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## The Non-Newtonian Microrheology of Hydrodynamically Interacting Colloids: Toward a Non-equilibrium Stokes-Einstein Relation

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**Author**

Chu, Henry Chi Wah

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**Abstract**

The understanding of the flow properties of complex fluids is central to the development of materials and technology as diverse as paint, polymer solutions, biofluids, foodstuffs, and many other industrial compounds, and was made possible by advances in theoretical and experimental methods in the second half of the $20^{th}$ century. Development of nonequilibrium statistical mechanics theory, computational methods, microscopy, and bulk rheometry produced detailed understanding of structure-property relationships in these ubiquitous materials. Complex fluid rheology typically focuses on suspensions subjected to bulk shearing or extensional flow. However, ever-increasing need to understand microscopically small samples of fluid, especially biofluids, has demanded development of techniques that can interrogate tiny volumes of fluid and detect heterogeneity over the micron length scale --- far outside the capability of bulk techniques that smear out such detail. Active microrheology has emerged as the premiere technique to fill this need.
In active microrheology, a microscopic probe, or set of probes, is driven by an external force through a complex fluid. As the probe moves through the suspension, it drives its configuration out of equilibrium; meanwhile Brownian motion of the suspended particles acts to recover their equilibrium configuration. These distortions and relaxations of the microstructure alter probe motion, and this interplay evolves with flow strength and with microscopic suspension forces. Changes in mean and fluctuating probe motion can be related constitutively to suspension and flow properties, in direct analogy to such approaches in macroscopic rheology of single and complex fluids and solids.
The focus of this dissertation work has been to expand the existing theory of microrheology to predict one of the most central flow properties, the suspension stress. Because the particles are small enough to give a vanishingly small Reynolds number, inertia is negligible and Stokes' equations govern fluid motion. We consider particles small enough to undergo Brownian diffusion, which acts to restore flow-induced distortions of the spatial arrangement of particles. The shape of the distorted microstructure is set by the strength of the external force relative to the entropic restoring force of the suspension, and by the balance of microscopic forces between the constituent particles. The former is given by the P\'{e}clet number, $Pe$, whereas the latter comprise the external, Brownian, and interparticle forces, and the sensitivity of each to flow strength $Pe$ is set by the dimensionless repulsion range, $\kappa$. To analyze the influence of these forces on the structure and suspension stress as they evolve with flow strength, we formulate and solve a Smoluchowski equation --- an advection-diffusion equation governing the stochastic distribution of particles, systematically tuning the relative strength of hydrodynamic and entropic forces. The resulting distribution is employed as a weighting function in a statistical mechanics framework to compute the suspension stress averaged over the probe particle phase, the primary contribution to the nonequilibrium stress. A colloidal dispersion of hard spheres serves as the model system.
To understand the fundamental influence of hydrodynamic and entropic forces on the structure and suspension stress as they evolve with flow strength, we first solve the Smoluchowski equation analytically in the dual limits of weak and strong external force and hydrodynamic interactions, and then numerically for arbitrary values of $Pe$ and $\kappa$. Nonequilibrium statistical mechanics are then utilized to compute elements of the stress tensor. Because geometry of the microstructure about the line of the external force is axisymmetric, only the diagonal elements are nonzero. When hydrodynamic interactions are negligibly weak, only the hard-sphere interparticle force matters regardless of the flow strength, where the normal stresses scale quadratically and linearly in the external force strength in the limits of weak and strong forcing, respectively. As the repulsion range $\kappa$ shrinks, hydrodynamic interactions begin to play a role: when forcing is weak, Brownian disturbance flows provide the dominant contribution to the suspension stress, but as $Pe$ increases, the external force-induced stress takes over to dominate the total stress. Interestingly, the total suspension stress decreases as the strength of hydrodynamic interactions increases, regardless of the value of $Pe$. That is, hydrodynamic interactions suppress suspension stress. Owing to the dependence of hydrodynamic interactions on particle configuration, this stress suppression varies with flow strength: At low $Pe$, the stress scales as $Pe^2$ and the suppression is quantitative, whereas at high $Pe$ the stress scales as $Pe^\delta$, where $1 \geq \delta \geq 0.799$ for hydrodynamic interactions spanning from weak to strong. We identify the origin of stress suppression via an analysis of pair trajectories: While entropic forces --- interparticle repulsion and Brownian motion --- destroy reversible trajectories, hydrodynamic interactions suppress structural asymmetry and this underlies the suppression of the nonequilibrium stress. We relate the stress to the energy density: Hydrodynamic interactions shield particles from direct collisions and promote fore-aft and structural symmetry, resulting in reduced storage.
The detailed discussion of the individual normal stresses offers a fundamental understanding of the role played by hydrodynamic and entropic forces in energy density in a suspension. Non-Newtonian rheology is, however, more commonly characterized by the combined effect of the normal stresses --- the normal stress differences and particle osmotic pressure; and the rich phenomena that they exhibit warrant a separate examination. In Chapter~\ref{sec:Sec3}, the micromechanical theory developed in Chapter~\ref{sec:Sec2} is utilized to compute and analyze the first and the second normal stress difference, $N_1$ and $N_2$, and the particle osmotic pressure $\Pi$ . As hydrodynamic interactions grow from weak to strong, the influence of couplings between the stress and the entrained motion on $N_1$ changes with the strength of flow. When flow is strong, hydrodynamic interactions suppress magnitude of $N_1$, owing to collision shielding that preserves structural symmetry. In contrast, when flow is weak, hydrodynamic interactions enhance disparity in normal stresses and, in turn, increase the magnitude of $N_1$. The first normal stress difference changes sign as flow strength increases from weak to strong, due strictly to the influence of elastic interparticle forces. Regardless of the strength of flow and hydrodynamic interactions, the second normal stress difference is identically zero owing to the axisymmetry of the microstructure around the probe. Hydrodynamic forces act to suppress the osmotic pressure for any strength of flow; when forcing is strong, this effect is qualitative, reducing the flow-strength dependence from linear to sublinear as hydrodynamic interactions grow from weak to strong. Non-Newtonian rheology persists as long as entropic forces play a role, {\em i.e.} in the presence of particle roughness or even very weak Brownian motion, but vanishes entirely in the pure-hydrodynamic limit.
The micromechanical theory presented in Chapter~\ref{sec:Sec2} and \ref{sec:Sec3} predicts the non-Newtonian response of a dilute suspension, provided that its particle microstructure is known. Nevertheless, measuring particle distribution is tedious in practice. In the second part of this dissertation, Chapter~\ref{sec:Sec4}, we derive a non-equilibrium Stokes-Einstein relation for predicting suspension stress following the original model of Zia and Brady \cite{zb-12}, now for a dispersion of hydrodynamically interacting colloids simply by tracking probe motion. The theory is an expansion of Einstein's equilibrium fluctuation-dissipation theory, where Cauchy's equation of motion, rather than an equation of state, serves as the fundamental conservation framework. We construct an anisotropic effective resistance tensor comprising microviscosity and flow-induced diffusivity to model the hydrodynamically coupled particle motion which, when coupled with particle flux, constitutes the advective and diffusive components of Cauchy's momentum balance. The resultant phenomenological relation between suspension stress, viscosity and diffusivity is a generalized non-equilibrium Stokes-Einstein relation which enables a full rheological characterization of a hydrodynamically interacting suspension by simply tracking the mean and mean-square motion of a single probe. The predictions by the new theory are compared with the results obtained from the micromechanical theory for dilute suspensions via the normal stresses, normal stress differences and the particle osmotic pressure, where we find excellent agreement between the two sets of results in all quantities for any strength of flow and hydrodynamic interactions.
The micromechanical theory and phenomenological model presented in previous chapters give accurate prediction to non-Newtonian rheology in a dilute suspension. In the third part of this dissertation, Chapter~\ref{sec:Sec5}, we extend the scope of the present research by investigating the influence of particle concentration and the associated many-body interactions on suspension rheology. Analogous to the unbound monodisperse colloidal system studied in Chapter~\ref{sec:Sec2} to \ref{sec:Sec4}, we conduct active microrheology simulations via the Accelerated Stokesian Dynamics framework for suspensions with various particle concentrations, ranging from dilute $\phi=0.05$ to $\phi=0.40$. We showed that the influence of particle concentration on the particle structure and rheology is qualitative. When flow is strong, a high particle concentration alters the fore-aft asymmetry of the microstructure around the probe by enhancing the particle accumulation inside the boundary layer and shortening the wake due to stronger Brownian drift. At equilibrium, the ring-like structure is recovered for a concentrated suspension, and it is attenuated as the suspension becomes dilute. The suspension rheology is studied via the microviscosity and the suspension stress, and they are both enhanced with increasing particle concentration. As concentration increases, microstructure is more closely-spaced, leading to stronger hydrodynamic dissipation and thus viscosity. The more compact structure also suggests that a probe moving in a concentrated suspension pushes through more bath particles and causes more configuration distortion as compared to one driven by the same force in a dilute suspension, giving rise to higher energy density in a suspension, or equivalently a higher suspension stress. To bridge the results of dilute and concentrated suspensions, we further utilize concentrated mobility functions to construct scaling theories to collapse the microviscosity and particle stress of suspensions of different particle concentrations, offering a robust predictive model for concentrated suspension rheology.
Overall, in this work the major accomplishments are a micromechanical theory and a generalized nonequilibrium Stokes-Einstein relation for obtaining the stress, diffusivity and viscosity from the motion of a single particle, and a scaling theory connecting the response of dilute and concentrated suspensions.

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**Date Issued**

2017-08-30#####
**Subject**

Chemical engineering; Mechanical engineering; Active microrheology; Hydrodynamic interactions; Non-Newtonian rheology; Statistical mechanics; Stokes-Einstein relation; Stokesian Dynamics simulation

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**Committee Chair**

Zia, Roseanna N.

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**Committee Member**

Kirby, Brian; Koch, Donald L.

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**Degree Discipline**

Mechanical Engineering

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**Degree Name**

PHD of Mechanical Engineering

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**Degree Level**

Doctor of Philosophy

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**Type**

dissertation or thesis