Spatial Stress and Strain Distributions of Viscoelastic Layers in Oscillatory Shear.

One of the standard experimental probes of a viscoelastic material is to measure the response of a layer trapped between parallel surfaces, imposing either periodic stress or strain at one boundary and measuring the other. The relative phase between stress and strain yields solid-like and liquid-like properties, called the storage and loss moduli, respectively, which are then captured over a range of imposed frequencies. Rarely are the full spatial distributions of shear and normal stresses considered, primarily because they cannot be measured except at boundaries and the information was not deemed of particular interest in theoretical studies. Likewise, strain distributions throughout the layer were traditionally ignored except in a classical protocol of Ferry, Adler and Sawyer, based on snapshots of standing shear waves. Recent investigations of thin lung mucus layers exposed to oscillatory stress (breathing) and strain (coordinated cilia), however, suggest that the wide range of healthy conditions and environmental or disease assaults lead to conditions that are quite disparate from the "surface loading" and "gap loading" conditions that characterize classical rheometers. In this article, we extend our previous linear and nonlinear models of boundary stresses in controlled oscillatory strain to the entire layer. To illustrate non-intuitive heterogeneous responses, we characterize experimental conditions and material parameter ranges where the maximum stresses migrate into the channel interior.

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.

Metachronal wave formation in a model of pulmonary cilia.

A three-dimensional simulation of the formation of metachronal waves in rows of pulmonary cilia is presented. The cilia move in a two-layer fluid model. The fluid layer adjacent to the cilia bases is purely viscous while the tips of the cilia move through a viscoelastic fluid. An overlapping fixed-moving grid formulation is employed to capture the effect of the cilia on the surrounding fluid. In contrast with immersed boundary methods, this technique allows a natural enforcement of boundary conditions without the need for smoothing of singular force distributions. The fluid domains are discretized using a finite volume method. The 9 + 2 internal microtubule structure of an individual cilium is modeled using large-deflection, curved, finite-element beams. The microtubule skeleton is cross-linked to itself and to the cilium membrane through spring elements which model nexin links. The cilium membrane itself is considered to be elastic and subject to fluid stresses computed from the moving grid formulation as well as internal forces transmitted from the microtubule skeleton. A cilium is set into motion by the action of dynein molecules exerting forces between adjacent microtubules. Realistic models of the forces exerted by dynein molecules are extracted from measurements of observed cilia shapes.