Active matter is a class of materials that has constituents capable of self-propulsion through the conversion of energy into mechanical motion. The origin or specific details of their method of locomotion is a rich area of research, but is not important for understanding some aspects of their dynamics. Our interest is in the single-particle dynamics and the large-scale collective motion observed in these systems. Thus, we use the minimal active Brownian particle (ABP) model to model self-propulsion. The ABP model consists of particles of radius *a* that swim with an intrinsic speed *U₀* in some direction **q** and reorients on a timescale *τ _{R}*. Active motion is persistent in that a particle will continue to swim in a direction until it reorients itself, giving rise to a swim or persistence length ℓ =

“Active matter” refers to a broad class of materials in which the constituent particles or organisms are able to self-propel (swim) by some internal physicochemical mechanism. Though the origin of this self-propulsive motion is a rich area of study, we are primarily interested in the collective effects of this motion on the physical properties — and in particular, the rheology — of the active material as a whole. As such we model self-propulsive motion using the minimal active Brownian particle (ABP) model: a particle of size * a *, swims in a direction * q* with a speed

On a macroscopic scale, active motion leads to unique hydrodynamic and mechanical stresses exerted by the particles on their embedding medium. These stresses arise from the microscopic force associated with particle locomotion — the swim force * F ^{swim}*. Though the idea of the swim force is widely recognized in the abstract, little attention has been given to the characterization and mechanical consequences of this force. In this work we are particularly interested the role of the swim force in the effective motion of passive constituents in active environments, and how the swim force affects long-ranged hydrodynamic interactions (HI) in active suspensions. We examine these issues through the lens of microrheology: tracking the motion of a colloidal probe particle through an active medium, and using its motions to infer the effective viscoelastic properties of the suspension.

Using generalized Taylor dispersion theory, we find an activity-driven enhancement to the diffusion of the probe in an active medium. This first-principles theory unites many experimental observations of tracer diffusion, and provides simple physical descriptions of the problem that do not rely on the specific self-propulsion mechanism of the swimmer. This same framework is then used to compute the suspension microviscosity (as measured by the drag on the probe particle), and the fluctuation-dissipation relation in an active system. We find that activity reduces the drag on the probe, but the drag is still larger than it would be in a Newtonian fluid; this stands in contrast to experimental measurements of reduced shear viscosities. We show that the microviscosity of a suspension is reduced — and may even become negative! — due to HI, and that this effect is *not* due to the fluid velocity disturbance associated with the swimmers’ self-propulsion.

In this thesis I formulate and present a novel and new framework for simulating the dynamics of arbitrarily shaped active or passive particles immersed in a Stokesian fluid and evolving under confinement by a porous container or in free space. I use a completed double layer boundary integral equation to model the particle’s dynamics and combine this with a new formulation that uses a second kind integral equation for describing the motion of the porous container. This newly formulated porous container model permits fluid to pass through the container’s surface at a velocity in proportion to a discontinuous jump in stress across the container’s surface. This jump in stress is induced by the active particle’s motion. The proposed porous container model is general in the sense that it allows fluid to pass through the membrane with finite tangential and normal velocity components. I obtain the exact analytical solution to this model when the active particle and porous container are perfectly concentric. In addition, I numerically solve this system of boundary integral equations for arbitrary particle positions, and fully characterize the particle and container dynamics by performing a vast number of trajectory studies. Both the container and particle are seen to move in complicated ways owing to their self and pairwise hydrodynamic interactions. This system is studied over a vast parameter space, for multiple container to particle size ratios, multiple types of active particles, and various permeability parameterizations. This coupled set of particle and container boundary integral equations is discretized using a novel interpretation and new extension of the Galerkin Boundary Element discretization to multi-body particle systems in Stokes flow. I also implement and extend an *h*-adaptive conformal mesh refinement algorithm to accurately resolve near-contact particle and container interactions. In addition, I perform all Galerkin BEM calculations on CUDA enabled GPUs, allowing for these simulations to be run on modern desktop computers in seconds. I combine all of these techniques in a modern C++ Galerkin Boundary Element Method computational framework called GPUGBEM.

This thesis is a computational investigation on several aspects of the constant stress and pressure rheology of dense polydisperse colloidal suspensions. Using bidisperse suspensions as a model, we first study the influences of size polydispersity on short-time transport properties. The hydrodynamic interactions are calculated using a polydisperse implementation of Stokesian Dynamics (SD) via a Monte-Carlo approach. We carefully compare the SD computations with existing theoretical and numerical results, and critically assess the strengths and weaknesses of the SD algorithm. For suspensions, we find that the Pairwise Additive (PA) approximations with the Percus-Yevick structural input is valid up to volume fraction φ=0.1. We also develop an semi-analytical approximation scheme to predict the wavenumber-dependent partial hydrodynamic functions based on the δγ-scheme of Beenakker & Mazur [Physica 120A (1983) 388 & 126A (1984) 349], which is shown to be valid up to φ=0.4.

To meet the computation requirements of dynamic simulations, we then developed the Spectral Ewald Accelerated Stokesian Dynamics (SEASD) based on the framework of SD with extension to compressible solvents. The SEASD uses the Spectral Ewald (SE) method [Lindbo & Tornberg, J. Comput. Phys. 229 (2010) 8994] for mobility computation with flexible error control, a novel block-diagonal preconditioner for the iterative solver, and the Graphic Processing Units (GPU) acceleration. For further speedup, we developed the SEASD-nf, a polydisperse extension of the mean-field Brownian approximation of Banchio & Brady [J. Chem. Phys. 118 (2003) 10323]. The SEASD and SEASD-nf are extensively validated with static and dynamic computations, and are found to scale as O(NlogN) with N the system size. The SEASD and SEASD-nf agree satisfactorily over a wide range of parameters for dynamic simulations.

Next, we investigate the colloidal film drying processes to understand the structural and mechanical implications when the constant pressure constraint is imposed by confining boundaries. The suspension is sandwiched between a stationary substrate and an interface moving either at a constant velocity or with constant imposed stress. Using Brownian Dynamics (BD) simulations without hydrodynamic interactions, we find that both fast and slow interface movement promote crystallization via distinct mechanisms. The most amorphous suspension structures occur when the interface moves at a rate comparable to particle Brownian motion. Imposing constant normal stresses leads to similar suspension behaviors, except that the interface stops moving when the suspension osmotic pressure matches the imposed stress. We also compare the simulation results with a continuum model. This work reveals the critical role of interface movement on the stress and structure of the suspension.

Finally, we study the constant shear stress and pressure rheology of dense colloidal suspensions using both BD and SEASD-nf to identify the role of hydrodynamic interactions. The constant pressure constraint is imposed by introducing a compressible solvent. We focus on the rheological, structural, and dynamical characteristics of flowing suspensions. Although hydrodynamic interactions profoundly affect the suspension structure and dynamics, they only quantitatively influence the behaviors of amorphous suspensions. The suspension becomes glassy, i.e., exhibits flow-arrest transitions, when the imposed pressure is high, and reveals the Shear Arrest Point (SAP) in the non-Brownian limit. From a granular perspective, we find that the suspensions move away from the arrested state in a universal fashion regardless of the imposed pressure, suggesting the critical role of the jamming physics. The hydrodynamic simulations quantitatively agree with the experiments of Boyer et al. [Phys. Rev. Lett. 107 (2011) 188301] with a volume fraction shift. The results at all imposed stresses and pressures reveal a generalized Stokes-Einstein-Sutherland relation with an effective temperature proportional to the pressure. We develop a model that accurately describes the rheology and diffusion of glassy suspensions. Our results show the critical role of pressure on the behaviors of dense colloidal suspensions.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/Z9RN35WF, author = {Takatori, Sho C.}, title = {Forces, Stresses, and the (Thermo?) Dynamics of Active Matter: The Swim Pressure}, school = {California Institute of Technology}, year = {2017}, doi = {10.7907/Z9RN35WF}, url = {https://resolver.caltech.edu/CaltechTHESIS:05142017-141527247}, abstract = {A core feature of many living systems is their ability to move, self-propel, and be active. From bird flocks to bacteria swarms, to even cytoskeletal networks, active matter systems exhibit collective and emergent dynamics owing to their constituents’ ability to convert chemical fuel into mechanical activity. Active matter systems generate their own internal stress, which drives them far from equilibrium and thus frees them from conventional thermodynamic constraints, and by so doing they can control and direct their own behavior and that of their surrounding environment. This gives rise to fascinating behaviors such as spontaneous self-assembly and pattern formation, but also makes the theoretical understanding of their complex dynamical behaviors a challenging problem in the statistical physics of soft matter.

In this thesis, I present a new principle that all active matter systems display—namely, through their self-motion they generate an intrinsic `swim pressure’ that impacts their dynamic and collective behavior. I combine experimental and computational methods to demonstrate how intrinsic activity imparts new behaviors to soft materials that explain a variety of complex phenomena, including the collective motion of self-propelled particles and the complete loss of shear viscosity in fluid suspensions. These nonequilibrium phenomena are driven fundamentally by the active constituent’s tendency to diffuse, undergo a random walk, and exert a mechanical force or a pressure on a confining wall. The swim pressure theory is conceptually similar to the kinetic theory of gases, where molecular collisions with the container walls exert a pressure, or to the Brownian osmotic pressure exerted by molecular or colloidal solutes in solution. In contrast to thermodynamic quantities such as the chemical potential and free energy, the mechanical pressure (or stress) is valid out of equilibrium because it comes directly from the micromechanical equations of motion. I apply this swim pressure framework in a broad context to interpret living matter as a material and understand its complex behavior using tools of hydrodynamics, kinetic theory, and nonequilibrium statistical mechanics. The present theory is applied to active systems that are driven by self-propulsion and motility, but there are other types of nonequilibrium driving work that may fit into this general theoretical framework, like driven autocatalytic reactions in electrochemical and biochemical systems.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/Z9Z60M1V, author = {Yan, Wen}, title = {Dynamics of Chemically Active Suspensions}, school = {California Institute of Technology}, year = {2016}, doi = {10.7907/Z9Z60M1V}, url = {https://resolver.caltech.edu/CaltechTHESIS:05242016-214836974}, abstract = {Chemically active particles may swim by self-diffusiophoresis in a concentration gradient of chemical solutes they created themselves by patterned surface catalytic reactions. Those particles can also interact via normal diffusiophoresis in the same solute concentration field. The interaction can be attractive or repulsive. This ‘field-driven’ nature of the system makes its dynamics different from a thermodynamic system and is analyzed with a new simulation method. Simulations show that attractive active particles exhibit coexistence of dense and dilute regions, but it is different from a liquid-gas phase equilibrium. To explain the behavior, a continuum mechanics theory is developed based on the minimal Active Brownian Particles (ABP) model. In the continuum description, the surface force is found to be the swim stress, which can be anisotropic. The body force includes the average swim force as an internal contribution and an ‘activity-gradient’ force contribution. Further, behaviors of active matter at the sub-continuum scale are also analyzed. The continuum mechanics theory is shown to accurately describe the behaviors of chemically active particles. Particle clustering is explained with a linear stability analysis, and the steady state is explained with a sedimentation-like mechanical force balance.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/XN4G-TB33, author = {Hoh, Nicholas Jeffrey}, title = {Effects of Particle Size Ratio on Single Particle Motion in Colloidal Dispersions}, school = {California Institute of Technology}, year = {2013}, doi = {10.7907/XN4G-TB33}, url = {https://resolver.caltech.edu/CaltechTHESIS:06072013-154128559}, abstract = {The motion of a single Brownian particle of arbitrary size through a dilute colloidal dispersion of neutrally buoyant bath spheres of another characteristic size in a Newtonian solvent is examined in two contexts. First, the particle in question, the probe particle, is subject to a constant applied external force drawing it through the suspension as a simple model for active and nonlinear microrheology. The strength of the applied external force, normalized by the restoring forces of Brownian motion, is the Péclet number, Pe. This dimensionless quantity describes how strongly the probe is upsetting the equilibrium distribution of the bath particles. The mean motion and fluctuations in the probe position are related to interpreted quantities of an effective viscosity of the suspension. These interpreted quantities are calculated to first order in the volume fraction of bath particles and are intimately tied to the spatial distribution, or microstructure, of bath particles relative to the probe. For weak Pe, the disturbance to the equilibrium microstructure is dipolar in nature, with accumulation and depletion regions on the front and rear faces of the probe, respectively. With increasing applied force, the accumulation region compresses to form a thin boundary layer whose thickness scales with the inverse of Pe. The depletion region lengthens to form a trailing wake. The magnitude of the microstructural disturbance is found to grow with increasing bath particle size – small bath particles in the solvent resemble a continuum with effective microviscosity given by Einstein’s viscosity correction for a dilute dispersion of spheres. Large bath particles readily advect toward the minimum approach distance possible between the probe and bath particle, and the probe and bath particle pair rotating as a doublet is the primary mechanism by which the probe particle is able to move past; this is a process that slows the motion of the probe by a factor of the size ratio. The intrinsic microviscosity is found to force thin at low Péclet number due to decreasing contributions from Brownian motion, and force thicken at high Péclet number due to the increasing influence of the configuration-averaged reduction in the probe’s hydrodynamic self mobility. Nonmonotonicity at finite sizes is evident in the limiting high-Pe intrinsic microviscosity plateau as a function of bath-to-probe particle size ratio. The intrinsic microviscosity is found to grow with the size ratio for very small probes even at large-but-finite Péclet numbers. However, even a small repulsive interparticle potential, that excludes lubrication interactions, can reduce this intrinsic microviscosity back to an order one quantity. The results of this active microrheology study are compared to previous theoretical studies of falling-ball and towed-ball rheometry and sedimentation and diffusion in polydisperse suspensions, and the singular limit of full hydrodynamic interactions is noted.

Second, the probe particle in question is no longer subject to a constant applied external force. Rather, the particle is considered to be a catalytically-active motor, consuming the bath reactant particles on its reactive face while passively colliding with reactant particles on its inert face. By creating an asymmetric distribution of reactant about its surface, the motor is able to diffusiophoretically propel itself with some mean velocity. The effects of finite size of the solute are examined on the leading order diffusive microstructure of reactant about the motor. Brownian and interparticle contributions to the motor velocity are computed for several interparticle interaction potential lengths and finite reactant-to-motor particle size ratios, with the dimensionless motor velocity increasing with decreasing motor size. A discussion on Brownian rotation frames the context in which these results could be applicable, and future directions are proposed which properly incorporate reactant advection at high motor velocities.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/2743-8W26, author = {Zia, Roseanna Nellie}, title = {Individual Particle Motion in Colloids: Microviscosity, Microdiffusivity, and Normal Stresses}, school = {California Institute of Technology}, year = {2011}, doi = {10.7907/2743-8W26}, url = {https://resolver.caltech.edu/CaltechTHESIS:05262011-141745737}, abstract = {Colloidal dispersions play an important role in nearly every aspect of life, from paint to biofuels to nano-therapeutics. In the study of these so-called complex ﬂuids, a connection is sought between macroscopic material properties and the micromechanics of the suspended particles. Such properties include viscosity, diffusivity, and the osmotic pressure, for example. But many such systems are themselves only microns in size overall; recent years have thus seen a dramatic growth in demand for exploring microscale systems at a much smaller length scale than can be probed with conventional macroscopic techniques. Microrheology is one approach to such microscale interrogation, in which 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. With no external forcing the probe and background 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 probe motion, depends on the strength with which the probe is forced, F ext , compared to thermal forces, kT/b, defining a P´eclet number, P e = F ext /(kT /b), where kT is the thermal energy and b the bath-particle size. Both the mean and the ﬂuctuating motion of the probe are of interest. Recent studies showed that the reduction in mean probe speed gives the eﬀective material viscosity. But the velocity of the probe also ﬂuctuates due to collisions with the suspended particles, causing the probe to undergo a random walk process. 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 P´eclet number. At small Pe Brownian motion dominates and the diffusive behavior of the probe characteristic of passive microrheology is recovered, but with an incremental ﬂow-induced “micro-diffusivity” that scales as Dmicro ∼ Da P e 2 φb , where viii φb is the volume fraction of bath particles and Da is the self-diffusivity of an isolated probe. At the other extreme of high P´eclet number the fuctuational motion is still diffusive, and the diffusivity becomes primarily force-induced , scaling as (F ext /η)φb , 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-P e limit.

Previous work in microrheology defined a scalar viscosity; however, a tensorial expression for the suspension stress in microrheology was still lacking. The notion that diffusive ﬂux is driven by gradients in particle-phase stress leads to the idea that the microdiffusivity can be related directly to the suspension stress. In consequence, the anisotropy of the diffusion tensor may reﬂect the presence of normal stress differences in non-linear microrheology. While the particle-phase stress tensor can be determined as the second moment of the deformed microstructure, in this study a connection is made between diffusion and stress gradients, and an analytical expression for particle-phase stress as a function of the microdiffusivity and microviscosity is obtained. The two approaches agree, suggesting that normal stresses and normal stress differences can be measured in active 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 P e for P e ≫ 1 and vanishes as P e 3 for P e ≪ 1. An additional important outcome is that the analytical expression obtained for stress-induced migration can be viewed as a generalized non-equilibrium Stokes-Einstein relation.

Studies of steady-state dispersion behavior reveal the hydrodynamic and microstructural mechanisms that underlie non-Newtonian behaviors (e.g. shear-thinning, shear-thickening, and normal stress differences). But an understanding of how the microstructures evolve from the equilibrium state, and how non-equilibrium properties develop in time is much less well understood. Transient suspension behavior in the near-equilibrium, linear response regime has 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 ix in a medium are captured by start-up and cessation of the imposed deformation in the non-linear regime, where the structure is driven far from equilibrium. Here we study the evolution of stress and microstructure in a colloidal dispersion by tracking transient probe motion during start-up and cessation of a strong ﬂow. For large P e, steady state is reached when a boundary layer (in which advection balances diffusion) forms at particle contact on the timescale of the ﬂow, a/U , where a is the probe size and U its speed. On the other hand, relaxation following cessation occurs over several timescales corresponding to distinct physical processes. For very short times, the timescale for relaxation is set by the diffusion over the boundary-layer thickness. Nearly all stress relaxation occurs during this process, owing to the dependence of the bath-particle drag on the contact value of the microstructure. At longer times the collective diffusion of the bath particles acts to close the wake. In this long-time limit as structural isotropy is restored, the majority of the microstructural relaxation occurs with very little change in suspension stress. Theoretical results are presented and compared with Brownian dynamics simulation. Two regimes of probe motion are studied: an externally applied constant force and an imposed constant velocity. The microstructural evolution is qualitatively different for the two regimes, with a longer transient phase and a thinner boundary layer and longer wake at steady state in the latter case. The work is also compared to analogous results for sheared suspensions undergoing start-up and cessation.

The study moves next to investigations of dual-probe microrheology. Motivated by the phenomenon of equilibrium depletion interactions, we study the interaction between a pair of probe particles translating with equal velocity through a dispersion with their line of centers transverse to the external forcing. The character of the microstructure surrounding the probes is determined both by the distance R by which the two probes are separated and by the strength of the external forcing, P e = U a/Db , where U is the constant probe velocity and Db the diffusivity of the bath particles. Osmotic pressure gradients develop as the microstructure is deformed, giving rise to an interactive force between the probes. This force is studied for a range of P e and R. For all separations R > 2a, the probes attract when P e is small. As the strength of the forcing increases, a qualitative change in the interactive force occurs: the probes repel each other. The probe separation R at which the x attraction-to-repulsion transition occurs decreases as P e increases, because the entropic depletion attraction becomes weak compared to the force-induced osmotic repulsion. The non-equilibrium interactive force is strictly repulsive for two separated probes.

But non-linear microrheology provides far more than a microscale technique for interrogating complex ﬂuids. In 1906, Einstein published the famous thought experiment in which he proposed that if a liquid were indeed composed of atoms, then the motion of a small particle suspended in the ﬂuid would move with the same random trajectories as the solvent atoms. Combining the theories of kinetics, diffusion, and thermodynamics, he showed that the diffusive motion of a small particle is indeed evidence of the existence of the atom. Perrin conﬁrmed the theory with measurement in 1909. This is a profound conclusion, drawn by simply watching a particle move in a liquid. Here, we follow this example and watch a particle move in a complex ﬂuid—but now for a system that is not at equilibrium. In equilibrium systems, the relationship between ﬂuctuation and dissipation is fundamental to our understanding of colloid physics. By studying ﬂuctuations away from equilibrium, we have discovered an analogous non-equilibrium relation between ﬂuctuation and dissipation—and that the balance between the two is stored in the material stress. A ﬁnal connection can be made between this stress and energy storage.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, month = {July}, advisor = {Brady, John F.}, } @phdthesis{10.7907/JJ28-7641, author = {Swan, James W.}, title = {Colloids in Confined Geometries: Hydrodynamics, Simulation and Rheology}, school = {California Institute of Technology}, year = {2010}, doi = {10.7907/JJ28-7641}, url = {https://resolver.caltech.edu/CaltechTHESIS:05262010-140604509}, abstract = {The hydrodynamics of colloids in confined geometries is studied hierarchically beginning with the exact solutions for a spherical particle translating, rotating and deforming in the presence of a plane wall at low Reynolds number. The many-bodied hydrodynamic interactions among a collection of spherical particles near a plane wall are computed and used to study the Brownian motion of confined suspensions. The method of reflections is used to describe the motion of a single spherical particle embedded in the fluid constrained by two, parallel plane walls. From this, tables which are independent of the channel width are generated describing the particle’s response to various force moments. This same approach is expanded to describe the hydrodynamic interactions among the particles comprising a colloidal dispersion confined in a channel. The simulations arising from this theory depict the short-time self-diffusivity, sedimentation rate and high frequency viscosity of suspensions of varying volume fractions in channels of varying widths. A theory for the scattering of evanescent waves by colloidal dispersions is developed and cast in the form of the diffusivity measured by classical light scattering. A series of simulations is conducted to predict the short- time self-diffusivity and the collective diffusivity measured by evanescent wave dynamic light scattering. The thesis concludes with a discussion of how the developed simulations and theories can be extended to make dynamic measurements as well as a brief consideration of some remaining, open questions.}, address = {1200 East California Boulevard, Pasadena, California 91125}, month = {July}, advisor = {Brady, John F.}, } @phdthesis{10.7907/HQGZ-DV22, author = {Swaroop, Manuj}, title = {The Bulk Viscosity of Suspensions}, school = {California Institute of Technology}, year = {2010}, doi = {10.7907/HQGZ-DV22}, url = {https://resolver.caltech.edu/CaltechTHESIS:05282010-012507201}, abstract = {Particles suspended in a fluid are known to undergo variations in the local concentration in many flow situations; essentially a compression or expansion of the particle phase. The modeling of this behavior on a macroscopic scale requires knowledge of the effective bulk viscosity of the suspension, which has not been studied before. The bulk viscosity of a pure compressible fluid is defined as the constant of proportionality that relates the difference between the mechanical pressure and the thermodynamic pressure to the rate of compression. The bulk viscosity of a suspension is defined analogous to that for a pure fluid as the constant of proportionality relating the deviation of the trace of the macroscopic stress from its equilibrium value to the average rate of compression. The compression flow drives the suspension microstructure out of equilibrium and the thermal motion of the particles tries to restore equilibrium. The Peclet number (Pe), defined as the expansion rate made dimensionless with the Brownian time-scale, governs the departure of the microstructure from equilibrium. The microstructural forcing in compression is monopolar for small Pe resulting in a significantly slower spatial and temporal response of the microstructure compared to shearing or diffusive motion.

We have determined the effective suspension bulk viscosity for all concentrations and all rates of compression, accounting for the full thermodynamic and hydrodynamic interactions that particles experience at the micro-scale. Current simulation techniques were enhanced to enable the dynamic simulation of compression flows in a suspension. A ‘compression thinning’ of the suspension is observed at small rates of compression and there is some ‘compression thickening’ at large compression rates. The bulk viscosity diverges as the volume fraction nears maximum packing and is in fact larger than the shear viscosity. Existing models for multiphase flows must therefore include the bulk viscosity term to properly simulate variations in particle concentration.

An understanding of bulk viscosity effects in suspensions will enable the modeling of certain aggregation and separation behavior and lead to more accurate models for multiphase flows where there are variations in the particle concentration, such as filtration or fluidization.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/W8X1-FQ17, author = {Cordova-Figueroa, Ubaldo M.}, title = {Directed Motion of Colloidal Particles via Chemical Reactions: Osmotic Propulsion}, school = {California Institute of Technology}, year = {2008}, doi = {10.7907/W8X1-FQ17}, url = {https://resolver.caltech.edu/CaltechETD:etd-05292008-200411}, abstract = {Recent experiments showing reaction-driven propulsion at nanoscales have appeared as a possible mechanism for the transport of particles that may help us to not only understand chemo-mechanical transduction in biological systems, but also to create novel artificial motors that mimic living organisms and which can be harnessed to perform desired tasks. Reaction-driven propulsion consists of the generation of a localized potential gradient by an on-board surface chemical reaction. In this study, we propose and investigate a model for self-propulsion of a colloidal particle — the osmotic motor — immersed in a dispersion of ``bath" particles. The non-equilibrium concentration of bath particles induced by a surface chemical reaction creates an osmotic pressure imbalance on the motor causing it to move. The departure of the bath particle concentration distribution from equilibrium is governed by the Damkohler number Da — the ratio of the speed of reaction to that of diffusion — which is employed to calculate the driving force on the motor, and from which the self-induced osmotic velocity is determined via application of Stokes drag law. To illustrate the significant physics in osmotic propulsion, a first-order surface reaction on a portion of the motor’s surface is assumed, for the most part, in this work. The implications of these features for different bath particle concentrations and motor sizes are discussed. Furthermore, we investigate the role played by the distribution of reactions on the motor’s surface. Different responses are expected depending on the amount of reactive surface in the limiting behaviors of the reaction speed. Lastly, we consider a motor with constant production of particles on a hemisphere as a model that resembles actin-based motility of biological cells and organelles.

This research demonstrates that such an osmotic motor is possible and addresses such questions as: How fast can the motor move? How large of a force can it generate? What is the efficiency of such an osmotic motor? All motor behaviors discussed in this work are shown, after appropriate scaling, to be in good agreement with Brownian dynamics simulations.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/RWKF-E288, author = {Khair, Aditya Satish}, title = {Particle Motion in Colloidal Dispersions: Applications to Microrheology and Nonequilibrium Depletion Interactions}, school = {California Institute of Technology}, year = {2007}, doi = {10.7907/RWKF-E288}, url = {https://resolver.caltech.edu/CaltechETD:etd-02122007-130627}, abstract = {Over the past decade, microrheology has burst onto the scene as a technique to interrogate and manipulate complex fluids and biological materials at the micro- and nano-meter scale. At the heart of microrheology is the use of colloidal ‘probe’ particles embedded in the material of interest; by tracking the motion of a probe one can ascertain rheological properties of the material. In this study, we propose and investigate a paradigmatic model for microrheology: an externally driven probe traveling through an otherwise quiescent colloidal dispersion. From the probe’s motion one can infer a ‘microviscosity’ of the dispersion via application of Stokes drag law. Depending on the amplitude and time-dependence of the probe’s movement, the linear or nonlinear (micro-)rheological response of the dispersion may be inferred: from steady, arbitrary-amplitude motion we compute a nonlinear microviscosity, while small-amplitude oscillatory motion yields a frequency-dependent (complex) microviscosity. These two microviscosities are shown, after appropriate scaling, to be in good agreement with their (macro)-rheological counterparts. Furthermore, we investigate the role played by the probe’s shape — sphere, rod, or disc — in microrheological experiments.

Lastly, on a related theme, we consider two spherical probes translating in-line with equal velocities through a colloidal dispersion, as a model for depletion interactions out of equilibrium. The probes disturb the tranquility of the dispersion; in retaliation, the dispersion exerts a entropic (depletion) force on each probe, which depends on the velocity of the probes and their separation. When moving ‘slowly’ we recover the well-known equilibrium depletion attraction between probes. For ‘rapid’ motion, there is a large accumulation of particles in a thin boundary layer on the upstream side of the leading probe, whereas the trailing probe moves in a tunnel, or wake, of particle-free solvent created by the leading probe. Consequently, the entropic force on the trailing probe vanishes, while the force on the leading probe approaches a limiting value, equal to that for a single translating probe.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/857a-gg43, author = {Carpen, Ileana Cristina}, title = {Studies of Suspension Behavior: I. Instabilities of Non-Brownian Suspensions. II. Microrheology of Colloidal Suspensions}, school = {California Institute of Technology}, year = {2005}, doi = {10.7907/857a-gg43}, url = {https://resolver.caltech.edu/CaltechETD:etd-06022005-131439}, abstract = {Complex fluids are present in a multitude of forms: polymers, foods, paints, inks, biological materials, pharmaceuticals, cosmetics, etc. Many of these are suspensions, which have a particulate phase suspended in a solvent phase. This multiphase character gives a rich variety of behaviors, making suspensions interesting and useful materials but difficult to process. We investigate two different aspects of suspension behavior: instabilities in suspension flows and the use of microrheology in colloidal suspensions.

We look at two different mechanisms that generate instabilities and pattern formation in suspension flows. In the first, a jump in normal stresses at the interface between two fluids may lead to growing perturbations of the interface that ultimately give rise to migration of the particle phase into enriched regions. Fluids with a negative second normal stress difference, such as suspensions, can be unstable with respect to transverse or spanwise perturbations. The mechanism appears to be generic, although the details will depend on the specific system. The second mechanism may affect suspensions whose particle phase is not density-matched to the fluid. In this case, a flow can be unstable to spanwise perturbations of the particle phase when the shearing motion generates a density profile that increases with height. This is a Rayleigh–Taylor-like instability, due to having heavier material over light. As with the first instability, this mechanism may play an important role in pattern formation in multiphase flows.

The second aspect of suspension behavior we examine is the application of microrheology to colloidal suspensions. Microrheology has great promise for the study of soft, heterogeneous materials, but is not as well understood as traditional rheology. Most methods use tracer particles to investigate a medium, sometimes passively—tracking random motion (well established but restricted to the linear viscoelastic regime)—and more rarely actively—applying an external force to drive the tracers and access the medium’s nonlinear response. Active microrheology is not well understood, and we study it by simulating a prototypical example, the motion of a particle due to an imposed force through a colloidal suspension. The deformation of the microstructure results in resistance to the tracer’s motion. This system displays ‘force-thinning’, analogous to the ‘shear-thinning’ in a macrorheologically sheared suspension, but the comparison is not exact, and care needs to be taken in the use and application of microrheological results. Comparable length scales between the measurement device (the tracer) and the medium lead to interesting effects and distinctions between types of microrheological methods.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/SX3A-8M59, author = {Sierou, Asimina}, title = {Accelerated Stokesian dynamics : development and application to sheared non-Brownian suspensions}, school = {California Institute of Technology}, year = {2002}, doi = {10.7907/SX3A-8M59}, url = {https://resolver.caltech.edu/CaltechETD:etd-01202009-160503}, 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 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 can be readily extended to other low-Reynolds-number flow problems. The new method is applied to the study of sheared non-Brownian suspensions.

The rheological behavior of a monodisperse suspension of non-Brownian particles in simple shear flow in the presence of a weak interparticle force is studied first. The availability of a faster numerical algorithm permits the investigation of larger systems (typically of N = 512 — 1000 particles), and accurate results for the suspension viscosity, first and second normal stress differences and the particle pressure are determined as a function of the volume fraction. The system microstructure, expressed through the pair-distribution function, is also studied and it is demonstrated how the resulting anisotropy in the microstructure is correlated with the suspension non-Newtonian behavior. The ratio of the normal to excess shear stress is found to be an increasing function of the volume fraction, suggesting different volume fraction scalings for different elements of the stress tensor. The relative strength and range of the interparticle force is varied and its effect on the shear and normal stresses is analyzed. Volume fractions above the equilibrium freezing volume fraction (ø ≈ 0.494) are also studied, and it is found that the system exhibits a strong tendency to order under flow for volume fractions below the hard-sphere glass transition; limited results for ø = 0.60, however, show that the system is again disordered under shear.

Self-diffusion is subsequently studied and accurate results for the complete tensor of the shear-induced self-diffusivities are determined. The finite, and oftentimes large, auto-correlation time requires the mean-square displacement curves to be followed for longer times than was previously thought necessary. Results determined from either the mean-square displacement or the velocity autocorrelation function are in excellent agreement. The longitudinal (in the flow direction) self-diffusion coefficient is also determined, and it is shown that the finite autocorrelation time introduces an additional coupled term to the longitudinal self-diffusivity, a term which previous theoretical and numerical results omitted. The longitudinal self-diffusivities for a system of non-Brownian particles are calculated for the first time as a function of the volume fraction.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/DSMP-HV88, author = {Subramanian, Ganesh}, title = {Inertial effects in suspension dynamics}, school = {California Institute of Technology}, year = {2002}, doi = {10.7907/DSMP-HV88}, url = {https://resolver.caltech.edu/CaltechETD:etd-07042002-114141}, abstract = {This work analyses the role of small but finite particle inertia on the microstructure of suspensions of heavy particles subjected to an external flow. The magnitude of particle inertia is characterized by the Stokes number, St, defined as the ratio of the inertial relaxation time of a particle to the flow time scale. Fluid inertia is neglected so that the fluid motion satisfies the quasi-steady Stokes equations. The statistics of the particles is governed by a Fokker-Planck equation in position and velocity space. For small St, a multiple scales formalism is developed to solve for the phase-space probability density of a single spherical Brownian particle in a linear flow. Though valid for an arbitrary flow field, the method fails for a spatially varying mass and drag coefficient. In all cases, however, a Chapman-Enskog-like formulation provides a valid multi-scale description of the dynamics both for a single Brownian particle and a suspension of interacting particles. For long times, the leading order solution simplifies to the product of a local Maxwellian in velocity space and a spatial density satisfying the Smoluchowski equation. The higher order corrections capture both short-time momentum relaxations and long-time deviations from the Maxwellian. The inertially corrected Smoluchowski equation includes a non-Fickian term at O(St).

The pair problem is solved to O(St) for non-Brownian spherical particles in simple shear flow. In contrast to the zero inertia case, the relative trajectories of two particles are asymmetric. Open trajectories in the plane of shear suffer a downward displacement in the velocity gradient direction. The surface of the reference sphere `repels’ nearby trajectories that spiral out onto a new stable limit cycle in the shearing plane. This limit cycle acts as a local attractor; in-plane trajectories from an initial offset of […] or less approach the limit cycle. The topology of the off-plane trajectories is more complicated because the gradient displacement changes sign away from the plane of shear. The ‘neutral’ off-plane trajectory with zero net gradient displacement acts to separate trajectories spiralling onto contact from those that go off to infinity. The aforementioned asymmetry leads to a non-Newtonian rheology and self-diffusivities in the gradient and voriticity directions that scale as […], respectively.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/g6s5-x267, author = {Foss, David R.}, title = {Rheological behavior of colloidal suspensions : the effects of hydrodynamic interactions}, school = {California Institute of Technology}, year = {1999}, doi = {10.7907/g6s5-x267}, url = {https://resolver.caltech.edu/CaltechETD:etd-02132007-143514}, abstract = {The rheological behavior of hard-sphere colloidal suspensions in simple shear flow is examined theoretically and by dynamic simulation. The Stokesian Dynamics and Brownian Dynamics simulation techniques are used to study suspensions with and without many-body hydrodynamic interactions, respectively. Suspensions near equilibrium, where Brownian motion dominates, and at high shear rates, where hydrodynamic forces dominate, are examined. Steady-state simulations are performed using both simulation algorithms. The Brownian Dynamics system is found to be shear-thinning at low shear rates and undergoes a phase transition at high shear rates to a phase of hexagonally-packed strings aligned in the flow direction. Inclusion of hydrodynamic interactions eliminates the phase transition at high shear rates. Instead, the suspension is found to shear thicken due to a boundary layer of high particle probability that forms near contact where convection balances Brownian diffusion. This increases the role of strong hydrodynamic lubrication forces. Shear thinning and thickening data from many different volume fractions are collapsed using scaling theories. A previous steady-state theoretical analysis of the boundary layer at high shear rates (Brady & Morris 1997) is extended to include unsteady flow conditions. Theoretical results are found to be in agreement with start-up and flow-cessation simulations. A relation between start-up flow at low shear rates and the relaxation of equilibrium fluctuations is made. Equilibrium fluctuations are characterized using the shear-stress autocorrelation function and Green-Kubo formulae. Behavior of this function at short times is related to the behavior in an oscillatory shear flow at high frequencies that is also well-described by a boundary layer where unsteady convection balances Brownian diffusion. A new method for determining the components of the long-time self-diffusion tensor is proposed. Self-diffusion is found to be a decreasing function of volume fraction near equilibrium and an increasing function of volume fraction at high shear rates. Data is in agreement with previous theory and experiment. Due to the buildup of particle probability along the compressional axis relative to the extensional axis in simple shear flow, there is a nonzero off-diagonal component to the long-time self-diffusion tensor which is proportional to the shear rate. This component is positive near equilibrium and negative at high shear rates}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/NEB8-NH16, author = {Vicic, Michael}, title = {Rheology and microstructure of complex fluids: dispersions, emulsions and polymer solutions}, school = {California Institute of Technology}, year = {1999}, doi = {10.7907/NEB8-NH16}, url = {https://resolver.caltech.edu/CaltechETD:etd-04072005-160903}, abstract = {The rheology and microstructure of complex fluids are intimately related, and this relationship is explored to gain a deeper understanding of the physics of colloidal dispersions, emulsions and polymer solutions.

The nonequilibrium microstructure and rheological properties of dispersions in steady, simple shear flow are calculated by solving the Smoluchowski equation as a function of dimensionless shear rate. The particles have a purely repulsive interaction with an hydrodynamic radius, a, and a thermodynamic radius, b. For hard spheres, b/a –> l, shear thinning is caused by a decrease in the Brownian contribution since Brownian motion becomes less important with increasing shear. Shear thickening occurs because of an increase in the hydrodynamic viscosity caused by the increased probability of finding particles near contact with increasing shear when particles hydrodynamically interact. The first normal stress difference changes sign since Brownian and hydrodynamic contributions have opposite signs, while the second normal stress difference is always negative. Scaling arguments are made to extend these dilute results for concentrated dispersions. Similar calculations and analyses are performed to study the effects of hydrodynamic interactions and varying b/a ratios on rheology and microstructure.

Scaling arguments for the volume-fraction dependence of the bulk stress of emulsions at the critical capillary number are presented along with experimental evidence using an unstabilized emulsion of polymerized castor oil dispersed in polydimethylsiloxane. It is shown that the droplet contribution to both the relative shear viscosity and first normal stress difference is linear in volume fraction for a given viscosity ratio for dilute to moderately-concentrated emulsions in steady, simple shear flow.

Stress jump measurements are performed for the first tune for (i) shear startup and (ii) polymer solutions in shear. The startup viscosity of a polymer solution of polyacrylamide in fructose-water at equilibrium is equal to the measured high frequency dynamic viscosity, as expected, since both methods measure the viscous contribution to the viscosity associated with the equilibrium microstructure. Since polymer solutions exhibit stress jumps different from the solvent viscosity, effects of shear on the hydrodynamic viscosity can be investigated.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/NMJQ-2X32, author = {Yurkovetsky, Yevgeny}, title = {I. Statistical mechanics of bubbly liquids. II. Behavior of sheared suspensions of non-Brownian particles}, school = {California Institute of Technology}, year = {1998}, doi = {10.7907/NMJQ-2X32}, url = {https://resolver.caltech.edu/CaltechETD:etd-06222005-110302}, abstract = {NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

I. Statistical mechanics of bubbly liquids. 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.

- Behavior of sheared suspensions of non-Brownian particles. Suspensions of non-Brownian particles in simple shear flow of a Newtonian solvent in the range of particle phase concentration, […], from 0.05 to 0.52, are studied numerically by Stokesian Dynamics. The simulations are a function of […] and the dimensionless shear rate, […], which measures the relative importance of the shear and short-ranged interparticle forces. The pair-distribution functions, shear viscosity, normal stress differences, suspension pressure, long-time self-diffusion coefficients, and mean square of the particle velocity fluctuations in the velocity-gradient and vorticity directions are computed, tabulated and plotted. In concentrated suspensions ([…] > 0.45), two distinct microstructural patterns are shown to exist at the highest and lowest shear rates. At […] = 0.1 the particles form hexagonally packed strings in the flow direction. As […] increases, the strings are gradually being replaced by non-compact clusters of particles kept together by strong lubrication forces while the particle pair-distribution displays a broken fore-aft symmetry. These changes in the microstructure are accompanied by increases in the shear viscosity, normal stress differences, suspension pressure, longtime self-diffusion coefficients, and fluctuational motion. Agreement is found between the simulation results and the theoretical predictions of Brady and Morris (1997).

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

A theory of self-diffusivity in sheared suspensions valid for any particle volume fraction [phi], Peclet number Pe, and lengthscale of disturbance in [phi] is developed. The theory is applied to the determination of the full tensor self-diffusivity in a weakly- sheared (Pe << 1) suspension of hydrodynamically-interacting hard spheres and a strongly-sheared (Pe >> 1) suspension of hard spheres without hydrodynamic interactions, both at [phi] << 1.

The influence of weak Brownian motion alone and in conjunction with a repulsive interparticle force of hard-sphere type upon the pair-distribution function, g(r) where r is the separation vector of a pair of particles, is analyzed for a suspension of spheres at Pe >> 1 and [phi] << 1. At large Pe, the radial fluxes of pair probability due to advection and Brownian diffusion balance in a thin […] boundary layer at contact, with a the sphere radius. The boundary-layer analyses demonstrate that Brownian diffusion renders g finite at contact in the absence of interparticle forces, and that within the boundary layer there is generally a large excess of pair probability along the compressional axes. By calculation of the bulk normal stress differences in the case with repulsive forces, it is shown how this asymmetry of the microstructure yields non-Newtonian constitutive behavior in the limit Pe[superscript -1] = 0.

Hydrodynamic resistance functions relating the particle and bulk motions to the bulk isotropic stress are developed. Application of these functions is demonstrated by calculations of the shear-induced correction to the osmotic pressure and the particle contribution to the pressure in a sheared lattice.

Pressure-driven flow in a channel at vanishing Reynolds number of a suspension of particles denser than the suspending fluid has been dynamically simulated by Stokesian Dynamics over ranges of the particle fraction, channel width, and a buoyancy parameter characterizing the relative strength of the buoyancy to shearing forces. The predictions of the flow by the suspension-balance model are in good agreement with simulation results.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/ZA49-9N69, author = {Phung, Thanh Ngoc}, title = {Behavior of concentrated colloidal suspensions by Stokesian dynamics simulation}, school = {California Institute of Technology}, year = {1993}, doi = {10.7907/ZA49-9N69}, url = {https://resolver.caltech.edu/CaltechETD:etd-02052008-140111}, abstract = {NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

The Stokesian dynamics simulation method is applied to study the behavior of concentrated suspensions of hydrodynamically interacting colloidal particles in a shear flow. The aim of this study is the prediction of suspension macroscopic properties from the microstructure - the temporal and spatial distribution of suspended particles. The macroscopic properties includes the shear viscosity, normal stress differences, short- and long-time self-diffusivities. Suspension macroscopic properties and the microstructure are modeled as functions of two parameters: particle volume fraction, […], and the Peclet number, Pe, which measures the relative importance of the imposed shear and Brownian forces. Stokesian dynamics accurately accounts for both the hydrodynamic and Brownian forces of a colloidal dispersion. The method, which is very general and computationally efficient, imposes no restriction on the particle displacements and allows simulation of flowing suspension with particle volume fractions from infinite dilution to dense packing and a continuous range of the Peclet number from pure Brownian motion (Pe 0) to pure hydrodynamics (Pe —> […]).

The method is first employed for the pure Brownian suspensions (Pe=0) at a volume fraction […]=0.45. The accuracy of Stokesian dynamics is demonstrated by an excellent comparison of the radial pair-distribution function obtained from dynamic simulation which captures the same isotropic hard-sphere distribution computed by the random Monte-Carlo method. The simulation method is then applied to study the dynamics of sheared SCC, BCC, and FCC periodic lattices of non-colloidal spheres (Pe […]) with particle volume fraction ranging from dilution to maximum close packing. Results of the resistivity and the shear viscosity of sheared periodic lattices are successfully determined as a function of the volume fraction.

The Stokesian dynamics simulation method is finally applied to the dynamic simulation of unbounded concentrated suspensions of force- and torque-free colloidal particles. The particle volume fractions are varied from 0.316 to 0.6 and the Peclet numbers are ranged from the strong Brownian limit (Pe=0.01) to the hydrodynamic dominated regime (Pe=[…]). Comparisons of simulation results for the steady shear viscosities, self-diffusivities, and the structure factors with experiments are remarkably good. For the first time, the flow of particles are probed with detail to illustrate the shearing deformation to suspension microstructure. This information provides a physical understanding of the fundamental mechanisms causing interesting shear thinning and shear thickening behavior and its important relation to the shear-induced microstructure. The simulation results reveal three distinct behaviors of hard-sphere suspensions in the regions of strong Brownian motion, balance of Brownian and hydrodynamic interactions, and hydrodynamic domination.

In the region of strong Brownian motion with small Peclet numbers (Pe < 1), the suspension shear thins due to a decrease of Brownian contribution to particle stress. The isotropic microstructure is slightly deformed, but the particles are very well dispersed. More importantly, simulation results do not reveal ordered microstructure in the shear thinning region. For the special plateau region with Pe […] 10, the suspension no longer shear thins and the shear viscosity is minimized. The balance of hydrodynamic and Brownian forces induce a strongly ordered flowing suspension with hexagonally packed strings of particles flowing with the bulk flow. The string formation is due to the Brownian forces which act as short-range springlike repulsive and random forces to counter the shearing deformation to the suspension by the imposed shear; the string formation does not relate to the shear thinning. In the region of hydrodynamic domination (Pe > […]), the suspension shear thickens due to formation of large, elongated clusters of particles. In this region, the hydrodynamics contribute all particle stress as the direct Brownian contribution has essentially vanished, but weak Brownian forces are seen to perturb and induce a local anisotropic microstructure. The complete relation of the steady shear viscosity to particle volume fraction and the Peclet number for concentrated hard-sphere suspensions is also given.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/q7ya-ex37, author = {Lovalenti, Philip Michael}, title = {Inertial effects on particle dynamics}, school = {California Institute of Technology}, year = {1993}, doi = {10.7907/q7ya-ex37}, url = {https://resolver.caltech.edu/CaltechETD:etd-08302007-112722}, abstract = {While the theory of suspension flows and particle dynamics is well understood under Stokes flow conditions when viscous forces dominate, little is known at finite Reynolds number when the inertial forces of the suspending fluid are important. In the present study, expressions are derived that allow for dynamic calculations of particle, drop, and bubble motion at finite Reynolds number. The results show a significant change in the temporal behavior of the force/velocity relationship from that derived from the unsteady Stokes equations, particularly as a body approaches its steady state. At finite Reynolds number, when the convective inertial effects are included, the hydrodynamic force on a body has much weaker history dependence on the past motion of the body and it reaches its steady state faster than what would be predicted if only the unsteady inertial effects are accounted for. When compared with numerical solutions of the Navier-Stokes equations, the analytical force expressions perform well up to a Reynolds number of 0.5.

A common theme to the derivations is the use of the reciprocal theorem which provides for an efficient and elegant means for computing inertial effects in suspension mechanics. Connections with past approaches are made in light of these new applications of the reciprocal theorem.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/W1HN-XC85, author = {Bauer, John Edward}, title = {Hydrodynamic interactions in polymer dynamics}, school = {California Institute of Technology}, year = {1992}, doi = {10.7907/W1HN-XC85}, url = {https://resolver.caltech.edu/CaltechETD:etd-07202007-145157}, abstract = {A modification to the traditional bead-spring model of polymers is proposed, which properly accounts for the full hydrodynamic interactions between the beads. The new model uses the Stokesian dynamics simulation technique to calculate far-field, many-body effects as well as near-field lubrication and excluded-volume effects. No preaveraging of the interactions is required. In addition to the xF, “spring” contribution to the stress, the Stokesian dynamics model calculates hydrodynamic and direct Brownian contributions to the stress.

Orientations and stresses obtained from the Stokesian dynamics dumbbell were compared to predictions of the Rouse dumbbell (no hydrodynamic interaction) and the Zimm dumbbell (Rotne-Prager hydrodynamic interaction). Infinite-dilution behaviors were examined in steady, simple shear and oscillatory shear flows. In steady shear the Stokesian dynamics model provides no improvement over the Zimm model. Both give qualitatively similar results for shear and normal stresses. The hydrodynamic stress is constant and equal to the Einstein viscosity contribution from each bead. The Brownian stress is negligible. The analysis reveals how hydrodynamic interaction causes shear thinning. The interaction between the beads tilts the dumbbell towards the shear axis, reducing the xF contribution to the shear stress. The oscillatory-shear results are similar to the steady-shear results, except that the hydrodynamic stress results in a non-zero high-frequency viscosity. Hydrodynamic and Brownian stresses will provide greater contributions to the rheology of multibead chains, in which many-body effects are more important. This is true of both steady shear and oscillatory shear.

Simulations of non-dilute suspensions of Stokesian dynamics dumbbells were compared with results for suspensions of spheres at the same volume fractions. The xF stress reaches a maximum at a bead volume fraction of 0.15, above which hydrodynamic forces dominate the solution rheology. The interparticle forces have little effect on the magnitudes of the hydrodynamic and Brownian stresses. The interparticle forces become very dependent upon initial configuration at high volume fractions. It is hypothesized that there exists a critical volume fraction, above which the polymer distribution function will always be dependent upon the initial configuration and the shear history.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/DYJ3-4223, author = {Claeys, Ivan Lode Andre Maria}, title = {Hydrodynamic Transport Properties of Suspensions of Non-Brownian Prolate Spheroids}, school = {California Institute of Technology}, year = {1991}, doi = {10.7907/DYJ3-4223}, url = {https://resolver.caltech.edu/CaltechETD:etd-01282005-165455}, abstract = {The methodology of “Stokesian dynamics,” an efficient and accurate simulation technique for suspensions of spheres, is extended to non-spherical particles. The model system consists of rigid, non-Brownian prolate spheroids suspended in an incompressible Newtonian fluid at zero Reynolds number. The method is applied to calculate the hydrodynamic transport properties of unbounded dispersions of ellipsoids. Both “random” configurations and very orderly arrangements of particles are considered in order to probe the relation between the microstructure of the suspension and its macroscopically observable properties.

The simulation method is based on a microstructurally detailed description of the two-phase system and explicitly takes into account hydrodynamic interactions between the particles. Non-local singularity solutions for ellipsoids in Stokes flow are combined with Faxen laws using pair-wise additivity of velocities to construct a far-field approximation to the mobility tensor. The convergence problems associated with the long-ranged nature of viscous interactions at zero Reynolds number are handled using O’Brien’s renormalization procedure. The Ewald summation technique is applied to accelerate the evaluation of the lattice sums generated by the periodic boundary conditions. Lubrication stresses between almost touching spheroids are added in a pair-wise manner to the mobility inverse. All the two-body resistance functions which diverge as the surface separation vanishes are computed to O(ε^{0}), with ε the gap width, so that the singular behavior of the lubrication interactions is captured correctly for arbitrary relative orientations and relative motions of the particles.

The method is first illustrated for a finite number of particles in an unbounded fluid domain, and shown to be accurate and efficient. It is then applied to crystalline geometries of spheroids over the full concentration range from 0 to closest packing (74% by volume). The dependence of the hydrodynamic transport properties (sedimentation rate, diffusion coefficient, stress/rate-of-strain relation, permeability and hindered diffusivity) on the density of the dispersion, the aspect ratio of the particles and the lattice type is investigated. Equilibrium structures of hard ellipsoids generated by a Monte Carlo procedure are also considered. The high frequency limit of the hydrodynamic transport properties is computed and compared to the results for crystalline configurations, and to available experimental measurements. A discontinuous jump in some suspension properties is observed at the isotropic to nematic transition.

As a prelude to dynamic simulations, the compatibility of unit cells with pure straining flows is examined. It is demonstrated that no self-reproducing lattices exist in axisymmetric extensional flows, but a set of compatible basis vectors is derived. Planar straining fields on the other hand possess an infinite number of strain-periodic lattices.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F. and Arnold, Frances Hamilton}, } @phdthesis{10.7907/bshd-xb74, author = {Bonnecaze, Roger Temor}, title = {Macroscopic properties of electrically interacting suspensions}, school = {California Institute of Technology}, year = {1991}, doi = {10.7907/bshd-xb74}, url = {https://resolver.caltech.edu/CaltechETD:etd-07092007-143213}, abstract = {Several bulk or macroscopic properties of electrically interacting suspensions -or at least electrically interacting by mathematical analogy- are studied using simulation methods. These bulk properties include the effective conductivity, percolation transitions, effective reaction rates, and the effective viscosity of an electrorheological fluid. In order to compute these properties, the detailed potential field in the suspension is not required, but rather only the linear relationship between the charge and dipole moments of the particles to their potentials and the applied electric field is needed. A method is developed to compute this relationship accurately and efficiently for arbitrary particle configuration and shape. The method includes both the many-body far-field and near-field particle interactions and properly accounts for the long-ranged interactions common to electrostatic problems.

The method is applied as part of a simulation to determine the effective conductivity of spherical particles. The simulation accurately reproduces the known values of the conductivity for cubic lattices of spheres for any volume fraction and particle-matrix conductivity ratio providing confidence in the method. It is then applied to determine the conductivity of mono-disperse, random hard-sphere suspensions for a variety of conductivity ratios of up to sixty volume percent particles. This is the first rigorous theoretical determination of the effective conductivity at such large volume fractions. The method is also used to study the percolation behavior of highly conducting spherical particles in close contact. It is found that such a system does exhibit a percolation transition, but only when the near-field effects are extremely large compared to the far-field interactions. This indicates that such a condition is a necessity for modeling a suspension as a percolating system.

The diffusion-limited reaction rate of a highly mobile reactant with spherical traps is also computed using the simulation method. Here the average “charge” on the particle is the effective consumption of the reactant by the spherical particle. The results are quite good up to a volume fraction of 30%, but then unphysically deviate from the expected increasing reaction rate with volume fraction and show a maximum rate at forty volume percent. The deviation is explained and serves as an example of the limitations of the method developed earlier. Using the effective conductivity results, however, a self-consistent Brinkman medium-like theory is developed that predicts the effective reaction rate quite accurately.

The method is finally applied to the dynamic simulation of electrorheological (ER) fluids -suspensions of dielectric particles with electric field tunable effective viscosities. Using the method developed in earlier chapters, the electrostatic interparticle forces in the ER suspension can be determined accurately and efficiently for arbitrary particle configurations, especially in capturing the strong near-field interactions. Coupled with Stokesian dynamics to account for the hydrodynamic interactions among the particles, the dynamics of the microstructure and its rheology can be determined. Dynamic simulations of an unbounded monolayer for a variety of electric field strengths and shear rates reproduce qualitatively and quantitatively experimental behavior. From the correlation of the dynamics and rheology, a theory for the dynamic yield stress observed in the suspensions is proposed and then successfully tested with the dynamic simulation. From the theory a simple model of the dynamic yield stress is created to predict its dependence on the volume fraction and the particle to fluid dielectric constant. The power of simulations to provide insight into the physics of multiphase materials and then to allow the testing of theories is particularly well illustrated with the study on ER fluids.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @phdthesis{10.7907/hv2d-rx78, author = {Lester, Julia Catherine}, title = {Hydrodynamic Dispersion in Concentrated Sedimenting Suspensions}, school = {California Institute of Technology}, year = {1988}, doi = {10.7907/hv2d-rx78}, url = {https://resolver.caltech.edu/CaltechETD:etd-08012006-113003}, abstract = {The hydrodynamic dispersion in concentrated sedimenting suspensions is investigated by numerical simulation. The particle Reynolds number is zero, and the Péclet number is infinite (the particles are non-Brownian). Particle trajectories are calculated by Stokesian dynamics. Stokesian dynamics is a molecular-dynamics-like simulation that provides an accurate representation of the suspension hydrodynamics. Detailed in this thesis is a technique that accelerates the convergence of the mobility interactions among particles in an infinite suspension. The simulations are of a monolayer of identical spheres sedimenting in the plane of the monolayer. Relative motion among the spheres arises from hydrodynamic interactions. The displacement related to this relative motion may constitute a random walk, giving rise to diffusive behavior of the spheres. This hydrodynamically induced self-diffusivity has been seen in sheared suspensions of non-Brownian, neutrally buoyant spheres.

Results of the numerical simulations show that the motion of spheres in sedimenting suspensions is also diffusive. The diffusion coefficient is relatively insensitive to the nature of the microstructure, as expressed by the pair-distribution function and the short-time, self-diffusion coefficient. The coefficient of diffusion decreases as the concentration increases for concentrated suspensions (it increases in the shear case). The ratio of the diffusion coefficient to the velocity variance of the spheres should be proportional to the time scale of the diffusive interactions. The diffusion time scale and the diffusion velocity scale (the square root of the velocity variance) both decrease as the concentration increases. In the shear case, the velocity scale (sphere radius multiplied by the shear rate) is independent of concentration, and the time scale (the product of the square of the concentration and the inverse of the shear rate) increases with increasing concentration. At the lowest concentrations, the spheres whose centers are separated by less than 2.05 radii prefer to align in the direction of sedimentation. At the highest concentrations, the preferred alignment is in the perpendicular direction.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, } @mastersthesis{10.7907/gypy-h364, author = {Heil, Ann Terese}, title = {Movement of a Spherical Brownian Particle Between Infinite Parallel Plates: Hindered Dispersion and Sedimentation}, school = {California Institute of Technology}, year = {1988}, doi = {10.7907/gypy-h364}, url = {https://resolver.caltech.edu/CaltechTHESIS:03192013-151818424}, abstract = {The dispersion of an isolated, spherical, Brownian particle immersed in a Newtonian fluid between infinite parallel plates is investigated. Expressions are developed for both a ‘molecular’ contribution to dispersion, which arises from random thermal fluctuations, and a ‘convective’ contribution, arising when a shear flow is applied between the plates. These expressions are evaluated numerically for all sizes of the particle relative to the bounding plates, and the method of matched asymptotic expansions is used to develop analytical expressions for the dispersion coefficients as a function of particle size to plate spacing ratio for small values of this parameter.

It is shown that both the molecular and convective dispersion coefficients decrease as the size of the particle relative to the bounding plates increase. When the particle is small compared to the plate spacing, the coefficients decrease roughly proportional to the particle size to plate spacing ratio. When the particle closely fills the space between the plates, the molecular dispersion coefficient approaches zero slowly as an inverse logarithmic function of the particle size to plate spacing ratio, and the convective dispersion coefficent approaches zero approximately proportional to the width of the gap between the edges of the sphere and the bounding plates.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Brady, John F.}, }