Our Research. The Zia group develops predictive theory and computational models for the far-from equilibrium behavior of complex fluids and other soft matter undergoing low-Reynolds number flow. Our research focuses on three primary themes: structural evolution and particle transport in 3D micro-confined suspensions; slow evolution during and sudden release from kinetic arrest in colloidal gels and glasses; and development of a broad non-equilibrium "equation of state", a generalization of Einstein's equilibrium theory. Our approach to developing predictive theory for far-from equilibrium material behavior is to relate it to the microscopic mechanics of the constituent particles. Our work allows us to interact with the dogma that colloids serve as a model system for molecular fluids, ranging from glassy transitions to mechanical transport in living cells. As a result, we have offered new micro-mechanical perspectives of colloidal gels, offered the first accurate hydrodynamical model of the cell interior, and led the way in establishing active microrheology as the modern tool of rheology. Our over-arching goal is to show where the motion of colloidal-scale particles in crowded, watery environments can both zoom in and zoom out: zooming out to predict macroscopic behavior, and zooming in, to reveal signatures of molecular motions---and as part of our long-range vision---perhaps to an extent that it can elucidate the mechanical aspects of the life process.

Current projects — follow a link to explore:

Aging of colloidal gels
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Delayed and Transient Yield of Colloidal Gels
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Collapse of colloidal gels
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Intracellular Transport: Active & Passive Transport of Hydrodynamically Interacting Colloids in Spherical Confinement
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Non-Equilibrium Depletion Interactions: First Things Attract, Then They Repel
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Non-equilibrium Fluctuation Dissipation and Active Microrheology
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Does Suspension Crowding Screen Hydrodynamic Interactions?
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Testing the paradigms of the colloidal glass transition: Novel concentration jump experiments and large scale computer modeling
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How wonderful that we have met with a paradox. Now we have some hope of making progress.
- Niels Bohr




Aging of colloidal gels

Microscopically small particles dispersed in a solvent interact with each other via a variety of mechanisms, including solvent-mediated (hydrodynamic) interactions and interparticle forces. The nature of the interparticle force may produce repulsion, attraction, or a combination of the two. Such forces can arise due to steric, electrostatic, van der Waals or depletion interactions, for example, and the overall behavior of a complex fluid can vary dramatically depending on the strength, direction, range, and time-variance of the interparticle forces.
In attractive colloidal dispersions, an approaching pair of particles experiences increasingly stronger attraction until a critical separation is reached, whereupon a repulsive force prevents further approach (and overlap). At atttractive ranges, the strength of the attraction potential V(r) relative to the thermal energy kT determines how strongly the pair is bonded. When V/kT is small (weak attractions), thermal fluctuations easily break the bond, and the particles spend much time separated. But as the system is "cooled" (attraction strength increased), V/kT is not small, and particles can stick together for long periods of time.
When such a system is "cooled", in order to minimize free energy, the disordered state may be sacrificed for a lower potential energy, and phase separation by nucleation or spinodal decomposition begins (see phase diagram, above left). But in some cases, the attractions between the particles hinder the structural rearrangements required to proceed fully to equilibrium; spinodal decomposition arrests before the condensed phase can fully separate from the dilute dispersed phase. Instead a bi-continuous network of dense, disordered regions or "strands" forms (see simulation image, right). The resulting space-spanning network is called a colloidal gel; sufficiently strong particle bonds lead to macroscopic elastic strength, and the gel may thus support its own weight under gravity.
Because the bonds are reversible, the solid-like structure may flow viscously when sheared. Of particular interest is the colloidal gel that supports its own weight under gravity for an extended period of time, but then suddenly collapses under its own weight. In some cases, the structure compacts slowly at first, followed later by a very sudden and dramatic collapse of the entire structure. While sedimentation of strong gels has been well characterized (e.g. Buscall & White, 1987), "collapse" is a distinct physical phenomenon, and no model or theory exists that fully explains this behavior. We show that the characteristics of gel collapse are rheologically and microstructurally distinct from gel sedimentation; the goal of our research is to develop a theoretical framework that predicts the sudden and rapid-collapse phase of weak colloidal gels, and to simulate this behavior with computational efficiency.

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Delayed and Transient Yield of Colloidal Gels

We study via dynamic simulation the nonlinear response of a reversible colloidal gel towards elucidating the micro-mechanical origins of two rheological behaviors: (1) delayed yield, as the gel undergoes deformation due to an applied stress, and (2) transient yield, as the gel responds to the sudden startup of shear flow. The gel network comprises microscopic, spherical particles that interact via a hard-sphere repulsion and a short-range attraction. The strength of attractions is on the order of a few kT and leads to a bi-continuous, space-spanning network that exhibits elastic and viscous behaviors and continues to evolve structurally over time due to thermal fluctuations.

Delayed yield: Under a constant shear stress, such gels may flow then regain solidlike behavior upon removal of the stress. The transition from solidlike to liquidlike behavior is a yielding process that is not instantaneous but rather occurs after a finite delay. The delay length decreases as stress increases, but the underlying microstructural origin is not clear. Recent experiments reveal two regimes, suggesting multiple yield mechanisms. Theories advanced to link gel structure to rheology aim to predict the ultimate state of a gel under an applied load. While these hypothesize a competition between bond breakage and reconnection rates, no such particle-scale dynamics have been directly observed, and it is not clear these theories reconcile with ongoing structural evolution.

Transient yield: Upon startup of an imposed strain rate, the transition from rest to steady flow is characterized by one or more “overshoots” in the shear stress. Experimental studies, in which the overshoots depend on gel age, strain rate, volume fraction, and attraction strength, suggest that the underlying microstructural origin is a two-step process of cage breaking and bond breaking. The location in strain at the onset of transient yield is postulated to reveal the length scales in the gel responsible and the strength of the stress response is supposedly set by the strengthening of pair bonds or coarsening gel features with gel age. To this end, careful tracking of the evolution of gel structure over many lengthscales, from pair bonds to dominant feature size, will be necessary to evaluate such claims.

To study these behaviors, we conduct large-scale dynamic simulation to model structural evolution and particle transport in colloidal gels subjected (1) to a step stress and (2) to a step strain rate. Detailed particle dynamics and structural evolution reveal key yielding mechanisms to enable the prediction and control of colloidal gel response under nonlinear forcing.

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Collapse of Colloidal Gels

Colloidal gels formed by arrested phase separation with O(kT) attractive potential exhibit elastic and viscous behaviors due to the temporary yet durable nature of the physical interparticle bonds. Because the bonds are reversible, the solid-like structure may flow viscously when sheared. When the colloidal particles of the gel and the surrounding solvent differ in density, gravitational forces play an important role in the dynamic behavior of the gels. Of particular interest is the colloidal gel that supports its own weight for an extended period of time, but then suddenly collapses under its own weight. In some cases, the structure compacts slowly at first, followed later by a very sudden and dramatic collapse of the entire structure. While sedimentation of strong gels has been well characterized (e.g. Buscall & White, 1987), "collapse" is a distinct physical phenomenon, and no model or theory exists that fully explains this behavior.

We investigate the microscopic origins of this "tipping point" behavior where the transition from slow to fast sedimentation occurs via dynamic simulation. Neglecting the complicating influences of complex container shapes, curved menisci and hydrodynamic back flow permits focus on the basic aspects of the gel yield under gravity. We track particle positions, velocities and particle-phase pressure throughout the evolution of the structure which reveal that a shift in bond dynamics plays an important role in the collapse behavior. We show that the characteristics of gel collapse are rheologically and microstructurally distinct from gel sedimentation; the goal of our research is to develop a theoretical framework that predicts the sudden and rapid-collapse phase of weak colloidal gels, and to simulate this behavior with computational efficiency.

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Intracellular Transport: Active & Passive Transport of Hydrodynamically Interacting Colloids in Spherical Confinement

The motion of particles within a eukaryotic cell is essential to many of its critical functions: metabolism, gene expression, and signaling pathways; pattern formation (e.g. establishment of polarity in C. elegans embryos); cargo transport via directed motors along cytoskeletal tracks. Intracellular transport is also an important factor in the efficacy of gene delivery. While the chemical signaling that orchestrates these processes is well studied, a clear understanding of the physical transport mechanics of such motion lags behind. The objective of this investigation is to develop a predictive model for the transport of particles inside a model eukaryotic cell, the C. elegans early embryo, and use this model to explore the role of particle fluctuations, active forcing, and effective temperature on intracellular transport. To study transport processes inside eukaryotic cells we focus on two modes: 1) Diffusion 2) Active towing by motor proteins. The initial thrust of the research is to develop a theoretical framework that captures the effects of crowding as well as fully 3D confinement and use this predictive model to help answer questions regarding the role of gradient diffusion in establishment of cell polarity. To this end, we have developed a new theoretical framework to rigorously model suspensions of hydrodynamically interacting particles confined inside a spherical boundary. Our framework is capable of rigorously accounting for many-body hydrodynamic interactions as well as electrostatic, steric and entropic forces between the confined particles. Furthermore, the framework is valid for arbitrary particle-to-cavity size ratios and volume fractions allowing for the study of a myriad of macromolecules and organelles found in the cell interior. We have used this framework to develop a computational model that faithfully captures particle motion, and thus the micro-structural evolution of the confined suspension. We used this model to study the short-time self-diffusivity of spherically confined colloidal suspensions as a function of both volume fraction and particle-to-cavity size ratio, and found the short-time self-diffusivity to be anisotropic and hindered as compared to that of an unbound colloidal suspension. Current research efforts focus in using this framework to study gradient and long-time self-diffusion as well as active motion of particles inside the confining boundary.

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Non-Equilibrium Depletion Interactions: First Things Attract, Then They Repel

Depletion interactions between colloid in suspensions occur in the presence of smaller depletant species. The depletants exert an osmotic pressure on the colloids, which is isotropic on an isolated probe. This symmetry is broken when colloid are close enough together to exclude the depletant species from the space between them, resulting in an attractive force. These interactions, first studied by Asakura and Oosawa in 1954, are well characterized in equilibrium systems. In this regime interactions are strictly attractive, and can lead to flocculation of colloidal particles. Much less is known about depletion flocculation away from equilibrium. We study non-equilibrium depletion interactions in colloidal dispersions via a combination of asymptotic and numerical solutions of the Smoluchowski equation. A pair of probes at arbitrary separation is driven by an external force at arbitrary orientation through a suspension, deforming the surrounding microstructure. The degree to which the structure is distorted, and the shape of this deformation, depends on the separation between the probes, on the orientation of the pair to the driving force, and on the strength with which the probes are forced relative to the entropic restoring force of the suspension particles, defined by the Péclet number, Pe = U(a+b)/Db, where U is the probes’ velocity, a is the probe size, Db is the diffusivity of the background particles of size b. The resultant non-equilibrium osmotic pressure gradients give rise to both drag and interactive forces between the probes. When the external force is zero, the depletion attraction of Asakura and Oosawa is recovered. When an external force is applied, the interactive force can lead either to attraction or repulsion, as well as deterministic re-orientation of the probes relative to the external force, depending on initial separation, orientation, and strength of forcing. Our current research focus on the use of this model for the interrogation and characterization of a broad class of complex fluids.

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Non-equilibrium Fluctuation Dissipation and Active Microrheology

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, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe = Fextb/kT, where kT is the thermal energy and b is the characteristic microstructural length scale.

Toward a Non-Equilibrium Fluctuation Dissipation Model
In this work, we extend our Non-Equilibrium Stokes-Einstein Relation / Equation of State for colloidal suspensions to include hydrodynamic interactions between particles, with a dual aim. First, a practical and powerful tool: we aim to enlarge on our current phenomenological model that allows measurement of suspension viscosity, flow-induced diffusion, and the full stress tensor by simply tracking the motion of one "probe" particle driven through the material. This active, nonlinear micro-rheology theoretical framework has proven a powerful technique for interrogating microscopically small soft-matter systems, ranging from actin networks to mucus to rare biological fluids to the interior of eukaryotic cells. We are completing the comparison of our phenomenological model to solutions obtained via statistical mechanics and Accelerated Stokesian Dynamics simulations, with excellent agreement, suggesting that full rheological characterization of hydrodynamically interacting suspensions can be obtained by tracking the motion of one externally force probe particle.

Insights into the fundamental connection between far-from-equilibrium fluctuation and dissipation have emerged from these theoretical studies. Our work shows a nuanced relationship between flow strength, Brownian motion, and hydrodynamic forces that leads to such interesting behavior as compressive-to-tensile transitions in the osmotic pressure, advective relaxation, and hydrodynamic stress suppression. A primary goal of this work is to elucidate the nature of the suspension stress itself, interpreted as a balance between fluctuation and dissipation.

Hydrodynamic Diffusion and the Role of Interparticle Forces
As the probe is forced through a complex fluid, microstructural constituents can scatter the probe from its intended trajectory, leading to a diffusive spread of probe motion. The flow-induced diffusivity of the probe quantifies this scattering, which we have studied in the model system of a probe driven through a colloidal dispersion. Flow-induced diffusivity is known to depend on the strength of external forcing (Pe). We are investigating its dependence on the strength of hydrodynamic interactions, and on the size of the probe relative to the background bath particles. Prior studies (Zia and Brady, 2010) have shown that the flow-induced diffusivity Dflow of non-hydrodynamically interacting colloids is anisotropic, with the probe preferentially diffusing in the direction of forcing. For weak external forcing, Dflow/Da ~ Pe2, which is negligible in comparison to the random walk caused by Brownian diffusion Da. For strong forcing, Dflow/Da ~ Pe, and trajectories scatter over length scales comparable to deterministic motion. Our current studies focus on how the anisotropy in the flow-induced diffusivity changes, if at all, with stronger hydrodynamic interactions. We also aim to connected flow-induced diffusion seen in active microrheology to prior experiments (Abbott, Graham, Mondy, and Brenner 1997) and theoretical investigations (Davis and Hill, 1992; Davis 1992) of falling ball rheometry, the macroscopic and non-colloidal analog to active microrheology.

Decoupling Microscopic Forces: Transient Microrheology
Prior theoretical work (Squires and Brady, 2005; Khair and Brady, 2006) showed that the mean steady motion of the probe can be related to the steady microstructure around the probe via the effective viscosity of the surrounding medium. This gives the steady state behavior; microstructural development and relaxation can be captured by studying evolution under flow startup and cessation. Our previous work (Zia and Brady, 2013) answered these in the absence of hydrodynamic interactions. However, the effect of hydrodynamics still remains an open question. Hydrodynamic interactions introduce an additional mode of energy dissipation, thereby slowing down the system response after flow startup. Upon flow cessation, some of the energy that has been stored entropically in the distorted microstructure is recovered elastically as the probe moves backwards. The remaining energy is dissipated viscously as the system relaxes. This time-dependent study can reveal the time scales over which various contributions to stress formation (hydrodynamic, Brownian, and interparticle) act and this can provide insight into the role played by each in the storage and dissipation of flow energy. Such studies also may be used to probe the viscoelastic memory of a flowing suspension.

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Does Suspension Crowding Screen Hydrodynamic Interactions?

Resistance and mobility functions describe linear couplings between moments of the hydrodynamic traction on a suspended particle and the motion of that or other particles. For two isolated spheres, these functions are well known and have been applied directly in the solution of many important problems for dilute colloidal dispersions. We have devised a new stochastic technique to calculate an analogous set of functions for two spheres immersed in a suspension that are then used to model the near-equilibrium dynamics of concentrated dispersions, including viscoelasticity and long-time diffusion. Of interest is the degree of screening of hydrodynamic interactions by the intervening medium. We find that the mobility is unscreened at the pair level, even in suspensions of high concentration, confirming that hydrodynamic interactions are an essential part of the dynamics of crowded systems and cannot be neglected in favor of simple renormalization schemes. We compare our results for the hydrodynamic interactions between suspended particles to predictions from two-point microrheology. This technique can be used to infer the complex viscosity from long-ranged decay of the pair mobility in viscoelastic materials. “Pair mobility functions for rigid spheres in concentrated colloidal dispersions: Force, torque, translation, and rotation.” J. Chem. Phys.

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Testing the paradigms of the colloidal glass transition: Novel concentration jump experiments and large scale computer modeling

Many molecular liquids transition from a liquid to a crystalline state upon cooling. If the cooling is sufficiently rapid, however, the material can undergo a glass transition, whereby it solidifies without crystallization, forming a glass in which the amorphous structure is retained but the dynamics become extremely slow. Similar behavior has been observed in colloidal fluids, whereby they form a colloidal glass upon a “concentration quench”. Much analysis in the colloidal-glass literature assumes that the behavior of colloidal systems at concentrations near the jamming transition is similar to that of a molecular glass near its glass transition, hence making the simpler colloid a good model for glass-like materials. Extensive studies of the equilibrium dynamics in both systems show that rheology and dynamics are well described by diverging viscosity and relaxation time as the glass transition is approached. Recent studies by McKenna and co-workers (Phys. Rev. Lett. 2011, J. Chem. Phys. 2014) reveal that non-equilibrium aging and structural recovery of colloidal systems differ qualitatively from molecular glassy dynamics when interrogated for the Kovacs signatures, viz., the intrinsic isotherm, the asymmetry of approach to equilibrium, and the memory experiment. The reasons for such differences have not yet been elucidated beyond the broad hypothesis that colloids may not be “true” glass-forming systems despite many shared dynamical and mechanical signatures. In this study, we are collaborating with the McKenna Lab (TTU) in a fundamental investigation of the structure, dynamics and rheology of thermo-sensitive colloidal dispersions near to the glass and jamming transitions in a novel combination of laboratory experiments utilizing thermo-sensitive colloids for which the glass transition can be triggered by temperature-induced volume fraction changes, alongside in silico investigations whereby we utilize large-scale dynamic simulation to model the same concentration jumps, and to study structural relaxation, dynamical behavior, and rheology during and after the transition. Broadly this work expands to colloidal glasses the previous investigation of Kovacs signatures of structural recovery in molecular glasses to develop a fundamental framework for understanding glassy behavior in both systems from a micromechanical perspective.

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