Equilibrium and nonequilibrium molecular dynamics methods to compute the first normal stress coefficient of a model polymer solution

The zero-shear first normal stress coefficient, ${\mathrm{\Psi }}_{1,0}$, is an important viscometric function used throughout rheology. It can be directly computed by using various molecular dynamics simulation methods. Homogeneous shear algorithms used in nonequilibrium molecular dynamics simulations require some type of synthetic thermostat to remove dissipated heat resulting from the work done by the shear. While it has been shown conclusively that the linear transport coefficients are unaffected by synthetic thermostats, some doubt remains for the first normal stress coefficient. This problem can be circumvented by performing inhomogeneous Poiseuille flow simulations where heat is removed by thermal conduction to thermostatted walls. Alternatively, we can use equilibrium computations to calculate the properties of interest. This is significantly more difficult for nonlinear properties than it is for the linear transport properties. Here we present three different methods of calculating the first normal stress coefficient for a model polymer solution; in one, the computations are performed in equilibrium conditions (Coleman-Markovitz equation) and in the other two they are carried out in nonequilibrium conditions (SLLOD and Poiseuille flow). We find that both nonequilibrium methods produce matching results, with ${\mathrm{\Psi }}_{1,0}=170\pm 2$ for SLLOD and ${\mathrm{\Psi }}_{1,0}=172\pm 3$ for Poiseuille flow, regardless of whether the fluid of interest is directly or indirectly thermostatted. The Coleman-Markovitz result is difficult to resolve, and much less accurate, with ${\mathrm{\Psi }}_{1,0}\approx 190$, although it still may not be fully converged, even after extensive computations. We suggest that the Coleman-Markovitz equation is unsuitable for routine determination of the first normal stress coefficient by molecular simulation.

Figure 1

Visualization of the simulated system. The explicit solvent atoms have been displayed as a grayed out continuum for clarity. The polymer is shown in green and the walls are shown in blue. In this image, the $x$ direction is vertical, the $y$ direction horizontal, and the $z$ direction in and out of the page.

Figure 2

The cumulative integral of the first moment of the stress autocorrelation function. Converged values correspond to the first normal stress coefficient, ${\mathrm{\Psi }}_{1,0}$, given by Eq. (6). Each line corresponds to a different value of the total averaging time, where the symbol terminating the line corresponds to the following values of the averaging time: $t_{\mathrm{av}}^{\square }=4\times {10}^7$, $t_{\mathrm{av}}^{\diamond }=6\times {10}^7$, and $t_{\mathrm{av}}^{◯}=8\times {10}^7$.

Figure 3

Value of the first normal stress coefficient as a function of shear rate, calculated from SLLOD simulations and Eq. (11). The dashed line shows a quadratic fit to the data.

Figure 4

The value of $P_{xx}$ across the channel for body forces of $F^e={10}^{-3}$ (circles), $5\times {10}^{-4}$ (diamonds), ${10}^{-4}$ (crosses), 0 (squares). Some data points have been omitted for visual clarity.

Figure 5

The value of $P_{yy}$ across the channel for body forces of $F^e={10}^{-3}$ (circles), $5\times {10}^{-4}$ (diamonds), ${10}^{-4}$ (crosses), 0 (squares). Some data points have been omitted for visual clarity.

Figure 6

The value of $P_{yy}-P_{xx}$ across the channel for body forces of $F^e={10}^{-3}$ (circles), $5\times {10}^{-4}$ (diamonds), ${10}^{-4}$ (crosses), 0 (squares).

Figure 7

The value of $\dot{\gamma }$ across the channel for a body forces of $F^e={10}^{-3}$ (circles), $5\times {10}^{-4}$ (triangles), ${10}^{-4}$ (crosses), 0 (squares). Some data points have been omitted for visual clarity.

Figure 8

Values of the first normal stress coefficient calculated from Eq. (11) for body forces of $F^e={10}^{-3}$, $5\times {10}^{-4}$, and ${10}^{-4}$. The dotted line gives the value of the first normal stress coefficient extracted from Fig. 3.