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Colloidal particles with active boundary layers -regions surrounding the particles where nonequilibrium processes produce large velocity gradients -are common in many physical, chemical and biological contexts. The velocity or stress at the edge of the boundary layer determines the exterior fluid flow and, hence, the many-body interparticle hydrodynamic interaction. Here, we present a method to compute the many-body hydrodynamic interaction between N spherical active particles induced by their exterior microhydrodynamic flow. First, we use a boundary integral representation of the Stokes equation to eliminate bulk fluid degrees of freedom. Then, we expand the boundary velocities and tractions of the integral representation in an infinite-dimensional basis of tensorial spherical harmonics and, on enforcing boundary conditions in a weak sense on the surface of each particle, obtain a system of linear algebraic equations for the unknown expansion coefficients. The truncation of the infinite series, fixed by the degree of accuracy required, yields a finite linear system that can be solved accurately and efficiently by iterative methods. The solution linearly relates the unknown rigid body motion to the known values of the expansion coefficients, motivating the introduction of propulsion matrices. These matrices completely characterize hydrodynamic interactions in active suspensions just as mobility matrices completely characterize hydrodynamic interactions in passive suspensions. The reduction in the dimensionality of the problem, from a three-dimensional partial differential equation to a two-dimensional integral equation, allows for dynamic simulations of hundreds of thousands of active particles on multi-core computational architectures. In our simulation of 10 4 active colloidal particle in a harmonic trap, we find that the necessary and sufficient ingredients to obtain steady-state convective currents, the so-called "selfassembled pump", are (a) one-body self-propulsion and (b) two-body rotation from the vorticity of the Stokeslet induced in the trap.

Active particles, including swimming microorganisms, autophoretic colloids, and droplets, are known to self-organize into ordered structures at fluid-solid boundaries. The entrainment of particles in the attractive parts of their spontaneous flows has been postulated as a possible mechanism underlying this phenomenon. Here, combining experiments, theory, and numerical simulations, we demonstrate the validity of this flow-induced ordering mechanism in a suspension of active emulsion droplets. We show that the mechanism can be controlled, with a variety of resultant ordered structures, by simply altering hydrodynamic boundary conditions. Thus, for flow in Hele-Shaw cells, metastable lines or stable traveling bands can be obtained by varying the cell height. Similarly, for flow bounded by a plane, dynamic crystallites are formed. At a no-slip wall, the crystallites are characterized by a continuous out-of-plane flux of particles that circulate and re-enter at the crystallite edges, thereby stabilizing them. At an interface where the tangential stress vanishes, the crystallites are strictly 2D, with no out-of-plane flux. We rationalize these experimental results by calculating, in each case, the slow viscous flow produced by the droplets and the long-ranged, many-body active forces and torques between them. The results of numerical simulations of motion under the action of the active forces and torques are in excellent agreement with experiments. Our work elucidates the mechanism of flow-induced phase separation in active fluids, particularly active colloidal suspensions, and demonstrates its control by boundaries, suggesting routes to geometric and topological phenomena in an active matter.

The lack of detailed balance in active colloidal suspensions allows dissipation to determine stationary states. Here we show that slow viscous flow produced by polar or apolar active colloids near plane walls mediates attractive hydrodynamic forces that drive crystallization. Hydrodynamically mediated torques tend to destabilize the crystal but stability can be regained through critical amounts of bottom heaviness or chiral activity. Numerical simulations show that crystallization is not nucleational, as in equilibrium, but is preceded by a spinodal-like instability. Harmonic excitations of the active crystal relax diffusively but the normal modes are distinct from an equilibrium colloidal crystal. The hydrodynamic mechanisms presented here are universal and rationalize recent experiments on the crystallization of active colloids.

The force per unit area on the surface of a colloidal particle is a fundamental dynamical quantity in the mechanics and statistical mechanics of colloidal suspensions. Here we compute it in the limit of slow viscous flow for a suspension of N spherical active colloids in which activity is represented by surface slip. Our result is best expressed as a set of linear relations, the 'generalized Stokes laws', between the coefficients of a tensorial spherical harmonic expansion of the force per unit area and the surface slip. The generalized friction tensors in these laws are many-body functions of the colloidal configuration and can be obtained to any desired accuracy by solving a system of linear equations. Quantities derived from the force per unit area-forces, torques and stresslets on the colloids and flow, pressure and entropy production in the fluid-have succinct expressions in terms of the generalized Stokes laws. Most notably, the active forces and torques have a dissipative, long-ranged, many-body character that can cause phase separation, crystallization, synchronization and a variety of other effects observed in active suspensions. We use the results above to derive the Langevin and Smoluchowski equations for Brownian active suspensions, to compute active contributions to the suspension stress and fluid pressure, and to relate the synchrony in a lattice of harmonically trapped active colloids to entropy production. Our results provide the basis for a microscopic theory of active Brownian suspensions that consistently accounts for momentum conservation in the bulk fluid and at fluid-solid boundaries.

Micronutrient malnutrition, resulting from dietary deficiency of important minerals such as zinc (Zn) and iron (Fe), is a widespread food-related health problem. Genetic enhancement of crops with elevated levels of these micronutrients is one of the most cost effective ways of solving global micronutrient malnutrition problem. Development and dissemination of high Zn and Fe containing high-yielding, disease-resistant wheat varieties by International Maize and Wheat Improvement Center (CIMMYT) is initially targeted for the Indo-Gangetic plains of South Asia, a region with high population density and high micronutrient malnutrition. The most promising sources for grain Zn and Fe concentrations are wild relatives, primitive wheats and landraces. Synthetic hexaploids were developed at CIMMYT by crossing Aegilops taushii and high Zn and Fe containing accessions of T. dicoccon. Current breeding efforts at CIMMYT have focused on transferring genes governing increased Zn and Fe from T. spelta, T. dicoccon based synthetics, land races and other reported high Zn and Fe sources to high yielding elite wheat backgrounds.

Suspensions of spherical active particles often show microphase separation. At a continuum level, coupling their scalar density to fluid flow, there are two distinct explanations. Each involves an effective interfacial tension: the first mechanical (causing flow) and the second diffusive (causing Ostwald ripening). Here we show how the negative mechanical tension of contractile swimmers creates, via a self-shearing instability, a steady-state life cycle of droplet growth interrupted by division whose scaling behavior we predict. When the diffusive tension is also negative, this is replaced by an arrested regime (mechanistically distinct, but with similar scaling) where division of small droplets is prevented by reverse Ostwald ripening.

The fluctuation-dissipation relation, a central result in non-equilibrium statistical physics, relates equilibrium fluctuations in a system to its linear response to external forces. Here we provide a direct experimental verification of this relation for viscously coupled oscillators, as realized by a pair of optically trapped colloidal particles. A theoretical analysis, in which interactions mediated by slow viscous flow are represented by non-local friction tensors, matches experimental results and reveals a frequency maximum in the amplitude of the mutual response which is a sensitive function of the trap stiffnesses and the friction tensors. This allows for its location and width to be tuned and suggests the utility of the trap setup for accurate two-point microrheology.The relation between the generalized susceptibility and equilibrium fluctuations of the generalized forces, first obtained for a linear resistive circuit by Nyquist [1] and then proved for any general linear dissipative system by Callen and Welton [2], is a central result in non-equilibrium statistical physics. The relation can be used to infer the intrinsic fluctuations of a system from measurements of its response to external perturbations or, perhaps more startlingly, to predict its response to external perturbations from the character of its intrinsic fluctuations [3]. The fluctuation-dissipation relation is the point of departure for several areas of current research including fluctuation relations [4], relaxation in glasses [5], and response and correlations in active [6] and driven systems [7,8].The first experimental verification of the relation between fluctuation and dissipation was due to Johnson [9], whose investigation of the "thermal agitation of electricity in conductors" provided the motivation for Nyquist's theoretical work [1]. Though the relation has been verified since in systems with conservative couplings, a direct verification in a system where the coupling is entirely dissipative is, to the best of our knowledge, not available. Colloidal particles in a viscous fluid interact through velocity-dependent many-body hydrodynamic forces whose strength, away from boundaries, is inversely proportional to the distance between the particles. The range of these dissipative forces can be made much greater than that of conservative forces such as the DLVO interaction [10,11]. Therefore, it is possible to engineer a situation where the dominant coupling between colloidal particles is the viscous hydrodynamic force and all other interactions are negligibly small. Such systems, then, are ideal for testing the fluctuation-dissipation relation when couplings are purely dissipative.In this Letter, we present a direct verification of the fluctuation-dissipation relation for a pair of optically trapped colloidal particles in water. We measure the equilibrium fluctuations of the distance between the particles and the response of one particle to the sinusoidal motion of another particle. Transforming both correlations and responses to the fr...

The multivariate Ornstein-Uhlenbeck process is used in many branches of science and engineering to describe the regression of a system to its stationary mean. Here we present an O(N) Bayesian method to estimate the drift and diffusion matrices of the process from N discrete observations of a sample path. We use exact likelihoods, expressed in terms of four sufficient statistic matrices, to derive explicit maximum a posteriori parameter estimates and their standard errors. We apply the method to the Brownian harmonic oscillator, a bivariate Ornstein-Uhlenbeck process, to jointly estimate its mass, damping, and stiffness and to provide Bayesian estimates of the correlation functions and power spectral densities. We present a Bayesian model comparison procedure, embodying Ockham's razor, to guide a data-driven choice between the Kramers and Smoluchowski limits of the oscillator. These provide novel methods of analyzing the inertial motion of colloidal particles in optical traps.

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