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Liquid Carbon: Freezing Line and Structure Near Freezing

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Computer-Based Modeling of Novel Carbon Systems and Their Properties

Part of the book series: Carbon Materials: Chemistry and Physics ((CMCP,volume 3))

Abstract

This chapter deals with the phase diagram of carbon with emphasis on the liquid phase occurring in extreme conditions of temperature and pressure. After presenting a critical review of the experimental results and still unresolved issues, the authors discuss the possibility of modeling carbon by use of empirical potentials. Also the techniques to evaluate numerically the free energy of each phase are presented in detail. The second part of the chapter discusses in detail the structure of the liquid in different ranges of pressure, the pressure–density equations of state at different temperatures and the possibility of a liquid–liquid phase transition.

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Notes

  1. 1.

    Equation 1.5then becomes:

    $$\frac{\beta {F}^{\textrm{ E(CM)}}} {N} \; = 3\textrm{ ln}\Lambda -\frac{3} {2}\textrm{ ln}\left ( \frac{2\pi } {\beta \alpha }\right ) -\frac{3{N}_{s}} {2N} \left (\textrm{ ln}\left (\frac{\alpha \beta } {2\pi } \right ) + \textrm{ ln}{\mathit{NN}}_{s}\right )$$

    where N s is the number of sheets. Equation 1.6becomes:

    $$\frac{\beta \Delta {F}^{\maltese \textrm{ (CM)}\rightarrow \,\maltese }} {N} = -\frac{{N}_{s}} {N} \,\textrm{ ln} \frac{V } {{N}_{ws}}$$

    where, in \({N}_{ws} = N/{n}_{ws}\), one has to define the Wigner–Seitz cell within a graphite sheet; this leads to n ws = 2. Equation 1.8becomes:

    $$\frac{\beta \Delta {F}^{ \frac{1} {N} }} {N} = -\frac{{N}_{s}} {N} \,\left [\frac{3} {2}\textrm{ ln}\left (N{N}_{s}\,\frac{\alpha \beta } {2\pi } \right ) + \textrm{ ln} \frac{V } {{N}_{ws}}\right ].$$
  2. 2.

    For the correct application of the method it is not needed to have the three states at the same P. It is only required that the phases share a broad stable region in pressure at the chosen T.

  3. 3.

    The distribution usually exhibits a bimodal shape in case of phase boundary crossing.

  4. 4.

    The difference between these two values gives a hint on the uncertainties related to the two different methods used for calculating coexistence, given that the DF-MD set-up is quite similar in the two works.

  5. 5.

    The transition in the stable liquid region is supercritical, thus continuous, but taking place in a short range of pressures around 6.5 GPa.

  6. 6.

    The authors adopted the graphite melting line measured by Togaya [16]. This melting line is reported in Fig. 1.4, together with our results. According to Glosli and Ree, from the maximum of that melting line would branch off the LLPT coexistence line.

  7. 7.

    The factor two multiplying the off-diagonal partial distribution functions (g ij (r), with i≠j) is needed when those distributions are calculated according to the literature (e.g. Refs. [71, 72]). The algorithm calculating the g ij (r) browses the pairs of particles only once, as is commonly done for the total g(r). If the algorithm browsed over all the neighbors of each particle, the factor two would clearly not be needed.

  8. 8.

    A fourfold coordinated liquid with a rather pronounced diamond-like structure in the first coordination shell [47]).

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Acknowledgements

We gratefully acknowledge F. Colonna, A. Fasolino, D. Frenkel, J. H. Los, and C. Valeriani for inspiring and useful discussions.

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Authors and Affiliations

  1. Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195, Berlin, Germany

    Luca M. Ghiringhelli

  2. Van’t Hoff Institute for Molecular Sciences & Amsterdam Center for Multiscale Modeling, Universiteit van Amsterdam, Nieuwe Achtergracht 166, NL-1018, WV, Amsterdam, The Netherlands

    Evert Jan Meijer

Authors
  1. Luca M. Ghiringhelli
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  2. Evert Jan Meijer
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Correspondence to Luca M. Ghiringhelli .

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Editors and Affiliations

  1. Dipartimento di Fisica, Università di Cagliari, Monserrato, 09042, Cagliari, Italy

    Luciano Colombo

  2. Inst. Molecules & Materials, Radboud University of Nijmegen, Toernooiveld 1, Nijmegen, 6525 ED, Netherlands

    Annalisa Fasolino

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Ghiringhelli, L.M., Meijer, E.J. (2010). Liquid Carbon: Freezing Line and Structure Near Freezing. In: Colombo, L., Fasolino, A. (eds) Computer-Based Modeling of Novel Carbon Systems and Their Properties. Carbon Materials: Chemistry and Physics, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9718-8_1

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