Testing the minimum thermal conductivity model for amorphous polymers using high pressure

Pressure dependence of thermal conductivity provides a critical test of the validity of the model of the minimum thermal conductivity for describing heat transport by molecular vibrations of an amorphous polymer. We measure the pressure dependence of the thermal conductivity Λ($P$) of poly(methyl methacrylate) using a combination of time-domain thermoreflectance and SiC anvil-cell techniques. We also determine Λ($P$) from a computational model of amorphous polystyrene. In both cases, Λ($P$) is accurately predicted by the minimum thermal conductivity model via the pressure dependence of the elastic constants and density.

Figure 1

Pressure dependence of the (a) Brillouin frequency and (b) $C_{11}$ of a spun-cast layer of PMMA. $C_{11}$ is derived from the Brillouin-frequency data using a self-consistent equation of state of PMMA and the assumptions that the Poisson ratio is constant and the refractive index follows the Lorentz-Lorenz equation. Data for Polydimethylsiloxane (PDMS)[40] and previously estimated $C_{11}$ $=$ 110 GPa $+$4$P$ of Al[41] are plotted for comparison.

Figure 2

Measurements of the thermal conductivity of PMMA brushes (solid symbols) and a spun-cast layer (open diamond) as a function of pressure. The uncertainty in the thermal conductivity and pressure measurements are ≈10% and 0.2 GPa, respectively. Ar was the pressure medium for all measurements except for those of 13-nm brushes and 10-nm spun-cast layer, where ${\mathrm{H}}_2$O was used. The dashed line shows the predicted thermal conductivity of PMMA based on the minimum thermal conductivity model, the pressure dependence of the atomic density $n$, and the elastic constant $C_{11}$ obtained by a polynomial fit. Data for bulk PMMA by Andersson and Ross (open circle)[10] and by Cahill and Pohl and Putnam et al. (open square)[30, 31] are included for comparison. Lines between Andersson's data are added to emphasize that the pressure dependence of the prior data is similar to that of our measurements in the low-pressure regime.

Figure 3

Pressure dependence of the thermal conductivity of PS by MD simulations (solid circles). The solid line shows the predicted thermal conductivity by the model of the minimum thermal conductivity. The pressure dependence of the thermal conductivity predicted by the Leibfried-Schlömann equation using the scaling ${\omega }_D\propto \sqrt{C_{11}}$ is shown as the dashed line.