A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales

In this work it is shown how the immersed boundary method of (Peskin2002) for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic tau^(-3/2) decay for long times. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method.Comment: 52 pages, 11 figures, published in journal of computational physic

Theoretical Framework for Microscopic Osmotic Phenomena

The basic ingredients of osmotic pressure are a solvent fluid with a soluble molecular species which is restricted to a chamber by a boundary which is permeable to the solvent fluid but impermeable to the solute molecules. For macroscopic systems at equilibrium, the osmotic pressure is given by the classical van't Hoff Law, which states that the pressure is proportional to the product of the temperature and the difference of the solute concentrations inside and outside the chamber. For microscopic systems the diameter of the chamber may be comparable to the length-scale associated with the solute-wall interactions or solute molecular interactions. In each of these cases, the assumptions underlying the classical van't Hoff Law may no longer hold. In this paper we develop a general theoretical framework which captures corrections to the classical theory for the osmotic pressure under more general relationships between the size of the chamber and the interaction length scales. We also show that notions of osmotic pressure based on the hydrostatic pressure of the fluid and the mechanical pressure on the bounding walls of the chamber must be distinguished for microscopic systems. To demonstrate how the theoretical framework can be applied, numerical results are presented for the osmotic pressure associated with a polymer of N monomers confined in a spherical chamber as the bond strength is varied

Mathematical and Computational Models of Fluctuating Vesicles in Time-Varying Flows

Modeling vesicle dynamics involves a complicated moving boundary problem while the shapes of vesicles are determined dynamically from a balance between interfacial forces and fluid stresses. In this thesis, we investigate the dynamics of a two-dimensional fluctuating vesicle in a viscous fluid.Firstly we develop a two-dimensional stochastic immersed boundary method (SIBM) and analyze thermal fluctuations by matching the numerical results with a theoretical solution. Then we apply the SIBM with fitted thermal fluctuations to study the long term dynamics of an impermeable vesicle in a periodically time-reversed flow. The wrinkling process contains three stages. In the first stage, high-order modes are excited by the negative surface tension and wrinkles appear. In the second stage, low Fourier modes increase, the high-order wrinkles decay, and the shapes of vesicles keep relatively stable. In the last stage, the second Fourier mode grows and dominates. The shapes of the vesicle are ellipse-like with inclination angle θ ≈ 45◦. Then by performing an asymptotic linear analysis of a quasi-circular vesicle, we derive and solve the deterministic and stochastic equations for the motion of membrane interface numerically. The linear theory also indicates this three stage process.Finally, we investigate the nonlinear wrinkling dynamics of a permeable vesicle using an extension of the SIBM. We observe the vesicle shrinkage and the wrinkles on the membrane caused by a large osmosis pressure. We extend the linear theory to account for permeability and find a good agreement between linear and fully nonlinear vesicle dynamics

Finite Element Methods in Smart Materials and Polymers

Functional polymers show unique physical and chemical properties, which can manifest as dynamic responses to external stimuli such as radiation, temperature, chemical reaction, external force, and magnetic and electric fields. Recent advances in the fabrication techniques have enabled different types of polymer systems to be utilized in a wide range of potential applications in smart structures and systems, including structural health monitoring, anti‐vibration, and actuators. The progress in these integrated smart structures requires the implementation of finite element modelling using a multiphysics approach in various computational platforms. This book presents finite element methods applied in modeling of the smart structures and materials with particular emphasis on hydrogels, metamaterials, 3D-printed and anti-vibration constructs, and fibers

Electrostatics of Colloidal Particles Confined in Nanochannels: Role of Double-Layer Interactions and Ion-Ion Correlations

We perform computational investigations of electrolyte-mediated interactions of charged colloidal particles confined within nanochannels. We investigate the role of discrete ion effects, valence, and electrolyte strength on colloid-wall interactions. We find for some of the multivalent charge regimes that the like-charged colloids and walls can have attractive interactions. We study in detail these interactions and the free energy profile for the colloid-wall separation. We find there are energy barriers and energy minima giving preferred colloid locations in the channel near the center and at a distance near to but separated from the channel walls. We characterize contributions from surface overcharging, condensed layers, and overlap of ion double-layers. We perform our investigations using Coarse-Grained Brownian Dynamics simulations (BD), classical Density Functional Theory (cDFT), and mean-field Poisson-Boltzmann Theory (PB). We discuss the implications of our results for phenomena in nanoscale devices.Comment: 23 figure

Incorporating Cellular Stochasticity in Solid–Fluid Mixture Biofilm Models

The dynamics of cellular aggregates is driven by the interplay of mechanochemical processes and cellular activity. Although deterministic models may capture mechanical features, local chemical fluctuations trigger random cell responses, which determine the overall evolution. Incorporating stochastic cellular behavior in macroscopic models of biological media is a challenging task. Herein, we propose hybrid models for bacterial biofilm growth, which couple a two phase solid/fluid mixture description of mechanical and chemical fields with a dynamic energy budget-based cellular automata treatment of bacterial activity. Thin film and plate approximations for the relevant interfaces allow us to obtain numerical solutions exhibiting behaviors observed in experiments, such as accelerated spread due to water intake from the environment, wrinkle formation, undulated contour development, and the appearance of inhomogeneous distributions of differentiated bacteria performing varied tasks

Continuum and discrete approach in modeling biofilm development and structure: a review

The scientific community has recognized that almost 99% of the microbial life on earth is represented by biofilms. Considering the impacts of their sessile lifestyle on both natural and human activities, extensive experimental activity has been carried out to understand how biofilms grow and interact with the environment. Many mathematical models have also been developed to simulate and elucidate the main processes characterizing the biofilm growth. Two main mathematical approaches for biomass representation can be distinguished: continuum and discrete. This review is aimed at exploring the main characteristics of each approach. Continuum models can simulate the biofilm processes in a quantitative and deterministic way. However, they require a multidimensional formulation to take into account the biofilm spatial heterogeneity, which makes the models quite complicated, requiring significant computational effort. Discrete models are more recent and can represent the typical multidimensional structural heterogeneity of biofilm reflecting the experimental expectations, but they generate computational results including elements of randomness and introduce stochastic effects into the solutions

Reconfigurable Periodic Porous Membranes & Nanoparticle Assemblies

The thesis here will cover two parts of my research. The focus of the first part of the thesis will be using responsive hydrogel materials to manipulate the pattern transformation at microscale (Chapter 3-5), and meanwhile using the finite element method (FEM) to guide new designs of the periodic porous structures that can undergo controlled pattern transformation processes (Chapter 6). In beginning, I design fabrication methods of micro-structures from responsive hydrogel materials via micro-/nano- imprinting. The responsiveness of the hydrogels is introduced by incorporating responsive monomers into the hydrogel precursors. Here, the responsiveness of the hydrogel leads to the tunable swelling ratio of the hydrogel under external stimuli, e.g. pH, temperature, and variation of humidity, so that the imprinted nano-/micro- structures can be dynamically controlled. Later, upon using FEM simulation, we design and experimentally test the deformation and mechanical properties of the periodic porous membranes based on different collapsing modes of kagome lattices. The experiments are performed at macroscopic scale taking advantage of powerful 3D printing prototyping. As the deformation phenomenon is scale independent, the observed phenomenon is applicable to predict the deformation of the micro-structures. In the second part of the thesis, we investigate two colloidal assembly systems. First (Chapter 7-8), we utilize the new form of silica nanoparticles with chain-like morphology to generate sharp nanostructures on the coating surface that minimize the contact between liquid and solid phase, and thus improve dramatically the water repellency on the coating surfaces. The stability test of the superhydrophobicity against hydrodynamic/hydrostatic pressure, low surface tension liquid, and vapor phase condensation, are also investigated for a complete interpretation of the wetting behavior. Secondly (Chapter 9), I design colloidal suspensions matching the inter-particle interactions with those used in theoretical study of colloidal assembly within the confined the space. The beauty of the system is that the colloidal suspension can be cross-linked and lock the assembled structures, so that the assembled structure can be observed under electron microscope and compare to theory and simulation. So far, a good consistence has been observed, indicating a validated design of the systems

Active elastohydrodynamics of vesicles in narrow, blind constrictions

Fluid-resistance limited transport of vesicles through narrow constrictions is a recurring theme in many biological and engineering applications. Inspired by the motor-driven movement of soft membrane-bound vesicles into closed neuronal dendritic spines, here we study this problem using a combination of passive three-dimensional simulations and a simplified semi-analytical theory for active transport of vesicles that are forced through such constrictions by molecular motors. We show that the motion of these objects is characterized by two dimensionless quantities related to the geometry and the strength of forcing relative to the vesicle elasticity. We use numerical simulations to characterize the transit time for a vesicle forced by fluid pressure through a constriction in a channel, and find that relative to an open channel, transport into a blind end leads to the formation of an effective lubrication layer that strongly impedes motion. When the fluid pressure forcing is complemented by forces due to molecular motors that are responsible for vesicle trafficking into dendritic spines, we find that the competition between motor forcing and fluid drag results in multistable dynamics reminiscent of the real system. Our study highlights the role of non-local hydrodynamic effects in determining the kinetics of vesicular transport in constricted geometries