# Estimating the density scaling exponent of viscous liquids from specific heat and bulk modulus data

## Abstract

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.

## .1 Introduction

A liquid is termed “strongly correlating” if its viral () and potential energy () equilibrium fluctuations correlate better than 90% [10]; [2]; [3] at constant volume and temperature,

(1) |

where , . The “slope” is defined as

(2) |

and the correlation coefficient is defined as

(3) |

We have previously shown that the fluctuations of such liquids are well described by those generated by soft-sphere potentials (inverse power law potentials) [2]; [3]. Strongly correlating liquids are approximate single-parameter liquids [11]; [6]; [1]. Moreover the density scaling exponent is one third of the exponent of the approximate inverse power law potentials [14]; [5]. 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 [10]; [2]; [3].

## .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 [9] by

(4) |

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

(5) |

Write the energy as potential plus kinetic energy, . Because the kinetic energy in the NVT ensemble fluctuates fast compared to one has . Thus,

(6) |

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

(7) |

Similarly one finds for the virial fluctuations (where is the isothermal bulk modulus)

(8) |

and for the virial / potential energy correlation (where is the pressure coefficient)

(9) |

## .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

(12) |

and

(13) |

Combining these we get

(14) |

and

(15) |

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:

(16) |

(17) |

and

(18) |

Using these equations we arrive at the numbers in tables 2 and 3.

## .6 Summary

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 ([12]). 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 [13]; similarly we find for pure OTP that compares favorably to the density scaling [13]. 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.

### References

- Nicholas P. Bailey, Tage Christensen, Bo Jakobsen, Kristine Niss, Niels Boye Olsen, Ulf R. Pedersen, Thomas B. Schrøder, and Jeppe C. Dyre. Glass-forming liquids: one or more ‘order’ parameters? Journal of Physics: Condensed Matter, 20(24), Jun 18 2008. European-Science-Foundation Exploratory Workshop On Glassy Liquids Under Pressure, Ustron, Poland, Oct 10-12, 2007.
- Nicholas P. Bailey, Ulf R. Pedersen, Nicoletta Gnan, Thomas B. Schrøder, and Jeppe C. Dyre. Pressure-energy correlations in liquids. I. Results from computer simulations. Journal of Chemical Physics, 129(18):184507, Nov 2008.
- Nicholas P. Bailey, Ulf R. Pedersen, Nicoletta Gnan, Thomas B. Schrøder, and Jeppe C. Dyre. Pressure-energy correlations in liquids. II. Analysis and consequences. Journal of Chemical Physics, 129(18):184508, Nov 2008.
- S. S. Chang and A. B. Bestul. Heat-Capacity and Thermodynamic Properties of Ortho-Terphenyl Crystal, Glass, and Liquid. Journal of Chemical Physics, 56(1):503–&, 1972.
- D. Coslovich and C. M. Roland. Pressure-energy correlations and thermodynamic scaling in viscous Lennard-Jones liquids. Journal of Chemical Physics, 130(1):014508, 7 Jan 2009.
- N. L. Ellegaard, T. E. Christensen, P. V. Christiansen, N. B. Olsen, U. R. Pedersen, T. Schrøder, and J. Dyre. Single-order-parameter description of glass-forming liquids: A one-frequency test. Journal of Chemical Physics, 126(074502), 2007.
- Tina Hecksher and Bo Jakobsen. unpublished data, 2009.
- M. Naoki and S. Koeda. Pressure Volume Temperature Relations of Liquid, Crystal, and Glass of ortho-Terphenyl - Excess Amorphous Entropies and Factors Determining Molecular Mobility. Journal of Physical Chemistry, 93(2):948–955, Jan 26 1989.
- J. K. Nielsen and J. C. Dyre. Fluctuation-dissipation theorem for frequency-dependent specific heat. Physical Review B: Condensed Matter and Materials Physics, 54(22):15754–15761, Dec 1 1996.
- Ulf R. Pedersen, Nicholas P. Bailey, Thomas B. Schrøder, and Jeppe C. Dyre. Strong pressure-energy correlations in van der waals liquids. Phys. Rev. Lett., 100(1):015701, Jan 11 2008.
- Ulf R. Pedersen, Tage Christensen, Thomas B. Schrøder, and Jeppe C. Dyre. Feasibility of a single-parameter description of equilibrium viscous liquid dynamics. Phys. Rev. E, 77(1, Part 1):011201, Jan 2008.
- I. Prigogine and R. Defay. Chemical thermodynamics. Longmans, Green and Co, New York, 1954.
- C. M. Roland, S. Hensel-Bielowka, M. Paluch, and R. Casalini. Supercooled dynamics of glass-forming liquids and polymers under hydrostatic pressure. Reports on Progress in Physics, 68(6):1405–1478, Jun 2005.
- Thomas B. Schrøder, Ulf R. Pedersen, and Jeppe C. Dyre. Density scaling as a property of strongly correlating viscous liquids. arXiv.org, 0803.2199v2 [cond-mat.soft], Jun 2008.
- S. Takahara, M. Ishikawa, O. Yamamuro, and T. Matsuo. Structural relaxations of glassy polystyrene and o-terphenyl studied by simultaneous measurement of enthalpy and volume under high pressure. Journal of Physical Chemistry B, 103(5):792–796, Feb 4 1999.