Matrix-free Brownian dynamics simulation technique for semidilute polymeric solutions

Evaluating the concentration dependence of static and dynamic properties of macromolecules in semidilute polymer solutions requires accurate calculation of long-range hydrodynamic interactions (HI) and short range excluded volume (EV) forces. In conventional Brownian dynamics simulations (BDS), computation of HI necessitates construction of a dense diffusion tensor commonly performed via Ewald summation. Krylov subspace techniques allow efficient decomposition of this tensor [computational cost scales as $O\left(N^2\right)$, where $N$ is the total number of beads in bead-spring representation of macromolecules in a simulation box] and computation of Brownian displacements in the box. In this paper, a matrix-free approach for calculation of HI is implemented which leads to $O\left(NlogN\right)$ scaling of computational expense. The fidelity of the algorithm is demonstrated by evaluating the asymptotic value of center-of-mass diffusivity of polymer molecules at very low concentrations and their radius of gyration scaling as a function of number of beads for dilute and semidilute solutions (with concentrations up to 5 times the overlap concentration). In turn, a favorable comparison between our results and the blob theory is shown.

Figure 1

Spreading the force on a regular mesh, where the cells represent the grid points. The filled cell is the nearest point to the particle and the cells bounded by the dash-dotted line are $p^3$ grid points at which the force on the particle is distributed.

Figure 2

(a) The mean-square displacement and (b) the diffusivity of center of mass in $\theta$ solvent obtained using both Ewald and matrix-free techniques. The filled and open symbols represent the results for the Ewald and the matrix-free algorithms, respectively.

Figure 3

Mean-square radius of gyration as a function of number of beads in a multichain system with $c/c^{*}=0.1$ and 5. The error bars are smaller than the symbols.

Figure 4

Execution time per time step for Ewald as well as matrix-free algorithms.