## Analysis of Brownian dynamics and unsteady particle-motion in viscoelastic fluids

##### Abstract

In recent times, micro-rheological applications involve determination of viscoelastic properties for samples that are either too precious and fragile or in a state (like inside a cell) where macroscopic experiments are impossible. In such cases, direct measurements using rheometers are not possible, because then the system can be structurally destroyed. One way to circumvent this problem is to predict fluid-rheology from the random motion of a Brownian sphere in the medium. Thus, many past attempts tried to relate viscoelastic properties to features of stochastic motion like time-dependent velocity correlation or mean square displacement. All such theories, however, invariably involve heuristic assumptions inherited from classical studies on purely viscous fluid. This is why in this thesis the classical theories of statistical mechanics for Brownian dynamics are first reevaluated and then modified to suit the new technological demand.
This research first focuses on the flow-analysis which describes hydrodynamic field inside a viscoelastic medium. Accordingly, a mathematically rigorous perturbation method is developed which isolates the leading order linear contributions from higher order non-linearities due to both convective acceleration and constitutive relation. As a result, the conditions for linearized analysis are identified, and the leading order fields as well as particle-motion are determined.
Then the analysis concentrates on the leading order linearized hydrodynamic equation only, and scrutinizes the relevance of classical theories of statistical mechanics for micro-rheological applications. In this context, three key conclusions are drawn revealing the errors in the earlier concepts. Firstly, the validity of fluctuation-dissipation theorem are questioned, as it requires Markovian condition only true for memory-less systems without viscoelasticity and flow-inertia. Secondly, well-known Langevin equation for Brownian dynamics is rectified by including the effect of fluid-inertia in the equation of motion of the suspended body in a density-matched liquid. Thirdly, the equipartition principle is reinterpreted to find the correct normalization for correlation of Brownian forces where energy associated with the translation of a Brownian particle is considered to have an additional contribution from the induced flow in the liquid. Thus, we discard the fluctuation-dissipation postulate, and recommend an inertia-corrected modified Langevin formulation to be used in micro-rheological problems.
We use our new theory to correctly describe the stochastic dynamics of a Brownian sphere in a viscoelastic liquid by relating its time-dependent velocity correlation function and mean square displacement to fluid-rheology. Resulting conclusions differ substantially from popular beliefs while maintaining agreements under the long-time or low-frequency limit under proper conditions. Thus, our alternative formulation can be used in microrheological measurements to predict large-frequency complex viscosity for which the failure of past theories are well-documented.
Moreover, we analyze the classical problem involving a Brownian sphere in a purely viscous liquid with density similar to the suspended solid. The errors in the original Langevin formulation are highlighted where the inertia of the fluid is ignored in both equation of particle-motion and equipartition principle. Our new theory with proper corrections is used to find the unsteady velocity correlation and mean square displacement of the sphere. The computed temporal variations of these quantities differ substantially from the results obtained from the classical Langevin equation. Curiously, however, the long-time diffusion coefficients in both cases exactly coincide. It seems that the earlier analysis calculates the correct diffusivity, because the error in equation of motion and misinterpretation in equipartition principle nullify each other. As long-time diffusivity is a quantity which has been experimentally verified over a century, the aforementioned agreement can be viewed as a further verification of the new theory.