Determination of the Thermodynamic Scaling Exponent for Relaxation in Liquids from Static Ambient-Pressure Quantities

An equation is derived that expresses the thermodynamic scaling exponent, $\gamma$, which superposes relaxation times $\tau$ and other measures of molecular mobility determined over a range of temperatures and densities, in terms of static physical quantities. The latter are available in the literature or can be measured at ambient pressure. We show for 13 materials, both molecular liquids and polymers, that the calculated $\gamma$ are equivalent to the scaling exponents obtained directly by superpositioning. The assumptions of the analysis are that the glass transition $T_g$ is isochronal (i.e., ${\tau }_{\alpha }$ is constant at $T_g$, which is true by definition) and that the pressure derivative of the glass temperature is given by the first Ehrenfest relation. The latter, derived assuming continuity of the entropy at the glass transition, has been corroborated for many glass-forming materials at ambient pressure. However, we find that the Ehrenfest relation breaks down at elevated pressure; this limitation is of no consequence herein, since the appeal of the new equation is its applicability to ambient-pressure data. The ability to determine, from ambient-pressure measurements, the scaling exponent describing the high-pressure dynamics extends the applicability of this approach to a broader range of materials. Since $\gamma$ is linked to the intermolecular potential, the new equation thus provides ready access to information about the forces between molecules.

Figure 1

Scaling exponent calculated from Eq. (7) versus the value of $\gamma$ obtained by superposition of relaxation times. PMMA, polymethymethacrylate (high polymer); o-PMMA, oligomeric PMMA; PS, polystyrene; PCHMA, polycyclohexylmethacrylate; PVAC, polyvinylacetate; PMA, polymethacrylate; OTP, ortho-terphenyl; PED, phenolphthalein dimethylether; DC704, tetramethyl tetraphenyl trisiloxane; OTP-OPP, mixture of 67% OTP and 33% ortho-phenylphenol; PCB, chlorinated biphenyl. The dashed line represents equivalence of the $\gamma$. References for the data are in Table 1.

Figure 2

Thermal pressure coefficient at $T_g$ as a function of pressure determined experimentally (filled symbols) and calculated using the first Ehrenfest relation (open symbols). The latter’s validity is limited to ambient pressure.

Figure 3

Configurational entropy of three liquids measured at various temperatures and pressures plotted versus the scaling variable with the indicated value of $\gamma$. The plotted values are at a reference point $S_{\text{ref}}$ after subtraction of $S_c$; thus, the ordinate can be negative (see Refs. [46, 47]).