(726h) New Computational Methods for Rapid Simulation of Hydrodynamic Interactions in Polymer Solutions

Coarse-grained simulations of soft materials and complex fluids such as polymer solutions require rapid calculation of the hydrodynamic forces exerted on suspended, microscopic constituents (particles). Because of the small length scales associated with polymers, low Reynolds number flows dictate these forces. Inertia of the fluid and of the constituents is negligible on time scales that are relevant for rheological measurements. Many coarse-grained simulation methods for modeling soft materials construct an explicit model of the solvent, determining both the position and momentum of the solvent and suspended particles as a function of time. These include Lattice-Boltzmann, Dissipative Particle Dynamics, Stochastic Rotation Dynamics, Multi-particle Collision Dynamics, and others. Always, degrees of freedom associated with the solvent are neglected, and a stochastic force satisfying the fluctuation dissipation theorem must be exerted on all components of the dispersion.

The explicit solvent models were developed in part because of difficulties performing rapid calculations with implicit solvent models, such as Brownian Dynamics, in which the hydrodynamic interactions between particles are modeled deterministically and depend only on the relative configuration of particles. The chief advantage of explicit solvent models comes from sampling of the stochastic forces, which is no more expensive computationally than generating a string of random numbers. For implicit solvent models, the stochastic forces must be drawn from a normal distribution whose covariance is a complicated function of the particle configuration. For a system of interacting N particles, drawing a single sample requires O(N^3) operations, if numerically exact techniques from linear algebra are employed. So-called “fast” methods can approximate the sampling with roughly O(N^m log N) computational complexity, where m is a coefficient greater than one which depends on the configuration of the particles. Typical values of m range from 1.25 to 1.5. This puts a serious limit on application of implicit solvent methods to large-scale simulation.

In the presented work, we will demonstrate an optimal and spectrally accurate, fast method for calculation of thermal forces in implicit solvent simulations of soft materials such as Brownian Dynamics. The computational complexity of this approach is O(N (log N)^(d/(d+3))), where d is the fractal dimension of the polymer microstructure being modeled. Remarkably, this new approach adapts to the structure of the material under study by properly balancing the computational effort spent evaluating the hydrodynamics on to different spatial scales. The separation between local and global scales is set by computational necessity and enables optimization of the algorithms performance. We term this algorithm optimal because in the periodic geometries needed to model viscometric flows of polymer solutions, log-linear computational scaling is the best available. Up to 10^6 discrete elements can be simulated with this methodology on a single consumer grade graphics processing unit. Applications of this approach to macromolecular solutions and systematic coarse-graining of suspended polymers are demonstrated.