Phoretic Motion of Colloids : Single Particle and Collective Behaviour
In this thesis we have studied systems that driven by mechanisms broadly known as phoresis. More specifically, in the second chapter we calculate the excess noise in electrophoresis of a colloid due to microion fluctuations. In the next three chapters we study in detail a system of self-phoretic colloids, propelled by the energy released when an ambient fuel molecule makes contact with a catalytic region on the particle’s surface. We start with the behaviour of a single particle in a linear substrate gradient, then go on to study interactions between two particles due to their diffusion clouds, and finally obtain the collective equations of motion by a systematic coarse-graining of the microscopic Langevin dynamics. To understand the role of nonequilibrium fluctuations in an electrophoretic system we have theoretically analyzed the dynamics of a single colloidal particle in an externally applied electric field. We have studied the colloidal dynamics in two scenarios: a particle free to move in an unbounded fluid and a colloid near a wall which is stationary due to a balance between gravity and the electric field. The thermal motions of microions lead to an anisotropic, nonequilibrium noise, proportional to the field, in the effective Langevin equation for the colloid. The fluctuation-dissipation ratio depends strongly on frequency, in contrast to an equilibrium system, and the colloid if displaced from its steady-state position relaxes with a velocity not proportional to the gradient of the logarithm of the steady-state probability. Other measurable effects of this noise are a superdiffusive peak at short times and an enhanced diffusity at long times. We have then studied the effective potential and obtained a non-dimensional measure of the size of the excess noise. Possible extensions of this study to include the behaviour of the mean and fluctuation properties in the case of an applied alternating potential, and the effect of the excess noise on electrohydrodynamic aggregation of colloids. We next turn to a phoretic system that has been much studied in the recent years – active Janus colloids . On one hand these colloids are an important contribution to the general class of problems on self-propulsion at low Reynolds number. On the other hand since their behaviour can be tuned at the level of single particle we can ask how their collective behaviour depends on the swimmer design. This makes it a very rich field with lots of challenging questions. We first study the single particle behaviour of an active Janus colloid in an imposed substrate gradient, then build the two-particle interactions and ultimately the collective equations of motion by a generalisation of these results. Our work presents a new approach to active matter. We show theoretically how to design particles that are not only motile but can reorient in response to gradients, thus mimicking chemotaxis. We outline the collective behaviour emerging from these single-particle properties, including colloidal realisations of gravitational collapse, plasma oscillations and spontaneously ringing states, and present a phase diagram, in terms of single particle parameters, that can be tested in experiments. This provides a template to design collective behaviours of interest by tuning the surface properties of the colloids. We can also control the range of the interaction by varying the concentration of reactant. Our coarse-grained equations of motion for the polar orientation and number density fields for a collection of colloids propelled by and interacting through long-ranged dif-fusion fields are novel in a number of ways. This is the first example in active matter literature of a microscopic derivation of collective dynamics for particles interacting via long-ranged diffusion fields. The instabilities and possible phases that we predict are different from those in traditional flocking models, which consider only short-ranged aligning interactions. The long-ranged interactions of interest here cannot produce a globally polar ordered state, and we work in a concentration regime where steric and collisional interactions are not important. Instabilities towards flocking, and the advective nonlinearities of the Toner-Tu model, although not ruled out by the symmetries of our model, do not play a significant role in our system. The collective behaviour we predict will not be seen in purely locally interacting active-particle systems. The mechanisms at work in the “saturated” case where reactant is abundant cannot be viewed as totally generic features of collections of self-driven particles; they require interactions mediated by the production or consumption of long-ranged diffusing solute fields. Earlier work on saturated systems resolved neither interactions mediated by the polarity of the objects nor chemotactic effects. Their treatment truncated the equations at the level of the concentration . In the “unsaturated” case more than one mechanism operates. One is related to the motility-induced phase separation discussed phenomenologically in refs. [2,3] (for which our system provides an important microscopic realisation). The other is due to chemo-taxis and phoresis which we report for the first time. Our expression of the various coefficients in the equaions of motion in terms of the single particle properties can also be used to design systems in which one or the other of these mechanisms dominate. We are now planning to study a collection of these particles in a fluid and examine the diffusion of a tracer particle as was done by Yeomans et al.  for hydrodynamic interactions. The Levy flights obtained in  is due to the long-ranged nature of the hydrodynamic fields, which cause effects like entrainment leading to interesting tracer dynamics. In this thesis we have considered colloids in which the symmetry axis of the colloid and the catalytic coat coincide. It might be of interest to consider cases when the axes are at an angle making the swimmer biaxial, or more complicated arrangements leading to chirality and thus rotation. Collective dynamics and two particle interaction between such swimmers can also be interesting. The formalism developed for the study of interaction between two active colloids through their diffusion fields and hydrodynamics can be extended to study their interaction with extended passive surfaces like walls or spheres. The collective dynamics of this class of active systems when it is confined between parallel walls is also of interest. Work in progress includes studies of the motion of the swimmer in a periodic array of passive colloids. In this study of collective dynamics, we have ignored the role of hydrodynamics, as the slowest decay of the field is 1/r3, which is subdominant to the decay of the chemical fields and in the dilute limit is expected to change things only qualitatively. However their role would be more important when we consider the stability of ordered structures like an aster in the saturated case. Another effect of hydrodynamics is to stir the fluid. It might be interesting to study the finite-P´eclet number regime [5, 6] of our system particularly in the unscreened region when advection of the scalar fields s and p by the velocity can affect clustering. We have derived the form of the nonlinear equations of motion in both the saturated and the unsaturated regimes. It will be interesting to investigate their relevance in the dynamics and phases that this extremely rich system can form. Even in the overdamped limit where we obtain an effective density equation it is not clear that the dynamics will resemble that of the Keller-Segel model due to the presence of the interesting nonlinear terms. Also, in this thesis, we have only looked at the fluid-like state of the system. We have just started exploring the high concentration regime where we can check the propensity of the system to develop crystalline order. In the screened limit where we obtain a condensation due a negative squared sound speed, it is posssible to study the condensation phenomenon in greater detail. In future we also plan to examine whether the tendency to condense at nonzero wavenumber (See Fig 5.1), i.e., microphase separation, can lead to liquid-crystalline phases like smectics. The systems described in this thesis are extremely rich and the few ideas mentioned above form just a small subset of the plethora of exciting theoretical and experimental explorations that can be performed with them. Since they can be “designed”, unlike biological substances, they can also become a test-bed for testing theoretical predictions of the nonequilibrium statistical mechanics of self-propelled systems.
- Physics (PHY) 
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