Variation of the dynamic susceptibility along an isochrone

Koperwas et al. showed in a recent paper [Phys. Rev. Lett. 111, 125701 (2013)] that the dynamic susceptibility ${\chi }_4$ as estimated by dielectric measurements for certain glass-forming liquids decreases substantially with increasing pressure along a curve of constant relaxation time. This observation is at odds with other measures of dynamics being invariant and seems to pose a problem for theories of glass formation. We show that this variation is in fact consistent with predictions for liquids with hidden scale invariance: Measures of dynamics at constant volume are invariant along isochrones, called isomorphs in such liquids, but contributions to fluctuations from long-wavelength fluctuations can vary. This is related to the known noninvariance of the isothermal bulk modulus. Considering the version of ${\chi }_4$ defined for the NVT ensemble, data from simulations of a binary Lennard-Jones liquid show in fact a slight increase with increasing density. This is a true departure from the formal invariance expected for this quantity.

Figure 1

Experimental estimates of ${\chi }_4^{\mathrm{NPT}}$ for ortho-terphenyl taken from supplemental material to Ref. [33] (Figure S2). (a) Shows ${\chi }_4$ as a function of relaxation time along three different thermodynamic paths and estimated in two different ways: One (open symbols) is the estimate based on Eq. (10), namely Eq. (15), while the other is based on an alternative decomposition into contributions from the NPH ensemble and a term involving the temperature derivative at constant pressure [9]. (b) Shows the pressure dependence of the peak value on the isochrone $\tau$ = 100 s (top curve), together with the contributions induced separately by energy and density fluctuations $(T$ and $\rho$ terms, circles and stars, respectively). The solid symbols indicate the estimate from the alternate decomposition. The lines are guides for the eyes.

Figure 2

Self-intermediate scattering function for the Kob-Andersen system (large particles) for several state points along an isochrone with lowest density 1.20 (temperature 0.55), plotted against reduced time $\tilde{t}$. The inset shows the relaxation time $\tilde{\tau }$ defined as the time when the correlation function has fallen to $1/e$ along to curves: a isochrone (squares) determined by Eq. (16) parametrized by matching its logarithmic derivative to Eq. (6) at density 1.6 and the configurational adiabat (diamonds) determined iteratively by fitting the logarithmic derivative of Eq. (16) to Eq. (6) for each step in density of size 0.05. The isochrone and adiabat share the state point $\left(\rho =1.20,T=0.55\right)$. Lines are to guide the eyes.

Figure 3

(a) ${\chi }_4^{\mathrm{NVE}}$ for the same points as in Fig. 2; (b) ${\chi }_4^{\mathrm{NVT}}$ for the same points; (c) temperature term corresponding to experimental estimate of ${\chi }_4^{\mathrm{NVT}}$; (d) density term used to estimate ${\chi }_4^{\mathrm{NPT}}$.

Figure 4

Maximum values of ${\chi }_4$ versus density for the isochrone in different ensembles also including different contributing terms. Absolute and percentage changes are indicated to the right of each set.

Figure 5

Maximum values of ${\chi }_4$ versus density for the adiabat in different ensembles also including different contributing terms.

Figure 6

Maxima of $S_4(k,t)$ for different wave numbers/wavelengths versus density on isochrone. The indicated wavelengths are in reduced units, and equal to $N^{1/3}/n$ for integer $n$. Data from a system with double the linear size are also shown, for the exact same state points. For wavelengths 10.08 and 20.16 a slight decrease of $S_{4,\max }$ is seen. Finite size effects cause the values of $S_4$ at wavelength 10.08 to differ slightly (the reduced relaxation time is 15% shorter for the larger system, data not shown; its variation along the isochrone is as small as for the shorter one). For the largest $k$ the maximum value differs little from the long-time random value coming from the mean of ${cos}^2$ (see Appendix pp2).

Figure 7

Effect of varying the thermostat relaxation time, $\rho =1.20,T=0.55$ $\left({\tau }_{\alpha }=27\right)$. For a thermostat relaxation time long compared to ${\tau }_{\alpha }$, and a simulation long enough to include of order 10 000 thermostat relaxation times, we have effectively Newtonian (energy-conserving) dynamics with an NVT ensemble. In this case Eq. (8) holds, as can be seen (a). For the case of a very short thermostat relaxation time, there is a small but significant difference (b).