Optimization of a Brownian-dynamics algorithm for semidilute polymer solutions

Abstract

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(N 1.8 ), 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 θ-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 optimisations should be possible.

Wavelet Monte Carlo dynamics: A new algorithm for simulating the hydrodynamics of interacting Brownian particles

“…Unlike the Ewald sum in BD simulations, which splits the workload so that the cost of the Fourier space calculation is comparable to the position space part, 8 there is no requirement for the total cost of WMCD Fourier moves to be comparable to the cost of wavelet moves. Indeed Figs.…”

“…3, the cost for the LB algorithm for identical semi-dilute systems is seen to be a factor of order 1 times faster than the WMCD algorithm, with the exact factor depending on both A 0 and N. (The equivalent BD costs are several orders of magnitude larger. 8 ) Here we note that a fair comparison would use the same hardware for each algorithm, which was not done here, and the ratio between the costs is therefore only a rough indicator of their relative performance. Moreover our code was far from optimised with particle neighbour lists (the dominant contribution to the cost) being recomputed every move rather than updated only as required; we did not exploit multiple processors, and we compiled with just the standard gcc compiler.…”

“…Finally, to match the systems in Ref. 8, whenever a system is called "semi-dilute" it consists of polymer chains of length N b = 10 beads and a global bead concentration N/L 3 = 0.625, where L is the side length of our simulation box.…”

“…This includes particles in the solvent as well as those of interest. For systems of fixed concentration, this leads to the cost scaling as N, 8 though the overhead cost of moving the solvent molecules limits the feasible system size, especially as the systems become more dilute. When considering a non-periodic system, the scaling rises.…”

“…Fixman's algorithm is well known to cut the scaling of this decomposition down to N 2.25 from the naive N 3 , 9 and several methods have since been put forward to reduce the scaling further. 8,10 It has even been reduced to or near N ln N in some cases, although these approaches are only valid for bounded systems, 11 or introduce errors to allow more efficient computation via particle mesh Ewald techniques 12,13 or sparse arrays. 14 In this work, we present an entirely different approach to BD, using a Monte Carlo (MC) algorithm to bypass explicit calculations with the mobility tensor altogether.…”