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Particle-tracking microrheology of dilute active (self-propelled) colloidal suspensions is studied by considering the external force required to maintain the steady motion of an immersed constant-velocity colloidal probe. If the probe speed is zero, the suspension microstructure is isotropic but exhibits a boundary accumulation of active bath particles at contact due to their self-propulsion. As the probe moves through the suspension, the microstructure is distorted from the non-equilibrium isotropic state, which allows us to define a microviscosity for the suspension using the Stokes drag law. For a slow probe, we show that active suspensions exhibit a swim-thinning behavior in which their microviscosity is gradually lowered from that of passive suspensions as the swim speed increases. When the probe speed is fast, the suspension activity is obscured by the rapid advection of the probe and the measured microviscosity is indistinguishable from that of passive suspensions. Generally for finite activity, the suspension exhibits a velocity-thinning behavior -- though with a zero-velocity plateau lower than passive suspensions -- as a function of the probe speed. These behaviors originate from the interplay between the suspension activity and the hard-sphere excluded volume interaction between the probe and a bath particle.

Snell's law, which encompasses both refraction and total internal reflection, provides the foundation for ray optics and all lens-based instruments, from microscopes to telescopes. Refraction results when light crosses the interface between media of different refractive index, the dimensionless number that captures how much a medium retards the propagation of light. In this work, we show that the motion of self-propelled particles moving across a drag discontinuity is governed by an analogous Snell's law, allowing for swimmer ray optics. We derive a variant of Snell's law for swimmers moving across media of different viscosities. Just as the ratio of refractive indexes sets the path of a light ray, the ratio of viscosities is shown to determine the trajectories of swimmers. We find that the magnitude of refraction depends on the swimmer's shape, specifically the aspect ratio, as analogous to the wavelength of light. This enables the demixing of a polymorphic, many-shaped, beam of swimmers into distinct monomorphic, single-shaped, beams through a viscosity prism. In turn, beams of monomorphic swimmers can be focused by spherical and gradient viscosity lenses. Completing the analogy, we show that the shape-dependence of the total internal reflection critical angle can be used to create swimmer traps. Such analogies to ray optics suggest a universe of new devices for sorting, concentrating, and analyzing microscopic swimmers is possible.

By introducing the notion of a dynamic overlap concentration scale, we identify universal and previously unreported features of the mechanical properties of active colloids. These features are codified by recognizing that the characteristic length scale of an active particle's trajectory, the run-length, introduces a new concentration scale ϕ∗. Large-scale simulations of repulsive active Brownian particles (ABPs) confirm that this new run-length dependent concentration, which is the trajectory-space analogue of the overlap concentration in polymer solutions, delineates distinct concentration regimes in which interparticle collisions alter particle trajectories. Using ϕ∗ and concentration scales associated with colloidal jamming, the mechanical equation-of-state for ABPs can be collapsed onto a set of principal curves that contain a number of previously overlooked features. The inclusion of these features qualitatively alters previous predictions of the behavior for active colloids as we demonstrate by computing the spinodal for a suspension of purely-repulsive ABPs. Our findings suggest that dynamic overlap concentration scales should be of great utility in unraveling the behavior of active and driven systems.

We demonstrate that deep learning techniques can be used to predict motility induced phase separation (MIPS) in suspensions of active Brownian particles (ABPs) by creating a notion of phase at the particle level. Using a fully connected network in conjunction with a graph neural network we use individual particle features to predict to which phase a particle belongs. From this, we are able to compute the fraction of dilute particles to determine if the system is in the homogeneous dilute, dense, or coexistence region. Our predictions are compared against the MIPS binodal computed from simulation. The strong agreement between the two suggests that machine learning provides an effective way to determine the phase behavior of ABPs and could prove useful for determining more complex phase diagrams.

Confinement of living microorganisms and self-propelled particles by an external trap provides a means of analysing the motion and behaviour of active systems. Developing a tweezer with a trapping radius large compared with the swimmers' size and run length has been an experimental challenge, as standard optical traps are too weak. Here we report the novel use of an acoustic tweezer to confine self-propelled particles in two dimensions over distances large compared with the swimmers' run length. We develop a near-harmonic trap to demonstrate the crossover from weak confinement, where the probability density is Boltzmann-like, to strong confinement, where the density is peaked along the perimeter. At high concentrations the swimmers crystallize into a close-packed structure, which subsequently 'explodes' as a travelling wave when the tweezer is turned off. The swimmers' confined motion provides a measurement of the swim pressure, a unique mechanical pressure exerted by self-propelled bodies.

We present a general theory for determining the force (and torque) exerted on a boundary (or body) in active matter. The theory extends the description of passive Brownian colloids to self-propelled active particles and applies for all ratios of the thermal energy kBT to the swimmer's activity k_sT_s = ζU^2_0T_R/6, where ζ is the Stokes drag coefficient, U_0 is the swim speed and T_R is the reorientation time of the active particles. The theory has a natural microscopic length scale over which concentration and orientation distributions are confined near boundaries, but the microscopic length does not appear in the force. The swim pressure emerges naturally and dominates the behavior when the boundary size is large compared to the swimmer's run length ℓ = U_0T_R. The theory is used to predict the motion of bodies of all sizes immersed in active matter.

We present a general theory for determining the force (and torque) exerted on a boundary (or body) in active matter. The theory extends the description of passive Brownian colloids to self-propelled active particles and applies for all ratios of the thermal energy kBT to the swimmer's activity k_sT_s = ζU^2_0T_R/6, where ζ is the Stokes drag coefficient, U_0 is the swim speed and T_R is the reorientation time of the active particles. The theory has a natural microscopic length scale over which concentration and orientation distributions are confined near boundaries, but the microscopic length does not appear in the force. The swim pressure emerges naturally and dominates the behavior when the boundary size is large compared to the swimmer's run length ℓ = U_0T_R. The theory is used to predict the motion of bodies of all sizes immersed in active matter.

We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a fractal dimension df. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same non-integer dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semi-analytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.

We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a fractal dimension df. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same non-integer dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semi-analytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.

A model for self-propulsion of a colloidal particle—the osmotic motor—immersed in a dispersion of “bath” particles is presented. The nonequilibrium concentration of bath particles induced by a surface chemical reaction creates an osmotic pressure imbalance on the motor causing it to move. The ratio of the speed of reaction to that of diffusion governs the bath particle distribution which is employed to calculate the driving force on the motor, and from which the self-induced osmotic velocity is determined. For slow reactions, the self-propulsion is proportional to the reaction velocity. When surface reaction dominates over diffusion the osmotic velocity cannot exceed the diffusive speed of the bath particles. Implications of these features for different bath particle volume fractions and motor sizes are discussed. Theoretical predictions are compared with Brownian dynamics simulations.

We consider a “probe” particle translating at constant velocity through an otherwise quiescent dispersion of colloidal “bath” particles, as a model for particle-tracking microrheology experiments in the active (nonlinear) regime. The probe is a body of revolution with major and minor semiaxes a and b, respectively, and the bath particles are spheres of radii b. The probe's shape is such that when its major or minor axis is the axis of revolution the excluded-volume, or contact, surface between the probe and a bath particle is a prolate or oblate spheroid, respectively. The moving probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the bath particles. For a prolate or oblate probe translating along its symmetry axis, we calculate the nonequilibrium microstructure to first order in the volume fraction of bath particles and over the entire range of the Péclet number (Pe), neglecting hydrodynamic interactions. Here, Pe is defined as the non-dimensional velocity of the probe. The microstructure is employed to calculate the average external force on the probe, from which one can infer a “microviscosity” of the dispersion via Stokes drag law. The microviscosity is computed as a function of the aspect ratio of the probe, â=a/b, thereby delineating the role of the probe's shape. For a prolate probe, regardless of the value of â, the microviscosity monotonically decreases, or “velocity thins,” from a Newtonian plateau at small Pe until a second Newtonian plateau is reached as Pe-->[infinity]. After appropriate scaling, we demonstrate this behavior to be in agreement with microrheology studies using spherical probes [Squires and Brady, “A simple paradigm for active and nonlinear microrheology,” Phys. Fluids 17(7), 073101 (2005)] and conventional (macro-)rheological investigations [Bergenholtz et al., “The non-Newtonian rheology of dilute colloidal suspensions,” J. Fluid. Mech. 456, 239–275 (2002)]. For an oblate

We consider a “probe” particle translating at constant velocity through an otherwise quiescent dispersion of colloidal “bath” particles, as a model for particle-tracking microrheology experiments in the active (nonlinear) regime. The probe is a body of revolution with major and minor semiaxes a and b, respectively, and the bath particles are spheres of radii b. The probe's shape is such that when its major or minor axis is the axis of revolution the excluded-volume, or contact, surface between the probe and a bath particle is a prolate or oblate spheroid, respectively. The moving probe drives the microstructure of the dispersion out of equilibrium; counteracting this is the Brownian diffusion of the bath particles. For a prolate or oblate probe translating along its symmetry axis, we calculate the nonequilibrium microstructure to first order in the volume fraction of bath particles and over the entire range of the Péclet number (Pe), neglecting hydrodynamic interactions. Here, Pe is defined as the non-dimensional velocity of the probe. The microstructure is employed to calculate the average external force on the probe, from which one can infer a “microviscosity” of the dispersion via Stokes drag law. The microviscosity is computed as a function of the aspect ratio of the probe, â=a/b, thereby delineating the role of the probe's shape. For a prolate probe, regardless of the value of â, the microviscosity monotonically decreases, or “velocity thins,” from a Newtonian plateau at small Pe until a second Newtonian plateau is reached as Pe-->[infinity]. After appropriate scaling, we demonstrate this behavior to be in agreement with microrheology studies using spherical probes [Squires and Brady, “A simple paradigm for active and nonlinear microrheology,” Phys. Fluids 17(7), 073101 (2005)] and conventional (macro-)rheological investigations [Bergenholtz et al., “The non-Newtonian rheology of dilute colloidal suspensions,” J. Fluid. Mech. 456, 239–275 (2002)]. For an oblate

A model for self-propulsion of a colloidal particle—the osmotic motor—immersed in a dispersion of “bath” particles is presented. The nonequilibrium concentration of bath particles induced by a surface chemical reaction creates an osmotic pressure imbalance on the motor causing it to move. The ratio of the speed of reaction to that of diffusion governs the bath particle distribution which is employed to calculate the driving force on the motor, and from which the self-induced osmotic velocity is determined. For slow reactions, the self-propulsion is proportional to the reaction velocity. When surface reaction dominates over diffusion the osmotic velocity cannot exceed the diffusive speed of the bath particles. Implications of these features for different bath particle volume fractions and motor sizes are discussed. Theoretical predictions are compared with Brownian dynamics simulations.

The dynamics of spherical particles near a single plane wall are computed using an extension of the Stokesian dynamics method that includes long-range many-body and pairwise lubrication interactions between the spheres and the wall in Stokes flow. Extra care is taken to ensure that the mobility and resistance tensors are symmetric, positive, and definite—something which is ineluctable for particles in low-Reynolds-number flows. We discuss why two previous simulation methods for particles near a plane wall, one using multipole expansions and the other using the Rotne-Prager tensor, fail to produce symmetric resistance and mobility tensors. Additionally, we offer some insight on how the Stokesian dynamics paradigm might be extended to study the dynamics of particles in any confining geometry.

The dynamics of spherical particles near a single plane wall are computed using an extension of the Stokesian dynamics method that includes long-range many-body and pairwise lubrication interactions between the spheres and the wall in Stokes flow. Extra care is taken to ensure that the mobility and resistance tensors are symmetric, positive, and definite—something which is ineluctable for particles in low-Reynolds-number flows. We discuss why two previous simulation methods for particles near a plane wall, one using multipole expansions and the other using the Rotne-Prager tensor, fail to produce symmetric resistance and mobility tensors. Additionally, we offer some insight on how the Stokesian dynamics paradigm might be extended to study the dynamics of particles in any confining geometry.

The motion of a single Brownian probe particle subjected to a constant external body force and immersed in a dispersion of colloidal particles is studied with a view to providing a simple model for particle tracking microrheology experiments in the active and nonlinear regime. The non-equilibrium configuration of particles induced by the motion of the probe is calculated to first order in the volume fraction of colloidal particles over the entire range of Pe, accounting for hydrodynamic and excluded volume interactions between the probe and dispersion particles. Here, Pe is the dimensionless external force on the probe, or Péclet number, and is a characteristic measure of the degree to which the equilibrium microstructure of the dispersion is distorted. For small Pe, the microstructure (in a reference frame moving with the probe) is primarily dictated by Brownian diffusion and is approximately fore–aft symmetric about the direction of the external force. In the large Pe limit, advection is dominant except in a thin boundary layer in the compressive region of the flow where it is balanced by Brownian diffusion, leading to a highly non-equilibrium microstructure. The computed microstructure is employed to calculate the average translational velocity of the probe, from which the ‘microviscosity’ of the dispersion may be inferred via application of Stokes drag law. For small departures from equilibrium (Pe), the microviscosity ‘force-thins’ proportional to $\hbox{\it Pe}^2$ from its Newtonian low-force plateau. For particles with long-range excluded volume interactions, force-thinning persists until a terminal Newtonian plateau is reached in the limit $\hbox{\it Pe}\,{\rightarrow}\,\infty$. In the case of particles with very short-range excluded volume interactions, the force-thinning ceases at $\hbox{\it Pe}\,{\sim}\, O(1)$, at which point the microviscosity attains a minimum value. Beyond $\hbox{\it Pe}\,{\sim}\, O(1)$, the microstructural boundary layer coincides wit

The pressure moment of a rigid particle is defined as the trace of the first moment of the surface stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication theory. Following the practice established in previous studies on two-body hydrodynamic interactions at low Reynolds number, the results are expressed in terms of a new (stresslet) resistance function.

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, $\mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p$, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-$\mbox{\textit{St}}$ expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in $\mbox{\textit{St}}$. The equations for relative trajectories are then solved, to $O(\mbox{\textit{St}})$, in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-$\mbox{\textit{St}}$ open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains $O(\mbox{\textit{St}})$, while the scaling of the gradient displacement increases from $O(\mbox{\textit{St}})$ for far-field open trajectories, to $O(\mbox{\textit{St}}^{{1}/{2}})$ for open trajectories with $O(\mbox{\textit{St}}^{{1}/{2}})$ upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is $O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2)$, while that in the gradient direction scales as $O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2)$ and $O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2)$ in the limits $\phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}}$ and $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi

The bulk viscosity of a suspension relates the deviation of the trace of the macroscopic or averaged stress from its equilibrium value to the average rate of expansion. For a suspension the equilibrium macroscopic stress is the sum of the fluid pressure and the osmotic pressure of the suspended particles. An average rate of expansion drives the suspension microstructure out of equilibrium and is resisted by the thermal motion of the particles. Expressions are given to compute the bulk viscosity for all concentrations and all expansion rates and shown to be completely analogous to the well-known formulae for the deviatoric macroscopic stress, which are used, for example, to compute the shear viscosity. The effect of rigid spherical particles on the bulk viscosity is determined to second order in volume fraction and to leading order in the Péclet number, which is defined as the expansion rate made dimensionless with the Brownian time scale. A repulsive hard-sphere-like interparticle force reduces the hydrodynamic interactions between particles and decreases the bulk viscosity.

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, $\mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p$, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-$\mbox{\textit{St}}$ expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in $\mbox{\textit{St}}$. The equations for relative trajectories are then solved, to $O(\mbox{\textit{St}})$, in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-$\mbox{\textit{St}}$ open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains $O(\mbox{\textit{St}})$, while the scaling of the gradient displacement increases from $O(\mbox{\textit{St}})$ for far-field open trajectories, to $O(\mbox{\textit{St}}^{{1}/{2}})$ for open trajectories with $O(\mbox{\textit{St}}^{{1}/{2}})$ upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is $O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2)$, while that in the gradient direction scales as $O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2)$ and $O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2)$ in the limits $\phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}}$ and $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi

The bulk viscosity of a suspension relates the deviation of the trace of the macroscopic or averaged stress from its equilibrium value to the average rate of expansion. For a suspension the equilibrium macroscopic stress is the sum of the fluid pressure and the osmotic pressure of the suspended particles. An average rate of expansion drives the suspension microstructure out of equilibrium and is resisted by the thermal motion of the particles. Expressions are given to compute the bulk viscosity for all concentrations and all expansion rates and shown to be completely analogous to the well-known formulae for the deviatoric macroscopic stress, which are used, for example, to compute the shear viscosity. The effect of rigid spherical particles on the bulk viscosity is determined to second order in volume fraction and to leading order in the Péclet number, which is defined as the expansion rate made dimensionless with the Brownian time scale. A repulsive hard-sphere-like interparticle force reduces the hydrodynamic interactions between particles and decreases the bulk viscosity.

The motion of a single Brownian probe particle subjected to a constant external body force and immersed in a dispersion of colloidal particles is studied with a view to providing a simple model for particle tracking microrheology experiments in the active and nonlinear regime. The non-equilibrium configuration of particles induced by the motion of the probe is calculated to first order in the volume fraction of colloidal particles over the entire range of Pe, accounting for hydrodynamic and excluded volume interactions between the probe and dispersion particles. Here, Pe is the dimensionless external force on the probe, or Péclet number, and is a characteristic measure of the degree to which the equilibrium microstructure of the dispersion is distorted. For small Pe, the microstructure (in a reference frame moving with the probe) is primarily dictated by Brownian diffusion and is approximately fore–aft symmetric about the direction of the external force. In the large Pe limit, advection is dominant except in a thin boundary layer in the compressive region of the flow where it is balanced by Brownian diffusion, leading to a highly non-equilibrium microstructure. The computed microstructure is employed to calculate the average translational velocity of the probe, from which the ‘microviscosity’ of the dispersion may be inferred via application of Stokes drag law. For small departures from equilibrium (Pe), the microviscosity ‘force-thins’ proportional to $\hbox{\it Pe}^2$ from its Newtonian low-force plateau. For particles with long-range excluded volume interactions, force-thinning persists until a terminal Newtonian plateau is reached in the limit $\hbox{\it Pe}\,{\rightarrow}\,\infty$. In the case of particles with very short-range excluded volume interactions, the force-thinning ceases at $\hbox{\it Pe}\,{\sim}\, O(1)$, at which point the microviscosity attains a minimum value. Beyond $\hbox{\it Pe}\,{\sim}\, O(1)$, the microstructural boundary layer coincides wit

The pressure moment of a rigid particle is defined as the trace of the first moment of the surface stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication theory. Following the practice established in previous studies on two-body hydrodynamic interactions at low Reynolds number, the results are expressed in terms of a new (stresslet) resistance function.

In microrheology, elastic and viscous moduli are obtained from measurements of the fluctuating thermal motion of embedded colloidal probes. In such experiments, the probe motion is passive and reflects the near-equilibrium (linear response) properties of the surrounding medium. By actively pulling the probe through the material, further information about material properties can be obtained, analogous to large-amplitude measurements in (macro-) rheology. We consider a simple model of such systems: a colloidal probe pulled through a suspension of neutrally buoyant bath colloids. We choose a system with hard-sphere interactions but neglect hydrodynamic interactions, which is simple enough to permit analytic solutions, but nontrivial enough to raise issues important for the interpretation of experiments in active and nonlinear microrheology. We calculate the microstructural deformation for arbitrary probe size and pulling rate (expressed as a dimensionless Péclet number Pe). From this, we determine the average retarding effect on the probe due to the microstructure, as well as fluctuations about this average. The high-Pe limit is singular, giving a finite Brownian contribution even in the limit of negligible diffusion. Significantly, different results are obtained for probes driven at constant velocity and constant force. Furthermore, we demonstrate that a probe pulled with an optical tweezer (roughly a harmonic well) can behave as fixed-force, fixed-velocity, or as a mixture of those modes, depending on the strength of the trap and on the pulling speed. More generally, we discuss how these results relate to previous work on the rheology of colloidal suspensions. Not surprisingly, the present theory (which ignores hydrodynamic interactions) gives shear thinning but no shear thickening; we expect that the incorporation of hydrodynamics would result in shear thickening as well. The effective micro- and macro-viscosities, when appropriately scaled, are in semi-quantitative

Diffusion of neutrally buoyant spherical particles in concentrated monodisperse suspensions under simple shear flow is investigated. We consider the case of non-Brownian particles in Stokes flow, which corresponds to the limits of infinite Péclet number and zero Reynolds number. Using an approach based upon ideas of dynamic light scattering we compute self- and gradient diffusion coefficients in the principal directions normal to the flow numerically from Accelerated Stokesian Dynamics simulations for large systems (up to 2000 particles). For the self-diffusivity, the present approach produces results identical to those reported earlier, obtained by probing the particles' mean-square displacements (Sierou & Brady, J. Fluid Mech. vol. 506, 2004 p. 285). For the gradient diffusivity, the computed coefficients are in good agreement with the available experimental results. The similarity between diffusion mechanisms in equilibrium suspensions of Brownian particles and in non-equilibrium non-colloidal sheared suspensions suggests an approximate model for the gradient diffusivity: ${\textsfbi D}^\triangledown\,{\approx}\,{\textsfbi D}^s/S^{eq}(0)$, where ${\textsfbi D}^s$ is the shear-induced self-diffusivity and $S^{eq}(0)$ is the static structure factor corresponding to the hard-sphere suspension at thermodynamic equilibrium.

In microrheology, elastic and viscous moduli are obtained from measurements of the fluctuating thermal motion of embedded colloidal probes. In such experiments, the probe motion is passive and reflects the near-equilibrium (linear response) properties of the surrounding medium. By actively pulling the probe through the material, further information about material properties can be obtained, analogous to large-amplitude measurements in (macro-) rheology. We consider a simple model of such systems: a colloidal probe pulled through a suspension of neutrally buoyant bath colloids. We choose a system with hard-sphere interactions but neglect hydrodynamic interactions, which is simple enough to permit analytic solutions, but nontrivial enough to raise issues important for the interpretation of experiments in active and nonlinear microrheology. We calculate the microstructural deformation for arbitrary probe size and pulling rate (expressed as a dimensionless Péclet number Pe). From this, we determine the average retarding effect on the probe due to the microstructure, as well as fluctuations about this average. The high-Pe limit is singular, giving a finite Brownian contribution even in the limit of negligible diffusion. Significantly, different results are obtained for probes driven at constant velocity and constant force. Furthermore, we demonstrate that a probe pulled with an optical tweezer (roughly a harmonic well) can behave as fixed-force, fixed-velocity, or as a mixture of those modes, depending on the strength of the trap and on the pulling speed. More generally, we discuss how these results relate to previous work on the rheology of colloidal suspensions. Not surprisingly, the present theory (which ignores hydrodynamic interactions) gives shear thinning but no shear thickening; we expect that the incorporation of hydrodynamics would result in shear thickening as well. The effective micro- and macro-viscosities, when appropriately scaled, are in semi-quantitative

Diffusion of neutrally buoyant spherical particles in concentrated monodisperse suspensions under simple shear flow is investigated. We consider the case of non-Brownian particles in Stokes flow, which corresponds to the limits of infinite Péclet number and zero Reynolds number. Using an approach based upon ideas of dynamic light scattering we compute self- and gradient diffusion coefficients in the principal directions normal to the flow numerically from Accelerated Stokesian Dynamics simulations for large systems (up to 2000 particles). For the self-diffusivity, the present approach produces results identical to those reported earlier, obtained by probing the particles' mean-square displacements (Sierou & Brady, J. Fluid Mech. vol. 506, 2004 p. 285). For the gradient diffusivity, the computed coefficients are in good agreement with the available experimental results. The similarity between diffusion mechanisms in equilibrium suspensions of Brownian particles and in non-equilibrium non-colloidal sheared suspensions suggests an approximate model for the gradient diffusivity: ${\textsfbi D}^\triangledown\,{\approx}\,{\textsfbi D}^s/S^{eq}(0)$, where ${\textsfbi D}^s$ is the shear-induced self-diffusivity and $S^{eq}(0)$ is the static structure factor corresponding to the hard-sphere suspension at thermodynamic equilibrium.

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed.

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed.

A new Stokesian dynamics (SD) algorithm for Brownian suspensions is presented. The implementation is based on the recently developed accelerated Stokesian dynamics (ASD) simulation method [Sierou and Brady, J. Fluid Mech. 448, 115 (2001)] for non-Brownian particles. As in ASD, the many-body long-range hydrodynamic interactions are computed using fast Fourier transforms, and the resistance matrix is inverted iteratively, in order to keep the computational cost O(N log N). A fast method for computing the Brownian forces acting on the particles is applied by splitting them into near- and far-field contributions to avoid the O(N3) computation of the square root of the full resistance matrix. For the near-field part, representing the forces as a sum of pairwise contributions reduces the cost to O(N); and for the far-field part, a Chebyshev polynomial approximation for the inverse of the square root of the mobility matrix results in an O(N1.25 log N) computational cost. The overall scaling of the method is thus roughly of O(N1.25 log N) and makes possible the simulation of large systems, which are necessary for studying long-time dynamical properties and/or polydispersity effects in colloidal dispersions. In this work the method is applied to study the rheology of concentrated colloidal suspensions, and results are compared with conventional SD. Also, a faster approximate method is presented and its accuracy discussed.

A new Stokesian dynamics (SD) algorithm for Brownian suspensions is presented. The implementation is based on the recently developed accelerated Stokesian dynamics (ASD) simulation method [Sierou and Brady, J. Fluid Mech. 448, 115 (2001)] for non-Brownian particles. As in ASD, the many-body long-range hydrodynamic interactions are computed using fast Fourier transforms, and the resistance matrix is inverted iteratively, in order to keep the computational cost O(N log N). A fast method for computing the Brownian forces acting on the particles is applied by splitting them into near- and far-field contributions to avoid the O(N3) computation of the square root of the full resistance matrix. For the near-field part, representing the forces as a sum of pairwise contributions reduces the cost to O(N); and for the far-field part, a Chebyshev polynomial approximation for the inverse of the square root of the mobility matrix results in an O(N1.25 log N) computational cost. The overall scaling of the method is thus roughly of O(N1.25 log N) and makes possible the simulation of large systems, which are necessary for studying long-time dynamical properties and/or polydispersity effects in colloidal dispersions. In this work the method is applied to study the rheology of concentrated colloidal suspensions, and results are compared with conventional SD. Also, a faster approximate method is presented and its accuracy discussed.

The gravity-driven flow of non-neutrally buoyant suspensions is shown to be unstable to spanwise perturbations when the shearing motion generates a density profile that increases with height. The instability is simply due to having heavier material over light – a Rayleigh–Taylor-like instability. The wavelength of the fastest growing disturbance is on the order of the thickness of the suspension layer. The parameters important to the problem are the angle of inclination of the layer relative to gravity, the relative density difference between the particles and the fluid, the ratio of the particle size to the thickness of the layer and the bulk volume fraction of particles. The instability is illustrated for a range of these parameters and shown to be most pronounced at intermediate values thereof. This instability mechanism may play an important role in pattern formation in multiphase flows.

The gravity-driven flow of non-neutrally buoyant suspensions is shown to be unstable to spanwise perturbations when the shearing motion generates a density profile that increases with height. The instability is simply due to having heavier material over light – a Rayleigh–Taylor-like instability. The wavelength of the fastest growing disturbance is on the order of the thickness of the suspension layer. The parameters important to the problem are the angle of inclination of the layer relative to gravity, the relative density difference between the particles and the fluid, the ratio of the particle size to the thickness of the layer and the bulk volume fraction of particles. The instability is illustrated for a range of these parameters and shown to be most pronounced at intermediate values thereof. This instability mechanism may play an important role in pattern formation in multiphase flows.

It is shown that the method of reflections in resistance form (with truncated multipoles) is one of many possible iterative methods to obtain the inverse of the mobility matrix (with the same truncation) in low-Reynolds-number hydrodynamics. Although the method of reflections in the mobility form is guaranteed to converge, it is found that in the resistance form the method may fail to converge. This breakdown is overcome by conjugate-gradient-type iterative methods, and the implications of the iterative method for low-Reynolds-number hydrodynamics are discussed.

A new implementation of the conventional Stokesian Dynamics (SD) algorithm, called accelerated Stokesian Dynamics (ASD), is presented. The equations governing the motion of N particles suspended in a viscous fluid at low particle Reynolds number are solved accurately and efficiently, including all hydrodynamic interactions, but with a significantly lower computational cost of O(N ln N). The main differences from the conventional SD method lie in the calculation of the many-body long-range interactions, where the Ewald-summed wave-space contribution is calculated as a Fourier transform sum and in the iterative inversion of the now sparse resistance matrix. The new method is applied to problems in the rheology of both structured and random suspensions, and accurate results are obtained with much larger numbers of particles. With access to larger N, the high-frequency dynamic viscosities and short-time self-diffusivities of random suspensions for volume fractions above the freezing point are now studied. The ASD method opens up an entire new class of suspension problems that can be investigated, including particles of non-spherical shape and a distribution of sizes, and the method can readily be extended to other low-Reynolds-number-flow problems.

A new implementation of the conventional Stokesian Dynamics (SD) algorithm, called accelerated Stokesian Dynamics (ASD), is presented. The equations governing the motion of N particles suspended in a viscous fluid at low particle Reynolds number are solved accurately and efficiently, including all hydrodynamic interactions, but with a significantly lower computational cost of O(N ln N). The main differences from the conventional SD method lie in the calculation of the many-body long-range interactions, where the Ewald-summed wave-space contribution is calculated as a Fourier transform sum and in the iterative inversion of the now sparse resistance matrix. The new method is applied to problems in the rheology of both structured and random suspensions, and accurate results are obtained with much larger numbers of particles. With access to larger N, the high-frequency dynamic viscosities and short-time self-diffusivities of random suspensions for volume fractions above the freezing point are now studied. The ASD method opens up an entire new class of suspension problems that can be investigated, including particles of non-spherical shape and a distribution of sizes, and the method can readily be extended to other low-Reynolds-number-flow problems.

It is shown that the method of reflections in resistance form (with truncated multipoles) is one of many possible iterative methods to obtain the inverse of the mobility matrix (with the same truncation) in low-Reynolds-number hydrodynamics. Although the method of reflections in the mobility form is guaranteed to converge, it is found that in the resistance form the method may fail to converge. This breakdown is overcome by conjugate-gradient-type iterative methods, and the implications of the iterative method for low-Reynolds-number hydrodynamics are discussed.

The non-equilibrium behaviour of concentrated colloidal dispersions is studied using Stokesian Dynamics, a molecular-dynamics-like simulation technique for analysing suspensions of particles immersed in a Newtonian fluid. The simulations are of a monodisperse suspension of Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of hydrodynamic and Brownian forces, over a range of volume fraction 0.316 [less-than-or-eq, slant] [phi] [less-than-or-eq, slant] 0.49. For Pe < 10, Brownian motion dominates the behaviour, the suspension remains well-dispersed, and the viscosity shear thins. The first normal stress difference is positive and the second negative. At higher Pe, hydrodynamics dominate resulting in an increase in the long-time self-diffusivity and the viscosity. The first normal stress difference changes sign when hydrodynamics dominate. Simulation results are shown to agree well with both theory and experiment.

The non-equilibrium behaviour of concentrated colloidal dispersions is studied using Stokesian Dynamics, a molecular-dynamics-like simulation technique for analysing suspensions of particles immersed in a Newtonian fluid. The simulations are of a monodisperse suspension of Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of hydrodynamic and Brownian forces, over a range of volume fraction 0.316 [less-than-or-eq, slant] [phi] [less-than-or-eq, slant] 0.49. For Pe < 10, Brownian motion dominates the behaviour, the suspension remains well-dispersed, and the viscosity shear thins. The first normal stress difference is positive and the second negative. At higher Pe, hydrodynamics dominate resulting in an increase in the long-time self-diffusivity and the viscosity. The first normal stress difference changes sign when hydrodynamics dominate. Simulation results are shown to agree well with both theory and experiment.

The behaviour of the long-time self-diffusion tensor in concentrated colloidal dispersions is studied using dynamic simulation. The simulations are of a suspension of monodisperse Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of shear and Brownian forces, and the volume fraction, [phi]. Here, Pe = &[gamma]dot;a^2/D0, where &[gamma]dot; is the shear rate, a the particle size and D0 = kT/6[pi][eta]a is the Stokes–Einstein diffusivity of an isolated particle of size a with thermal energy kT in a solvent of viscosity [eta]. Two simulations algorithms are used: Stokesian Dynamics for inclusion of the many-body hydrodynamic interactions, and Brownian Dynamics for suspensions without hydrodynamic interactions. A new procedure for obtaining high-quality diffusion data based on averaging the results of many short simulations is presented and utilized. At low shear rates, low Pe, Brownian diffusion due to a random walk process dominates and the characteristic scale for diffusion is the Stokes–Einstein diffusivity, D0. At zero Pe the diffusivity is found to be a decreasing function of [phi]. As Pe is slowly increased, O(Pe) and O(Pe^3/2) corrections to the diffusivity due to the flow are clearly seen in the Brownian Dynamics system in agreement with the theoretical results of Morris & Brady (1996). At large shear rates, large Pe, both systems exhibit diffusivities that grow linearly with the shear rate by the non-Brownian mechanism of shear-induced diffusion. In contrast to the behaviour at low Pe, this shear-induced diffusion mode is an increasing function of [phi]. Long-time rotational self-diffusivities are of interest in the Stokesian Dynamics system and show similar behaviour to their translational analogues. An off-diagonal long-time self-diffusivity, Dxy, is reported for both systems. Results for both the translational and rotational Dxy show a sign change from low Pe to high Pe due to different

The behaviour of the long-time self-diffusion tensor in concentrated colloidal dispersions is studied using dynamic simulation. The simulations are of a suspension of monodisperse Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of shear and Brownian forces, and the volume fraction, [phi]. Here, Pe = &[gamma]dot;a^2/D0, where &[gamma]dot; is the shear rate, a the particle size and D0 = kT/6[pi][eta]a is the Stokes–Einstein diffusivity of an isolated particle of size a with thermal energy kT in a solvent of viscosity [eta]. Two simulations algorithms are used: Stokesian Dynamics for inclusion of the many-body hydrodynamic interactions, and Brownian Dynamics for suspensions without hydrodynamic interactions. A new procedure for obtaining high-quality diffusion data based on averaging the results of many short simulations is presented and utilized. At low shear rates, low Pe, Brownian diffusion due to a random walk process dominates and the characteristic scale for diffusion is the Stokes–Einstein diffusivity, D0. At zero Pe the diffusivity is found to be a decreasing function of [phi]. As Pe is slowly increased, O(Pe) and O(Pe^3/2) corrections to the diffusivity due to the flow are clearly seen in the Brownian Dynamics system in agreement with the theoretical results of Morris & Brady (1996). At large shear rates, large Pe, both systems exhibit diffusivities that grow linearly with the shear rate by the non-Brownian mechanism of shear-induced diffusion. In contrast to the behaviour at low Pe, this shear-induced diffusion mode is an increasing function of [phi]. Long-time rotational self-diffusivities are of interest in the Stokesian Dynamics system and show similar behaviour to their translational analogues. An off-diagonal long-time self-diffusivity, Dxy, is reported for both systems. Results for both the translational and rotational Dxy show a sign change from low Pe to high Pe due to different

The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe[rightward arrow][infty infinity] under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe=[gamma][ogonek]a2/2D is the Péclet number with [gamma][ogonek] the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe[minus sign]1) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe0.78) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a3Pe[minus sign]1) layer volume, vanish as Pe[minus sign]0.22 as Pe[rightward arrow][infty infinity].

The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe[rightward arrow][infty infinity] under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe=[gamma][ogonek]a2/2D is the Péclet number with [gamma][ogonek] the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe[minus sign]1) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe0.78) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a3Pe[minus sign]1) layer volume, vanish as Pe[minus sign]0.22 as Pe[rightward arrow][infty infinity].

The dynamics of bubbles at high Reynolds numbers is studied from the viewpoint of statistical mechanics. Individual bubbles are treated as dipoles in potential flow. A virtual mass matrix of the system of bubbles is introduced, which depends on the instantaneous positions of the bubbles, and is used to calculate the energy of the bubbly flow as a quadratic form of the bubbles' velocities. The energy is shown to be the system's Hamiltonian and is used to construct a canonical ensemble partition function, which explicitly includes the total impulse of the suspension along with its energy. The Hamiltonian is decomposed into an effective potential due to the bubbles' collective motion and a kinetic term due to the random motion about the mean. An effective bubble temperature-a measure of the relative importance of the bubbles' relative to collective motion-is derived with the help of the impulse-dependent partition function. Two effective potentials are shown to operate: one due to the mean motion of the bubbles, dominates at low bubble temperatures, where it leads to their grouping in flat clusters normal to the direction of the collective motion, while the other, temperature-invariant, is due to the bubbles' position-dependent virtual mass and results in their mutual repulsion. Numerical evidence is presented for the existence of the effective potentials, the condensed and dispersed phases, and a phase transition.

The dynamics of bubbles at high Reynolds numbers is studied from the viewpoint of statistical mechanics. Individual bubbles are treated as dipoles in potential flow. A virtual mass matrix of the system of bubbles is introduced, which depends on the instantaneous positions of the bubbles, and is used to calculate the energy of the bubbly flow as a quadratic form of the bubbles' velocities. The energy is shown to be the system's Hamiltonian and is used to construct a canonical ensemble partition function, which explicitly includes the total impulse of the suspension along with its energy. The Hamiltonian is decomposed into an effective potential due to the bubbles' collective motion and a kinetic term due to the random motion about the mean. An effective bubble temperature-a measure of the relative importance of the bubbles' relative to collective motion-is derived with the help of the impulse-dependent partition function. Two effective potentials are shown to operate: one due to the mean motion of the bubbles, dominates at low bubble temperatures, where it leads to their grouping in flat clusters normal to the direction of the collective motion, while the other, temperature-invariant, is due to the bubbles' position-dependent virtual mass and results in their mutual repulsion. Numerical evidence is presented for the existence of the effective potentials, the condensed and dispersed phases, and a phase transition.

A modification to the O(Re)-accurate expression for the hydrodynamic force acting on a body in arbitrary time-dependent motion, determined by Lovalenti & Brady (1993), is presented. This simple modification captures the O(Re^2) transient behaviour of the force, which has been recently shown to dominate at large time (Lawrence & Mei 1995), while maintaining the overall O(Re) accuracy.

A modification to the O(Re)-accurate expression for the hydrodynamic force acting on a body in arbitrary time-dependent motion, determined by Lovalenti & Brady (1993), is presented. This simple modification captures the O(Re^2) transient behaviour of the force, which has been recently shown to dominate at large time (Lawrence & Mei 1995), while maintaining the overall O(Re) accuracy.

Dynamic simulations of the pressure-driven flow in a channel of a non-Brownian suspension at zero Reynolds number were conducted using Stokesian Dynamics. The simulations are for a monolayer of identical particles as a function of the dimensionless channel width and the bulk particle concentration. Starting from a homogeneous dispersion, the particles gradually migrate towards the centre of the channel, resulting in an homogeneous concentration profile and a blunting of the particle velocity profile. The time for achieving steady state scales as (H/a)3a/[left angle bracket]u[right angle bracket], where H is the channel width, a the radii of the particles, and [left angle bracket]u[right angle bracket] the average suspension velocity in the channel. The concentration and velocity profiles determined from the simulations are in qualitative agreement with experiment. A model for suspension flow has been proposed in which macroscopic mass, momentum and energy balances are constructed and solved simultaneously. It is shown that the requirement that the suspension pressure be constant in directions perpendicular to the mean motion leads to particle migration and concentration variations in inhomogeneous flow. The concept of the suspension ‘temperature’ – a measure of the particle velocity fluctuations – is introduced in order to provide a nonlocal description of suspension behaviour. The results of this model for channel flow are in good agreement with the simulations.

A reply to the comments of Cichocki and Felderhof concerning my paper on the rheological behavior of concentrated colloidal dispersions [J. Chem. Phys. 99, 567 (1993)] is given. Differences exist between my scaling predictions as maximum packing is approached for the reduced dynamic viscosity and the dilute analysis of Cichocki and Felderhof.