Thinning Approximation for Calculating Two-Dimensional Scattering Patterns in Dissipative Particle Dynamics Simulations under Shear Flow

Abstract

Modifications to improve thinning approximation (TA) were considered in order to calculate two-dimensional scattering patterns (2DSPs) for dissipative particle dynamics (DPD) simulations of polymer melts under a shear flow. We proposed multipoint TA and adaptive TA because the bond lengths in DPD chains vary widely when compared to those in Kremer⁻Grest (KG) chains, and the effectiveness of these two types of TA for the two major DPD parameter sets were investigated. In this paper, we report our findings on the original DPD model with soft bonds and that with rigid bonds. Based on the behavior of the 2DSPs and the distribution of orientations of the bond vectors, two spot patterns originating from the oriented chain correlations were observed when distinct distributions of the highly oriented bond vectors in the shear direction were obtained. For multipoint TA, we concluded that at least two additional midpoints ( n mid ≥ 2 ) are required to clearly observe the two spot patterns. For adaptive TA, a dividing distance of l ATA ≤ 0.4 is sufficient for clear observation, which is consistent with the requirement of n mid ≥ 2 for multipoint TA.

Keywords: dissipative particle dynamics (DPD) simulations; shear deformation; thinning approximation (TA); two-dimensional scattering patterns (2DSPs).

Figure 1

Schematic illustrations of (a) the thinning approximation (TA) process; (b) the velocity profile obtained at a constant shear rate; and (c) the circular average; two-dimensional scattering patterns (2DSPs) of Kremer–Grest (KG) chains under shear flow (d) with TA and (e) without TA. The presented 2DSPs were partially reprinted from an earlier paper, Hagita, K.; Murashima, T.; Takano, H.; Kawakatsu, T. J. Phys, Soc. Jpn. 2017, 86, 124803. [44], with the permission.

Figure 2

Probability, $P\left(\left|\theta \right|\right),$ versus $⎣5\left|\theta \right|/\mathsf{\pi }⎦$ for dense polymer melts with (a) rigid and (b) soft bonds. Here, $⎣X⎦$ is the floor function of $X$.

Figure 3

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with rigid bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.002,6\right)$. The 2DSPs were calculated with and without thinning approximation (TA): (a) Original method; (b–e) $n_{\mathrm{mid}}$ additional points, where $n_{\mathrm{mid}}=1$ to 4; (f) azimuthal intensities averaged over from q = 6.0 to 8.0 with 0.1 steps.

Figure 4

Schematic illustrations of (a) DPD chains and (b) additional midpoints for thinning approximation of a dense polymer melt. The circles denote the overlapping DPD particles. The red bold lines in (b) denote the series of additional midpoints. Based on the approximation, a clear picture of the chains was obtained.

Figure 5

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with rigid bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.004,12\right)$. The 2DSPs were calculated with and without thinning approximation (TA): (a) Original method; (b–e) $n_{\mathrm{mid}}$ additional points, where $n_{\mathrm{mid}}=1$ to 4; (f) azimuthal intensities averaged over from q = 6.0 to 8.0 with 0.1 steps.

Figure 6

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with rigid bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.01,30\right)$. The 2DSPs were calculated with and without thinning approximation (TA): (a) Original method; (b–e) $n_{\mathrm{mid}}$ additional points, where $n_{\mathrm{mid}}=1$ to 4; (f) azimuthal intensities averaged over from q = 6.0 to 8.0 with 0.1 steps.

Figure 7

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with soft bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.005,6\right)$. The 2DSPs were calculated with and without thinning approximation (TA): (a) Original method; (b–e) $n_{\mathrm{mid}}$ additional points, where $n_{\mathrm{mid}}=1$ to 4; (f) azimuthal intensities averaged over from q = 6.0 to 8.0 with 0.1 steps.

Figure 8

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with the soft bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.01,12\right)$. The 2DSPs were calculated with and without thinning approximation (TA): (a) Original method; (b–e) $n_{\mathrm{mid}}$ additional points, where $n_{\mathrm{mid}}=1$ to 4; (f) azimuthal intensities averaged over from q = 6.0 to 8.0 with 0.1 steps.

Figure 9

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with rigid bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.004,12\right)$. The 2DSPs were calculated for $l_{\mathrm{ATA}}=0.2,0.3,0.4,0.5,0.6$, and $0.7$.

Figure 10

Two-dimensional scattering patterns (2DSPs) of DPD simulations of a dense polymer melt with soft bonds for the cases with $\left(\dot{\gamma },Wi\right)=\left(0.01,12\right)$. The 2DSPs were calculated for $l_{\mathrm{ATA}}=0.2,0.3,0.4,0.5,0.6$, and $0.7$.