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A complete propagation of error procedure for passive microrheology is illustrated using synthetic data from generalized Brownian dynamics. Moreover, measurement errors typical of bead tracking done with laser interferometry are employed. We use the blocking transformation method of Flyvbjerg and Petersen (J Chem Phys 91(1):461–466 1989) applicable to estimating statistical uncertainty in autocorrelations for any time series data, to account properly for the correlation in the bead position data. These contributions to uncertainty in correlations have previously been neglected when calculating the error in the mean-squared displacement of the probe bead (MSD). The uncertainty in the MSD can be underestimated by a factor of about 20 if the correlation in the bead position data is neglected. Using the generalized Stokes-Einstein relation, the uncertainty in the MSD is then propagated to the dynamic modulus. Uncertainties in the bead radius and the trap stiffness are also taken into account. A simple code used to aid in the calculations is provided.

The linear viscoelastic behavior of three well-entangled linear monodisperse polystyrene melts and their blends is investigated. The monodisperse melts are blended in a 1:1 weight ratio to obtain three polystyrene bidisperse blends for which the linear viscoelastic behavior is also measured. Special attention is paid to controlling sample size and solvent content, and checking for consistency in the high-frequency regime. We also attempt to estimate uncertainty quantitatively. The experimental results agree well with the discrete slip-link model, a robust mesoscopic theory that has been successful in predicting the rheology of flexible entangled polymer liquids and gels. Using recently developed analytic expressions for the relaxation modulus, predictions of the monodisperse samples are made. The parameters for the model are obtained from the low-frequency crossover of one experiment. Using this parameter set without adjustment, predictions over the fully accessible experimental frequency range are obtained for the monodisperse samples and their blends with very good agreement.

The discrete slip-link model (DSM) is a robust mesoscopic theory that has great success predicting the rheology of flexible entangled polymer liquids and gels. In the most coarse-grained version of the DSM, we exploit heavily the universality observed in the shape of the relaxation modulus of linear monodisperse melts. For this type of polymer, we present here analytic expressions for the relaxation modulus. The high-frequency dynamics which are typically coarse-grained out from the DSM are added back into these expressions by using a Rouse chain with fixed ends to represent the fast motion of Kuhn steps between entanglements. We find consistency in the friction used for both fast and slow modes. We test these expressions against experimental data for three chemistries and molecular weights with good agreement. Using these analytic expressions, the polymer density, the molecular weight of a Kuhn step, MK, and the low-frequency cross-over between the storage and loss moduli, \(G^{\prime }\) and \(G^{\prime \prime }\), it is now straightforward to estimate model parameter values and obtain predictions over the experimentally accessible frequency range without performing expensive numerical calculations.

We study the effects of fluctuations of cross-linking points, or junctions, on the dynamic mechanical and viscoelastic properties of polymer networks formed by multisticker associating polymers on the basis of a single-chain approach. Fluctuations of junctions are implemented by introducing virtual springs. We consider two possible cases for the treatment of virtual springs: the direct contribution from virtual springs is either neglected or included in the stress. We show that, if neglected, the fluctuation of junctions decreases (or softens) the dynamic modulus over a wide range of frequencies. This result agrees qualitatively with the result of several multichain models that predicts the decrease of the static or plateau modulus. We also show that the fluctuation accelerates the associative Rouse mode at low frequencies originating from the association/dissociation process of stickers. These results are apparently reasonable, but it is expected that there are some errors arising from thermodynamical inconsistency due to the neglect of virtual-spring contribution from the total stress against the virtual work principle and the second law of thermodynamics. On the other hand, if the direct contribution from the virtual springs is included in the stress, thermodynamics is satisfied but the plateau modulus does not change, contrary to the multichain prediction. The softening occurs only at the low-frequency regime. Thus, each method has both merits and demerits, and hence the treatment of junction fluctuations in the framework of single-chain approaches requires care and further investigation.

Linear viscoelastic (LVE) measurements as well as non-linear elongation measurements have been performed on stoichiometrically imbalanced polymeric networks to gain insight into the structural influence on the rheological response (Jensen et al., Rheol Acta 49(1):1–13, 2010). In particular, we seek knowledge about the effect of dangling ends and soluble structures. To interpret our recent experimental results, we exploit a molecular model that can predict LVE data and non-linear stress–strain data. The slip-link model has proven to be a robust tool for both LVE and non-linear stress–strain predictions for linear chains (Khaliullin and Schieber, Phys Rev Lett 100(18):188302–188304, 2008, Macromolecules 42(19):7504–7517, 2009; Schieber, J Chem Phys 118(11):5162–5166, 2003), and it is thus used to analyze the experimental results. Initially, we consider a stoichiometrically balanced network, i.e., all strands in the ensemble are attached to the network in both ends. Next we add dangling strands to the network representing the stoichiometric imbalance, or imperfections during curing. By considering monodisperse network strands without dangling ends, we find that the relative low-frequency plateau, $G_0/G_N^0$ , decreases linearly with the average number of entanglements. The decrease from $G_N^0$ to G 0 is a result of monomer fluctuations between entanglements, which is similar to “longitudinal modes” in tube theory. It is found that the slope of G′ is dependent on the fraction of network strands and the structural distribution of the network. The power-law behavior of G ″ is not yet captured quantitatively by the model, but our results suggest that it is a result of polydisperse dangling and soluble structures.

There are several models for the determination of molecular weight distributions (MWDs) of linear, entangled, polymer melts via rheometry. Typically, however, models require a priori knowledge of the critical molecular weight, the plateau modulus, and parameters relating relaxation time and molecular weight (e.g., k and α in τ=kMα). Also, in an effort to obtain the most general MWD or to describe certain polymer relaxation mechanisms, models often rely on the inversion of integral equations via regularization. Here, the inversion of integral equations is avoided by using a simple double-reptation model and assuming that the MWD can be described by an analytic function. Moreover, by taking advantage of dimensionless variables and explicit analytic relations, we have developed an unambiguous and virtually parameter-free methodology for the determination of MWDs via rheometry. Unimodal MWDs have been determined using only a priori knowledge of the exponent α and dynamic moduli data. In addition, the uncertainty in rheological MWD determinations has been quantified, and it is shown that the reliability of the predictions is greater for the high-molecular-weight portion of the distribution.

The pom-pom rheological constitutive equation for branched polymers proposed by McLeish and Larson is evaluated in step shear strain flows. Semianalytic expressions for the shear-stress relaxation modulus are derived for both the integral and approximate differential versions of the pom-pom model. Predictions from the thermodynamically motivated differential pompon model of Ottinger are also examined. Single-mode integral and differential pom-pom models are found to give qualitatively different predictions, the former displays time–strain factorability after the backbone stretch is relaxed, while the latter does not. We also find that the differential pompon model gives quantitatively similar predictions to the integral pom-pom model in step strain flows. Predictions from multimode integral and differential pom-pom models are compared with experimental data on a widely characterized, low-density polyethylene known as 1810H. The experiments strongly support time–strain factorability, while the multimode pom-pom model predictions show deviations from this behavior over the entire range of time that is experimentally accessible.

A reptation model for the primitive chain that does not assume independent alignment or consistent-averaging for the retraction process, or equilibrium relaxation for the reptation process is proposed and compared to the analytical expressions of Doi and Edwards in single-step, double-step strains and steady-state shear flow. The Doi and Edwards model with independent alignment approximation underpredicts the magnitude of the relaxation modulus by 25%, and consistently overpredicts the magnitude of the damping function; for steady shear flow, it predicts the correct shape for the steady-state viscosity and the first normal stress difference coefficient, although the magnitude is incorrect. The analytical expressions of Doi and Edwards without independent alignment approximation are excellent approximations to the damping function. In double-step strains, the expressions of Doi assuming consistent averaging, but no independent alignment, predict well the stress decay following the second strain. Linear response theory is found to be invalid for describing the stress relaxation following single-step strain for the models considered. Similar to the Doi and Edwards model, no overshoot for the first normal stress difference is observed for the simulation model. Unlike the Doi equation derived without the independent alignment approximation but restricted to double-step strains, the simulation model proposed here can be easily generalized to complex flow fields. No contour length fluctuation or constraint release is considered in this model, and chain retraction is assumed to be instantaneous.

The linear viscoelastic modulus G(t) predicted by the analytical formulations of Schieber (1993), Wedgewood (1993), Dasbach et al. (1992), and Booij and van Wiechen (1970) for the free-draining Hookean dumbbell with internal viscosity (IV) are compared with exact analytical results and exact numerical results obtained from Brownian dynamics simulations. All of these analytical formulations employ the Booij and van Wiechen expression for the IV force, thereby eliminating errors associated with linearization of the “deformational” velocity, however the theories differ in the approximations employed to solve configuration moment equations. Comparison with the exact G(t) results provides a means of testing these approximations. The approximate theories all correctly predict the singular part of G(t) at t = 0, providing correct predictions of η′∞, however deviations from the exact G(t) are seen in all cases for t > 0.