.2G. Franzese, G. Malescio, A. Skibinsky, S.V. Buldyrev, and HStanley, Nature~London! 409, 692 ~2001!.3A. Ferraz and N.H. March, Phys. Chem. Liq.8, 289 ~1979!.4M. van Thiel and F.H. Ree, Phys. Rev. B48, 3591~1993!.5D.W. Brenner, Phys. Rev. B42, 9458~1990!.6D.W. Brenner, J.A. Harrison, C.T. White, and R.J. Colton, ThSolid Films206, 220 ~1991!.7J.N. Glosli and F.H. Ree, Phys. Rev. Lett.82, 4659~1999!.8C.J. Wu, J.N. Glosli, G. Galli, and F.H. Ree, Phys. Rev. Lett.89,135701~2002!.9J.H. Los and A. Fasolino, Phys. Rev. B68, 024107~2003!.10D.W. Brenneret al., J. Phys.: Condens. Matter14, 783 ~2002!.11F.P. Bundy, J. Chem. Phys.38, 618 ~1963!.12M. Togaya, Phys. Rev. Lett.79, 2474~1997!.13J.N. Glosli and F.H. Ree, J. Chem. Phys.110, 441 ~1999!.14M.P. Tosi, J. Phys.: Condens. Matter6, A13 ~1994!.15M.P. Grumbach and R.M. Martin, Phys. Rev. B54, 15 730~1996!.16R. Car and M. Parrinello, Phys. Rev. Lett.55, 2471~1985!..17CPMD, J. Hutter, A. Alavi, T. Deutsch, M. Bernasconi, SGoedecker, D. Marx, M. Tuckerman, and M. Parrinello, MPI f¨rFestkörperforschung and IBM Zurich Research Laborato1995–1999.18W.G. Hoover, Phys. Rev. A31, 1695~1985!.19A.D. Becke, Phys. Rev. A38, 3098~1988!.20J.P. Perdew, Phys. Rev. B33, 8822 ~1986!; Phys. Rev. B34,7406~E! ~1986!.21P.E. Blöchl and M. Parrinello, Phys. Rev. B45, 9413~1992!.22J.H. Loset al. ~unpublished!.23K. Raghavachari and J.S. Binkley, J. Chem. Phys.87, 2191~1987!.24K. Kobayashi, N. Kurita, H. Kumahora, and K. Tago, Phys. RB 45, 11 299~1992!.25P.G. Dacosta, O.H. Nielsen, and K. Kunc, J. Phys. C19, 3163~1986!.26J.R. Morris, C.Z. Wang, and K.M. Ho, Phys. Rev. B52, 4138~1995!.27N.A. Marks, Phys. Rev. B63, 035401~2001!. The EDIP potentialtakes into account interactions within a radius of 3.2 Å.28N. Marks, J. Phys.: Condens. Matter14, 2901~2002!.1-4

High pressure diamond-like liquid carbon

Utrecht University Repository

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                  60494.pdf ( ) (Open Access)We report density-functional based molecular-dynamics simulations, which show that, with increasing pressure, liquid carbon undergoes a gradual transformation from a liquid with local threefold coordination to a "diamondlike" liquid. We demonstrate that this unusual structural change is well reproduced by an empirical bond-order potential with isotropic long-range interactions, supplemented by torsional terms. In contrast, state-of-the-art short-range bond-order potentials do not reproduce this diamond structure. This suggests that a correct description of long-range interactions is crucial for a unified description of the solid and liquid phases of carbon

Radboud Repository

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)High pressure diamond-like liquid carbonGhiringhelli, L.M.; Los, J.H.; Meijer, E.J.; Fasolino, A.; Frenkel, D.Publication date2004Published inPhysical Review BLink to publicationCitation for published version (APA):Ghiringhelli, L. M., Los, J. H., Meijer, E. J., Fasolino, A., & Frenkel, D. (2004). High pressurediamond-like liquid carbon. Physical Review B, 69(10), 100101/1-100101/4.General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:11 Mar 2023RAPID COMMUNICATIONSPHYSICAL REVIEW B 69, 100101~R! ~2004!High-pressure diamondlike liquid carbonLuca M. Ghiringhelli,1 Jan H. Los,2 Evert Jan Meijer,1 A. Fasolino,1,2 and Daan Frenkel1,31van ’t Hoff Institute for Molecular Sciences, Universiteit van Amsterdam, Nieuwe Achtergracht 166,1018 WV Amsterdam, The Netherlands2Theoretical Physics, NSRIM, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands3FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ, Amsterdam, The Netherlands~Received 31 October 2003; published 12 March 2004!We report density-functional based molecular-dynamics simulations, which show that, with increasing pres-sure, liquid carbon undergoes a gradual transformation from a liquid with local threefold coordination to a‘‘diamondlike’’ liquid. We demonstrate that this unusual structural change is well reproduced by an empiricalbond-order potential with isotropic long-range interactions, supplemented by torsional terms. In contrast,state-of-the-art short-range bond-order potentials do not reproduce this diamond structure. This suggests that acorrect description of long-range interactions is crucial for a unified description of the solid and liquid phasesof carbon.DOI: 10.1103/PhysRevB.69.100101 PACS number~s!: 61.20.Ja, 34.20.Cf, 71.15.PdtherorrdnhntutooeleneenauerintlindnTtod-u--ithh-ledh-stthere-tethethetheitsWhen a simple fluid, such as argon, is cooled belowcritical temperature, it can form a liquid phase. Howevmore complex fluids, such as molten phosphorus, can ftwo distinct liquid phases that are separated by a first-oliquid-liquid phase transition~LLPT!.1 There is increasingevidence that a variety of network-forming liquids can udergo an LLPT~see, e.g., Franzeseet al.2!. In particular, ithas long been suspected that elemental carbon may exan LLPT.3,4As direct experiments are difficult in the relevatemperature regime~4000–6000 K!, much of the evidencefor such a transition comes from simulations. Classical simlations based on the Brenner bond-order potential withsional terms5,6 predicted a coexistence line between a twfold and a fourfold coordinated liquids starting from thmelting line of graphite end ending in a critical point~at;8800 K)7 as shown in Fig. 1. Recent first-principinvestigations8 have ruled out the occurrence of such a trasition in the range of specific volumes 6.59–15.6 Å3/atom at6000 K. More importantly, theseab initio results found amajority of three-fold coordination over the studied rangWu et al.8 pointed out that the presence ofp bonds in theliquid phase makes it difficult to represent the torsion pottial adequately. They concluded that empirical potentialsunable to capture these subtle electronic effects.In this Rapid Communication, we reexamine these issby extending the range of studied densities while compadensity-functional theory based molecular dynamics~DF-MD! simulations to Monte Carlo~MC! results based on twostate-of-the-art empirical potentials. The first is the recenproposed long-range carbon bond-order potential~LCBOP!~Ref. 9! extended with torsional interactions; the secondthe reactive empirical bond-order~REBO! ~Ref. 10! potentialwhich improves the earlier version due to Brenner.5 LCBOPintroduces long-range terms to account for interplanar biing of graphite which is not described by the REBO potetial.On the basis of DF-MD we find that, while no true LLPoccurs even for higher densities than in Wuet al.,8 a welldefined switching of the dominant coordination from threefour is found at;5.8 Å3/atom, just beyond the range stu0163-1829/2004/69~10!/100101~4!/$22.50 69 1001e,mer-ibit-r---.-resgys--ied in Wu et al.8 This ‘‘diamondlike’’ fourfold coordinatedliquid is highly structured, with a strongly anisotropic anglar distribution of the first neighbors. Interestingly, the empirical potential LCBOP gives an impressive agreement wthe DF-MD results for the structural properties of this higdensity liquid whereas the short-ranged REBO potentia10yields a graphitelike liquid in this range. It should be notthat both the low-density threefold coordinated and higdensity diamondlike liquid are very unusual. In fact mocovalent semiconductors become highly coordinated inliquid phase.14 Highly coordinated phases have been pdicted by Grumbach and Martin15 also for carbon albeit apressures of TPa.FIG. 1. ~Color online! Equation of state of liquid carbon atT56000 K. The inset shows theP-T phase diagram, with diamond~D!, graphite~G!, and liquid ~L! domains: the dotted curve is thsampled isotherm, the solid curves are experimental data—diamond-graphite coexistence curve from Bundy Ref. 11,graphite melting curve from Togaya~Ref. 12!. The dot-dashed lineis the estimated diamond melting line from Glosli and Ree~R f.13!, obtained with Brenner potential, the dashed curves areliquid-liquid coexistence curve ending in a critical point andgraphite melting curve from Glosli and Ree~Ref. 7!. Thel in theinset indicates the switching in coordination shown in Fig. 2.©2004 The American Physical Society01-1ngoathcfolypfoopoxhemmrlusinardstaiaicathhacethuenglth-mnsehdnoheelaythemeofereedDntairnialinldngdn-derht-tOPureofb-ghri-nsi-entdin-easRAPID COMMUNICATIONSGHIRINGHELLI, LOS, MEIJER, FASOLINO, AND FRENKEL PHYSICAL REVIEW B69, 100101~R! ~2004!We performed constant volume DF-MD simulations usithe Car-Parrinello method16 as implemented inCPMDpackage.17 The system consisted of 128 atoms in a cubic bwith periodic boundary conditions at nine densities andtemperature T56000 K, imposed by means ofNosé-Hoover18 thermostat. We used the Becke19 exchangeand Perdew20 correlation gradient corrected functional~BP!with a plane-wave basis set cut off at 35 Ry and sampledBrillouin zone only in the gamma point. BP gives a corredescription of bulk diamond. Each state point was studied5 ps, starting from a sample equilibrated via LCBOP; onminor structural changes occurred in the first tenths ofSince liquid carbon is metallic, we imposed a thermostatthe electronic degrees of freedom in order to ensure a primplementation of the Car-Parrinello scheme.21 We per-formed the MC simulations of 128 particles in a cubic bwith periodic boundary conditions with the LCBOP and tREBO potential. We sampled at 6000 K the constant voluensemble for specific volumes larger than 8 Å3/atom and theconstant pressure ensemble for smaller specific voluwhere the increase of pressure is steeper, with an ovebetween the two regions to check for consistency. MC simlations at selected volumes with 512 and 4096 particles uthe LCBOP showed negligible differences with the 128 pticle results for the local structure and equation of state.In order to reproduce theab initio results, we needed torefine the LCBOP.9 The detailed procedure will be describeelsewhere.22 Here we give a brief summary of the changeFollowing the strategy of the REBO potential10 we have in-troduced a correction to the angular function in order to sbilize small clusters in shapes other than chainlike. It isfact known ~see, e.g., Raghavachari and Binkley23! that,while odd numbered clusters prefer to arrange in a cheven numbered ones can also have metastable ring orstructures. We focused attention on a planarC4 cluster withC2v symmetry~a rhombus with two;60° angles23! and acubic C8.24 It turned out that the angular dependence ofbond order had to be weakened for coordination lower tthree. More important is the addition of torsional interations, in such a way as to describe the conjugation depdence of the torsional energy barrier in agreement withab initio calculations for double and conjugated bonds of Wet al.8 This dependence is not captured by the REBO pottial. There, the barrier as a function of the dihedral anv i jkl is always described by sin2(vijkl) and differs only by ascale factor between these two bonding situations. Inoriginal LCBOP, a variableNi jcon j was already defined, assuming values 1 or 0, respectively, in these two extrecases, to account for the presence or absence of ap bond.We fit the results of Wuet al.8 for Ni jcon j50,1 by two poly-nomials in cos2(vijkl) and interpolate between them by meaof a function decaying rather quickly away fromNi jcon j51.The torsional energy is activated only for a bond betwethreefold coordinated atoms, as for the REBO potential. Tsmaller torsional energy of a bond between fourfold coornated atoms is already taken into account by the long-rainteractions between the third neighbors.In Fig. 1 we compare the pressure-volume isothermsour DF-MD simulations with our MC results based on t10010xaetrs.rereesap-g-.-nn,geen-n-e-eeenei-gefempirical REBO and LCBOP potentials, and with thDF-MD data of Wuet al.,8 all at a temperature of 6000 K. Inview of the relatively low cutoff~35 Ry!, we had to correctthe pressures for the spurious contribution due to Puforces.25 In the density range where we can compare withresults of Wuet al.,8 the pressures that we compute are so15% lower than those reported by Wuet al. This relativelysmall difference is probably due to the different choicedensity functionals. We have checked that our samples windeed liquid. Over the whole isotherm we have observdiffusive behavior in both the MC-LCBOP and the DF-Msimulations, the latter indicating a self-diffusion coefficieat least of order 1025 cm2/s.Up to 60 GPa both empirical potentials seem to be in fagreement withab initio data. However, the coordinatiofractions shown in Fig. 2 indicate that the REBO potentfails to reproduce the correct liquid structure. It predicts,the range between 6 and 7.5 Å3/atom, mainly twofold coor-dination, whereas the present DF-MD results~and those ofWu et al.8!, as well as the LCBOP, show dominant threefocoordination and low twofold coordination.At higher pressures, DF-MD predicts a marked switchiof dominant coordination from three to four aroun5.8 Å3/atom. Judging from the isotherm of Fig. 1, the trasition seems to be continuous with no sign of a vanWaals loop. These results are consistent with the tigbinding MD simulations of Morriset al.26 In contrast, be-tween 5.6 and 6.0 Å3/atom, where the switch of dominancoordination takes place, the MC results based on LCBdisplay large fluctuations in density at the imposed pressof 100 GPa, resulting in a slight bending of the isothermFig.1. Again, qualitatively the coordination fractions otained by LCBOP agree with the DF-MD results even thouwith a more pronounced switch to coordination four. Cuously, the REBO potential predicts a first-order phase tration in the same density range but to a completely differstructure, a threefold graphitelike liquid with well orderesliding sheets which eventually get stuck upon furtherFIG. 2. ~Color online! Coordination fractions vs specific volumof liquid carbon. Note that atV;6 Å3/atom DF-MD and LCBOPpredict a switch from threefold to fourfold coordination, whereREBO evolves from a twofold to a graphitelike liquid.1-2iveo-iotinl-firfbu-nd–5rabodyentheaToid,gh-rstborthetri-d antheheq-o-m-OPtedre-akeon-ce,forc-de-isn-r--tdforfcc6.RAPID COMMUNICATIONSHIGH-PRESSURE DIAMONDLIKE LIQUID CARBON PHYSICAL REVIEW B69, 100101~R! ~2004!creasing of the pressure. The REBO potential never grise to fourfold coordination.If the threefold to fourfold coordination change extraplates to lower temperatures, it might provide an explanatfor the sudden change of the slope of the graphite melline.12 The origin of the latter is still an open question, athough it has been suggested that it is associated with aorder phase transition.4,12 Our results provide no evidence oa first-order transition but rather indicate a pronouncedcontinuous change of dominant coordination.Both the DF-MD and LCBOP simulations of the highdensity fourfold coordinated liquid show a strong diamolike positional and orientational order, as shown in Figs. 3In Fig. 3, we compare the pair correlationsg(r ) of this high-density liquid with that of a~meta!stable bulk diamond neathe estimated coexistence point, at 5000 K and 150 GPa~alsothe LCBOP and DF-MD liquid samples were equilibratedthat temperature!. One can see that up to the second neighshell the liquid has a structure almost as pronounced asFIG. 3. ~Color online! Pair-correlation functions at 5000 K. Diamond~dotted curve! is at 150 GPa; the LCBOP liquid~solid curve!is at 150 GPa and 5.14 Å3/atom; DF-MD liquid~dashed curve! is at5.14 Å3/atom.FIG. 4. ~Color online! Angular correlation functions of firsneighbors with state parameters as in Fig. 3.10010sngst-t.tria-mond. Theg(r )’s obtained by LCBOP globally agree fairlwell with those obtained by DF-MD,22 except around 2 Å,where the LCBOP minimum is too deep. In Fig. 4 we presthe calculated angular correlationg(3)(u) for first neighbors,i.e., those atoms that fall within the short-range cutoff of tLCBOP. Again, the first shell of neighbors in the liquid hasstrong tetrahedral ordering, comparable to bulk diamond.test how far the local diamond structure persists in the liquwe define the angular correlation function for second neibors~i.e., all those particles that are first neighbors of the fineighbors, excluding the central atom and the first-neighshell!. Figure 5 shows that, in the second-neighbor shell,diamond structure is completely lost. Yet, the angular disbution is not structureless: we find a peak around 60° anshoulder at;35°; in a diamond lattice, the latter feature cabe attributed to cross correlations between the first- andsecond-neighbor shells.It is rather surprising that the LCBOP reproduces ttransformation to a predominantly fourfold coordinated liuid, while the REBO potential does not. Apparently, the istropic long-ranged interactions play a crucial role. To deonstrate this, we verified that the short-ranged CBreproduces the behavior of the REBO potential.9 This behav-ior is rather puzzling as long-range interactions are expecto play a negligible role at these high densities~see, e.g.,Glosli and Ree13!. Moreover, long-ranged interactions weintroduced in Los and Fasolino9 to describe threefold coordinated graphitic phases, and no attempt was made to mthe long-range interactions dependent on the local envirment. Torsional interactions appear to be important, sinwithout them, the calculated pressures would be too highhigh densities and too low at low densities.We conjecture that the combination of torsional interations, and long-range forces is required to give the bestscription of the liquid. It would be interesting to check thconjecture by studying the liquid with other empirical potetials, in particular with Marks’ environment-dependent inteaction potential~EDIP!.27 The study of the liquid with thisFIG. 5. ~Color online! Angular correlation functions for seconneighbors~see text! with state parameters as in Fig. 3. The peaksthe diamond are spread around the theoretical values for anlattice of21, 20.5, 0, 0.5, with weights 6/66, 24/66, 12/66, 24/61-3aWreordlavnsnangheheasoldatiesseh-ni-ci-Owl-cil-n-rRAPID COMMUNICATIONSGHIRINGHELLI, LOS, MEIJER, FASOLINO, AND FRENKEL PHYSICAL REVIEW B69, 100101~R! ~2004!potential presented in Marks28 is limited to volumes of6.2 Å3/atom where it gives coordination fractions comprable to those obtained here by DF and by LCBOP and inet al.8In summary, on the basis of DF-MD simulations, we pdict that liquid carbon should exhibit a continuous transfmation from a threefold to a mostly fourfold coordinateliquid at high pressure. This liquid has a strong local anguordering, reminiscent of the diamond structure. We hacompared this finding with the results of MC simulatiobased on two empirical bond-order potentials with torsioterms, the short-range REBO potential and the long-raLCBOP. We find that the latter reproduces quite well tstructure of both low-density and high-density liquid. TREBO simulations instead display a marked first-order phtransition between a mostly twofold and a mostly threef-.Ein10010-u--releegraphite fluid in this range. It is tempting to speculate ththe existence of a fourfold coordinate liquid at high densitwill greatly facilitate the nucleation of diamonds from denliquid carbon.This work is part of the research program of the ‘‘Sticting voor Fundamenteel Onderzoek der Materie~FOM!,’’which is financially supported by the ‘‘Nederlandse Orgasatie voor Wetenschappelijk Onderzoek~NWO!.’’ E.J.M. ac-knowledges the Royal Netherlands Academy of Art and Sences for financial support. J.L. and A.F. acknowledge NWProject No. 015.000.031 for financial support. We acknoedge support from the Stichting Nationale Computerfaiteiten ~NCF! and the Nederlandse Organisatie voor Weteschappelijk Onderzoek~NWO! for the use of supercomputefacilities..ury,ev.1Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata, and K. Funakoshi, Nature~London! 403, 170 ~2000!.2G. Franzese, G. Malescio, A. Skibinsky, S.V. Buldyrev, and HStanley, Nature~London! 409, 692 ~2001!.3A. Ferraz and N.H. March, Phys. Chem. Liq.8, 289 ~1979!.4M. van Thiel and F.H. Ree, Phys. Rev. B48, 3591~1993!.5D.W. Brenner, Phys. Rev. B42, 9458~1990!.6D.W. Brenner, J.A. Harrison, C.T. White, and R.J. Colton, ThSolid Films206, 220 ~1991!.7J.N. Glosli and F.H. Ree, Phys. Rev. Lett.82, 4659~1999!.8C.J. Wu, J.N. Glosli, G. Galli, and F.H. Ree, Phys. Rev. Lett.89,135701~2002!.9J.H. Los and A. Fasolino, Phys. Rev. B68, 024107~2003!.10D.W. Brenneret al., J. Phys.: Condens. Matter14, 783 ~2002!.11F.P. Bundy, J. Chem. Phys.38, 618 ~1963!.12M. Togaya, Phys. Rev. Lett.79, 2474~1997!.13J.N. Glosli and F.H. Ree, J. Chem. Phys.110, 441 ~1999!.14M.P. Tosi, J. Phys.: Condens. Matter6, A13 ~1994!.15M.P. Grumbach and R.M. Martin, Phys. Rev. B54, 15 730~1996!.16R. Car and M. Parrinello, Phys. Rev. Lett.55, 2471~1985!..17CPMD, J. Hutter, A. Alavi, T. Deutsch, M. Bernasconi, SGoedecker, D. Marx, M. Tuckerman, and M. Parrinello, MPI f¨rFestkörperforschung and IBM Zurich Research Laborato1995–1999.18W.G. Hoover, Phys. Rev. A31, 1695~1985!.19A.D. Becke, Phys. Rev. A38, 3098~1988!.20J.P. Perdew, Phys. Rev. B33, 8822 ~1986!; Phys. Rev. B34,7406~E! ~1986!.21P.E. Blöchl and M. Parrinello, Phys. Rev. B45, 9413~1992!.22J.H. Loset al. ~unpublished!.23K. Raghavachari and J.S. Binkley, J. Chem. Phys.87, 2191~1987!.24K. Kobayashi, N. Kurita, H. Kumahora, and K. Tago, Phys. RB 45, 11 299~1992!.25P.G. Dacosta, O.H. Nielsen, and K. Kunc, J. Phys. C19, 3163~1986!.26J.R. Morris, C.Z. Wang, and K.M. Ho, Phys. Rev. B52, 4138~1995!.27N.A. Marks, Phys. Rev. B63, 035401~2001!. The EDIP potentialtakes into account interactions within a radius of 3.2 Å.28N. Marks, J. Phys.: Condens. Matter14, 2901~2002!.1-4

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