Browsing by Subject "Statistical mechanics"
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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 Efficient computational strategies for predicting homogeneous fluid structure(2014-08) Hollingshead, Kyle Brady; Truskett, Thomas Michael, 1973-A common challenge in materials science is the "inverse design problem," wherein one seeks to use theoretical models to discover the microscopic characteristics (e.g., interparticle interactions) of a system which, if fabricated or synthesized, would yield a targeted material property. Inverse design problems are commonly addressed by stochastic optimization strategies like simulated annealing. Such approaches have the advantage of being general and easy to apply, and they can be effective as long as material properties required for evaluating the objective function of the optimization are feasible to accurately compute for thousands to millions of different trial interactions. This requirement typically means that "exact" yet computationally intensive methods for property predictions (e.g., molecular simulations) are impractical for use within such calculations. Approximate theories with analytical or simple numerical solutions are attractive alternatives, provided that they can make sufficiently accurate predictions for a wide range of microscopic interaction types. We propose a new approach, based on the fine discretization (i.e., terracing) of continuous pair interactions, that allows first-order mean-spherical approximation theory to predict the equilibrium structure and thermodynamics of a wide class of complex fluid pair interactions. We use this approach to predict the radial distribution functions and potential energies for systems with screened electrostatic repulsions, solute-mediated depletion interactions, and ramp-shaped repulsions. We create a web applet for introductory statistical mechanics courses using this approach to quickly estimate the equilibrium structure and thermodynamics of a fluid from its pair interaction. We use the applet to illustrate two fundamental fluid phenomena: the transition from ideal gas-like behavior to correlated-liquid behavior with increasing density in a system of hard spheres, and the water-like tradeoff between dominant length scales with changing temperature in a system with ramp-shaped repulsions. Finally, we test the accuracy of our approach and several other integral equation theories by comparing their predictions to simulated data for a series of different pair interactions. We introduce a simple cumulative structural error metric to quantify the comparison to simulation, and find that according to this metric, the reference hypernetted chain closure with a semi-empirical bridge function is the most accurate of the tested approximations.Item Inverse design methods for targeted self-assembly(2014-12) Jain, Avni; Truskett, Thomas Michael, 1973-In this thesis, we study the problem of what microscopic thermodynamic driving forces can stabilize target macroscopic structures. First, we demonstrate that inverse statistical mechanical optimization can be used to rationally design inter-particle interactions that display target open structures as ground states over a wide range of thermodynamic conditions. We focus on designing simple interactions (e.g., isotropic, convex-repulsive) that drive the spontaneous assembly of material constituents to low-coordinated ground states of diamond and simple cubic lattices. This is significant because these types of phases are typically accessible given more complex systems (e.g., particles with orientation-dependent attractive interactions) and for a narrow range of conditions. We subject the optimal interactions to stringent stability tests and also observe assembly of the target structures from disordered fluid states. We then use extensive free energy based Monte Carlo simulation techniques to construct the equilibrium phase diagrams for the model materials with interactions designed to feature diamond and simple cubic ground states, i.e., at zero temperatures. We find that both model materials, despite the largely featureless interaction form, display rich polymorphic phase behavior featuring not only thermally stable target ground state structures, but also a variety of other crystalline (e.g., hexagonal and body-centered cubic) phases. Next, we investigate whether isotropic interactions designed to stabilize given two-dimensional (2D) lattices (e.g., honeycomb or square) will favor their analogous three-dimensional (3D) structures (e.g., diamond or simple cubic), and vice versa. We find a remarkable transferability of isotropic potentials designed to stabilize analogous morphologies in 2D and 3D, irrespective of the exact interaction form, and we discuss the basis of this cross-dimensional behavior. Our results suggest that computationally inexpensive 2D material optimizations can assist in isolating rare isotropic interactions that drive the assembly of materials into 3D open lattice structures.Item Modeling the interaction and energetics of biological molecules with a polarizable force field(2013-05) Shi, Yue, active 21st century; Ren, PengyuAccurate prediction of protein-ligand binding affinity is essential to computational drug discovery. Current approaches are limited by the accuracy of the underlying potential energy model that describes atomic interactions. A more rigorous physical model is critical for evaluating molecular interactions to chemical accuracy. The objective of this thesis research is to develop a polarizable force field with an accurate representation of electrostatic interactions, and apply this model to protein-ligand recognition and to ultimately solve practical problems in computer aided drug discovery. By calculating the hydration free energies of a series of organic small molecules, an optimal protocol is established to develop the electrostatic parameters from quantum mechanics calculations. Next, the systematical development and parameterization procedure of AMOEBA protein force field is presented. The derived force field has gone through extensive validations in both gas phase and condensed phase. The last part of the thesis involves the application of AMOEBA to study protein-ligand interactions. The binding free energies of benzamidine analogs to trypsin using molecular dynamics alchemical perturbation are calculated with encouraging accuracy. AMOEBA is also used to study the thermodynamic effect of constraining and hydrophobicity on binding energetics between phosphotyrosine(pY)-containing tripeptides and the SH2 domain of growth receptor binding protein 2 (Grb2). The underlying mechanism of an "entropic paradox" associated with ligand preorganization is explored.Item Ordering in dense packings(2011-05) Aristoff, David Gregory; Radin, Charles, 1945-; Koch, Hans; de la Llave, Rafael; Gamba, Irene; Gonzalez, Oscar; Raizen, MarkWe examine various models of soft matter, and one model of quasicrystals, focusing on abrupt changes as density is varied. We consider in detail two models, one of granular matter and another of confined wires, showing that the models become ordered as density is increased, with crystalline order observed in the former and nematic order observed in the latter. We associate the phenomenon of random close packing with the onset of crystalline order in our granular model, and we conjecture that crumpled wires should exhibit a nematic transition with increasing compaction. We also consider two other models of granular matter: one which describes dilatancy onset as a second order phase transition, and one which describes random loose packing as a precise, well- defined density. Finally, we examine an equilibrium model of quasicrystals with a first order phase transition to a solid phase without any crystalline order.Item Prediction of critical properties for mixtures of carbon dioxide and reservoir fluids(Texas Tech University, 2001-08) Ortiz, AldoThe main objective of this project is to establish two binary interaction parameters (ttij, pij) for the attractive pressure term of the Lawal-Lake Silberberg (LLS) Equation of State (EOS). Three iterative techniques for establishing aij and pij values in the mixing rules ofthe LLS EOS are developed for that purpose. By applying the critical constraint criteria to the LLS EOS, two sets of closed-form equations are developed for resolving the van der Waals critical point. The closed-form equations are used in the expressions developed for critical pressure, critical temperature and critical volume to predict the critical points (Pc, Tc and Vc) of asymmetric binary mixtures. The prediction of critical properties compared with the measured data are generally within 0.2 % ofthe reported data. The mixture parameters (ocm, Pm) obtained from the binary critical property prediction results by the LLS EOS are used to deduce the binary interaction parameters for mixture of polar and non-polar systems. The deduced interaction parameters are used to predict critical properties of ternary and multi-component mixtures and the results are generally within 1.5% ofthe measured data. The refined interaction parameters are utilized in the prediction of critical properties of reservoir fluids (natural gases, volatile oil, gas-condensate and crude oil) and mixture of carbon dioxide and reservoir fluids. The average absolute percent error of the predicted critical temperatures and pressures compared to the experimental critical properties ofthe mixtures of CO2 and reservoir fluid systems is within 2.5% ofthe measured data.Item Statistical mechanics of 2-D fluids(1994) Padhye, Nikhil Subhash, 1970-; Morrison, Philip J.Item Statistics of turbulence in a rapidly rotating system(2005) Jung, Sunghwan; Swinney, H. L., 1939-; Morrison, Philip J.