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Isomorph invariance of the structure and dynamics of classical crystals

Dan E. Albrechtsen, Andreas E. Olsen, Ulf R. Pedersen, Thomas B. Schrøder, and Jeppe C. Dyre
Phys. Rev. B 90, 094106 – Published 5 September 2014
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  • INTRODUCTION
  • SIMULATION DETAILS
  • POTENTIAL ENERGIES OF SCALED…
  • ISOMORPH INVARIANCE OF STRUCTURE AND…
  • ISOMORPH JUMPS OF THE LJ CRYSTAL
  • RESULTS FOR FIVE OTHER MODEL CRYSTALS
  • DISCUSSION
    • References

    Abstract

    This paper shows by computer simulations that some crystalline systems have curves in their thermodynamic phase diagrams, so-called isomorphs, along which structure and dynamics in reduced units are invariant to a good approximation. The crystals are studied in a classical-mechanical framework, which is generally a good description except significantly below melting. The existence of isomorphs for crystals is validated by simulations of particles interacting via the Lennard-Jones pair potential arranged into a face-centered cubic (fcc) crystalline structure; the slow vacancy-jump dynamics of a defective fcc crystal is also shown to be isomorph invariant. In contrast, a NaCl crystal model does not exhibit isomorph invariances. Other systems simulated, though in less detail, are the Wahnström binary Lennard-Jones crystal with the MgZn2 Laves crystal structure, monatomic fcc crystals of particles interacting via the Buckingham pair potential and via a purely repulsive pair potential diverging at a finite separation, an ortho-terphenyl molecular model crystal, and SPC/E hexagonal ice. Except for NaCl and ice, the crystals simulated all have isomorphs. Based on previous simulations of liquid models, we conjecture that crystalline solids with isomorphs include most or all formed by atoms or molecules interacting via metallic or van der Waals forces, whereas covalently bonded or hydrogen-bonded crystals are not expected to have isomorphs; crystals of ions or dipolar molecules constitute a limiting case for which isomorphs are only expected when the Coulomb interactions are relatively weak. We briefly discuss the consequences of the findings for theories of melting and crystallization.

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    • Received 7 April 2014
    • Revised 5 August 2014

    DOI:https://doi.org/10.1103/PhysRevB.90.094106

    ©2014 American Physical Society

    Authors & Affiliations

    Dan E. Albrechtsen, Andreas E. Olsen, Ulf R. Pedersen, Thomas B. Schrøder, and Jeppe C. Dyre

    • DNRF Center “Glass and Time”, IMFUFA, Department of Sciences, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark

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    Vol. 90, Iss. 9 — 1 September 2014

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    • Figure 1
      Figure 1

      (a) and (b) Scatter plots of the potential energies of scaled configurations. (a) Results from NVT simulations of a single-component Lennard-Jones (LJ) 10×10×10 face-centered cubic (fcc) crystal at density ρ1=1.2 and temperature T1=1.0. The figure shows on the y-axis a scatter plot of the potential energy per particle when each configuration is scaled uniformly to density ρ2=2.4, versus the potential energy of the original configuration denoted by R at density ρ1. Note that upon the compression the potential energies change from small negative to large positive values. The blue dashed line of slope 16=(2−1/3)−12 gives the prediction of the repulsive r−12 part of the LJ potential plus a constant; this does not fit the data that have slope 21.14. (This figure first appeared in Ref. [42].) (b) Similar scatter plot for simulations of a model NaCl crystal [36] in which density is also doubled. (c) Plot of virial and potential energy as functions of time in argon units for the LJ crystal in equilibrium at the state point (ρ1,T1), demonstrating very strong correlations; (d) similar plot for the NaCl model, showing much weaker correlations. The correlation coefficient R in (c) and (d) is defined in Eq. (4).

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    • Figure 2
      Figure 2

      Radial distribution functions (RDFs) of the LJ crystal as functions of the reduced pair distance r̃ (Eq. (6)) along various lines in the thermodynamic phase diagram. Panel (a) (upper subpanel) RDFs calculated for state points along an isomorph, involving more than a factor of 2 density change. The data collapse demonstrates structural invariance. For comparison, the bottom two subpanels of (a) give the RDFs from state points along an isochore and an isotherm, respectively, for the same temperature/density variation. (This figure first appeared in Ref. [42].) Panel (b) A zoom-in on the first peak of the RDF for isomorph (left) and r−12 inverse-power-law scaling implying invariance along the line of constant ρ4/T (right), demonstrating that isomorph invariance is not merely a trivial consequences of the repulsive r−12 term of the LJ pair potential.

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    • Figure 3
      Figure 3

      RDFs of the NaCl crystal for prospective isomorphic state points identified by the direct isomorph check method [Fig. 1]. The figure shows the sodium-chloride, chloride-chloride, and sodium-sodium partial RDFs. This crystal does not have isomorphs (at the highest-density state point the systems has in fact melted).

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    • Figure 4
      Figure 4

      (a) Normalized velocity autocorrelation functions of the LJ crystal along an isomorph, an isochore, and an isotherm (same state points as in Fig. 2). (b) Normalized velocity autocorrelation functions for the NaCl crystal along a prospective isomorph, showing no data collapse (same state points as in Fig. 3).

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    • Figure 5
      Figure 5

      Reduced-unit mean-square displacement (MSD) of the particles of an 8×8×8 fcc LJ crystal from which eight particles were removed, making vacancy-jump dynamics possible. The top figure shows the MSD as a function of reduced time for state points along an isomorph identified by the direct-isomorph-check method. The two bottom figures show MSDs calculated for state points along the isochore and isotherm with the same temperature/density variation. The figure shows that vacancy jump dynamics depends strongly on temperature and density, but along an isomorph these two effects compensate each other to a good approximation. (This figure first appeared in Ref. [42].)

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    • Figure 6
      Figure 6

      Potential energy per particle after jumps from three different state points of the LJ crystal to the state point (ρ,T)=(1.5,2.81). Only the jump from a state point isomorphic to the final state point (green curve) leads to instantaneous equilibration. Time is given in argon units.

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    • Figure 7
      Figure 7

      (a) Direct isomorph check for the Wahnström binary LJ crystal, which is a MgZn2 Laves phase structure with a unit cell of twelve atoms. (b) All-particle RDF for the two isomorphic state points identified on the basis of the direct isomorph check in (a). The inset shows the crystal structure [49].

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    • Figure 8
      Figure 8

      (a) Direct isomorph check for the Buckingham potential fcc crystal. This system does not have a repulsive IPL term, but instead an exponentially repulsive term [Eq. (7)]. (b) RDFs along an isomorph and the corresponding isochore and isotherm, demonstrating isomorph invariance of the structure.

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    • Figure 9
      Figure 9

      (a) Direct isomorph check for an fcc crystal of particles interacting via the purely repulsive “sum-IPL” pair potential of Eq. (8). (b) RDFs along an isomorph, isochore, and isotherm, demonstrating isomorph invariance of the structure. (c) The viral potential-energy correlation coefficient (upper subpanel) and the so-called density-scaling exponent γ=〈ΔWΔU〉/〈(ΔU)2〉 [22] (lower subpanel) as functions of density along the isomorph simulated in (b). The density-scaling exponent γ is quite different from 2, the value for an r−6 IPL potential; this shows that the logarithmic term is important in the density range studied.

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    • Figure 10
      Figure 10

      RDFs along an isomorph and the corresponding isochore and isotherm of the Lewis-Wahnström OTP crystal in which each molecule consists of three LJ spheres connected by rigid bonds with a 75o angle between the bonds (the RDFs refer to the LJ particles of different molecules). The isomorph was generated in the usual way from direct isomorph checks (not shown); the uniform scaling of the molecules for the direct isomorph checks generating the isomorphs keeps the bond angles and lengths fixed. The figure demonstrates isomorph invariance of the structure, though not as accurately as for the LJ crystal.

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    • Figure 11
      Figure 11

      Oxygen-oxygen RDF along a prospective isomorph generated from direct isomorph checks (not shown) for a 20% density change of SPC/E hexagonal ice. The figure demonstrates that ice does not have isomorphs, a finding that is consistent with the fact that water has almost zero virial potential-energy correlation coefficient [39].

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