A self-consistent reptation theory that accounts for chain-tube interactions, segment connectivity, chain-length breathing, segmental stretch, and constraint release is proposed. Simulation results are compared semiquantitatively to experimental observations in single-step strain flows. Since stochastic simulation techniques are used, no approximations, such as independent alignment or consistent averaging are needed to obtain results. The simulation results show excellent agreement with experimental trends in shear and normal stress relaxations, including the second normal stress difference, well into the nonlinear regime. For most of these experiments, the original Doi and Edwards theory, which incorporates independent alignment or consistent averaging, is not satisfactory. In the following companion paper, we show how the model is capable of describing double-step-strain flows for all stress components. Subsequent papers show excellent agreement for the inception of steady shear flow, and steady shear flow.

A full-chain, temporary network model is proposed for nonlinear flows of linear, entangled polymeric liquids. The model is inspired by the success of a recent reptation model, but contains no beads or tubes. Instead, each chain uses a different (and smaller) set of dynamic variables: the location of each entanglement and the number of Kuhn steps in chain strands between entanglements. As before, the model requires only a single phenomenological parameter that is fit by linear viscoelasticity. The number of Kuhn steps varies stochastically from imbalances in chemical potential and Brownian forces. In the language of reptation, the model exhibits chain connectivity, chain-length fluctuations, chain stretching, and tube dilation. The current implementation in this framework does not include constraint release, although its addition is possible. The entanglements are assumed to move affinely. Because of the affinity assumption and lack of constraint release, the model should be expected to approximate well a linear chain in a matrix of fixed obstacles, and somewhat less accurately a polymer melt. Straightforward modifications to these assumptions allow us to consider chains of any architecture in concentrated solutions or melts. A simulation algorithm for the model is described in detail. Stress results are encouraging, since the model performs at least as well as a complete tube model without constraint release at much lower computational cost. Finally, we consider possible generalizations of the proposed model, to include additional physics, such as constraint release, branching, and nonaffine motion.

Collagen fibers are the main components of the extra cellular matrix and the primary contributors to the mechanical properties of tissues. Here we report a novel approach to measure the longitudinal component of the elastic moduli of biological fibers under conditions close to those found in vivo and apply it to type I collagen from rat tail tendon. This approach combines optical tweezers, atomic force microscopy, and exploits Euler-Bernoulli elasticity theory for data analysis. This approach also avoids drying for measurements or visualization, since samples are freshly extracted. Importantly, strains are kept below 0.5%, which appear consistent with the linear elastic regime. We find, surprisingly, that the longitudinal elastic modulus of type I collagen cannot be represented by a single quantity but rather is a distribution that is broader than the uncertainty of our experimental technique. The longitudinal component of the single-fiber elastic modulus is between 100 MPa and 360 MPa for samples extracted from different rats and/or different parts of a single tail. Variations are also observed in the fibril-bundle / fibril diameter with an average of 325±40 nm. Since bending forces depend on the diameter to the fourth power, this variation in diameter is important for estimating the range of elastic moduli. The remaining variations in the modulus may be due to differences in composition of the fibril-bundles, or the extent of the proteoglycans constituting fibril-bundles, or that some single fibrils may be of fibril-bundle size.

A method for incorporating reptation ideas in a temporary network model is used to predict the fluctuations of entanglement number and monomer density between entanglements in entangled polymeric liquids. The dynamic variables are chosen to be the number of Kuhn steps and the position of the entanglements. A chain of fixed number of Kuhn steps in a bath that fixes the chemical potential conjugate to the number of entanglements is considered. The static equilibrium statistics of such a model can be calculated analytically. Since the dynamics of the model may also be simulated, these analytic expressions may be used to check the algorithm. Also, the damping function is calculated analytically from these distributions, as well as normal stresses following a step shear.

The dynamic modulus $G^{*}$ of a viscoelastic medium is often measured by following the trajectory of a small bead subject to Brownian motion in a method called “passive microbead rheology.” This equivalence between the positional autocorrelation function of the tracer bead and $G^{*}$ is assumed via the generalized Stokes-Einstein relation (GSER). However, inertia of both bead and medium are neglected in the GSER so that the analysis based on the GSER is not valid at high frequency where inertia is important. In this paper we show how to treat both contributions to inertia properly in one-bead passive microrheological analysis. A Maxwell fluid is studied as the simplest example of a viscoelastic fluid to resolve some apparent paradoxes of eliminating inertia. In the original GSER, the mean-square displacement (MSD) of the tracer bead does not satisfy the correct initial condition. If bead inertia is considered, the proper initial condition is realized, thereby indicating an importance of including inertia, but the MSD oscillates at a time regime smaller than the relaxation time of the fluid. This behavior is rather different from the original result of the GSER and what is observed. What is more, the discrepancy from the GSER result becomes worse with decreasing bead mass, and there is an anomalous gap between the MSD derived by naïvely taking the zero-mass limit in the equation of motion and the MSD for finite bead mass as indicated by McKinley et al. [ J. Rheol. 53, 1487 (2009)]. In this paper we show what is necessary to take the zero-mass limit of the bead safely and correctly without causing either the inertial oscillation or the anomalous gap, while obtaining the proper initial condition. The presence of a very small purely viscous element can be used to eliminate bead inertia safely once included in the GSER. We also show that if the medium contains relaxation times outside the window where the single-mode Maxwell behavior is observed, the oscillation can be attenuated inside the window. This attenuation is realized even in the absence of a purely viscous element. Finally, fluid inertia also affects the bead autocorrelation through the Basset force and the fluid dragged around with the bead. We show that the Basset force plays the same role as the purely viscous element in high-frequency regime, and the oscillation of MSD is suppressed if fluid density and bead density are comparable.