Isomorph theory prediction for the dielectric loss variation along an isochrone

This paper derives a prediction for the variation of the amplitude of the dielectric loss from isomorph theory, and presents an experimental test of the prediction performed by measuring the dielectric-relaxation behavior of the van der Waals liquid 5-phenyl-4-ether (5PPE). The liquid is studied at isochronal states in the temperature range $266-333$ K and pressure range $0.1-300$ MPa, for relaxation times around $10^{-3}$ s and $10^{-4}$ s. From the isomorph statement that there is structural and dynamic invariance of isomorph states in reduced units for Roskilde simple liquids we derive four equivalent isomorph-invariant terms, one of which is used in analyzing our data. It is the frequency-dependent term $\chi_{e}(f) \rho^{\gamma-1}$, with electric susceptibility $\chi_{e}$, density $\rho$, and density-scaling factor $\gamma$. Due to the unique design of our experimental setup, we obtain dielectric loss data where the amplitude is reproducible $\pm 0.1 \%$. We moreover find that the empty capacitance of the capacitor cell is stable within $\pm 0.3 \%$ in our measuring range and can be assumed to be constant. Using this we predict for two isomorph states there is $-C_{2}"(f)= -C_{1}"(f) \left(\rho_1 / \rho_2 \right)^{\gamma-1}$ to scale the negative imaginary capacitance, where $C_{1}$ is the capacitance measurement at ambient pressure and $C_{2}$ is the predicted capacitance at elevated pressure. We visually compare the predicted and measured plots and there is good match between the two plots among the 42 pairs of isochronal states from the measurement.