Optimization of a Brownian-dynamics algorithm for semidilute polymer solutions

Simulating the static and dynamic properties of semidilute polymer solutions with Brownian dynamics (BD) requires the computation of a large system of polymer chains coupled to one another through excluded-volume and hydrodynamic interactions. In the presence of periodic boundary conditions, long-ranged hydrodynamic interactions are frequently summed with the Ewald summation technique. By performing detailed simulations that shed light on the influence of several tuning parameters involved both in the Ewald summation method, and in the efficient treatment of Brownian forces, we develop a BD algorithm in which the computational cost scales as $O\left(N^{1.8}\right)$, where $N$ is the number of monomers in the simulation box. We show that Beenakker's original implementation of the Ewald sum, which is only valid for systems without bead overlap, can be modified so that $\theta$ solutions can be simulated by switching off excluded-volume interactions. A comparison of the predictions of the radius of gyration, the end-to-end vector, and the self-diffusion coefficient by BD, at a range of concentrations, with the hybrid lattice Boltzmann–molecular dynamics (LB-MD) method shows excellent agreement between the two methods. In contrast to the situation for dilute solutions, the LB-MD method is shown to be significantly more computationally efficient than the current implementation of BD for simulating semidilute solutions. We argue, however, that further optimizations should be possible.

Figure 1

Periodic boundary conditions in two dimensions: demonstration of the distance vector between two beads.

Figure 2

Execution time scaling for the real space and the Fourier space sums for (a) constant $r_{\mathrm{c}}$ and (b) constant $L/r_{\mathrm{c}}$.

Figure 3

(a) Total execution time vs exponent $x$ for various $N$. (b) Power law scaling for $T_R$, $T_F$, and $T_{\mathrm{opt}}$ at ${\left(r_{\mathrm{c}}^{\mathrm{E}}\right)}_{\mathrm{opt}}$.

Figure 4

CPU time scalings for (a) HMA and (b) LMA. Symbols represent simulations results, and the lines are analytical estimates for the total time.

Figure 5

Comparison between HMA and LMA for (a) computer memory requirement and (b) CPU time required for a single time-step computation.

Figure 6

Validation of static properties under $\theta$ conditions: (a) mean-square end-to-end distance $\langle R_e^2\rangle$ and (b) mean-square radius of gyration $\langle R_g^2\rangle$. Symbols indicate simulation data, while solid lines represent the analytical results given by Eqs. (38) and (39).

Figure 7

Long-time self-diffusion coefficient under $\theta$-solvent conditions. Symbols indicate simulation data obtained with the current multiparticle algorithm, while the solid line represents the value obtained by simulating the dynamics of a single chain in a dilute solution. The circle symbol on the $y$ axis is the value obtained by extrapolating the finite concentration results to the limit of zero concentration.

Figure 8

Comparison of the CPU time required by the LB and BD systems for a wide range of system sizes $N$, at concentration $c/c^{☆}=1.2$, for the equivalent of one LB time unit.