Estimating the density scaling exponent of viscous liquids from specific heat and bulk modulus data
It was recently shown by computer simulations that a large class of liquids exhibits strong correlations in their thermal fluctuations of virial and potential energy [Pedersen et al., Phys. Rev. Lett. 100, 015701 (2008)]. Among organic liquids the class of strongly correlating liquids includes van der Waals liquids, but excludes ionic and hydrogen-bonding liquids. The present note focuses on the density scaling of strongly correlating liquids, i.e., the fact their relaxation time at different densities and temperatures collapses to a master curve according to the expression [Schrøder et al., arXiv:0803.2199]. We here show how to calculate the exponent from bulk modulus and specific heat data, either measured as functions of frequency in the metastable liquid or extrapolated from the glass and liquid phases to a common temperature (close to the glass transition temperature). Thus an exponent defined from the response to highly nonlinear parameter changes may be determined from linear response measurements.
where , . The “slope” is defined as
and the correlation coefficient is defined as
We have previously shown that the fluctuations of such liquids are well described by those generated by soft-sphere potentials (inverse power law potentials) ; . Strongly correlating liquids are approximate single-parameter liquids ; ; . Moreover the density scaling exponent is one third of the exponent of the approximate inverse power law potentials ; . Recall that density (thermodynamic) scaling applies whenever the relaxation time at different densities and temperatures collapse to a master curve according to the expression . It is generally reported now that density scaling applies for van der Waals liquids, but e.g. not for hydrogen-bonding liquids. This is consistent with our finding that the class of strongly correlating liquids includes van der Waals and metallic liquids, but excludes covalent, ionic, or hydrogen-bonding liquids – the latter three classes of liquids have competing interactions that spoil the correlation ; ; .
.2 Fluctuation expressions
Consider a viscous liquid with slow structural relaxation, i.e., with a relaxation time that is much larger than one picosecond. The fluctuation-dissipation (FD) theorem for the frequency-dependent specific heat per unit volume, , is given  by
Relaxation takes place over a limited range of frequencies – typically 3-5 decades. By subtracting the responses at high (“”) and low (“”) frequencies well outside the relaxation frequency range, it follows that if is a time much shorter than those of the relaxations, but much longer than one picosecond, then
Write the energy as potential plus kinetic energy, . Because the kinetic energy in the NVT ensemble fluctuates fast compared to one has . Thus,
where is the slow part of the potential energy fluctuations, i.e., slow compared to the picosecond time scale.
Since the low-frequency limit gives the ordinary (dc) liquid specific heat and the high-frequency limit gives the “glassy” specific heat corresponding to perturbations that probe a frozen structure, this result may be written
Similarly one finds for the virial fluctuations (where is the isothermal bulk modulus)
and for the virial / potential energy correlation (where is the pressure coefficient)
.3 Calculating and from data
.4 Calculating for the commercial silicone oil DC704
The slope may be estimated from the high- and low-frequency limits of and (and a rough estimate of ) by proceeding as follows. Recall the identities
Combining these we get
The value is calculated from these expressions using our unpublished linear response data (table 1). Unfortunately no experimental density scaling ’s are available for this liquid to compare to.
.5 OTP/OPP mixture and pure OTP
The high- and low-frequency limits of the dynamic response can be estimated by extrapolation of static response functions of the glass and liquid phases to a temperature close to the glass transition temperature . Table 2 lists extrapolated values of , and for a mixture of o-terphenyl (OTP) and o-phenylphenol (OPP), Table 3 lists values for pure OTP.
From the natural response functions of the constant ensemble it is straightforward to calculate the natural response functions of the constant ensemble:
We have shown that it is possible to calculate the density scaling exponent from linear response measurements of specific heat and bulk modulus data. There are two ways to do this: Either by measuring broad-range frequency-dependent linear responses in the equilibrium metastable liquid phase or by extrapolations as done when evaluating the Prigogine-Defay ratio (). Using the first method for DC704 we find . To the best of our knowledge there are yet no density scaling data for this liquid. Using the second method for the OTP-OPP mixture we find which compares favorably to the density scaling ; similarly we find for pure OTP that compares favorably to the density scaling . This good agreement may well be fortuitous given the uncertainties associated with our estimates. Nevertheless these preliminary findings suggest that for strongly correlating liquids (“single-parameter liquids”) the density scaling exponent – which refers to highly nonlinear parameter changes – may be determined from linear response measurements. This is consistent with a general hypothesis of ours that strongly correlating liquids have simpler physics than liquids in general.
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