Computer simulation of thermal conductivity in vulcanized polyisoprene at variable strain and temperature

We study the thermal conductivity tensor in an atomistic model of vulcanized cis-1,4-polyisoprene (PI) rubber via molecular dynamics simulations. Our polymer force field is based on V. A. Harmandaris et al. [J. Chem. Phys. 116, 436 (2002)], whereas the polymerization algorithm follows the description in J. Hager et al. [Macromolecules 48, 9039 (2015)]. The polymer chains are chemically cross linked via sulfur bridges of adjustable cross-link density. A volume-conserving uniaxial strain of up to 200% is applied to the systems. The widely used GROMACS simulation package is adapted to allow using the Green-Kubo approach to calculate the thermal conductivity tensor components. Our analysis of the heat flux autocorrelation functions leads to the conclusion that the thermal conductivity in PI is governed by short-lived phonon modes at low wave numbers due to deformation of the monomers along the polymer backbone. Applying uniaxial strain causes increased orientation of monomers along the strain direction, which enhances the attendant thermal conductivity component. We find an exponential increase of the conductivity in stretch direction in terms of an attendant orientation order parameter. This is accompanied by a simultaneous decline of thermal conductivity in the orthogonal directions. Increase of the cross-link density only has a weak effect on thermal conductivity in the unstrained system, even at high cross-link density. In the strained system we do observed a rising thermal conductivity in the limit of high stress. This increase is attributed to enhanced coupling between chains rather than to their orientation.

Figure 1

Top: Type and position of the sulfur bridges in this work. Bottom: Bulk-polymer density vs concentration of sulfur in phr [parts (by weight) per hundred rubber] at $T=300$ K and $P=1$ bar. This systems consists of 40 polymer chains containing 200 monomers each. The experimental densities are taken from McPherson [28].

Figure 2

Example showing one particular system in the undeformed state (left) and at 150% strain (right). Selected polymer chains are highlighted in different colors. Yellow dumbbells indicate sulfur cross links. The system consists of 20 PI chains containing 250 monomers each and 500 sulfur cross links.

Figure 3

(Double) bond orientation distribution functions at different strain values, $u=[0.0,0.5,1.0,1.5,2.0]$. Notice that $\theta$ refers to the angle between the double bond and the direction of stretch.

Figure 4

Top: Root mean square of $R_{\alpha }$, divided by the attendant box dimension, $L_{\alpha }$, vs strain, $u$. Bottom: $c_{\alpha }$ vs $u$ corresponding to the panel above.

Figure 5

Top: Diagonal components of the thermal conductivity tensor at $T=300$ K vs strain, $u$. The experimental data measured along the long direction are taken from Tautz [21]. The number of cross links is about 100 or about 1.9 phr. Bottom: Thermal conductivity along the stretch direction for systems with very different cross-link concentrations. Notice that the over bars indicate averages over four independent measurements.

Figure 6

Time autocorrelation functions of the heat current components $J_{\alpha }\left(t\right)$ for different strain values. Top: $\alpha =x$. Bottom: $\alpha =z$, where $z$ is the direction of stretch. The temperature is 300 K in all cases. Notice that the curves are all normalized to $c_0=\langle J_z\left(0\right)J_z\left(0\right)\rangle$ at $u=1.625$.

Figure 7

${\mathrm{\Lambda }}_{xx}\left(t^{\prime }\right)$ (top) and ${\mathrm{\Lambda }}_{zz}\left(t^{\prime }\right)$ (bottom) vs $t^{\prime }$, corresponding to the autocorrelation functions of the previous figure. The quantity $t^{\prime }$ is the upper limit of the integral in Eq. (A21) and the temperature is 300 K.

Figure 8

The function of Eq. (1) for $\omega =1$ and $\tau =1$.

Figure 9

Heat flux autocorrelation functions obtained in the LJ system at two different state points, i.e., $T=1.39$, $\rho =0.484$ and $T=1.39$, $\rho =0.0186$. The index $N$ means that a factor $\sqrt{V/\left(3T^2\right)}$ is absorbed into each $\vec{J}$; i.e., the area under the curves is the respective isotropic thermal conductivity in LJ units.

Figure 10

Normalized heat flux autocorrelation functions obtained for the three indicated rigid water models at $T=300$ K and ambient pressure using molecular dynamics simulations.

Figure 11

${\mathrm{\Lambda }}_{zz}\left(u\right)$ vs $n_{\parallel }\left(u\right)/n_o$ (right side) and $[{\mathrm{\Lambda }}_{xx}\left(u\right)+{\mathrm{\Lambda }}_{yy}\left(u\right)]/2$ vs $n_{\perp }\left(u\right)/n_o$ (left side) based on the simulation results shown in Figs. 3 and 5 (upper panel). The lines through the data are explained in the text.

Figure 12

Temperature dependence of the thermal conductivity for $u=0$. The three simulated systems are distinguished by the sulfur content. The experimental data [20] encompass hard and soft rubber, where hard rubber (open circles) corresponds to about 47 phr of sulfur and soft rubber (filled circles) means about 2.5 phr of sulfur, respectively.

Figure 13

Relation between frequency and temperature of the segment relaxation peak ($\alpha$ process) for polyisoprene. The simulated points are compared to results obtained via dielectric spectroscopy [(i), (ii), (iii)] as well as mechanical measurements (iii). Experimental molecular weights are between $97.0\mathrm{kg}\mathrm{mol}{}^{-1}$ and $504.0\mathrm{kg}\mathrm{mol}{}^{-1}$. This figure is taken from Meyer et al. [39].