Abstract: Active Brownian particles (ABPs) distribute non-homogeneously near surfaces, and understanding how this depends on system properties—size, shape, activity level, etc.—is essential for predicting and exploiting the behavior of active matter systems. Active particles accumulate at no-flux surfaces owing to their persistent swimming, which depends on their intrinsic swim speed and reorientation time, and are subject to confinement effects when their run or persistence length is comparable to the characteristic size of the confining geometry. It has been observed in simulations that two parallel plates experience a “Casimir effect” and attract each other when placed in a dilute bath of ABPs. In this work, we provide a theoretical model based on the Smoluchowski equation and a macroscopic mechanical momentum balance to analytically predict this attractive force. We extend this method to describe the concentration partitioning of active particles between a confining channel and a reservoir, showing that the ratio of the concentration in the channel to that in the bulk increases as either run length increases or channel height decreases. The theoretical results agree well with Brownian dynamics simulations and finite element calculations.

Publication: Soft Matter Vol.: 17 No.: 3 ISSN: 1744-683X

ID: CaltechAUTHORS:20201124-120529448

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Abstract: We demonstrate that the mechanically defined “isothermal” compressibility behaves as a thermodynamic-like response function for suspensions of active Brownian particles. The compressibility computed from the active pressure—a combination of the collision and unique swim pressures—is capable of predicting the critical point for motility induced phase separation, as expected from the mechanical stability criterion. We relate this mechanical definition to the static structure factor via an active form of the thermodynamic compressibility equation and find the two to be equivalent, as would be the case for equilibrium systems. This equivalence indicates that compressibility behaves like a thermodynamic response function, even when activity is large. Finally, we discuss the importance of the phase interface when defining an active chemical potential. Previous definitions of the active chemical potential are shown to be accurate above the critical point but breakdown in the coexistence region. Inclusion of the swim pressure in the mechanical compressibility definition suggests that the interface is essential for determining phase behavior.

Publication: Journal of Chemical Physics Vol.: 154 No.: 1 ISSN: 0021-9606

ID: CaltechAUTHORS:20201019-100840969

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Abstract: Locomotion of self-propelled particles such as motile bacteria or phoretic swimmers often takes place in the presence of applied flows and confining boundaries. Interactions of these active swimmers with the flow environment are important for the understanding of many biological processes, including infection by motile bacteria and the formation of biofilms. Recent experimental and theoretical works have shown that active particles in a Poiseuille flow exhibit interesting dynamics including accumulation at the wall and upstream swimming. Compared to the well-studied Taylor dispersion of passive Brownian particles, a theoretical understanding of the transport of active Brownian particles (ABPs) in a pressure-driven flow is relatively less developed. In this paper, employing a small wave-number expansion of the Smoluchowski equation describing the particle distribution, we explicitly derive an effective advection-diffusion equation for the cross-sectional average of the particle number density in Fourier space. We characterize the average drift (specifically upstream swimming) and effective longitudinal dispersion coefficient of active particles in relation to the flow speed, the intrinsic swimming speed of the active particles, their Brownian diffusion, and the degree of confinement. In contrast to passive Brownian particles, both the average drift and the longitudinal dispersivity of ABPs exhibit a nonmonotonic variation as a function of the flow speed. In particular, the dispersion of ABPs includes the classical shear-enhanced (Taylor) dispersion and an active contribution called the swim diffusivity. In the absence of translational diffusion, the classical Taylor dispersion is absent and we observe a giant longitudinal dispersion in the strong flow limit. Our continuum theory is corroborated by a direct Brownian dynamics simulation of the Langevin equations governing the motion of each ABP.

Publication: Physical Review Fluids Vol.: 5 No.: 7 ISSN: 2469-990X

ID: CaltechAUTHORS:20200708-093747683

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Abstract: In nonequilibrium active matter systems, a spatial variation in activity can lead to a spatial variation in concentration of active particles satisfying, at steady state, the condition nU = const [Schnitzer, Phys. Rev. E 48, 2553 (1993); Tailleur and Cates, Phys. Rev. Lett. 100, 218103 (2008)], where n is the number density and U is the active (swim) speed. We show that this condition holds even when the variation is abrupt and when thermal Brownian motion is present provided that the Péclet number is large. This spatial variation in swim speed and concentration produces a fluid pressure distribution that drives a reverse osmotic flow—fluid flows from regions of high concentration to low.

Publication: Physical Review E Vol.: 101 No.: 6 ISSN: 2470-0045

ID: CaltechAUTHORS:20200611-144523625

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Abstract: We highlight the unique wavelike character observed in the relaxation dynamics of active systems via a Smoluchowski based theoretical framework and Brownian dynamic simulations. Persistent swimming motion results in wavelike dynamics until the advective swim displacements become sufficiently uncorrelated, at which point the motion becomes a random walk process characterized by a swim diffusivity, D^(swim) = U²₀τ_R/[d(d−1)], dependent on the speed of swimming U₀, reorientation time τ_R, and reorientation dimension d. This change in behavior is described by a telegraph equation, which governs the transition from ballistic wavelike motion to long-time diffusive motion. We study the relaxation of active Brownian particles from an instantaneous source, and provide an explanation for the nonmonotonicity observed in the intermediate scattering function. Using our simple kinetic model we provide the density distribution for the diffusion of active particles released from a line source as a function of time, position, and the ratio of the activity to thermal energy. We extend our analysis to include the effects of an external field on particle spreading to further understand how reorientation events in the active force vector affect relaxation. The strength of the applied external field is shown to be inversely proportional to the decay of the wavelike structure. Our theoretical description for the evolution of the number density agrees with Brownian dynamic simulation data.

Publication: Physical Review E Vol.: 101 No.: 5 ISSN: 2470-0045

ID: CaltechAUTHORS:20200518-095827341

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Abstract: Restricted sliding or rotational motion of colloidal particles plays a key role in the emergence of discontinuous shear thickening (DST). From viscometric functions to the number of contacting neighbors under an applied deformation, a hindrance to sliding motion significantly changes the behavior of dense suspensions on all scales. In this work, implicitly by using a modified hydrodynamic model based on Stokesian dynamics and explicitly by solving for the hydrodynamics of nonsmooth colloids, we show that lubrication forces that arise from surface asperities effectively provide such constraints to tangential particle motion. A transition from continuous shear thickening to DST is observed as the surface roughness of the particles is systematically increased. In this hydrodynamic model for DST, normal stress differences remain negative in the shear-thickened state (STS). Study of the spatial stress distribution indicates the onset of DST to be a highly localized event; however, particle self-diffusivity and the microstructural network suggest a rather uniform structure in the STS.

Publication: Journal of Rheology Vol.: 64 No.: 2 ISSN: 0148-6055

ID: CaltechAUTHORS:20200305-144824608

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Abstract: The rheological properties of active suspensions are studied via microrheology: tracking the motion of a colloidal probe particle in order to measure the viscoelastic response of the embedding material. The passive probe particle with size R is pulled through the suspension by an external force F^(ext), which causes it to translate at some speed U^(probe). The bath is comprised of a Newtonian solvent with viscosity η_s and a dilute dispersion of active Brownian particles (ABPs) with size a, characteristic swim speed U₀, and a reorientation time τ_R. The motion of the probe distorts the suspension microstructure, so the bath exerts a reactive force on the probe. In a passive suspension, the degree of distortion is governed by the Péclet number, Pe = F^(ext)/(k_BT/a), the ratio of the external force to the thermodynamic restoring force of the suspension. In active suspensions, however, the relevant parameter is L^(adv)/ℓ = U^(probe)τ_R/U₀τ_R ∼ F^(ext)/F^(swim), where F^(swim) = ζU₀ is the swim force that propels the ABPs (ζ is the Stokes drag on a swimmer). When the external forces are weak, L^(adv) ≪ ℓ, the autonomous motion of the bath particles leads to “swim-thinning,” though the effective suspension viscosity is always greater than η_s. When advection dominates, L^(adv) ≫ ℓ, we recover the familiar behavior of the microrheology of passive suspensions. The non-Newtonian behavior for intermediate values of L^(adv)/ℓ is determined by ℓ/R_c = U₀τ_R/R_c—the ratio of the swimmer's run length l to the geometric length scale associated with interparticle interactions R_c = R + a. The results in this manuscript are approximate as they are based on numerical solutions to mean-field equations that describe the motion of the active bath particles.

Publication: Soft Matter Vol.: 16 No.: 4 ISSN: 1744-683X

ID: CaltechAUTHORS:20191219-112735845

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Abstract: Thermal motion of particles and molecules in liquids underlies many chemical and biological processes. Liquids, especially in biology, are complex due to structure at multiple relevant length scales. While diffusion in homogeneous simple liquids is well understood through the Stokes–Einstein relation, this equation fails completely in describing diffusion in complex media. Modeling, understanding, engineering and controlling processes at the nanoscale, most importantly inside living cells, requires a theoretical framework for the description of viscous response to allow predictions of diffusion rates in complex fluids. Here we use a general framework with the viscosity η(k) described by a function of wave vector in reciprocal space. We introduce a formulation that allows one to relate the rotational and translational diffusion coefficients and determine the viscosity η(k) directly from experiments. We apply our theory to provide a database for rotational diffusion coefficients of proteins/protein complexes in the bacterium E. coli. We also provide a database for the diffusion coefficient of proteins sliding along major grooves of DNA in E. coli. These parameters allow predictions of rate constants for association of proteins. In addition to constituting a theoretical framework for description of diffusion of probes and viscosity in complex fluids, the formulation that we propose should decrease substantially the cost of numerical simulations of transport in complex media by replacing the simulation of individual crowding particles with a continuous medium characterized by a wave-length dependent viscosity η(k).

Publication: Soft Matter Vol.: 16 No.: 1 ISSN: 1744-683X

ID: CaltechAUTHORS:20191108-080923889

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Abstract: The unique pressure exerted by active particles—the “swim” pressure—has proven to be a useful quantity in explaining many of the seemingly confounding behaviors of active particles. However, its use has also resulted in some puzzling findings including an extremely negative surface tension between phase separated active particles. Here, we demonstrate that this contradiction stems from the fact that the swim pressure is not a true pressure. At a boundary or interface, the reduction in particle swimming generates a net active force density—an entirely self-generated body force. The pressure at the boundary, which was previously identified as the swim pressure, is in fact an elevated (relative to the bulk) value of the traditional particle pressure that is generated by this interfacial force density. Recognizing this unique mechanism for stress generation allows us to define a much more physically plausible surface tension. We clarify the utility of the swim pressure as an “equivalent pressure” (analogous to those defined from electrostatic and gravitational body forces) and the conditions in which this concept can be appropriately applied.

Publication: Physical Review E Vol.: 101 No.: 1 ISSN: 2470-0045

ID: CaltechAUTHORS:20200116-094602547

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Abstract: A consensus has emerged that a constraint to rotational or sliding motion of particles in dense suspensions under flow is the genesis of the discontinuous shear thickening (DST) phenomenon. We show that tangential fluid lubrication interactions due to finite-sized asperities on particle surfaces effectively provide these constraints, changing the dynamics of particle motion. By explicitly resolving for the surface roughness of particles, we show that, while smooth particles exhibit continuous shear thickening, purely hydrodynamic interactions in rough particles result in DST. In contrast to the frictional contact model, the hydrodynamic model predicts negative first and second normal stress differences for dense suspensions in the shear thickened state.

Publication: Physical Review Letters Vol.: 123 No.: 13 ISSN: 0031-9007

ID: CaltechAUTHORS:20190925-102153055

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Abstract: In a colloidal suspension at equilibrium, the diffusive motion of a tracer particle due to random thermal fluctuations from the solvent is related to the particle’s response to an applied external force, provided this force is weak compared to the thermal restoring forces in the solvent. This is known as the fluctuation-dissipation theorem (FDT) and is expressed via the Stokes-Einstein-Sutherland (SES) relation D = k_BT/ζ, where D is the particle’s self-diffusivity (fluctuation), ζ is the drag on the particle (dissipation), and k_BT is the thermal Boltzmann energy. Active suspensions are widely studied precisely because they are far from equilibrium—they can generate significant nonthermal internal stresses, which can break the detailed balance and time-reversal symmetry—and thus cannot be assumed to obey the FDT a priori. We derive a general relationship between diffusivity and mobility in generic colloidal suspensions (not restricted to near equilibrium) using generalized Taylor dispersion theory and derive specific conditions on particle motion required for the FDT to hold. Even in the simplest system of active Brownian particles (ABPs), these conditions may not be satisfied. Nevertheless, it is still possible to quantify deviations from the FDT and express them in terms of an effective SES relation that accounts for the ABPs conversion of chemical into kinetic energy.

Publication: Journal of Chemical Physics Vol.: 150 No.: 18 ISSN: 0021-9606

ID: CaltechAUTHORS:20190520-073652187

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Abstract: External fields can decidedly alter the free energy landscape of soft materials and can be exploited as a powerful tool for the assembly of targeted nanostructures and colloidal materials. Here, we use computer simulations to demonstrate that nonequilibrium internal fields or forces—forces that are generated by driven components within a system—in the form of active particles can precisely modulate the dynamical free energy landscape of a model soft material, a colloidal gel. Embedding a small fraction of active particles within a gel can provide a unique pathway for the dynamically frustrated network to circumvent the kinetic barriers associated with reaching a lower free energy state through thermal fluctuations alone. Moreover, by carefully tuning the active particle properties (the propulsive swim force and persistence length) in comparison to those of the gel, the active particles may induce depletion-like forces between the constituent particles of the gel despite there being no geometric size asymmetry between the particles. These resulting forces can rapidly push the system toward disparate regions of phase space. Intriguingly, the state of the material can be altered by tuning macroscopic transport properties such as the solvent viscosity. Our findings highlight the potential wide-ranging structural and kinetic control facilitated by varying the dynamical properties of a remarkably small fraction of driven particles embedded in a host material.

Publication: ACS Nano Vol.: 13 No.: 1 ISSN: 1936-0851

ID: CaltechAUTHORS:20190102-092234094

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Abstract: We study the motion of a spherical active Brownian particle (ABP) of size a, moving with a fixed speed U0, and reorienting on a time scale τ_R in the presence of a confining boundary. Because momentum is conserved in the embedding fluid, we show that the average force per unit area on the boundary equals the bulk mechanical pressure P∞ = p∞_f + Π∞, where p∞_f is the fluid pressure and Π∞ is the particle pressure; this is true for active and passive particles alike regardless of how the particles interact with the boundary. As an example, we investigate how hydrodynamic interactions (HI) change the particle-phase pressure at the wall, and find that Π^(wall) = n∞(k_BT+ ζ(Δ)U_0ℓ(Δ)/6), where ζ is the (Stokes) drag on the swimmer, ℓ = U_0τ_R is the run length, and Δ is the minimum gap size between the particle and the wall; as Δ → ∞ this is the familiar swim pressure [Takatori et al., Phys. Rev. Lett., 2014, 113, 1–5].

Publication: Soft Matter Vol.: 14 No.: 18 ISSN: 1744-683X

ID: CaltechAUTHORS:20180423-105343832

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Abstract: Active Brownian particles (ABPs) transmit a swim pressure II^(swim) = n ζD^(swim) to the container boundaries, where ζ is the drag coefficient, D^(swim) is the swim diffusivity and n is the uniform bulk number density far from the container walls. In this work we extend the notion of the isotropic swim pressure to the anisotropic tensorial swim stress σ^(swim) = - n ζD^(swim), which is related to the anisotropic swim diffusivity D^(swim). We demonstrate this relationship with ABPs that achieve nematic orientational order via a bulk external field. The anisotropic swim stress is obtained analytically for dilute ABPs in both 2D and 3D systems. The anisotropy, defined as the ratio of the maximum to the minimum of the three principal stresses, is shown to grow exponentially with the strength of the external field. We verify that the normal component of the anisotropic swim stress applies a pressure II^(swim) = -( σ^(swim) · n) · n on a wall with normal vector n, and, through Brownian dynamics simulations, this pressure is shown to be the force per unit area transmitted by the active particles. Since ABPs have no friction with a wall, the difference between the normal and tangential stress components—the normal stress difference—generates a net flow of ABPs along the wall, which is a generic property of active matter systems.

Publication: New Journal of Physics Vol.: 20 No.: 5 ISSN: 1367-2630

ID: CaltechAUTHORS:20180524-132438946

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Abstract: Suspensions of motile bacteria or synthetic microswimmers, termed active matter, manifest a remarkable propensity for self-organization, and formation of large-scale coherent structures. Most active matter research deals with almost homogeneous in space systems and little is known about the dynamics of strongly heterogeneous active matter. Here we report on experimental and theoretical studies on the expansion of highly concentrated bacterial droplets into an ambient bacteria-free fluid. The droplet is formed beneath a rapidly rotating solid macroscopic particle inserted in the suspension. We observe vigorous instability of the droplet reminiscent of a violent explosion. The phenomenon is explained in terms of continuum first-principle theory based on the swim pressure concept. Our findings provide insights into the dynamics of active matter with strong density gradients and significantly expand the scope of experimental and analytic tools for control and manipulation of active systems.

Publication: Nature Communications Vol.: 9ISSN: 2041-1723

ID: CaltechAUTHORS:20180409-080942319

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Abstract: A body submerged in active matter feels the swim pressure through a kinetic accumulation boundary layer on its surface. The boundary layer results from a balance between translational diffusion and advective swimming and occurs on the microscopic length scale λ^(-1) = δ/√2[1+1/6(ℓ/δ)^2] . Here λ = √D_Tτ_R, D_T is the Brownian translational diffusivity, τ_R is the reorientation time and ℓ = U0τR is the swimmer's run length, with U_0 the swim speed [Yan and Brady, J. Fluid. Mech., 2015, 785, R1]. In this work we analyze the swim pressure on arbitrary shaped bodies by including the effect of local shape curvature in the kinetic boundary layer. When δ ≪ L and ℓ ≪ L, where L is the body size, the leading order effects of curvature on the swim pressure are found analytically to scale as JSλδ^2/L, where JS is twice the (non-dimensional) mean curvature. Particle-tracking simulations and direct solutions to the Smoluchowski equation governing the probability distribution of the active particles show that λδ^2/L is a universal scaling parameter not limited to the regime δ, ℓ ≪ L. The net force exerted on the body by the swimmers is found to scale as F^(net)/(n^∞k_sT_sL^2) = f(λδ^2/L), where f(x) is a dimensionless function that is quadratic when x ≪ 1 and linear when x ∼ 1. Here, k_sT_s= ζU0^2τ_R/6 defines the ‘activity’ of the swimmers, with ζ the drag coefficient, and n^∞ is the uniform number density of swimmers far from the body. We discuss the connection of this boundary layer to continuum mechanical descriptions of active matter and briefly present how to include hydrodynamics into this purely kinetic study.

Publication: Soft Matter Vol.: 14 No.: 2 ISSN: 1744-683X

ID: CaltechAUTHORS:20171201-130807253

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Abstract: Chemically active Brownian particles with surface catalytic reactions may repel each other due to diffusiophoretic interactions in the reaction and product concentration fields. The system behavior can be described by a “chemical” coupling parameter Γ_c that compares the strength of diffusiophoretic repulsion to Brownian motion, and by a mapping to the classical electrostatic one component plasma (OCP) system. When confined to a constant-volume domain, body-centered cubic (bcc) crystals spontaneously form from random initial configurations when the repulsion is strong enough to overcome Brownian motion. Face-centered cubic (fcc) crystals may also be stable. The “melting point” of the “liquid-to-crystal transition” occurs at Γ_c ≈ 140 for both bcc and fcc lattices.

Publication: Physical Review E Vol.: 96 No.: 6 ISSN: 2470-0045

ID: CaltechAUTHORS:20171201-095649069

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Abstract: We use Brownian Dynamics (BD) simulations and continuum models to study the microstructures and mechanics in the colloidal film drying process. Colloidal suspensions are compressed between a planar moving interface and a stationary substrate. In the BD simulations, we develop a new Energy Minimization Potential-Free (EMPF) algorithm to enforce the hard-sphere potential in confined systems and to accurately measure the stress profile. The interface moves either at a constant velocity U_w or via a constant imposed normal stress Σ_e. Comparing the interface motions to the particle Brownian motion defines the Péclet numbers Pe_U = U_wa/d_0 and Pe_Σ = Σ_ea^3/k_BT, respectively, where d_0 = k_BT/ζ with k_BT the thermal energy scale, ζ the single-particle resistance, and a the particle radius. With a constant interface velocity, thermodynamics drives the suspension behavior when Pe_U ≪ 1, and homogeneous crystallization appears when the gap spacing between the two boundaries pushes the volume fraction above the equilibrium phase boundary. In contrast, when Pe_U ≫ 1, local epitaxial crystal growth appears adjacent to the moving interface even for large gap sizes. Interestingly, the most amorphous film microstructures are found at moderate Pe_U. The film stress profile develops sharp transitions and becomes step-like with growing Péclet number. With a constant imposed stress, the interface stops moving as the suspension pressure increases and the microstructural and mechanical behaviors are similar to the constant velocity case. Comparison with the simulations shows that the model accurately captures the stress on the moving interface, and quantitatively resolves the local stress and volume fraction distributions for low to moderate Péclet numbers. This work demonstrates the critical role of interface motion on the film microstructures and stresses.

Publication: Soft Matter Vol.: 13 No.: 44 ISSN: 1744-683X

ID: CaltechAUTHORS:20171030-084645065

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Abstract: Suspensions of self-propelled bodies generate a unique mechanical stress owing to their motility that impacts their large-scale collective behavior. For microswimmers suspended in a fluid with negligible particle inertia, we have shown that the virial swim stress is a useful quantity to understand the rheology and nonequilibrium behaviors of active soft matter systems. For larger self-propelled organisms such as fish, it is unclear how particle inertia impacts their stress generation and collective movement. Here we analyze the effects of finite particle inertia on the mechanical pressure (or stress) generated by a suspension of self-propelled bodies. We find that swimmers of all scales generate a unique swim stress and Reynolds stress that impact their collective motion. We discover that particle inertia plays a similar role as confinement in overdamped active Brownian systems, where the reduced run length of the swimmers decreases the swim stress and affects the phase behavior. Although the swim and Reynolds stresses vary individually with the magnitude of particle inertia, the sum of the two contributions is independent of particle inertia. This points to an important concept when computing stresses in computer simulations of nonequilibrium systems: The Reynolds and the virial stresses must both be calculated to obtain the overall stress generated by a system.

Publication: Physical Review Fluids Vol.: 2 No.: 9 ISSN: 2469-990X

ID: CaltechAUTHORS:20170929-141132324

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Abstract: We study the diffusion of a Brownian probe particle of size R in a dilute dispersion of active Brownian particles of size a, characteristic swim speed U_0, reorientation time τ_R, and mechanical energy k_sT_s=ζ_aU^2_0τ_R/6, where ζ_a is the Stokes drag coefficient of a swimmer. The probe has a thermal diffusivity D_P=k_BT/ζ_P, where k_BT is the thermal energy of the solvent and ζ_P is the Stokes drag coefficient for the probe. When the swimmers are inactive, collisions between the probe and the swimmers sterically hinder the probe's diffusive motion. In competition with this steric hindrance is an enhancement driven by the activity of the swimmers. The strength of swimming relative to thermal diffusion is set by Pe_s=U_0a/D_P. The active contribution to the diffusivity scales as Pe^2_s for weak swimming and Pe_s for strong swimming, but the transition between these two regimes is nonmonotonic. When fluctuations in the probe motion decay on the time scale τ_R , the active diffusivity scales as k_sT_s/ζ_P: the probe moves as if it were immersed in a solvent with energy k_sT_s rather than k_BT.

Publication: Physical Review E Vol.: 95 No.: 5 ISSN: 2470-0045

ID: CaltechAUTHORS:20170512-092926729

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Abstract: The rheology during the start-up and cessation of simple shear flow has been investigated for near hard-sphere colloidal suspensions. Simulations augmented by theoretical analysis are used to determine how the non-Newtonian stress development and relaxation depend on the microstructure. Accelerated Stokesian dynamics (ASD) and Brownian dynamics (BD) simulations are used for 0.05 ≤ Pe ≤ 500 in concentrated freely flowing suspensions; the Péclet number defining the ratio of shear to thermal motion is Pe=3πηγ ̇a^3/kT with η the suspending fluid viscosity, γ ̇ the shear rate, and kT the thermal energy. Theoretical predictions based on the Smoluchowski equation for dilute suspensions are made, and these are primarily used for comparison with results from BD simulations in which hydrodynamic interactions are neglected. For suspensions with hydrodynamics, simulations by ASD are used to probe start-up and flow cessation over a large range of Pe; these studies focus on solid volume fraction ϕ=0.4, with more limited examinations at other ϕ. The use of both BD and ASD simulations allows us to discriminate hydrodynamic interaction effects on the suspension rheology. The Brownian stresses computed by either method exhibit overshoots of their steady state value during the start-up of shear flow. The overshoots occur at strain amplitudes which depend on Pe, and the overshoot is described by a model based on extension of the concept of cage-breaking from glass dynamics. Results from the relaxation of a sheared suspension show that the distortion of the pair distribution function from its equilibrium form has a fast radial relaxation and a slow angular relaxation. The various rheometric functions (relative viscosity; first and second normal stress differences) are found to respond on different timescales, reflecting their different dependences on the flow-induced structure. A re-examination of steady shear flow allows us to find normal stress differences which tend properly toward zero at small Pe, unlike prior work; the discrepancy is found to be due to finite size scaling, as small simulations used in prior work resulted in excessively large normal stress responses at small Pe.

Publication: Journal of Rheology Vol.: 61 No.: 3 ISSN: 0148-6055

ID: CaltechAUTHORS:20170405-095705528

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Abstract: The current understanding is that the non-Newtonian rheology of active matter suspensions is governed by fluid-mediated hydrodynamic interactions associated with active self-propulsion. Here we discover an additional contribution to the suspension shear stress that predicts both thickening and thinning behavior, even when there is no nematic ordering of the microswimmers with the imposed flow. A simple micromechanical model of active Brownian particles in homogeneous shear flow reveals the existence of off-diagonal shear components in the swim stress tensor, which are independent of hydrodynamic interactions and fluid disturbances. Theoretical predictions from our model are consistent with existing experimental measurements of the shear viscosity of active suspensions, but also suggest new behavior not predicted by conventional models.

Publication: Physical Review Letters Vol.: 118 No.: 1 ISSN: 0031-9007

ID: CaltechAUTHORS:20170106-100314156

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Abstract: Diffusiophoresis is the process by which a colloidal particle moves in response to the concentration gradient of a chemical solute. Chemically active particles generate solute concentration gradients via surface chemical reactions which can result in their own motion — the self-diffusiophoresis of Janus particles — and in the motion of other nearby particles — normal down-gradient diffusiophoresis. The long-range nature of the concentration disturbance created by a reactive particle results in strong interactions among particles and can lead to the formation of clusters and even coexisting dense and dilute regions often seen in active matter systems. In this work, we present a general method to determine the many-particle solute concentration field allowing the dynamic simulation of the motion of thousands of reactive particles. With the simulation method, we first clarify and demonstrate the notion of “chemical screening,” whereby the long-ranged interactions become exponentially screened, which is essential for otherwise diffusiophoretic suspensions would be unconditionally unstable. Simulations show that uniformly reactive particles, which do not self-propel, form loosely packed clusters but no coexistence is observed. The simulations also reveal that there is a stability threshold — when the “chemical fuel” concentration is low enough, thermal Brownian motion is able to overcome diffusiophoretic attraction. Janus particles that self-propel show coexistence, but, interestingly, the stability threshold for clustering is not affected by the self-motion.

Publication: Journal of Chemical Physics Vol.: 145 No.: 13 ISSN: 0021-9606

ID: CaltechAUTHORS:20161005-090742307

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Abstract: Hard sphere colloidal particles are a basic model system to study phase transitions, self-assembly and out-equilibrium states. Experimentally it has been shown that oscillatory shearing of a monodisperse hard sphere glass, produces two different crystal orientations; a face centered cubic (FCC) crystal with the close packed direction parallel to shear at high strains and an FCC crystal with the close packed direction perpendicular to shear at low strains. Here, using Brownian dynamics simulations of hard sphere particles, we have examined high volume fraction shear-induced crystals under oscillatory shear as well their glass counterparts at the same volume fraction. While particle displacements under shear in the glass are almost isotropic, the sheared FCC crystal structures oriented parallel to shear, are anisotropic due to the cooperative motion of velocity–vorticity layers of particles sliding over each other. These sliding layers generally result in lower stresses and less overall particle displacements. Additionally, from the two crystal types, the perpendicular crystal exhibits less stresses and displacements at smaller strains, however at larger strains, the sliding layers of the parallel crystal are found to be more efficient in minimizing stresses and displacements, while the perpendicular crystal becomes unstable. The findings of this work suggest that the process of shear-induced ordering for a colloidal glass is facilitated by large out of cage displacements, which allow the system to explore the energy landscape and find the minima in energy, stresses and displacements by configuring particles into a crystal oriented parallel to shear.

Publication: Journal of Non-Newtonian Fluid Mechanics Vol.: 233ISSN: 0377-0257

ID: CaltechAUTHORS:20160120-151644666

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Abstract: The transient response of model hard sphere glasses is examined during the application of steady rate start-up shear using Brownian dynamics simulations, experimental rheology and confocal microscopy. With increasing strain, the glass initially exhibits an almost linear elastic stress increase, a stress peak at the yield point and then reaches a constant steady state. The stress overshoot has a nonmonotonic dependence with Peclet number, Pe, and volume fraction, φ, determined by the available free volume and a competition between structural relaxation and shear advection. Examination of the structural properties under shear revealed an increasing anisotropic radial distribution function, g(r), mostly in the velocity-gradient (xy) plane, which decreases after the stress peak with considerable anisotropy remaining in the steady-state. Low rates minimally distort the structure, while high rates show distortion with signatures of transient elongation. As a mechanism of storing energy, particles are trapped within a cage distorted more than Brownian relaxation allows, while at larger strains, stresses are relaxed as particles are forced out of the cage due to advection. Even in the steady state, intermediate super diffusion is observed at high rates and is a signature of the continuous breaking and reformation of cages under shear.

Publication: Journal of Rheology Vol.: 60 No.: 4 ISSN: 0148-6055

ID: CaltechAUTHORS:20160616-070638625

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Abstract: We thank Ikeda, Berthier, and Sollich (IBS) [1] for their comments which question our interpretation of the universal viscosity divergence near the flow-arrest transition in constant stress and pressure rheology of hard-sphere colloidal suspensions [2]. IBS introduced two Péclet numbers: Pe_0 = γa^2/d^0 and Pe = γa^2/d(ϕ), with γ the strain rate, a the particle size, d_0 the isolated single-particle diffusivity, and d(ϕ) the long-time at-rest self-diffusivity. And they considered three regimes: (i) Pe_0 < Pe ≪ 1, (ii) Pe_0 ≪ 1 ≪ Pe, and (iii) 1 ≪ Pe_0 < Pe.

Publication: Physical Review Letters Vol.: 116 No.: 17 ISSN: 0031-9007

ID: CaltechAUTHORS:20160525-104339165

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Abstract: 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.

Publication: Nature Communications Vol.: 7ISSN: 2041-1723

ID: CaltechAUTHORS:20160315-100739165

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Abstract: In this work we develop the Spectral Ewald Accelerated Stokesian Dynamics (SEASD), a novel computational method for dynamic simulations of polydisperse colloidal suspensions with full hydrodynamic interactions. SEASD is based on the framework of Stokesian Dynamics (SD) with extension to compressible solvents, and uses the Spectral Ewald (SE) method [Lindbo and Tornberg (2010) [29]] for the wave-space mobility computation. To meet the performance requirement of dynamic simulations, we use Graphic Processing Units (GPU) to evaluate the suspension mobility, and achieve an order of magnitude speedup compared to a CPU implementation. For further speedup, we develop a novel far-field block-diagonal preconditioner to reduce the far-field evaluations in the iterative solver, and SEASD-nf, a polydisperse extension of the mean-field Brownian approximation of Banchio and Brady (2003) [39]. We extensively discuss implementation and parameter selection strategies in SEASD, and demonstrate the spectral accuracy in the mobility evaluation and the overall O(N log N) computation scaling. We present three computational examples to further validate SEASD and SEASD-nf in monodisperse and bidisperse suspensions: the short-time transport properties, the equilibrium osmotic pressure and viscoelastic moduli, and the steady shear Brownian rheology. Our validation results show that the agreement between SEASD and SEASD-nf is satisfactory over a wide range of parameters, and also provide significant insight into the dynamics of polydisperse colloidal suspensions.

Publication: Journal of Computational Physics Vol.: 306ISSN: 0021-9991

ID: CaltechAUTHORS:20150720-091417561

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Abstract: The statistical mechanics and microhydrodynamics of active matter systems have been studied intensively during the past several years, by various soft matter physicists, chemists, engineers, and biologists around the world. Recent attention has focused on the fascinating nonequilibrium behaviors of active matter that cannot be observed in equilibrium thermodynamic systems, such as spontaneous collective motion and swarming. Even minimal kinetic models of active Brownian particles exhibit self-assembly that resembles a gas–liquid phase separation from classical equilibrium systems. Self-propulsion allows active systems to generate internal stresses that enable them to control and direct their own behavior and that of their surroundings. In this Review we discuss the forces that govern the motion of active Brownian microswimmers, the stress (or pressure) they generate, and the implication of these concepts on their collective behavior. We focus on recent work involving the unique ‘swim pressure’ exerted by active systems, and discuss how this perspective may be the basic underlying physical mechanism responsible for self-assembly and pattern formation in all active matter. We discuss the utility of the swim pressure concept to quantify the forces, stresses, and the (thermo?) dynamics of active matter.

Publication: Current Opinion in Colloid and Interface Science Vol.: 21ISSN: 1359-0294

ID: CaltechAUTHORS:20160119-095455048

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Abstract: 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 k_BT to the swimmer’s activity k_sT_s=ζU^2_0τ_R/6, where ζ is the Stokes drag coefficient, U0_ is the swim speed and τ_R is the reorientation time of the active particles. The theory, which is valid on all length and time scales, 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 behaviour when the boundary size is large compared to the swimmer’s run length ℓ=U_0τ_R. The theory is used to predict the motion of bodies of all sizes immersed in active matter.

Publication: Journal of Fluid Mechanics Vol.: 785ISSN: 0022-1120

ID: CaltechAUTHORS:20160328-102204704

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Abstract: Systems at equilibrium like molecular or colloidal suspensions have a well-defined thermal energy k_BT that quantifies the particles' kinetic energy and gauges how “hot” or “cold” the system is. For systems far from equilibrium, such as active matter, it is unclear whether the concept of a “temperature” exists and whether self-propelled entities are capable of thermally equilibrating like passive Brownian suspensions. Here we develop a simple mechanical theory to study the phase behavior and “temperature” of a mixture of self-propelled particles. A mixture of active swimmers and passive Brownian particles is an ideal system for discovery of the temperature of active matter and the quantities that get shared upon particle collisions. We derive an explicit equation of state for the active/passive mixture to compute a phase diagram and to generalize thermodynamic concepts like the chemical potential and free energy for a mixture of nonequilibrium species. We find that different stability criteria predict in general different phase boundaries, facilitating considerations in simulations and experiments about which ensemble of variables are held fixed and varied.

Publication: Soft Matter Vol.: 11 No.: 40 ISSN: 1744-683X

ID: CaltechAUTHORS:20150908-133503718

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Abstract: We study the constant stress and pressure rheology of dense hard-sphere colloidal suspensions using Brownian dynamics simulation. Expressing the flow behavior in terms of the friction coefficient—the ratio of shear to normal stress—reveals a shear arrest point from the collapse of the rheological data in the non-Brownian limit. The flow curves agree quantitatively (when scaled) with the experiments of Boyer et al. [Phys. Rev. Lett. 107, 188301 (2011)]. Near suspension arrest, both the shear and the incremental normal viscosities display a universal power law divergence. This work shows the important role of jamming on the arrest of colloidal suspensions and illustrates the care needed when conducting and analyzing experiments and simulations near the flow-arrest transition.

Publication: Physical Review Letters Vol.: 115 No.: 15 ISSN: 0031-9007

ID: CaltechAUTHORS:20151030-143145753

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Abstract: We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a noninteger dimension d_l. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same 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 semianalytical 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.

Publication: Physical Review Letters Vol.: 115 No.: 9 ISSN: 0031-9007

ID: CaltechAUTHORS:20150917-110132241

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Abstract: Net (as opposed to random) motion of active matter results from an average swim (or propulsive) force. It is shown that the average swim force acts like a body force – an internal body force. As a result, the particle-pressure exerted on a container wall is the sum of the swim pressure [Takatori et al., Phys. Rev. Lett., 2014, 113, 028103] and the ‘weight’ of the active particles. A continuum description is possible when variations occur on scales larger than the run length of the active particles and gives a Boltzmann-like distribution from a balance of the swim force and the swim pressure. Active particles may also display ‘action at a distance’ and accumulate adjacent to (or be depleted from) a boundary without any external forces. In the momentum balance for the suspension – the mixture of active particles plus fluid – only external body forces appear.

Publication: Soft Matter Vol.: 11 No.: 31 ISSN: 1744-683X

ID: CaltechAUTHORS:20150714-134052439

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Abstract: Using a powerful combination of experiments and simulations we demonstrate how the microstructure and its time evolution are linked with mechanical properties in a frustrated, out-of-equilibrium, particle gel under shear. An intermediate volume fraction colloid–polymer gel is used as a model system, allowing quantification of the interplay between interparticle attractions and shear forces. Rheometry, confocal microscopy and Brownian dynamics reveal that high shear rates, fully breaking the structure, lead after shear cessation to more homogeneous and stronger gels, whereas preshear at low rates creates largely heterogeneous weaker gels with reduced elasticity. We find that in comparison, thermal quenching cannot produce structural inhomogeneities under shear. We argue that external shear has strong implications on routes towards metastable equilibrium, and therefore gelation scenarios. Moreover, these results have strong implications for material design and industrial applications, such as mixing, processing and transport protocols coupled to the properties of the final material.

Publication: Soft Matter Vol.: 11 No.: 23 ISSN: 1744-683X

ID: CaltechAUTHORS:20150625-111520006

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Abstract: We present a comprehensive computational study of the short-time transport properties of bidisperse hard-sphere colloidal suspensions and the corresponding porous media. Our study covers bidisperse particle size ratios up to 4 and total volume fractions up to and beyond the monodisperse hard-sphere close packing limit. The many-body hydrodynamic interactions are computed using conventional Stokesian Dynamics (SD) via a Monte-Carlo approach. We address suspension properties including the short-time translational and rotational self-diffusivities, the instantaneous sedimentation velocity, the wavenumber-dependent partial hydrodynamic functions, and the high-frequency shear and bulk viscosities and porous media properties including the permeability and the translational and rotational hindered diffusivities. We carefully compare the SD computations with existing theoretical and numerical results. For suspensions, we also explore the range of validity of various approximation schemes, notably the pairwise additive approximations with the Percus-Yevick structural input. We critically assess the strengths and weaknesses of the SD algorithm for various transport properties. For very dense systems, we discuss in detail the interplay between the hydrodynamic interactions and the structures due to the presence of a second species of a different size.

Publication: Journal of Chemical Physics Vol.: 142 No.: 9 ISSN: 0021-9606

ID: CaltechAUTHORS:20150111-125624009

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Abstract: Self-propulsion allows living systems to display self-organization and unusual phase behavior. Unlike passive systems in thermal equilibrium, active matter systems are not constrained by conventional thermodynamic laws. A question arises, however, as to what extent, if any, can concepts from classical thermodynamics be applied to nonequilibrium systems like active matter. Here we use the new swim pressure perspective to develop a simple theory for predicting phase separation in active matter. Using purely mechanical arguments we generate a phase diagram with a spinodal and critical point, and define a nonequilibrium chemical potential to interpret the “binodal.” We provide a generalization of thermodynamic concepts like the free energy and temperature for nonequilibrium active systems. Our theory agrees with existing simulation data both qualitatively and quantitatively and may provide a framework for understanding and predicting the behavior of nonequilibrium active systems.

Publication: Physical Review E Vol.: 91 No.: 3 ISSN: 1539-3755

ID: CaltechAUTHORS:20141208-083141228

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Abstract: Diffusion in bidisperse Brownian hard-sphere suspensions is studied by Stokesian Dynamics (SD) computer simulations and a semi-analytical theoretical scheme for colloidal short-time dynamics, based on Beenakker and Mazur's method [Physica 120A, 388 (1983) & 126A, 349 (1984)]. Two species of hard spheres are suspended in an overdamped viscous solvent that mediates the salient hydrodynamic interactions among all particles. In a comprehensive parameter scan that covers various packing fractions and suspension compositions, we employ numerically accurate SD simulations to compute the initial diffusive relaxation of density modulations at the Brownian time scale, quantified by the partial hydrodynamic functions. A revised version of Beenakker and Mazur's δγ-scheme for monodisperse suspensions is found to exhibit surprisingly good accuracy, when simple rescaling laws are invoked in its application to mixtures. The so-modified δγ scheme predicts hydrodynamic functions in very good agreement with our SD simulation results, for all densities from the very dilute limit up to packing fractions as high as 40%.

Publication: Journal of Chemical Physics Vol.: 142 No.: 6 ISSN: 0021-9606

ID: CaltechAUTHORS:20141117-080952740

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Abstract: We analyze the stress, dispersion, and average swimming speed of self-propelled particles subjected to an external field that affects their orientation and speed. The swimming trajectory is governed by a competition between the orienting influence (i.e., taxis) associated with the external (e.g., magnetic, gravitational, thermal, nutrient concentration) field versus the effects that randomize the particle orientations (e.g., rotary Brownian motion and/or an intrinsic tumbling mechanism like the flagella of bacteria). The swimmers' motion is characterized by a mean drift velocity and an effective translational diffusivity that becomes anisotropic in the presence of the orienting field. Since the diffusivity yields information about the micromechanical stress, the anisotropy generated by the external field creates a normal stress difference in the recently developed “swim stress” tensor [Takatori, Yan, and Brady, Phys. Rev. Lett., 2014]. This property can be exploited in the design of soft, compressible materials in which their size, shape, and motion can be manipulated and tuned by loading the material with active swimmers. Since the swimmers exert different normal stresses in different directions, the material can compress/expand, elongate, and translate depending on the external field strength. Such an active system can be used as nano/micromechanical devices and motors. Analytical solutions are corroborated by Brownian dynamics simulations.

Publication: Soft Matter Vol.: 10 No.: 47 ISSN: 1744-683X

ID: CaltechAUTHORS:20141023-082715390

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Abstract: We calculate the effective shear viscosity of a dilute dispersion of n-dimensional non no-slip hyperspheres, first for hyperdrops, second for slippery hyperspheres, and third for porous hyperspheres.

Publication: Journal of Colloid and Interface Science Vol.: 430ISSN: 0021-9797

ID: CaltechAUTHORS:20140703-133719813

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Abstract: We discover a new contribution to the pressure (or stress) exerted by a suspension of self-propelled bodies. Through their self-motion, all active matter systems generate a unique swim pressure that is entirely athermal in origin. The origin of the swim pressure is based upon the notion that an active body would swim away in space unless confined by boundaries—this confinement pressure is precisely the swim pressure. Here we give the micromechanical basis for the swim stress and use this new perspective to study self-assembly and phase separation in active soft matter. The swim pressure gives rise to a nonequilibrium equation of state for active matter with pressure-volume phase diagrams that resemble a van der Waals loop from equilibrium gas-liquid coexistence. Theoretical predictions are corroborated by Brownian dynamics simulations. Our new swim stress perspective can help analyze and exploit a wide class of active soft matter, from swimming bacteria to catalytic nanobots to molecular motors that activate the cellular cytoskeleton.

Publication: Physical Review Letters Vol.: 113 No.: 2 ISSN: 0031-9007

ID: CaltechAUTHORS:20140926-105955901

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Abstract: The behaviour of a non-spherical osmotic motor – an axisymmetric catalytic particle self-propelling in a dilute dispersion of reactant particles – is considered. In contrast to a conventional osmotic motor that creates differences in concentration, and hence in osmotic pressure, due to asymmetry in reaction rate along its surface (e.g. a Janus particle with reactive and non-reactive patches), a non-spherical particle is able to move even with uniform chemical activity on its surface. For small departures from a sphere the velocity of self-propulsion is proportional to the square of the non-sphericity or distortion of the particle shape. It is shown that the inclusion of hydrodynamic interactions (HI) may drastically change the self-propulsion. Except for very slow chemical reactions, even the direction of self-propulsion changes with and without HI. Numerical calculations at finite non-sphericity suggest that the maximum velocity of self-propulsion is obtained by a sail-like motor shape, leading to the name ‘chemical sailing’. Moreover, no saturation in the speed of propulsion is found; the motor velocity increases as the area of this ‘sail’ grows and its thickness decreases. The self-propulsion of a non-spherical particle releasing products of a chemical reaction – a constant flux motor – is also considered.

Publication: Journal of Fluid Mechanics Vol.: 748ISSN: 0022-1120

ID: CaltechAUTHORS:20140708-134520862

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Abstract: We study the motion of a colloidal particle as it is driven by an oscillating external force of arbitrary amplitude and frequency through a colloidal dispersion. Large amplitude oscillatory flows (LAOFs) are examined predominantly from a phenomenological perspective in which experimental measurements inform constitutive models. Here, we investigate a LAOF from a microstructural perspective by connecting motion of the probe particle to the material response while making no assumptions a priori about how stress relaxes in the material. The suspension exerts nonconservative, hydrodynamic forces on the probe, while distortions in the particle configuration exert conservative forces: Brownian and interparticle forces, for example. The relative importance of each of these contributions to particle motion evolves with the degree of displacement from equilibrium. When the force on the probe is weak, the linear microviscoelasticity of the suspension is probed [see, e.g., Khair and Brady, J. Rheol. 49, 1449–1481 (2005)]. When oscillation rate is slow, the steady microrheology is probed [see, e.g., Squires and Brady, Phys. Fluids 17, 073101 (2005); Khair and Brady, J. Fluid Mech. 557, 73–117 (2006)]. This article develops a micromechanical model that recovers these limiting cases and then uses the same model to reveal the microrheology of colloidal dispersions deformed by a probe driven with arbitrary force amplitude and frequency. A chief result of this work is the discovery of a regime in which the resistance to motion of the probe particle is on average weaker than the resistance the probe experiences when deformed by high frequency oscillation. This hypoviscous effect arises when the reciprocating motion of the probe particle opens a channel free of other particles which is thus less resistive to probe motion. This effect is most apparent under the conditions of strong forces, rapid oscillation, and large extent of deformation.

Publication: Journal of Rheology Vol.: 58 No.: 1 ISSN: 0148-6055

ID: CaltechAUTHORS:20140130-110432252

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Abstract: Evanescent wave dynamic light scattering and Stokesian dynamics simulations were employed to study the dynamics of hard-sphere colloidal particles near a hard wall in concentrated suspensions. The evanescent wave averaged short-time diffusion coefficients were determined from experimental correlation functions over a range of scattering wave vectors and penetration depths. Stokesian dynamics simulations performed for similar conditions allow a direct comparison of both the short-time self- and collective diffusivity. As seen earlier [V. N. Michailidou, G. Petekidis, J. W. Swan, and J. F. Brady, Phys. Rev. Lett.102, 068302 (2009)] while the near wall dynamics in the dilute regime slow down compared to the free bulk diffusion, the reduction is negligible at higher volume fractions due to an interplay between the particle-wall and particle-particle hydrodynamic interactions. Here, we provide a comprehensive comparison between experiments and simulations and discuss the interplay of particle-wall and particle-particle hydrodynamics in the self- and cooperative dynamics determined at different scattering wave vectors and penetration depths.

Publication: Journal of Chemical Physics Vol.: 136 No.: 16 ISSN: 0021-9606

ID: CaltechAUTHORS:20131213-102410397

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Abstract: We propose a model for the self-propulsion of a small motor particle that generates a nonuniform concentration distribution of solute in the surrounding fluid via a constant solute flux asymmetrically from the motor surface. The net osmotic driving force and motor speed are investigated in the limits of slow and fast product particle flux (relative to the diffusive flux of the product species). When the only solute species in solution is that produced by the motor, the motor's speed is shown to be proportional to the solute flux for slow flux rates and to the square root of the solute flux for large flux rates. When solute species are already present in solution at concentration high compared to that generated by the motor, the motor speed at high flux rates saturates and scales as the diffusivity of the solute divided by the motor size. The analytical results compare well with Brownian dynamics simulations. Full hydrodynamic interactions are taken into account in the theoretical analysis.

Publication: Soft Matter Vol.: 9 No.: 28 ISSN: 1744-683X

ID: CaltechAUTHORS:20130805-101058789

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Abstract: The yielding behavior of hard sphere glasses under large-amplitude oscillatory shear has been studied by probing the interplay of Brownian motion and shear-induced diffusion at varying oscillation frequencies. Stress, structure and dynamics are followed by experimental rheology and Brownian dynamics simulations. Brownian-motion-assisted cage escape dominates at low frequencies while escape through shear-induced collisions at high ones, both related with a yielding peak in G′′. At intermediate frequencies a novel, for hard sphere glasses, double peak in G′′ is revealed reflecting both mechanisms. At high frequencies and strain amplitudes a persistent structural anisotropy causes a stress drop within the cycle after strain reversal, while higher stress harmonics are minimized at certain strain amplitudes indicating an apparent harmonic response.

Publication: Physical Review Letters Vol.: 110 No.: 17 ISSN: 0031-9007

ID: CaltechAUTHORS:20130531-101843545

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Abstract: The motion of a single Brownian particle in a complex fluid can reveal material behavior both at and away from equilibrium. In active microrheology, a probe particle is driven by an external force through a complex medium and its motion studied in order to infer properties of the embedding material. Most work in microrheology has focused on steady behavior and established the relationship between the motion of the probe, the microstructure, and the effective microviscosity of the medium. Transient behavior in the near-equilibrium, linear-response regime has also been studied via its connection to low-amplitude oscillatory probe forcing and the complex modulus; at very weak forcing, the microstructural response that drives viscosity is indistinguishable from equilibrium fluctuations. But important information about the basic physical aspects of structural development and relaxation in a medium is captured by startup and cessation of the imposed deformation in the nonlinear regime, where the structure is driven far from equilibrium. Here, we study theoretically and by dynamic simulation the transient behavior of a colloidal dispersion undergoing nonlinear microrheological forcing. The strength with which the probe is forced, Fext, compared to thermal forces, kT/b, governs the dynamics and defines a Péclet number, Pe = F^ext/(kT/b), where kT is the thermal energy and b is the colloidal bath particle size. For large Pe, a boundary layer (in which unsteady advection balances diffusion) forms at particle contact on the time scale of the flow, a/U, where a is the probe size and U its speed, whereas the wake forms over O(Pe) diffusive time steps. Similarly, relaxation following cessation occurs over several time scales corresponding to distinct physical processes. For very short times, the time scale for relaxation is set by a boundary layer of thickness δ ∼ (a+b)/Pe, and so τ ∼ δ^2/D_r, where Dr is the relative diffusivity between the probe of size a and a bath particle. Nearly all stress relaxation occurs during this time. At longer times, the Brownian diffusion of the bath particles acts to close the wake on a time scale set by how long it takes a bath particle to diffuse laterally across it, τ ∼ (a+b)^2/D_r. Although the majority of the microstructural relaxation occurs during this wake-healing process, it does so with little change in the stress. Also during relaxation, the probe travels backward in the suspension; this recovered strain is proportional to the free energy stored in the compressed particle configuration, an indicator that the stress is proportional to the free energy density stored entropically in the microstructure. Theoretical results are compared with Brownian dynamics simulation where it is found that the dilute theory captures the correct behavior even for concentrated suspensions. Two modes of forcing are studied: Constant force and constant velocity. Results are compared to analogous macrorheology results for suspensions undergoing simple shear flow.

Publication: Journal of Rheology Vol.: 57 No.: 2 ISSN: 0148-6055

ID: CaltechAUTHORS:20130307-111828320

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Abstract: A combination of confocal microscopy and rheology experiments, Brownian dynamics (BD) and molecular dynamics (MD) simulations and mode coupling theory (MCT) have been applied in order to investigate the effect of shear rate on the transient dynamics and stress–strain relations in supercooled and glassy systems under shear. Immediately after shear is switched on, the microscopic dynamics display super-diffusion and the macroscopic rheology a stress overshoot, which become more pronounced with increasing shear rate. MCT relates both to negative sections of the generalized shear modulus, which grow with increasing shear rate. When the inverse shear rate becomes much smaller than the structural relaxation time of the quiescent system, relaxation through Brownian motion becomes less important. In this regime, larger stresses are accumulated before the system yields and the transition from localization to flow occurs earlier and more abruptly.

Publication: Journal of Physics: Condensed Matter Vol.: 24 No.: 46 ISSN: 0953-8984

ID: CaltechAUTHORS:20121211-151943836

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Abstract: In active, nonlinear microrheology, a Brownian “probe” particle is driven through a complex fluid and its motion tracked in order to infer the mechanical properties of the embedding material. In the absence of external forcing, the probe and background particles form an equilibrium microstructure that fluctuates thermally. Probe motion through the medium distorts the microstructure; the character of this deformation, and hence its influence on probe motion, depends on the strength with which the probe is forced, F^(ext), compared to thermal forces, kT/b, defining a Péclet number, Pe = F^(ext)/(kT/b), where kT is the thermal energy and b is the characteristic microstructural length scale. Recent studies showed that the mean probe speed can be interpreted as the effective material viscosity, whereas fluctuations in probe velocity give rise to an anisotropic force-induced diffusive spread of its trajectory. The viscosity and diffusivity can thus be obtained by two simple quantities—mean and mean-square displacement of the probe. The notion that diffusive flux is driven by stress gradients leads to the idea that the stress can be related directly to the microdiffusivity, and thus the anisotropy of the diffusion tensor reflects the presence of normal stress differences in nonlinear microrheology. In this study, a connection is made between diffusion and stress gradients, and a relation between the particle-phase stress and the diffusivity and viscosity is derived for a probe particle moving through a colloidal dispersion. This relation is shown to agree with two standard micromechanical definitions of the stress, suggesting that the normal stresses and normal stress differences can be measured in nonlinear microrheological experiments if both the mean and mean-square motion of the probe are monitored. Owing to the axisymmetry of the motion about a spherical probe, the second normal stress difference is zero, while the first normal stress difference is linear in Pe for Pe≫1 and vanishes as Pe^4 for Pe≪1. The expression obtained for stress-induced migration can be viewed as a generalized nonequilibrium Stokes–Einstein relation. A final connection is made between the stress and an “effective temperature” of the medium, prompting the interpretation of the particle stress as the energy density, and the expression for osmotic pressure as a “nonequilibrium equation of state.”

Publication: Journal of Rheology Vol.: 56 No.: 5 ISSN: 0148-6055

ID: CaltechAUTHORS:20120913-100621344

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Abstract: Concentrated hard-sphere suspensions and glasses are investigated with rheometry, confocal microscopy, and Brownian dynamics simulations during start-up shear, providing a link between microstructure, dynamics, and rheology. The microstructural anisotropy is manifested in the extension axis where the maximum of the pair-distribution function exhibits a minimum at the stress overshoot. The interplay between Brownian relaxation and shear advection as well as the available free volume determine the structural anisotropy and the magnitude of the stress overshoot. Shear-induced cage deformation induces local constriction, reducing in-cage diffusion. Finally, a superdiffusive response at the steady state, with a minimum of the time-dependent effective diffusivity, reflects a continuous cage breakup and reformation.

Publication: Physical Review Letters Vol.: 108 No.: 9 ISSN: 0031-9007

ID: CaltechAUTHORS:20120402-131312090

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Abstract: A method is proposed for computing the low-Reynolds-number hydrodynamic forces on particles comprising a suspension confined by two parallel, no-slip walls. This is constructed via the two-dimensional analogue of Hasimoto’s solution (J. Fluid Mech., vol. 5, 1959, pp. 317–328) for a periodic array of point forces in a viscous, incompressible fluid, and, like Hasimoto, the summation of interactions is accelerated by substitution and superposition of ‘Ewald-like’ forcing. This method is akin to the accelerated Stokesian dynamics technique (J. Fluid Mech., vol. 448, 2001, pp. 115–146) and models the suspension dynamics with log–linear computational scaling. The effectiveness of this approach is demonstrated with a calculation of the high-frequency dynamic viscosity of a colloidal dispersion as function of volume fraction and channel width. Similarly, the short-time self-diffusivity for and the sedimentation rate of spherical particles in a confined suspension are determined. The results demonstrate the influence of confining geometry on the transport of small particles, which is becoming increasingly important for micro- and biofluidics.

Publication: Journal of Fluid Mechanics Vol.: 687ISSN: 0022-1120

ID: CaltechAUTHORS:20120117-100915152

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Abstract: A method is proposed for computing the low-Reynolds-number hydrodynamic forces on particles comprising a suspension confined by two parallel, no-slip walls. This is constructed via the two-dimensional analogue of Hasimoto’s solution (J. Fluid Mech., vol. 5, 1959, pp. 317–328) for a periodic array of point forces in a viscous, incompressible fluid, and, like Hasimoto, the summation of interactions is accelerated by substitution and superposition of ‘Ewald-like’ forcing. This method is akin to the accelerated Stokesian dynamics technique (J. Fluid Mech., vol. 448, 2001, pp. 115–146) and models the suspension dynamics with log–linear computational scaling. The effectiveness of this approach is demonstrated with a calculation of the high-frequency dynamic viscosity of a colloidal dispersion as function of volume fraction and channel width. Similarly, the short-time self-diffusivity for and the sedimentation rate of spherical particles in a confined suspension are determined. The results demonstrate the influence of confining geometry on the transport of small particles, which is becoming increasingly important for micro- and biofluidics.

Publication: Journal of Fluid Mechanics Vol.: 687ISSN: 0022-1120

ID: CaltechAUTHORS:20120628-135421071

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Abstract: We employ an analogy to traditional dynamic light scattering to describe the inhomogeneous and anisotropic diffusion of colloid particles near a solid boundary measured via evanescent wave dynamic light scattering. Following this approach, we generate new expressions for the short-time self- and collective diffusivities of colloidal dispersions with arbitrary volume fraction. We use these expressions in combination with accelerated Stokesian dynamics simulations to calculate the diffusivities in the limit of large and small scattering wave numbers for evanescent penetration depths ranging from four particle radii to one-fifth of a particle radius and volume fractions from 10% to 40%. We show that at high volume fractions, and larger penetration depths, the boundaries have little effect on the dynamics of the suspension parallel to the wall since, to a first approximation, the boundary acts hydrodynamically much as another nearby particle. However, near and normal to the wall, the diffusivity shows a strong dependence on penetration depth for all volume fractions. This is due to the lubrication interactions between the particles and the boundary as the particle moves relative to the wall. These results are novel and comprehensive with respect to the range of penetration depth and volume fraction and provide a complete determination of the effect of hydrodynamic interactions on colloidal diffusion adjacent to a rigid boundary.

Publication: Journal of Chemical Physics Vol.: 135 No.: 1 ISSN: 0021-9606

ID: CaltechAUTHORS:20110802-135414917

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Abstract: We develop a general framework for modeling the hydrodynamic self-propulsion (i.e., swimming) of bodies (e.g., microorganisms) at low Reynolds number via Stokesian Dynamics simulations. The swimming body is composed of many spherical particles constrained to form an assembly that deforms via relative motion of its constituent particles. The resistance tensor describing the hydrodynamic interactions among the individual particles maps directly onto that for the assembly. Specifying a particular swimming gait and imposing the condition that the swimming body is force- and torque-free determine the propulsive speed. The body’s translational and rotational velocities computed via this methodology are identical in form to that from the classical theory for the swimming of arbitrary bodies at low Reynolds number. We illustrate the generality of the method through simulations of a wide array of swimming bodies: pushers and pullers, spinners, the Taylor=Purcell swimming toroid, Taylor’s helical swimmer, Purcell’s three-link swimmer, and an amoeba-like body undergoing large-scale deformation. An open source code is a part of the supplementary material and can be used to simulate the swimming of a body with arbitrary geometry and swimming gait.

Publication: Physics of Fluids Vol.: 23 No.: 7 ISSN: 1070-6631

ID: CaltechAUTHORS:20110906-073613861

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Abstract: Diffusiophoresis, the motion of a particle in response to an externally imposed concentration gradient of a solute species, is analysed from both the traditional coarse-grained macroscopic (i.e. continuum) perspective and from a fine-grained micromechanical level in which the particle and the solute are treated on the same footing as Brownian particles dispersed in a solvent. It is shown that although the two approaches agree when the solute is much smaller in size than the phoretic particle and is present at very dilute concentrations, the micromechanical colloidal perspective relaxes these restrictions and applies to any size ratio and any concentration of solute. The different descriptions also provide different mechanical analyses of phoretic motion. At the continuum level the macroscopic hydrodynamic stress and interactive force with the solute sum to give zero total force, a condition for phoretic motion. At the colloidal level, the particle's motion is shown to have two contributions: (i) a ‘back-flow’ contribution composed of the motion of the particle due to the solute chemical potential gradient force acting on it and a compensating fluid motion driven by the long-range hydrodynamic velocity disturbance caused by the chemical potential gradient force acting on all the solute particles and (ii) an indirect contribution arising from the mutual interparticle and Brownian forces on the solute and phoretic particle, that contribution being non-zero because the distribution of solute about the phoretic particle is driven out of equilibrium by the chemical potential gradient of the solute. At the colloidal level the forces acting on the phoretic particle – both the statistical or ‘thermodynamic’ chemical potential gradient and Brownian forces and the interparticle force – are balanced by the Stokes drag of the solvent to give the net phoretic velocity.

Publication: Journal of Fluid Mechanics Vol.: 667ISSN: 0022-1120

ID: CaltechAUTHORS:20110318-144327024

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Abstract: The hydrodynamic interactions between colloidal particles in small ensembles are measured at varying distances from a no-slip surface over a range of inter-particle separations. The diffusion tensor for motion parallel to the wall of each ensemble is calculated by analyzing thousands of particle trajectories generated by blinking holographic optical tweezers and by dynamic simulation. The Stokesian Dynamics simulations predict similar particle dynamics. By separating the dynamics into three classes of modes: self, relative and collective diffusion, we observe qualitatively different behavior depending on the relative magnitudes of the distance of the ensemble from the wall and the inter-particle separation. A simple picture of the pair-hydrodynamic interactions is developed, while many-body-hydrodynamic interactions give rise to more complicated behavior. The results demonstrate that the effect of many-body hydrodynamic interactions in the presence of a wall is much richer than the single particle behavior and that the multiple-particle behavior cannot be simply predicted by a superposition of pair interactions.

Publication: Soft Matter Vol.: 7 No.: 15 ISSN: 1744-683X

ID: CaltechAUTHORS:20110808-093428080

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Abstract: The low-Reynolds-number motion of a single spherical particle between parallel walls is determined from the exact reflection of the velocity field generated by multipoles of the force density on the particle’s surface. A grand mobility tensor is constructed and couples these force multipoles to moments of the velocity field in the fluid surrounding the particle. Every element of the grand mobility tensor is a finite, ordered sum of inverse powers of the distance between the walls. These new expressions are used in a set of Stokesian dynamics simulations to calculate the translational and rotational velocities of a particle settling between parallel walls and the Brownian drift force on a particle diffusing between the walls. The Einstein correction to the Newtonian viscosity of a dilute suspension that accounts for the change in stress distribution due to the presence of the channel walls is determined. It is proposed how the method and results can be extended to computations involving many particles and periodic simulations of suspensions in confined geometries.

Publication: Physics of Fluids Vol.: 22 No.: 10 ISSN: 1070-6631

ID: CaltechAUTHORS:20101129-135303247

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Abstract: We study the fluctuating motion of a Brownian-sized probe particle as it is dragged by a constant external force through a colloidal dispersion. In this nonlinear-microrheology problem, collisions between the probe and the background bath particles, in addition to thermal fluctuations of the solvent, drive a long-time diffusive spread of the probe's trajectory. The influence of the former is determined by the spatial configuration of the bath particles and the force with which the probe perturbs it. With no external forcing the probe and bath particles form an equilibrium microstructure that fluctuates thermally with the solvent. Probe motion through the dispersion distorts the microstructure; the character of this deformation, and hence its influence on the probe's motion, depends on the strength with which the probe is forced, F^(ext), compared to thermal forces, kT/b, defining a Péclet number, Pe = F^(ext)/(kT/b), where kT is the thermal energy and b the bath particle size. It is shown that the long-time mean-square fluctuational motion of the probe is diffusive and the effective diffusivity of the forced probe is determined for the full range of Péclet number. At small Pe Brownian motion dominates and the diffusive behaviour of the probe characteristic of passive microrheology is recovered, but with an incremental flow-induced ‘microdiffusivity’ that scales as D^(micro) ~ D_aPe^2φ_b, where φ_b is the volume fraction of bath particles and D_a is the self-diffusivity of an isolated probe. At the other extreme of high Péclet number the fluctuational motion is still diffusive, and the diffusivity becomes primarily force induced, scaling as (F^(ext)/η)φ_b, where η is the viscosity of the solvent. The force-induced microdiffusivity is anisotropic, with diffusion longitudinal to the direction of forcing larger in both limits compared to transverse diffusion, but more strongly so in the high-Pe limit. The diffusivity is computed for all Pe for a probe of size a in a bath of colloidal particles, all of size b, for arbitrary size ratio a/b, neglecting hydrodynamic interactions. The results are compared with the force-induced diffusion measured by Brownian dynamics simulation. The theory is also compared to the analogous shear-induced diffusion of macrorheology, as well as to experimental results for macroscopic falling-ball rheometry. The results of this analysis may also be applied to the diffusive motion of self-propelled particles.

Publication: Journal of Fluid Mechanics Vol.: 658ISSN: 0022-1120

ID: CaltechAUTHORS:20101011-112128191

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Abstract: The popular interest in cornstarch and water mixtures known as “oobleck” after the complex fluid in one of Dr. Seuss's classic children's books arises from their transition from fluid-like to solid-like behavior when stressed. The viscous liquid that emerges from a roughly 2-to-1 (by volume) combination of starch to water can be poured into one's hand. When squeezed, the liquid morphs into a doughy paste that can be formed into shapes, only to “melt” into a puddle when the applied stress is relieved. Internet videos show people running across a large pool of the stuff, only to sink once they stop in place, and “monsters” that grow out of the mixture when it's acoustically vibrated (for an example, see the online version of this article).

Publication: Physics Today Vol.: 62 No.: 10 ISSN: 0031-9228

ID: CaltechAUTHORS:20091102-121110772

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Abstract: We investigate the Brownian motion of hard-sphere colloids near a solid wall by Evanescent Wave Dynamic Light Scattering (EWDLS). We carried out measurements for various volume fractions of sterically stabilized poly(methyl methacrylate) (PMMA) particles over a range of scattering wave vectors, q. While in the dilute regime, the near wall short-time diffusion is significantly slowed down due to particle-wall hydrodynamic interactions (HI); as volume fraction increases, the wall effect is progressively diminished at all q's. We present a new analysis for the EWDLS short-time self- and collective diffusivities applicable to all volume fractions and a simple model for the self-diffusion describing the interplay between particle-wall and particle-particle HI. Moreover, a weaker decay of the near-wall self-diffusion coefficient with volume fraction is predicted by Stokesian dynamics simulations.

Publication: Physical Review Letters Vol.: 102 No.: 6 ISSN: 0031-9007

ID: CaltechAUTHORS:20090828-104119071

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Abstract: 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.

Publication: Physical Review Letters Vol.: 100 No.: 15 ISSN: 0031-9007

ID: CaltechAUTHORS:CORprl08

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Abstract: Collective diffusivity in a suspension of rigid particles in steady linear viscous flows is evaluated by investigating the dynamics of the time correlation of long-wavelength density fluctuations. In the absence of hydrodynamic interactions between suspended particles in a dilute suspension of identical hard spheres, closed-form asymptotic expressions for the collective diffusivity are derived in the limits of low and high Péclet numbers, where the Péclet number Pe = gamma-dot a^2/D0 with gamma-dot being the shear rate and D0 = kB T/6πη a is the Stokes–Einstein diffusion coefficient of an isolated sphere of radius a in a fluid of viscosity η. The effect of hydrodynamic interactions is studied in the analytically tractable case of weakly sheared (Pe « 1) suspensions. For strongly sheared suspensions, i.e. at high Pe, in the absence of hydrodynamics the collective diffusivity Dc = 6 Ds∞, where Ds∞ is the long-time self-diffusivity and both scale as φ gamma-dot a^2$, where φ is the particle volume fraction. For weakly sheared suspensions it is shown that the leading dependence of collective diffusivity on the imposed flow is proportional to D0 φPe Ê, where Ê is the rate-of-strain tensor scaled by gamma-dot, regardless of whether particles interact hydrodynamically. When hydrodynamic interactions are considered, however, correlations of hydrodynamic velocity fluctuations yield a weakly singular logarithmic dependence of the cross-gradient-diffusivity on k at leading order as ak → 0 with k being the wavenumber of the density fluctuation. The diagonal components of the collective diffusivity tensor, both with and without hydrodynamic interactions, are of O(φPe2), quadratic in the imposed flow, and finite at k = 0. At moderate particle volume fractions, 0.10 ≤ φ ≤ 0.35, Brownian Dynamics (BD) numerical simulations in which there are no hydrodynamic interactions are performed and the transverse collective diffusivity in simple shear flow is determined via time evolution of the dynamic structure factor. The BD simulation results compare well with the derived asymptotic estimates. A comparison of the high-Pe BD simulation results with available experimental data on collective diffusivity in non-Brownian sheared suspensions shows a good qualitative agreement, though hydrodynamic interactions prove to be important at moderate concentrations.

Publication: Journal of Fluid Mechanics Vol.: 597ISSN: 0022-1120

ID: CaltechAUTHORS:LESjfm08

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Abstract: 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 probe, the microviscosity again transitions between two Newtonian plateaus: for â<3.52 (to two decimal places) the microviscosity at small Pe is greater than at large Pe (again, velocity thinning); however, for â>3.52 the microviscosity at small Pe is less than at large Pe, which suggests it “velocity thickens” as Pe is increased. This anomalous velocity thickening—due entirely to the probe shape—highlights the care needed when designing microrheology experiments with non-spherical probes.

Publication: Journal of Rheology Vol.: 52 No.: 1 ISSN: 0148-6055

ID: CaltechAUTHORS:KHAjor08

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Abstract: 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.

Publication: Physics of Fluids Vol.: 19 No.: 11 ISSN: 1070-6631

ID: CaltechAUTHORS:WApof07

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Abstract: 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 $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi \,{\ll}\, 1$, respectively. Further, the region of zero-Stokes closed trajectories is destroyed, and there exists a new attracting limit cycle whose location in the shearing plane is, at leading order, independent of $\mbox{\textit{St}}$. The extension of the present analysis to include a generic linear flow, and the implications of the finite-$\mbox{\textit{St}}$ trajectory modifications for coagulating systems are discussed.

Publication: Journal of Fluid Mechanics Vol.: 559ISSN: 0022-1120

ID: CaltechAUTHORS:SUBjfm06

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Abstract: 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 with the lubrication range of hydrodynamic interactions causing the microviscosity to enter a continuous ‘force-thickening’ regime. The qualitative picture of the microviscosity variation with Pe is in good agreement with theoretical and computational investigations on the ‘macroviscosity’ of sheared colloidal dispersions, and, after appropriate scaling, we are able to make a direct quantitative comparison. This suggests that active tracking microrheology is a valuable tool with which to explore the rich nonlinear rheology of complex fluids.

Publication: Journal of Fluid Mechanics Vol.: 557ISSN: 0022-1120

ID: CaltechAUTHORS:KHAjfm06b

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 554ISSN: 0022-1120

ID: CaltechAUTHORS:BRAjfm06

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Abstract: 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.

Publication: Physics of Fluids Vol.: 18 No.: 4 ISSN: 1070-6631

ID: CaltechAUTHORS:KHApof06

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Abstract: Systems governed by time reversible equations of motion often give rise to irreversible behaviour. The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions. Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science, are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility6 fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk. By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.

Publication: Nature Vol.: 438 No.: 7070 ISSN: 0028-0836

ID: CaltechAUTHORS:20150331-092329031

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Abstract: We investigate active particle-tracking microrheology in a colloidal dispersion by Brownian dynamics simulations. A probe particle is dragged through the dispersion with an externally imposed force in order to access the nonlinear viscoelastic response of the medium. The probe’s motion is governed by a balance between the external force and the entropic “reactive” force of the dispersion resulting from the microstructural deformation. A “microviscosity” is defined by appealing to the Stokes drag on the probe and serves as a measure of the viscoelastic response. This microviscosity is a function of the Péclet number (Pe=Fa∕kT)(Pe=Fa∕kT)—the ratio of “driven” (F)(F) to diffusive (kT∕a)(kT∕a) transport—as well as of the volume fraction of the force-free bath particles making up the colloidal dispersion. At low Pe—in the passive microrheology regime—the microviscosity can be directly related to the long-time self-diffusivity of the probe. As Pe increases, the microviscosity “force-thins” until another Newtonian plateau is reached at large Pe. Microviscosities for all Péclet numbers and volume fractions can be collapsed onto a single curve through a simple volume fraction scaling and equate well to predictions from dilute microrheology theory. The microviscosity is shown to compare well with traditional macrorheology results (theory and simulations).

Publication: Journal of Rheology Vol.: 49 No.: 6 ISSN: 0148-6055

ID: CaltechAUTHORS:20180725-105809395

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Abstract: 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 agreement. This seems remarkable, given the rather significant difference in the two methods of measurement. However, for more complicated or unknown materials, where such scaling relations may not be known in advance, the comparison between micro- and macro may not be so favorable, which raises important questions about the relation between micro- and macrorheology. Finally, by analogy with previous work on macrorheology, we propose methods to scale up the present (dilute) theory to account for more concentrated suspensions, and suggest new active microrheological experiments to probe different aspects of suspension behavior.

Publication: Physics of Fluids Vol.: 17 No.: 7 ISSN: 1070-6631

ID: CaltechAUTHORS:SQUpof05

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 527ISSN: 0022-1120

ID: CaltechAUTHORS:LESjfm05

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 506ISSN: 0022-1120

ID: CaltechAUTHORS:SIEjfm04

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Abstract: An approach based on the generalized reciprocal theorem is presented to derive the well-known result for the drag force exerted on a rigid sphere translating in a viscous fluid in an arbitrary manner. The use of generalized reciprocal theorem allows one to bypass the calculation of stress distribution over the particle surface in order to compute the force.

Publication: Physics of Fluids Vol.: 16 No.: 3 ISSN: 1070-6631

ID: CaltechAUTHORS:LESpof04

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Abstract: 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.

Publication: Journal of Chemical Physics Vol.: 118 No.: 22 ISSN: 0021-9606

ID: CaltechAUTHORS:BANjcp03

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 472ISSN: 0022-1120

ID: CaltechAUTHORS:CARjfm02

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Abstract: The non-Newtonian rheology is calculated numerically to second order in the volume fraction in steady simple shear flows for Brownian hard spheres in the presence of hydrodynamic and excluded volume interactions. Previous analytical and numerical results for the low-shear structure and rheology are confirmed, demonstrating that the viscosity shear thins proportional to Pe2, where Pe is the dimensionless shear rate or Péclet number, owing to the decreasing contribution of Brownian forces to the viscosity. In the large Pe limit, remnants of Brownian diffusion balance convection in a boundary-layer in the compressive region of the flow. In consequence, the viscosity shear thickens when this boundary-layer coincides with the near-contact lubrication regime of the hydrodynamic interaction. Wakes are formed at large Pe in the extensional zone downstream from the reference particle, leading to broken symmetry in the pair correlation function. As a result of this asymmetry and that in the boundary-layer, finite normal stress differences are obtained as well as positive departures in the generalized osmotic pressure from its equilibrium value. The first normal stress difference changes from positive to negative values as Pe is increased when the hard-sphere limit is approached. This unusual effect is caused by the hydrodynamic lubrication forces that maintain particles in close proximity well into the extensional quadrant of the flow. The study demonstrates that many of the non-Newtonian effects observed in concentrated suspensions by experiments and by Stokesian dynamics simulations are present also in dilute suspensions.

Publication: Journal of Fluid Mechanics Vol.: 456ISSN: 0022-1120

ID: CaltechAUTHORS:BERjfm02

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Abstract: An instability mechanism is shown to operate in complex, non-Newtonian fluids in which a jump in normal stresses occurs between two fluids. Fluids with a negative second normal stress difference can be unstable with respect to transverse or spanwise perturbations. The mechanism appears to be generic, although the details will depend on the specific flow and the nature of the complex fluid. The instability mechanism is illustrated for two-layer Couette and falling film flows of viscous suspensions.

Publication: Journal of Non-Newtonian Fluid Mechanics Vol.: 102 No.: 2 ISSN: 0377-0257

ID: CaltechAUTHORS:20180725-105809518

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 448ISSN: 0022-1120

ID: CaltechAUTHORS:SIEjfm01

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Abstract: The linear viscoelasticity of semiflexible polymers is studied through Brownian Dynamics simulations covering a broad range of chain stiffness and time scales. Our results agree with existing theoretical predictions in the flexible and stiff limits; however, we find that over a wide intermediate-time window spanning several decades, the stress relaxation is described by a single power law t^(-alpha), with the exponent alpha apparently varying continuously from 1/2 for flexible chains, to 5/4 for stiff ones. Our study identifies the limits of validity of the t^(-3/4) power law at short times predicted by recent theories. An additional regime is identified, the "ultrastiff" chains, where this behavior disappears. In the absence of Brownian motion, the purely mechanical stress relaxation produces a t^(-3/4) power law for both short and intermediate times.

Publication: Physical Review E Vol.: 64 No.: 5 ISSN: 1063-651X

ID: CaltechAUTHORS:DIMpre01

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Abstract: 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.

Publication: Physics of Fluids Vol.: 13 No.: 1 ISSN: 1070-6631

ID: CaltechAUTHORS:ICHpof01

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 407ISSN: 0022-1120

ID: CaltechAUTHORS:FOSjfm00

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Abstract: 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 mechanisms in the two regimes. A physical explanation for the off-diagonal diffusivities is proposed.

Publication: Journal of Fluid Mechanics Vol.: 401ISSN: 0022-1120

ID: CaltechAUTHORS:FOSjfm99

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Abstract: 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].

Publication: Journal of Fluid Mechanics Vol.: 348ISSN: 0022-1120

ID: CaltechAUTHORS:BRAjfm97

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Abstract: The non-equilibrium behaviour of concentrated colloidal dispersions is studied by Stokesian Dynamics, a general molecular-dynamics-like technique for simulating particles suspended in a viscous fluid. 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. Three clearly defined regions of behaviour are revealed. There is first a Brownian-motion-dominated regime (Pe ≤ 1) where departures from equilibrium in structure and diffusion are small, but the suspension viscosity shear thins dramatically. When the Brownian and hydrodynamic forces balance (Pe ≈ 10), the dispersion forms a new ‘phase’ with the particles aligned in ‘strings’ along the flow direction and the strings are arranged hexagonally. This flow-induced ordering persists over a range of Pe and, while the structure and diffusivity now vary considerably, the rheology remains unchanged. Finally, there is a hydrodynamically dominated regime (Pe > 200) with a dramatic change in the long-time self-diffusivity and the rheology. Here, as the Péclet number increases the suspension shear thickens owing to the formation of large clusters. The simulation results are shown to agree well with experiment.

Publication: Journal of Fluid Mechanics Vol.: 313ISSN: 0022-1120

ID: CaltechAUTHORS:20120208-152602570

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Abstract: Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either raacroscopically quiescent or in linear flow at arbitrary Peclet number Pe = ẏa^2/2D, where ẏ is the shear rate, a is the particle radius, and D = k_BT/6πηa is the diffusion coefficient of an isolated particle. Here, k_B is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by k_BT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction ø = 4/3πa^3n and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, D_∞^s, is given by the sum of D_0^s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D_0^s and D_∞^s are anisotropic, in general, with the anisotropy of D_0^s due solely to that of the steady microstructure. The influence of flow upon D_∞^s is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations. The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(φPe^(3/2)) for Pe « 1 and ø « 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D_0^s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with ẏ. Regardless of whether or not the particles interact hydrodynamically, flow influences D_∞^s at O(øPe) and O(øPe^(3/2)). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of O(øPe^(3/2)), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe^(3/2)DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor. At high ø a scaling theory based on the approach of Brady (1994) is used to approximate D_∞^s. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of Pe = ẏa^2/2D^s_0(φ)$. At small Pe the dependence on Pe is the same as at low ø.

Publication: Journal of Fluid Mechanics Vol.: 312ISSN: 0022-1120

ID: CaltechAUTHORS:20120208-141206043

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Abstract: 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.

Publication: Physics of Fluids Vol.: 8 No.: 4 ISSN: 1070-6631

ID: CaltechAUTHORS:YURpof96

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Abstract: The steady reaction rate in a bed of spherical catalyst particles immersed in a flowing Newtonian fluid is studied. The dependence of the rate on the particle volume fraction φ = 4/3 πn a^3, where n is the number density of spheres and a is the sphere radius, and the Péclet number Pe of the uniform bulk flow through the bed is determined analytically for φ << 1 and Pe << 1. The Péclet number is the ratio of transport by advection to diffusion; Pe = Ua/D, where U is the bulk velocity through the bed and D is the molecular diffusivity of the reactant in the fluid. In the absence of a bulk flow, it is shown that a bulk reaction in the bed results in a screening of the concentration disturbance beyond a distance of O(aφ^(-1/2)) from a particle. In this case, the nondimensional bulk reaction rate or Nusselt number is Nu ~ 1 + (3φ)^(1/2). For an isolated particle, it is well-known that weak uniform advection balances diffusion at a length scale of O(aPe^(-1)) and Nu ~ 1 + Pe/2. When there is weak flow through a bed of catalytic particles, the influence of flow strength and particle fraction upon the reaction rate depends on the ratio of the two length scales, Pe/(φ^2), characteristic of these two limiting situations. For Pe/φ^(1/2) = O(1), there is a balance of advection, diffusion, and reaction at a distance of O(aφ^(-1/2)) from a particle. The first correction to Nu for all Pe/φ^(1/2) is shown to be (3φ + (Pe^2)/4)^(1/2), in which the dependence of Nu on Pe and φ is not, in general, separable, although it becomes separable in the limits Pe/φ^(1/2) → 0 and e/φ^(1/2) → ∞. We also discuss the case of Pe >> 1, in which Nu has the form of A(φ)Pe^(1/3), where A(φ) is an unknown function which is apparently O(1) for all φ.

Publication: Industrial & Engineering Chemistry Research Vol.: 34 No.: 10 ISSN: 0888-5885

ID: CaltechAUTHORS:20180524-155601923

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 293ISSN: 0022-1120

ID: CaltechAUTHORS:LOVjfm95

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 275ISSN: 0022-1120

ID: CaltechAUTHORS:NOTjfm94

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Abstract: The long-time self-diffusivity in concentrated colloidal dispersions is determined from a consideration of the temporal decay of density fluctuations. For hydrodynamically interacting Brownian particles the long-time self-diffusivity, D^s_∞, is shown to be expressible as the product of the hydrodynamically determined short-time self-diffusivity, D^s_0(φ), and a contribution that depends on the distortion of the equilibrium structure caused by a diffusing particle. An argument is advanced to show that as maximum packing is approached the long-time self-diffusivity scales as D^s_∞(φ)~ D^s_0(φ)/g(2;φ), where g(2;φ) is the value of the equilibrium radial-distribution function at contact and φ is the volume fraction of interest. This result predicts that the longtime self-diffusivity vanishes quadratically at random close packing, φ_m ≈ 0.63, i.e. D^s_∞D_0(1-φ/φ_m)^2 as φ → φ_m, where D_0 = kT/6πηα is the diffusivity of a single isolated particle of radius α in a fluid of viscosity η. This scaling occurs because Ds_0(φ) vanishes linearly at random close packing and the radial-distribution function at contact diverges as (1 -φ/φ_m)^(−1). A model is developed to determine the structural deformation for the entire range of volume fractions, and for hard spheres the longtime self-diffusivity can be represented by: D^s_∞(φ) = D^s_∞(φ)/[1 + 2φg(2;φ)]. This formula is in good agreement with experiment. For particles that interact through hard-spherelike repulsive interparticle forces characterized by a length b(> α), the same formula applies with the short-time self-diffusivity still determined by hydrodynamic interactions at the true or ‘hydrodynamic’ volume fraction φ, but the structural deformation and equilibrium radial-distribution function are now determined by the ‘thermodynamic’ volume fraction φ_b based on the length b. When b » α, the long-time self-diffusivity vanishes linearly at random close packing based on the ‘thermodynamic’ volume fraction φ_(bm). This change in behaviour occurs because the true or ‘hydrodynamic’ volume fraction is so low that the short-time self-diffusivity is given by its infinite-dilution value D_0. It is also shown that the temporal transition from short- to long-time diffusive behaviour is inversely proportional to the dynamic viscosity and is a universal function for all volume fractions when time is nondimensionalized by α^2/D^s_∞(φ).

Publication: Journal of Fluid Mechanics Vol.: 272ISSN: 0022-1120

ID: CaltechAUTHORS:20120229-143202470

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Abstract: 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.

Publication: Journal of Chemical Physics Vol.: 101 No.: 2 ISSN: 0021-9606

ID: CaltechAUTHORS:BRAjcp93bcomm

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Abstract: The unsteady force acting on a sphere that is held fixed in a steady uniform flow with small-amplitude oscillations is evaluated to O(Re) for small Reynolds number Re. Good agreement is shown with the numerical results of Mei, Lawrence & Adrian (1991) up to Re [approximate] 0.5. The analytical result is transformed by Fourier inversion to allow for an arbitrary time-dependent motion which is small relative to the steady uniform flow. This yields a history-dependent force which has an integration kernel that decays exponentially for large time.

Publication: Journal of Fluid Mechanics Vol.: 256ISSN: 0022-1120

ID: CaltechAUTHORS:LOVjfm93b

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Abstract: The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a^2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t^-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t^-2.

Publication: Journal of Fluid Mechanics Vol.: 256ISSN: 0022-1120

ID: CaltechAUTHORS:LOVjfm93a

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Abstract: The pressure moment of a rigid particle is defined to be the trace of the first moment of the surface stress acting on the particle. A Faxén law for the pressure moment of one spherical particle in a general low-Reynolds-number flow is found in terms of the ambient pressure, and the pressure moments of two rigid spheres immersed in a linear ambient flow are calculated using multipole expansions and lubrication theory. The results are expressed in terms of resistance functions, following the practice established in other interaction studies. The osmotic pressure in a dilute colloidal suspension at small Péclet number is then calculated, to second order in particle volume fraction, using these resistance functions. In a second application of the pressure moment, the suspension or particle-phase pressure, used in two-phase flow modeling, is calculated using Stokesian dynamics and results for the suspension pressure for a sheared cubic lattice are reported.

Publication: Physics of Fluids A Vol.: 5 No.: 10 ISSN: 0899-8213

ID: CaltechAUTHORS:JEFpofa93

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Abstract: The hydrodynamic force on a body that undergoes translational acceleration in an unbounded fluid at low Reynolds number is considered. The results extend the prior analysis of Lovalenti and Brady [to appear in J. Fluid Mech. (1993)] for rigid particles to drops and bubbles. Similar behavior is shown in that, with the inclusion of convective inertia, the long-time temporal decay of the force (or the approach to steady state) at finite Reynolds number is faster than the t-1/2 predicted by the unsteady Stokes equations.

Publication: Physics of Fluids A Vol.: 5 No.: 9 ISSN: 0899-8213

ID: CaltechAUTHORS:LOVpofa93

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Abstract: A simple model for the rheological behavior of concentrated colloidal dispersions is developed. For a suspension of Brownian hard spheres there are two contributions to the macroscopic stress: a hydrodynamic and a Brownian stress. For small departures from equilibrium, the hydrodynamic contribution is purely dissipative and gives the high-frequency dynamic viscosity. The Brownian contribution has both dissipative and elastic parts and is responsible for the viscoelastic behavior of colloidal dispersions. An evolution equation for the pair-distribution function is developed and from it a simple scaling relation is derived for the viscoelastic response. The Brownian stress is shown to be proportional to the equilibrium radial-distribution function at contact, g(2;phi), divided by the short-time self-diffusivity, D0(s)(phi), both evaluated at the volume fraction phi of interest. This scaling predicts that the Brownian stress diverges at random close packing, phi(m), with an exponent of -2, that is, eta0' approximately eta(1 - phi/phi(m))-2, where eta0' is the steady shear viscosity of the dispersion and eta is the viscosity of the suspending fluid. Both the scaling law and the predicted magnitude are in excellent accord with experiment. For viscoelastic response, the theory predicts that the proper time scale is a2/D0s, where a is the particle radius, and, when appropriately scaled, the form of the viscoelastic response is a universal function for all volume fractions, again in agreement with experiment. In the presence of interparticle forces there is an additional contribution to the stress analogous to the Brownian stress. When the length scale characterizing the interparticle forces is comparable to the particle size, the theory predicts that there is only a quantitative contribution from the interparticle forces to the stress; the qualitative behavior, particularly the singular scaling of the viscosity and the form of the viscoelastic response, remains unchanged from the Brownian case. For strongly repulsive interparticle forces characterized by a length scale b (much greater than a), however, the theory predicts that the viscosity diverges at the random close packing volume fraction, phi(bm), based on the length scale b, with a weaker exponent of -1. The viscoelastic response now occurs on the time scale b2/D0s(phi), but is of the same form as for Brownian dispersions.

Publication: Journal of Chemical Physics Vol.: 99 No.: 1 ISSN: 0021-9606

ID: CaltechAUTHORS:BRAjcp93b

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Abstract: A new simulation method is presented for low-Reynolds-number flow problems involving elongated particles in an unbounded fluid. The technique extends the principles of Stokesian dynamics, a multipole moment expansion method, to ellipsoidal particle shapes. The methodology is applied to prolate spheroids in particular, and shown to be efficient and accurate by comparison with other numerical methods for Stokes flow. The importance of hydrodynamic interactions is illustrated by examples on sedimenting spheroids and particles in a simple shear flow.

Publication: Journal of Fluid Mechanics Vol.: 251ISSN: 0022-1120

ID: CaltechAUTHORS:CLAjfm93a

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Abstract: The simulation method for prolate spheroids in Stokes flow introduced in a companion paper (Claeys & Brady 1993a) is extended to handle statistically homogeneous unbounded dispersions. The convergence difficulties associated with the slow decay of velocity disturbances at zero Reynolds number are overcome by applying O';Brien's renormalization procedure. The Ewald summation technique is employed to accelerate the evaluation of all mobility interactions. As a first application of this new method, the hydrodynamic transport properties of equilibrium hard-ellipsoid structures are calculated for aspect ratios ranging from 3 to 50. Calculated viscosities in the isotropic phase agree reasonably well with published experimental measurements.

Publication: Journal of Fluid Mechanics Vol.: 251ISSN: 0022-1120

ID: CaltechAUTHORS:CLAjfm93b

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Abstract: The short-time limit of the hydrodynamic transport properties is calculated for crystalline dispersions of parallel prolate spheroids using a moment expansion technique similar in concept to the simulation method known as Stokesian dynamics. The concentration dependence of the sedimentation rate, the hindered diffusivity and the Theological behaviour of face-centred lattices are examined for concentrations up to regular close packing (74% by volume). The influence of the detailed microstructure of the dispersion is also investigated by considering different arrangements of parallel ellipsoids. Useful reference configurations are proposed as standard geometries for regular arrays of prolate spheroids.

Publication: Journal of Fluid Mechanics Vol.: 251ISSN: 0022-1120

ID: CaltechAUTHORS:CLAjfm93c

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Abstract: It is shown that the osmotic pressure of a colloidal dispersion can be interpreted as the isotropic part of the macroscopic particle stress in the suspension. The particle stress is in turn expressible in terms of hydrodynamic interactions among the suspended particles. Thus, there is a completely mechanical definition of the osmotic pressure, just as there is for the pressure in a molecular fluid. For an equilibrium suspension of colloidal particles subjected to thermal Brownian forces, this mechanical definition is shown to be exactly equal to the usual ``thermodynamic'' one. The derivation given here places the equilibrium and nonequilibrium properties of macroparticle fluids on the same mechanical foundation that underlies the statistical mechanics of simple liquids. Furthermore, through this development the relationship between hydrodynamics and kinetic-theory-like descriptions of colloids is explained.

Publication: Journal of Chemical Physics Vol.: 98 No.: 4 ISSN: 0021-9606

ID: CaltechAUTHORS:BRAjcp93a

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Abstract: A molecular-dynamics-like method is presented for the simulation of a suspension of dielectric particles in a nonconductive solvent forming an electrorheological fluid. The method accurately accounts for both hydrodynamic and electrostatic interparticle interactions from dilute volume fractions to closest packing for simultaneous shear and electric fields. The hydrodynamic interactions and rheology are determined with the Stokesian dynamics methodology, while the electrostatic interactions, in particular, the conservative electrostatic interparticle forces, are determined from the electrostatic energy of the suspension. The energy of the suspension is computed from the induced particle dipoles by a method previously developed [R. T. Bonnecaze and J. F. Brady, Proc. R. Soc. London, Ser. A 430, 285 (1990)]. Using the simulation, the dynamics can be directly correlated to the observed macroscopic rheology of the suspension for a range of the so-called Mason number, Ma, the ratio of viscous to electrostatic forces. The simulation is specifically applied to a monolayer of spherical particles of areal fraction 0.4 with a particle-to-fluid dielectric constant ratio of 4 for Ma=10^−4 to [infinity]. The effective viscosity of the suspension increases as Ma^−1 or with the square of the electric field for small Ma and has a plateau value at large Ma, as is observed experimentally. This rheological behavior can be interpreted as Bingham plastic-like with a dynamic yield stress. The first normal stress difference is negative, and its magnitude increases as Ma^−1 at small Ma with a large Ma plateau value of zero. In addition to the time averages of the rheology, the time traces of the viscosities are presented along with selected "snapshots" of the suspension microstructure. In particular, at small Ma, the suspension dynamics exhibit two distinct motions: a slow elastic-body-like deformation where electrostatic energy is stored, followed by a rapid microstructural rearrangement where energy is viscously dissipated. It is suggested that the observed dynamic yield stress is associated with these dynamics.

Publication: Journal of Chemical Physics Vol.: 96 No.: 3 ISSN: 0021-9606

ID: CaltechAUTHORS:BONjcp92

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Abstract: We calculate the average hydrodynamic stress on fractal aggregates of spheres using Stokesian dynamics. We find that for fractal aggregates of force-free particles, the stress does not grow as the cube of the radius of gyration, but rather as the number of particles in the aggregate. This behavior is only found for random aggregates of force-free particles held together by hydrodynamic lubrication forces. The stress on aggregates of particles rigidly connected by interparticle forces grows as the radius of gyration cubed. We explain this behavior by examining the transmission of the tension along connecting lines in an aggregate and use the concept of a persistance length in order to characterize this stress transmission within an aggregate.

Publication: Journal of Chemical Physics Vol.: 94 No.: 7 ISSN: 0021-9606

ID: CaltechAUTHORS:BOSjcp91

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Abstract: The effective conductivity of an infinite, random, mono-disperse, hard-sphere suspension is reported for particle to matrix conductivity ratios of ∞, 10 and 0.01 for sphere volume fractions, c, up to 0.6. The conductivities are computed with a method previously described by the authors, which includes both far- and near-field interactions, and the particle configurations are generated via a Monte Carlo method. The results are consistent with the previous theoretical work of D. J. Jeffrey to O(c^2) and the bounds computed by S. Torquato and F. Lado. It is also found that the Clausius-Mosotti equation is reasonably accurate for conductivity ratios of 10 or less all the way up to 60 % (by volume). The calculated conductivities compare very well with those of experiments. In addition, percolation-like numerical experiments are performed on periodically replicated cubic lattices of N nearly touching spheres with an infinite particle to matrix conductivity ratio where the conductivity is computed as spheres are removed one by one from the lattice. Under suitable normalization of the conductivity and volume fraction, it is found that the initial volume fraction must be extremely close to maximum packing in order to observe a percolation transition, indicating that the near-field effects must be very large relative to far-field effects. These percolation transitions occur at the accepted values for simple (sc), bodycentred (BCC) and face-centred (FCC) cubic lattices. Also, the vulnerability of the lattices computed here are exactly those of previous investigators. Due to limited data above the percolation threshold, we could not correlate the conductivity with a power law near the threshold; however, it can be correlated with a power law for large normalized volume fractions. In this case the exponents are found to be 1.70, 1.75 and 1.79 for sc, BCC and FCC lattices respectively.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 432 No.: 1886 ISSN: 1364-5021

ID: CaltechAUTHORS:20141210-110725362

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Abstract: The effective reaction rate is calculated for a random array of reactive, stationary spherical traps in a medium containing a highly mobile reactant. Multipole scattering up to the quadrupole level, properly accounting for the conditionally convergent long-range interactions, plus direct addition of exact two-body interactions is employed. It is found that the addition of two-body interactions has a negligible effect on the effective reaction rates computed, in contrast to the case of the effective conductivity. Our results closely match the random walker simulation results of Lee, Kim, Miller, and Torquato [Phys. Rev. B 39, 11833 (1989)] up to 30% trap volume fraction, after which they underpredict the effective reaction rate. To accurately compute the effective reaction rate at high volume fractions, higher order many-body multipole interactions are required.

Publication: Journal of Chemical Physics Vol.: 94 No.: 1 ISSN: 0021-9606

ID: CaltechAUTHORS:BONjcp91

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Abstract: A general method is developed to predict the effective conductivity of an infinite, statistically homogeneous suspension of particles in an arbitrary (ordered or disordered) configuration. The method follows closely that of 'stokesian dynamics', and captures both far-field and near-field particle interactions accurately with no convergence difficulties. This is accomplished by forming a capacitance matrix, the electrostatic analogue of the low-Reynolds-number resistance matrix, which relates the monopole (charge), dipole and quadrupole of the particles to the potential field of the system. A far-field approximation to the capacitance matrix is formed via a moment expansion of the integral equation for the potential. The capacitance matrix of the infinite system is limited to a finite number of equations by using periodic boundary conditions, and the Ewald method is used to form rapidly converging lattice sums of particle interactions. To include near-field effects, exact two-body interactions are added to the far-field approximation of the capacitance matrix. The particle dipoles are then calculated directly to determine the effective conductivity of the system. The Madelung constant of cohesive energy of ionic crystals is calculated for simple and body-centred cubic lattices as a check on the method formulation. The results are found to be in excellent agreement with the accepted values. Also, the effective conductivities of spherical particles in cubic arrays are calculated for particle to matrix conductivity ratios of infinity, 10 and 0.01.

Publication: Proceedings of the Royal Society of London. Series A, Mathematical, Physical and Engineering Sciences Vol.: 430 No.: 1879 ISSN: 0962-8444

ID: CaltechAUTHORS:20141215-102858372

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Abstract: The viscosity of a suspension of spherical Brownian particles is determined by Stokesian dynamics as a function of the Péclet number. Several new aspects concerning the theoretical derivation of the direct contribution of the Brownian motion to the bulk stress are given, along with the results obtained from a simulation of a monolayer. The simulations reproduce the experimental behavior generally observed in dense suspensions, and an explanation of this behavior is given by observing the evolution of the different contributions to the viscosity with shear rate. The shear thinning at low Péclet numbers is due to the disappearance of the direct Brownian contribution to the viscosity; the deformation of the equilibrium microstructure is, however, small. By contrast, at very high Péclet numbers the suspension shear thickens due to the formation of large clusters.

Publication: Journal of Chemical Physics Vol.: 91 No.: 3 ISSN: 0021-9606

ID: CaltechAUTHORS:BOSjcp89

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Abstract: The dispersion of a tracer in a heterogeneous medium in which the tracer's velocity has zero mean and a covariance that decays as x^−γ with distance x is studied using nonlocal advection–diffusion theory. If the velocity covariance decays slowly, γ ≤ 2, the tracer's dispersive motion is non-Fickian even at long times after its release. Under these circumstances, it is not possible to predict the dispersion by assuming that the tracer samples the velocity fluctuations primarily by molecular diffusion, even if the fluctuations are weak. Instead, we develop a self-consistent theory in which the tracer samples each velocity fluctuation by the motion resulting from the other fluctuations. It is shown that in cases of anomalous diffusion, the tracer's mean-square displacement grows faster than linearly with time—as t^[4/(2+ γ)] for 0 < γ < 2 and as t(ln t)1/2 for γ = 2 as t→∞.

Publication: Physics of Fluids A Vol.: 1 No.: 1 ISSN: 0899-8213

ID: CaltechAUTHORS:KOCpofa89

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Abstract: The hydrodynamic transport properties of hard-sphere dispersions are calculated for volume fractions (φ) spanning the dilute limit up to the fluid–solid transition at φ=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite-size effects are derived. The effects of using various levels of approximation in computing both the far- and near-field hydrodynamic interactions are also examined. The transport properties associated with freely mobile suspensions—sedimentation velocities, self-diffusion coefficients, and effective viscosities—are determined here, while the corresponding properties of porous media are determined in a companion paper [Phys. Fluids 31, 3473 (1988)]. Comparison of the simulation results is made with both experiment and theory. In particular, the short-time self-diffusion coefficients and the suspension viscosities are in excellent agreement with experiment.

Publication: Physics of Fluids Vol.: 31 No.: 12 ISSN: 0031-9171

ID: CaltechAUTHORS:PHIpof88a

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Abstract: The hydrodynamic transport properties of hard-sphere dispersions are calculated for volume fractions (φ) spanning the dilute limit up to the fluid–solid transition at φ=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite-size effects are determined. The effects of using various levels of approximation in computing both the far- and near-field hydrodynamic interactions are also examined. The transport properties associated with porous media—permeabilities and hindered diffusion coefficients—are determined here. The corresponding properties of freely mobile suspensions are determined in a companion paper [Phys. Fluids 31, 3462 (1988)].

Publication: Physics of Fluids Vol.: 31 No.: 12 ISSN: 0031-9171

ID: CaltechAUTHORS:PHIpof88b

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Abstract: A general method for computing the hydrodynamic interactions among an infinite suspension of particles, under the condition of vanishingly small particle Reynolds number, is presented. The method follows the procedure developed by O'Brien (1979) for constructing absolutely convergent expressions for particle interactions. For use in dynamic simulation, the convergence of these expressions is accelerated by application of the Ewald summation technique. The resulting hydrodynamic mobility and/or resistance matrices correctly include all far-field non-convergent interactions. Near-field lubrication interactions are incorporated into the resistance matrix using the technique developed by Durlofsky, Brady & Bossis (1987). The method is rigorous, accurate and computationally efficient, and forms the basis of the Stokesian-dynamics simulation method. The method is completely general and allows such diverse suspension problems as self-diffusion, sedimentation, rheology and flow in porous media to be treated within the same formulation for any microstructural arrangement of particles. The accuracy of the Stokesian-dynamics method is illustrated by comparing with the known exact results for spatially periodic suspensions.

Publication: Journal of Fluid Mechanics Vol.: 195ISSN: 0022-1120

ID: CaltechAUTHORS:20120531-135717062

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Abstract: The dispersion of a passive tracer resulting from flow through heterogeneous porous media is studied. When the correlation length for the permeability fluctuations is finite, a normal diffusive process, with the mean-square displacement of the tracer growing linearly with time, is obtained at long times. However, it is shown that, when the correlation length diverges, anomalous diffusion occurs in which the mean-square displacement grows faster than linearly with time. The space–time evolution of the tracer's concentration is calculated and shown to be universal—uniquely related to the covariance of the permeability field—in the anomalous regime.

Publication: Physics of Fluids Vol.: 31 No.: 5 ISSN: 0031-9171

ID: CaltechAUTHORS:KOCpof88

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Abstract: An explicit expression for the sedimentation velocity at low particle Reynolds number in a concentrated suspension is derived and evaluated for two different approximations to the hydrodynamic interactions: a strict pairwise additive approximation and a far-field, or Rotne–Prager, approximation. It is shown that the simple Rotne–Prager approximation gives a very accurate prediction for the sedimentation velocity of random suspensions from the dilute limit all the way up to close packing. The pairwise additive approximation, however, fails completely, predicting an aphysical negative sedimentation velocity above a volume fraction φ ≈ 0.23. The explanation for these different behaviors is shown to be linked to the "effective medium" behavior of the suspensions. It is shown analytically and by Stokesian dynamics simulation that a suspension of neutrally buoyant particles may be modeled as a homogeneous fluid with an effective viscosity, but a sedimenting suspension cannot. As a result, the Rotne–Prager approximation actually captures the correct features of the many-body interactions in sedimentation. An analytical expression for the sedimentation rate, which is in good agreement with experiment, is obtained using the Percus–Yevick hard-sphere distribution function.

Publication: Physics of Fluids Vol.: 31 No.: 4 ISSN: 0031-9171

ID: CaltechAUTHORS:BRApof88

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Abstract: Particles suspended or dispersed in a fluid medium occur in a wide variety of natural and man-made settings, e.g. slurries, composite materials, ceramics, colloids, polymers, proteins, etc. The central theoretical and practical problem is to understand and predict the macroscopic equilibrium and transport properties of these multiphase materials from their microstructural mechanics. The macroscopic properties might be the sedimentation or aggregation rate, self-diffusion coefficient, thermal conductivity, or rheology of a suspension of particles. The microstructural mechanics entails the Brownian, interparticle, external, and hydrodynamic forces acting on the particles, as well as their spatial and temporal distribution, which is commonly referred to as the microstructure. If the distribution of particles were given, as well as the location and motion of any boundaries and the physical properties of the particles and suspending fluid, one would simply have to solve (in principle, not necessarily in practice) a well-posed boundary-value problem to determine the behavior of the material. Averaging this solution over a large volume or over many different configurations, the macroscopic or averaged properties could be determined. The two key steps in this approach, the solution of the many-body problem and the determination of the microstructure, are formidable but essential tasks for understanding suspension behavior. This article discusses a new, molecular-dynamics-like approach, which we have named Stokesian dynamics, for dynamically simulating the behavior of many particles suspended or dispersed in a fluid medium. Particles in suspension may interact through both hydrodynamic and nonhydrodynamic forces, where the latter may be any type of Brownian, colloidal, interparticle, or external force. The simulation method is capable of predicting both static (i.e. configuration-specific) and dynamic microstructural properties, as well as macroscopic properties in either dilute or concentrated systems. Applications of Stokesian dynamics are widespread; problems of sedimentation, flocculation, diffusion, polymer rheology, and transport in porous media all fall within its domain. Stokesian dynamics is designed to provide the same theoretical and computational basis for multiphase, dispersed systems as does molecular dynamics for statistical theories of matter. This review focuses on the simulation method, not on the areas in which Stokesian dynamics can be used. For a discussion of some of these many different areas, the reader is referred to the excellent reviews and proceedings of topical conferences that have appeared (e.g. Batchelor 1976a, Dickinson 1983, Faraday Discussions 1983, 1987, Family & Landau 1984). Before embarking on a description of Stokesian dynamics, we pause here to discuss some of the relevant theoretical literature on suspensions, and dynamic simulation in general, in order to put Stokesian dynamics in perspective.

Publication: Annual Review of Fluid Mechanics Vol.: 20ISSN: 0066-4189

ID: CaltechAUTHORS:BRAjfm88

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Abstract: The self-diffusivity of Brownian hard spheres in a simple shear flow is studied by numerical simulation. Particle trajectories are calculated by Stokesian dynamics, with an accurate representation of the suspension hydrodynamics that includes both many-body interactions and lubrication. The simulations are of a monolayer of identical spheres as a function of the Péclet number: Pe =gamma-dot a^2/D0, which measures the relative importance of shear and Brownian forces. Here gamma-dot is the shear rate, a the particle radius, and D0 the diffusion coefficient of a single sphere at infinite dilution. In the absence of shear, using only hydrodynamic interactions, the simulations reproduce the pair-distribution function of the equivalent hard-disk system. Both short- and long-time self-diffusivities are determined as a function of the Péclet number. The results show a clear transition from a Brownian motion dominated regime (Pe<1) to a hydrodynamically dominated regime (Pe>10) with a dramatic change in the behavior of the long-time self-diffusivity.

Publication: Journal of Chemical Physics Vol.: 87 No.: 9 ISSN: 0021-9606

ID: CaltechAUTHORS:BOSjcp87

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Abstract: The fundamental solution or Green's function for flow in porous media is determined using Stokesian dynamics, a molecular-dynamics-like simulation method capable of describing the motions and forces of hydrodynamically interacting particles in Stokes flow. By evaluating the velocity disturbance caused by a source particle on field particles located throughout a monodisperse porous medium at a given value of volume fraction of solids ø, and by considering many such realizations of the (random) porous medium, the fundamental solution is determined. Comparison of this fundamental solution with the Green's function of the Brinkman equation shows that the Brinkman equation accurately describes the flow in porous media for volume fractions below 0.05. For larger volume fractions significant differences between the two exist, indicating that the Brinkman equation has lost detailed predictive value, although it still describes qualitatively the behavior in moderately concentrated porous media. At low ø where the Brinkman equation is known to be valid, the agreement between the simulation results and the Brinkman equation demonstrates that the Stokesian dynamics method correctly captures the screening characteristic of porous media. The simulation results for ø ≥ 0.05 may be useful as a basis of comparison for future theoretical work.

Publication: Physics of Fluids Vol.: 30 No.: 11 ISSN: 1070-6631

ID: CaltechAUTHORS:20120626-112949432

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Abstract: A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.

Publication: Journal of Fluid Mechanics Vol.: 180ISSN: 0022-1120

ID: CaltechAUTHORS:20120625-142119475

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Abstract: The effect of anisotropic grain structures on the average transport properties resulting from flow through porous media is derived using methods that involve ensemble averaging the basic conservation equations and a multiple-scale analysis. The Lagrangian or "moments" effective diffusivity tensor, defined as the long-time limit of the time rate of change of the mean-squared displacement of a tracer particle, is equivalent to the symmetric part of the Eulerian diffusivity, defined as the mass flux induced by a linear concentration gradient in the limit of long times. In an anisotropic medium there may be a component of the mass flux perpendicular to the imposed concentration gradient, resulting in an effective diffusivity tensor that is not symmetric. Such an effect is not detected by the moments or Lagrangian approach.

Publication: Physics of Fluids Vol.: 30 No.: 3 ISSN: 0031-9171

ID: CaltechAUTHORS:KOCpof87

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Abstract: When the lengthscales and timescales on which a transport process occur are not much larger than the scales of variations in the velocity field experienced by a tracer particle, a description of the transport in terms of a local, averaged macroscale version of Fick's law is not applicable. Here, a non-local transport theory is developed in which the average mass flux is not simply proportional to the average local concentration gradient, but is given by a convolution integral over space and time of the average concentration gradient times a spatial- and temporal-wavelength-dependent diffusivity. The non-local theory is applied to the transport of a passive tracer in the advective field that arises in the bulk fluid of a porous medium, and the complete residence-time distribution - space-time response to a unit source input - of the tracer is determined. It is also shown how the method of moments may be simply recovered as a special case of the non-local theory. While developed in the context of and applied to tracer dispersion in porous media, the non-local theory presented here is applicable to the general problem of determining the average transport behaviour in advection-diffusion-type systems in which spatial and temporal variations are occurring on scales comparable with the scale of interest.

Publication: Journal of Fluid Mechanics Vol.: 180ISSN: 0022-1120

ID: CaltechAUTHORS:20120626-123241319

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Abstract: A general method is presented for simulating the dynamical behavior of a suspension of particles which interact through both hydrodynamic and nonhydrodynamic forces. In the molecular-dynamics-like simulation there are two different procedures for computing the interactions among particles: a pairwise additivity of forces and a pairwise additivity of velocities. The pairwise additivity of forces is the preferred method as it preserves the hydrodynamic lubrication forces which prevent particles from overlapping. The two methods are compared in a simulation of a monolayer of identical rigid non-Brownian spherical particles in a simple shear flow. Periodic boundary conditions are used to model an infinite suspension. Both methods predict the presence of a shear induced anisotropic local structure whose form and strength depend on the concentration of particles, the nonhydrodynamic forces, and the shear rate. Increasing the particle concentration up to near the maximum fraction that can still flow results in a transition to a layered structure in which planes of particles slide relative to one another. The anisotropic local structure and transition to a layered structure predict a non-Newtonian suspension rheology.

Publication: Journal of Chemical Physics Vol.: 80 No.: 10 ISSN: 0021-9606

ID: CaltechAUTHORS:BOSjcp84

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Abstract: The spatial development of the inlet velocity profile in a uniformly porous channel and tube is investigated. For tubes which are very long compared with their radius, it is shown that above a critical Reynolds number of 2.3 the inlet velocity profile does not necessarily decay into the fully developed, similarity profile for an infinite tube. Rather, the structure of the flow throughout the entire tube is influenced by the inlet profile. This loss of validity of the similarity solution is due to the fact that the tube is of finite length and the inlet profile is not of the similarity form. The actual length of the tube is, however, unimportant. Analogous results hold for the flow in a porous channel, with a critical Reynolds number of approximately 6.

Publication: Physics of Fluids Vol.: 27 No.: 5 ISSN: 0031-9171

ID: CaltechAUTHORS:BRApof84

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Abstract: The spatial stability of a class of exact similarity solutions of the Navier–Stokes equations whose longitudinal velocity is of the form xf′(y), where x is the streamwise coordinate and f′(y) is a function of the transverse, cross‐streamwise, coordinate y only, is determined. These similarity solutions correspond to the flow in an infinitely long channel or tube whose surface is either uniformly porous or moves with a velocity linear in x. Small perturbations to the streamwise velocity of the form x^λg′(y) are assumed, resulting in an eigenvalue problem for λ which is solved numerically. For the porous wall problem, it is shown that similarity solutions in which f′(y) is a monotonic function of y are spatially stable, while those that are not monotonic are spatially unstable. For the accelerating‐wall problem, the interpretation of the stability results is not unambiguous and two interpretations are offered. In one interpretation the conclusions are the same as for the porous problem—monotonic solutions are stable; the second interpretation is more restrictive in that some of the monotonic as well as the nonmonotonic solutions are unstable.

Publication: Physics of Fluids Vol.: 27 No.: 5 ISSN: 1070-6631

ID: CaltechAUTHORS:20120702-094236176

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