Equilibrium atomic configuration of simulation cell at T = 1800 K. Atoms are colored by potential energy.
Potential energy per atom versus temperature for bulk Ni. The heating rate in current simulations are 2 × 1011 K/s where the potential energy in inset is averaged over every 5 ps.
The non-Gaussian parameter α2 as a function of time interval Δt for a range of T. Temperatures are shown in the bottom left of figure. The inset shows the van Hove function G s (r,Δt) describing the probability for particle displacement from a particle position at the origin (r = 0) at Δt = 150 ps for T = 1800 K and T = 1880 K while at Δt = 5 ps for T = 2100 K. As T m h is approached, G s (r,Δt) develops multiple peaks above T = 1800 K, reflecting a “hopping” motion of the atoms (see text). At T = 1800 K, α 2 is rather small, indicating a nearly harmonic localization of the Ni atoms in the crystal. The minimum in the van Hove function occurs near 0.6 (see the supplementary material for further details and illustration 79 ).
String-like collective atomic motion in the superheated bulk Ni crystal at T = 1880 K. The lines denote Ni atoms that belong to the same collective atom movement and the colors are introduced to discriminate between different string events.
The string size distribution at T = 1920 K and T = 1970 K. Inset shows the T dependence of the average string length ⟨n⟩.
Scaling of string radius of gyration R g with its length L as R g ∼ L ν, where the inset shows ν = 1/d f as a function of T. The exponent ν is about 0.6, corresponding to a fractal dimension d f ≈ 5/3 of self-avoiding walks. 87
The number of open strings and close strings as a function of T.
Mean square displacement of bulk Ni at seven different temperatures. The bottom inset shows the temperature dependence of the Lindemann parameter, δ = ⟨Δr 2⟩1/2/r 0, describing the root-mean-square particle displacement relative to the average interatomic distance, the “Debye-Waller factor” (DWF), ⟨u 2⟩, and the fraction of “Lindemann particles” 59 as a function of T ( ; the magnitude 0.2 cut-off is prescribed by the peak position of the van Hove function in Figure 3 and Fig. S1 of the supplementary material). The top inset shows, D/T ∼ 1/t*, where D is the Ni atom diffusion coefficient and t* is the time at which α2 exhibits a maximum. This is a general result that holds as well for many GF liquids. The characteristic relaxation time t* has the significance of a diffusive relaxation time, a quantity that in general can have a qualitatively different T dependence from the inverse structural relaxation time τs from the self-intermediate scattering function (see Fig. 9 ).
The self-intermediate scattering function F s(q, t) in bulk Ni at T = 1840 K, 1880 K, 1920 K, and 1950 K and the collective intermediate scattering function (the black solid curve) in bulk Ni at T = 1950 K. The dashed black curves are fits using where the apparent value of β varies between 0.92 and 1.01. Inset shows D rescaled by the temperature, D/T, as a function of the structural relaxation time, τ obtained from F s(q, t). Note that the collective intermediate scattering function F c(q,t) (solid line) does not decay to 0 in the crystal state and that this quantity exhibits collective density oscillations on a timescale on the order of a ps. We discuss this “boson peak” feature below (see Fig. 10 ).
Reduced vibrational density of states and boson peak for bulk Ni at five different temperatures. The variation of the position of the boson peak with T is hard to resolve in our simulations, but is clear that the peak position blue shifts to smaller frequencies upon going from 1800 K to 1880 K, a trend that is opposite to the trend found in GF liquids. 137,138 Our impression from the limited data, however, is that the trend is non-monotonic when a larger T range is considered, as in the string length L data in Fig. 5 .
Defects exhibited in Ni crystal following energy minimization at different temperatures. Only non-fcc atoms are displayed in the figure for clarity. The defects observed correspond to simple interstitials. The density of these defects does increase with heating, but their concentration is rather small and their positions seem to be relatively uncorrelated in space. Stillinger and Weber 141 have provided insightful visualizations and discussion of this type of defect.
The self-interstitial concentration as a function of T. Inset shows the correlation between self-interstitial and Lindemann particle concentrations at different T.
(a) A typical string with a member of 15 atoms at T = 1840 K. (b) Potential energy fluctuations of atom that is involved in collective string-like motion over the time scale of 120 ps. (b) Relative displacements (Δr/r 0) fluctuation of atoms within string.
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