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Oliver Dyer

PhD student, theory group

Supervisor: Robin C. Ball

email: o dot dyer at warwick dot ac dot uk

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PhD Research

Motivations and Overview

Brownian dynamics algorithms simulate mesoscopic soft-matter systems by accounting for the uninteresting solvent molecules with hydrodynamic equations. This greatly reduces the degrees of freedom simulations. However, calculating the evolution of the system can still be computationally expensive when long range hydrodynamic interactions (HIs) are included, and large systems are still out of reach.

The focus of my PhD is to develop an algorithm that replaces the explicit calculation of HIs with a Monte Carlo simulation which obeys hydrodynamics implicitly. This approach leads to a significantly simpler and faster algorithm with a cost that scales to large systems as N\ln(N).

Summary of Wavelet Monte Carlo Dynamics (WMCD)

Basic theory

In WMCD the HI tensor that correlates particle motions, the so called Oseen tensor, is written as the integral over wavelets

\frac{1}{8\pi\eta r_{ij}} \left(\textbf{I}+ \hat{\textbf{r}}_{ij}\otimes\hat{\textbf{r}}_{ij}\right) =\int \frac{d\lambda}{\lambda} \frac{d^{3}\textbf{b}}{\lambda^{3}} d^{2}\hat{\textbf{p}} \frac{1}{\sqrt{\lambda}}\textbf{w} \left(\frac{\textbf{r}_{i}-\textbf{b}}{\lambda},\hat{\textbf{p}}\right) \otimes \frac{1}{\sqrt{\lambda}} \textbf{w} \left(\frac{\textbf{r}_{j}-\textbf{b}}{\lambda},\hat{\textbf{p}}\right).

(\textbf{w} = wavelet, \textbf{b} = wavelet centre, \lambda = radius, \hat{\textbf{p}} = orientation.)

Monte Carlo integration translates to a simulation using these wavelet displacements as Monte Carlo moves (see right for an example with particles inside the wavelet rotated around its axis) that evolves the system while ensuring the particle motions are correlated by the Oseen tensor.

 moveimage.png
Periodic boundary conditions
Periodic systems must account for HIs of all images because of the long range nature of the Oseen tensor. In place of the expensive Ewald summation used in Brownian dynamics algorithms, we simply need to add rare plane wave moves at negligible extra cost (see cost plots below). Example plane wave move
Computational cost

The big success of WMCD is how its cost (time to run a simulation) scales with system size, with a significant improvement on established algorithms in fractal (dilute) systems with linear scaling, and a very competitive N\ln N in periodic systems.

Fractal systems (e.g. isolated polymer) Homogeneous systems
fractal cost homogenous cost

Note: A_{0}^{2}~time step; 'w' refers to a pure wavelet simulation, while 'w+F' includes plane wave moves.

LB = Lattice Boltzmann and BD = Brownian dynamics.

Smart WMCD

WMCD can now use a 'smart Monte Carlo' algorithm, biasing moves in the direction of forces present. The effects of this change include:

  • lower rate of move rejection;
  • more accurate data at a given time step, or
  • larger time step for a given accuracy, decreasing computational cost;
  • easy implementation of non-conservative forces.

Applications of WMCD

Time dependence of polymer diffusivity

The diffusion of isolated polymer chains is a complex problem thanks to long-range hydrodynamic interactions coupling all internal forces and thermal fluctuations to the motion of all particles along the chain.

We have undertaken the most comprehensive investigation into this problem to date, with data able to observe universal short- and long-time behavior in the centre of mass velocity autocorrelation (= time derivative of diffusivity). Here is shown the data for Gaussian chains:

Short-time Long-time
ata0_a2plot.png Long-time collapse of velocity autocorrelation
Rheological systems

WMCD is applicatble beyond equilibrium systems. For example externally imposed flows can be introduced via affine deformations of the simulation coordinates (follow this link for an example: Shear flow)

In collaboration with J. Ravi Prakash, WMCD is being used to look at the rheological properties of semi-dilute DNA (i.e. long polymer) solutions.

Simulations of micro-swimmers

Multipolar flow fields are commonly used to approximate the HIs between micro-swimmers and their surroundings. With the smart Monte Carlo adaptation these flow fields can be simulated in WMCD, extending the scope of the algorithm to simulations in this exciting field.

The scope of WMCD

Beyond the examples above, it is worth highlighting the full scope of systems WMCD is able to simulate in its current state.

Within the realm of systems of Brownian particles in low Reynolds number fluids on diffusive time-scales:

  • periodic boundary conditions or `unbounded';
  • range of concentrations from ultra-dilute to the dense end of semi-dilute;
  • equilibrium or active systems, including
    • shear and extensional solvent flows and
    • micro-swimmers (with simple multipole flow fields or approximations to the squirmer model);
  • Brownian particles may or may not be connected into macromolecules, with soft or hard bond and excluded volume potentials between them;
    • in the Oseen tesnor is regularised (close to Rotne-Prager), increasing the accuracy of the beahvour of nearby particles
  • Brownian particles may be mono- or polydisperse.

The most notable shortcoming is with hard walls, for which WMCD is currently unable to implement the correct mobility tensor. It remains possible for future work to find a solution to this.

Posters and presentations

Poster Feb 2016

Poster Summer 2017 (Awarded the Best Poster prize at the Statistical Physics (Sigma Phi) conference 2017)

Talk slides from Open Statistical Physics 2018

Publications