Timo Nieminen

Honours project: Single and multiple scattering and the Rayleigh hypothesis

Hilbert space methods, such as the T-matrix method, for the scattering of electromagnetic or other waves by particles typically involve representing the fields as sums or integrals of a basis set of modes. Two mathematical issues need to be considered in these methods. First, the sum or integral over the of modes used is only guaranteed to converge to the fields in certain regions. For example, the scattered field represented in spherical wave modes is only guaranteed to converge outside the circumscribing sphere enclosing the scattering particle, and not between the circumscribing sphere and particle surface. Second, the infinite set of modes is truncated for practical computations, and might not converge subject to such truncation, even if convergence is guaranteed given infinite modes. Despite these two issues, the scattering problem can often be solved, giving a correct and convergence result for the far field. Some methods assume that the fields converge everywhere outside the scattering particle, even though such convergence is not guaranteed - this is the "Rayleigh hypothesis". Other methods will give essentially identical results in the far field without making such assumptions.

You will: Compare the near fields for single and multiple scattering using Hilbert space methods and other, Rayleigh hypothesis free, methods, Determine conditions under which we can obtain good far field result field without convergence of the near field.

Project type: Computational and mathematical

Honours project: Physical versus behavioural interactions in collective motion in active matter

Self-propelled active matter particles take energy from their environment and use it for motion and/or other purposes. Interaction between the active matter particles can result in collective motion such as flocking, schooling, and swarming, as seen with birds, fish, insects, and bacteria. The interactions can be behavioural ("Which way are my neighbours flying? How close are they?") or physical (e.g., bacteria). One important question is to what extent can artificial active matter particles, with purely physical interactions between them, mimic the complex collective motion driven by behaviour. Light can be uses as the energy source for artificial active matter particles, with optical and thermal forces producing motion. Interaction can be optical, hydrodynamic, or thermal.

You will: Develop models of physical interactions between active matter particles that provide both realistic accuracy and computational simplicity Use these models to compare collective behaviour in active matter based on simple behavioural models and physical models

Project type: Computational and mathematical

Honours project: Energy considerations in bacterial locomotion

General principles of motion, such as driving and resistive forces, and energy requirements, can be used study the scaling of the motion of organisms with size, fluid properties, etc. Such models can apply across many orders of magnitude of size, etc., from bacteria to macroscopic animals.

You will:

• Review existing models, including those developed bacterial for motion, and other organisms
• Use suitable methods, modified as appropriate, to study the effect of interactions with surfaces (and other bacteria? on the motion of bacteria such as E. coli
• Compare cases such as the swimming of single-flagellated E. coli and multi-flagellated E. coli, motion in bulk fluids vs motion next to surfaces, and motion in thin films, etc.

Project type: Computational and mathematical, can include experimental work

Honours project: Computational methods for multiple scattering

In principle, brute-force methods such as finite-difference time-doman (FDTD), the finite element method (FEM), and the discrete dipole approximation (DDA) allow us to computationally model multiple scattering problems. However, the computational demands of solving such problems for many particles can make them thoroughly infeasible. Methods making use of the single-scattering solutions for the individual particles can be much faster. However, the convergence and correctness of some of those memthods are unknown.

You will:

• Explore the computational and mathematical behaviour of methods for multiple scattering that build on single-scattering models, with an emphasis on T-matrix methods in spherical wavefunctions
• Compare results with methods such as DDA

Project type: Computational

Honours project: Probe microscopy for surface characterisation with optical tweezers

Optical tweezers-based probe microscopy of surfaces has an already-long history. One aspect that has been little-explored is to measure the change in the trap potential occupied by the particle, including the effect of the surface being probed. In this way, Brownian motion becomes a source of information, rather than a source of uncertainty. This can allow weaker traps to be used, enabling the characterisation of softer surfaces without damage. Deformable surfaces can also be studied.

You will: Use computational modelling for a feasibility study of surface characterisation based on measuring the potential confining an optically-trapped particle, as modified by the surface. Model the measurement of deformable surfaces and structures, with free particles and with attached particles Compare the use of 2D-only position measurements of the probe particle vs 3D measurements.

Project type: Computational

Honours project: Thermal forces in optical tweezers

Since the particles usually trapped with optical tweezers are highly transparent, thermal effects are often ignored. However, even in this case, there will be some absorption, and consequent heating. If absorbing particles are trapped, temperature rises can be very high. Thermal effects such as convective flow, thermophoresis (propulsive forces on particles due to temperature gradients), bubble formation, and Marangoni convection can be important.

You will:

• Survey thermal effects likely to be important in optical trapping with both high absorption and low absorption
• Identify conditions under which these effects will be important
• Review exact and approximate models for these phenomena
• Test computational implementations of appropriate models

Project type: Computational and mathematical

Honours project: Optical forces on soft particles

Optical tweezers have revolutionised biophysics, offering non-contact micromanipulation, and the measurement of forces in biophysical systems down to single-molecule levels. Computational light scattering gives us means of calculating the optical forces, and relating these to the size, shape, and composition of the trapped particles. This is useful for designing experiments, understanding measurements and observations, and more. However, many biological (and other) particles are soft, and will be deformed by the optical forces acting on them. This is a much more complex problem, as it involves the simultaneous modelling of the optical forces acting on the particle, which are affected by the particle's shape, and the particles shape, which is affected by the optical forces.

You will:

• Develop methods for calculating the optical stress on the surface of soft particles
• Develop iterative methods using alternating calculations of optical force and deformation
• Determine the accuracy and applicability of simple models for the deformation and/or optical stress

Project type: Computational and mathematical