Viscoelasticity at microscopic and macroscopic scales: characterization and prediction

In this dissertation, we build mathematical tools for applications to the transport properties of human lung mucus. The first subject is the microscopic diffusive transport of micron-scale particles in viscoelastic fluid. Inspired by the technique of passive microrheology, we model the motion of Brownian beads in general viscoelastic fluids by the generalized Langevin equation (GLE) with a memory kernel (the diffusive transport modulus). The GLE is a stochastic differential equation, which admits a discrete formulation as an autoregressive (AR) process. We further use exponential series for the memory kernel in the GLE, in which case the GLE has an explicit formulation as a vector Ornstein-Uhlenbeck process. In this framework, we can develop fast and accurate direct algorithm for pathogen transport in viscoelastic fluids, and the Kalman filter and maximum likelihood method give a new method for inversion of the memory kernel from experimental position time series. The framework is illustrated with multimode Rouse and Zimm chain models. In the second topic, we revisit the classical oscillatory shear wave model of Ferry et al., and extend the theory for active microrheology of small volume samples of viscoelastic fluids. In Ferry's original setup, oscillatory motion of the bottom plate generates uni-directional shear waves propagating in the viscoelastic fluid. Our colleague David Hill built a device to handle small volume viscoelastic samples. We extend the Ferry analysis to include finite depth and wave reflection off the top plate. We further consider nonlinear viscoelastic constitutive laws. The last problem considered is the numerical simulation of viscoelastic fluid flow, which will eventually be used to predict bulk transport of mucus layers. We start with the analysis of the system of model equations and demonstrate the difficulty of a robust numerical scheme. We develop an extension of projection method, which involves a new treatment of stress evolution based on stress splitting in the numerical scheme and show the advantage over previous work

Time-Domain Methods for Diffusive Transport in Soft Matter

Passive microrheology [12] utilizes measurements of noisy, entropic fluctuations (i.e., diffusive properties) of micron-scale spheres in soft matter to infer bulk frequency-dependent loss and storage moduli. Here, we are concerned exclusively with diffusion of Brownian particles in viscoelastic media, for which the Mason-Weitz theoretical-experimental protocol is ideal, and the more challenging inference of bulk viscoelastic moduli is decoupled. The diffusive theory begins with a generalized Langevin equation (GLE) with a memory drag law specified by a kernel [7, 16, 22, 23]. We start with a discrete formulation of the GLE as an autoregressive stochastic process governing microbead paths measured by particle tracking. For the inverse problem (recovery of the memory kernel from experimental data) we apply time series analysis (maximum likelihood estimators via the Kalman filter) directly to bead position data, an alternative to formulas based on mean-squared displacement statistics in frequency space. For direct modeling, we present statistically exact GLE algorithms for individual particle paths as well as statistical correlations for displacement and velocity. Our time-domain methods rest upon a generalization of well-known results for a single-mode exponential kernel [1, 7, 22, 23] to an arbitrary M-mode exponential series, for which the GLE is transformed to a vector Ornstein-Uhlenbeck process

Transient anomalous diffusion of tracer particles in soft matter

Synopsis This paper is motivated by experiments in which time series of tracer particles in viscoelastic liquids are recorded using advanced microscopy. The experiments seek to infer either viscoelastic properties of the sample Mason and Weitz, Phys. Rev. Lett. 74, 1250–1253 1995 or diffusive properties of the specific tracer particle in the host medium Suh et al. 2009. Our focus is the latter. Experimentalists often fit data to transient anomalous diffusion: a sub-diffusive power law scaling t , with 0 1 of mean-squared displacement MSD over a finite time interval, with longtime viscous scaling t 1 . The time scales of sub-diffusion and exponents are observed to vary with particle size, particle surface chemistry, and viscoelastic properties of the host material. Until now, explicit models for transient sub-diffusive MSD scaling behavior Doi and Edwards, The Theory of Polymer Physics Oxford University Press, New York, 1986; Kremer and Grest, J. Chem. Phys. 92, 5057–5086 1990; Rubinstein and Colby, Polymer Physics Oxford University Press, New York, 2003 are limited to precisely three exponents: monomer diffusion in Rouse chain melts t 1/2 , in Zimm chain solutions t 2/3 , and in reptating chains t 1/4 . In this paper, we present an explicit parametrized family of stochastic processes generalized Langevin equations with prescribed memory kernels and derive their closed-form solutions which 1 span the complete range of transient sub-diffusive behavior and 2 possess the flexibility to tune both the time window of sub-diffusive scaling and the power law exponent . These results establish a robust family of sub-diffusive models, for which the inverse problem of parameter inference from experimental data Fricks et al., SIAM J. Appl. Math.

Stability analysis of flow of active extensile fibers in confined domains

In this article, we study shear flow of active extensile filaments confined in a narrow channel. They behave as nematic liquid crystals that we assumed are governed by the Ericksen-Leslie equations of balance of linear and angular momentum. The addition of an activity source term in the Leslie stress captures the role of the biofuel prompting the dynamics. The dimensionless form of the governing system includes the Ericksen, activity, and Reynolds numbers together with the aspect ratio of the channel as the main driving parameters affecting the stability of the system. The active system that guides our analysis is composed of microtubules concentrated in bundles, hundreds of microns long, placed in a narrow channel domain, of aspect ratios in the range between 10(-2) and 10(-3) dimensionless units, which are able to align due to the combination of adenosine triphosphate-supplied energy and confinement effects. Specifically, this work aims at studying the role of confinement on the behavior of active matter. It is experimentally observed that, at an appropriately low activity and channel width, the active flow is laminar, with the linear velocity profile and the angle of alignment analogous to those in passive shear, developing defects and becoming chaotic, at a large activity and a channel aspect ratio. The present work addresses the laminar regime, where defect formation does not play a role. We perform a normal mode stability analysis of the base shear flow. A comprehensive description of the stability properties is obtained in terms of the driving parameters of the system. Our main finding, in addition to the geometry and magnitude of the flow profiles, and also consistent with the experimental observations, is that the transition to instability of the uniformly aligned shear flow occurs at a threshold value of the activity parameter, with the transition also being affected by the channel aspect ratio. The role of the parameters on the vorticity and angular profiles of the perturbing flow is also analyzed and found to agree with the experimentally observed transition to turbulent regimes. A spectral method based on Chebyshev polynomials is used to solve the generalized eigenvalue problems arising in the stability analysis

Extensions of the Ferry shear wave model for active linear and nonlinear microrheology

The classical oscillatory shear wave model of Ferry et al. [J. Polym. Sci. 2:593-611, (1947)] is extended for active linear and nonlinear microrheology. In the Ferry protocol, oscillation and attenuation lengths of the shear wave measured from strobe photographs determine storage and loss moduli at each frequency of plate oscillation. The microliter volumes typical in biology require modifications of experimental method and theory. Microbead tracking replaces strobe photographs. Reflection from the top boundary yields counterpropagating modes which are modeled here for linear and nonlinear viscoelastic constitutive laws. Furthermore, bulk imposed strain is easily controlled, and we explore the onset of normal stress generation and shear thinning using nonlinear viscoelastic models. For this paper, we present the theory, exact linear and nonlinear solutions where possible, and simulation tools more generally. We then illustrate errors in inverse characterization by application of the Ferry formulas, due to both suppression of wave reflection and nonlinearity, even if there were no experimental error. This shear wave method presents an active and nonlinear analog of the two-point microrheology of Crocker et al. [Phys. Rev. Lett. 85: 888 - 891 (2000)]. Nonlocal (spatially extended) deformations and stresses are propagated through a small volume sample, on wavelengths long relative to bead size. The setup is ideal for exploration of nonlinear threshold behavior

Effective Force Generation During Mammalian Cell Migration Under Different Molecular and Physical Mechanisms

We have developed much understanding of actin-driven cell migration and the forces that propel cell motility. However, fewer studies focused on estimating the effective forces generated by migrating cells. Since cells in vivo are exposed to complex physical environments with various barriers, understanding the forces generated by cells will provide insights into how cells manage to navigate challenging environments. In this work, we use theoretical models to discuss actin-driven and water-driven cell migration and the effect of cell shapes on force generation. The results show that the effective force generated by actin-driven cell migration is proportional to the rate of actin polymerization and the strength of focal adhesion; the energy source comes from the actin polymerization against the actin network pressure. The effective force generated by water-driven cell migration is proportional to the rate of active solute flux and the coefficient of external hydraulic resistance; the energy sources come from active solute pumping against the solute concentration gradient. The model further predicts that the actin network distribution is mechanosensitive and the presence of globular actin helps to establish a biphasic cell velocity in the strength of focal adhesion. The cell velocity and effective force generation also depend on the cell shape through the intracellular actin flow field

A Portable Waist-Loaded Soft Exosuit for Hip Flexion Assistance with Running

The soft exosuit is an emerging robotics, which has been proven to considerably reduce the metabolic consumption of human walking and running. However, compared to walking, relatively few soft exosuits have been studied for running. Many soft exosuits used for running are worn on the back and with a heavy weight load, which may cause instability while running and potentially increase metabolic consumption. Therefore, reducing the weight of the whole soft exosuit system as much as possible and keeping the soft exosuit close to the center of gravity, may improve running stability and further reduce metabolic consumption. In this paper, a portable waist-loaded soft exosuit, the weight of which is almost entirely concentrated at the waist, is shown to assist hip flexion during running, and justifies choosing to assist hip flexion while running. As indicated by the experiments of motion flexibility, wearing the waist-loaded soft exosuit can assist in performing many common and complex motions. The metabolic consumption experiments proved that the portable waist-loaded soft exosuit reduces the metabolic consumption rate of wearers when jogging on the treadmill at 6 km per hour by 7.79% compared with locomotion without the exosuit. Additionally, at the running speed of 8 km per hour, using the waist-loaded soft exosuit can reduce metabolic consumption rate by 4.74%. Similarly, at the running speed of 10 km per hour, it also can be reduced by 6.12%. It is demonstrated that assisting hip flexion for running is also a reasonable method, and wearing the waist-loaded soft exosuit can keep human motion flexibility and reduce metabolic consumption