Condensed Matter Theory

Abstract:

We derive effective equations of motion governing the dynamics of sharp interfaces in phase-separated binary mixtures driven by spatio-temporal modulations of their material properties. We demonstrate, in particular, that spatial heterogeneities in the surface tension induce an effective capillary force that drives the motion of interfaces, even in the absence of hydrodynamics. Applying our sharp interface model to quantify the dynamics of thermophoretic droplets, we find that their deformation and transport properties are controlled by a combination of bulk and capillary forces, whose relative strength depends on droplet size. Strikingly, we show that small thermophobic droplets - composed of a material with a positive Soret coefficient - can spontaneously migrate towards high-temperature regions as a result of capillary forces.

Abstract:

We analyze a generalization of Zöttl and Stark's model of active spherical particles [Phys. Rev. Lett. 108, 218104 (2012)0031-900710.1103/PhysRevLett.108.218104] and prolate spheroidal particles [Eur. Phys. J. E 36, 4 (2013)1292-894110.1140/epje/i2013-13004-5] suspended in cylindrical Poiseuille flow, to particle dynamics in an arbitrary unidirectional steady laminar flow through a straight duct geometry. Our primary contribution is to describe a Hamiltonian formulation of these systems and provide explicit forms of the constants of motions in terms of the arbitrary fluid velocity field. The Hamiltonian formulation provides a convenient and robust approach to the computation of particle orbits while also providing new insights into the dynamics, specifically the way in which orbits are trapped within basins defined by a potential well. In addition to considering spherical and prolate spheroidal particles, we also illustrate that the model can be adapted to oblate spheroidal particles.

Abstract:

Differential equations containing memory terms that depend nonlinearly on past states model a variety of non-Markovian processes. In this study, we present a Markovian embedding procedure for a subclass of such equations with distributed delay by utilising an exact spectral representation of the nonlinear memory function. This allows us to transform the nonlocal system to an equivalent local-in-time system in an abstract extended space. We demonstrate our embedding procedure for two one-dimensional physical models: (i) the walking droplet and (ii) the single-phase Stefan problem. In addition to providing an alternative representation of the underlying physical system, the local representation finds applications in designing efficient time-integrators with time-independent computational costs for memory-dependent systems which typically suffer from growing-in-time costs.

Abstract:

Most living things on Earth - from bacteria to humans - must migrate in some way to find favourable conditions. Therefore, they nearly all use chemotaxis, in which their movement is steered by a gradient of chemicals. Chemotaxis is fundamental to many processes that control our well-being, including inflammation, neuronal patterning, wound healing, tumour spread in cancer, even embryogenesis. Understanding it is a key goal for biologists. Despite the fact that many basic principles appear to have been conserved throughout evolution, most research has focused on understanding the molecular mechanisms that control signal processing and locomotion. Cell signaling - cells responding to time-varying external signals - underlies almost all biological processes at the cellular scale. Chemotaxis of single cells provides particularly amenable model systems for quantitative cell signaling studies, even in the presence of noise and fluctuations, because the output, the cell's motility response, is directly observable. However, the different scientific disciplines involved in chemotaxis research rarely overlap, so biologists, physicists and mathematicians interact far too infrequently, methodologies and models differ and commonalities are often overlooked, such as the possible influence of physical or environmental conditions, which has been largely neglected.

Abstract:

We develop a continuum theory to describe the collective dynamics of deformable epithelial cells, distinguishing the force-generating active filaments in the cells from their shape. The theory demonstrates how active flows driven by active filaments can create nematic domains and topological defects in the cell shape field. We highlight the role of the filament flow-aligning parameter, λQ, a rheological quantity that determines the response of the filaments to velocity gradients in the active flows, and plays a significant, to date unappreciated, role in determining the pattern of extensional and compressional active flows. In a contractile cell layer, local flows are expected to align elongated cells perpendicular to the active filaments. However, with increasing λQ, long-range correlations in the active turbulent flow field lead to extended regions where this alignment is parallel, consistent with recent experiments on confluent Madin-Darby canine kidney (MDCK) cell layers. Further, we distinguish defects in the filament director field, which contribute to the active driving, and those in the shape director field, measured in experiments, which are advected by the active flows. By considering the shape-filament orientation, we explain the unexpected motion of +1/2 defects towards their head in contractile cell layers, consistent with recent experiments on epithelial layers examining stress around shape defects.