Browsing by Subject "Brownian motion processes"
Now showing 1 - 6 of 6
Results Per Page
Sort Options
Item Analysis of Brownian dynamics and unsteady particle-motion in viscoelastic fluids(2012-05) Azese, Martin; Bhattacharya, Sukalyan; Blawzdziewicz, Jerzy; Ibragimov, Akif; Christopher, Gordon; He, ZhaomingIn 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.Item Brownian motion at fast time scales and thermal noise imaging(2008-12) Huang, Rongxin, 1978-; Florin, Ernst-LudwigThis dissertation presents experimental studies on Brownian motion at fast time scales, as well as our recent developments in Thermal Noise Imaging which uses thermal motions of microscopic particles for spatial imaging. As thermal motions become increasingly important in the studies of soft condensed matters, the study of Brownian motion is not only of fundamental scientific interest but also has practical applications. Optical tweezers with a fast position-sensitive detector provide high spatial and temporal resolution to study Brownian motion at fast time scales. A novel high bandwidth detector was developed with a temporal resolution of 30 ns and a spatial resolution of 1 °A. With this high bandwidth detector, Brownian motion of a single particle confined in an optical trap was observed at the time scale of the ballistic regime. The hydrodynamic memory effect was fully studied with polystyrene particles of different sizes. We found that the mean square displacements of different sized polystyrene particles collapse into one master curve which is determined by the characteristic time scale of the fluid inertia effect. The particle’s inertia effect was shown for particles of the same size but different densities. For the first time the velocity autocorrelation function for a single particle was shown. We found excellent agreement between our experiments and the hydrodynamic theories that take into account the fluid inertia effect. Brownian motion of a colloidal particle can be used to probe three-dimensional nano structures. This so-called thermal noise imaging (TNI) has been very successful in imaging polymer networks with a resolution of 10 nm. However, TNI is not efficient at micrometer scale scanning since a great portion of image acquisition time is wasted on large vacant volume within polymer networks. Therefore, we invented a method to improve the efficiency of large scale scanning by combining traditional point-to-point scanning to explore large vacant space with thermal noise imaging at the proximity of the object. This method increased the efficiency of thermal noise imaging by more than 40 times. This development should promote wider applications of thermal noise imaging in the studies of soft materials and biological systems.Item Contribution of electrostatic interaction to the image formation in 3D thermal noise imaging(2006-12) Qiu, Jinze; Florin, Ernst-LudwigThree dimensional structures are able to be imaged by scanning the volume with a nanometer-size Brownian particle. The contribution of electrostatic interaction to the image formation in thermal noise imaging was studied. The problem was simplified to one dimension by replacing the complex three-dimensional structure with a planar coverslip. A simple fluorescence experiment was designed first to calibrate the distance from the trapping center to the planar surface. Strong electrostatic repulsion between like charged surfaces was observed in the experiments at low ionic strength as expected. Further fluorescence experiments shows that minimum separation decreases as salt concentration increases. It was found that 0.01mol/l is the optimal salt concentration for the given experimental condition. Higher concentrations lead to a permanent adhesion of particles to the surface making thermal noise imaging impossible.Item On the linear Vlasov-Fokker-Planck equation in kinetic theory(Texas Tech University, 1989-05) O'Dwyer, Brian PatrickNot availableItem Random perturbation of a self-adjoint operator with a multiple eigenvalue(2012-05) Gaines, George; Ruymgaart, Frits; Gilliam, David S.; Trindade, A. AlexandreWe first consider a bounded self-adjoint operator on a Hilbert space with a multiple eigenvalue as its largest eigenvalue. We perturb the operator and study the resulting cluster of eigenvalues of the perturbed operator. We study the convergence of the scattered eigenvalues to the original. We then do computer simulations. We also show an approximation for Brownian motion.Item Wiener filtering of images degraded by film grain noise(Texas Tech University, 1973-08) Choens, Robert CharlesNot available