# Oliver Dyer

#### PhD student, theory group

Supervisor: Robin C. Ball

email: o dot dyer at warwick dot ac dot uk

## 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.
##### 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).
##### 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

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
##### 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 active matter

Multipolar flow fields are commonly used to approximate the HIs between micro-swimmers and their surroundings. At simplest pusher- and puller-type micro-swimmers can be represented with two equal and opposite forces places asymmetrically about their centre:

Such forces and flow fields are simple to include with the smart Monte Carlo adaptation of WMCD, extending the scope of the algorithm to simulations in this exciting field. Furthermore:

• Hydrodynamic coupling of rotations (relevent for oriented swimmers) can be included in a way anologous to the HIs of translations.
• Thermal diffusion of both position and orientation - often omitted in active matter simulations - is included by default with hydrodynamic coupling.
##### The (current!) 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:

• a regularisation of the Oseen tensor (close to Rotne-Prager) is generated implicitly by WMCD moves;
• regularised translation-rotation and rotation-rotation coupling can also be generated within the same wavelet moves;
• boundary conditions can be either periodic or `unbounded';
• hard walls/surfaces are yet to be implemented, but can be included by placing particles at the wall;
• 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 multipole flow fields);
• particles may or may not be connected into macromolecules, with soft or hard bond and excluded volume potentials between them;
• systems may be mono- or polydisperse;

## Posters and presentations

Poster Feb 2016

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

Poster May 2018

Talk slides from Open Statistical Physics 2018