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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.

In single-particle microrheology, the viscoelastic properties of a complex fluid can be extracted using the generalized Stokes–Einstein (GSE) relation by embedding a micrometer-sized particle and tracking its motion through the fluid. Applying the same analysis to molecular dynamics simulations can result in overestimated G∗ values because of the hydrodynamic interaction between the probe bead and its periodic images. We derive a simple correction to the GSE equation by implementing an analytical solution for Stokes drag on a periodic array of spheres, which allows smaller box sizes to be simulated while still retaining accuracy. Fluid and particle inertia are neglected, although the approach used here might be generalized to include them. The correction is applied to molecular dynamics simulations of a coarse-grained polymer melt. For several bead-to-box-size ratios R/L and two probe sizes, we measure the mean squared displacement and calculate G∗ using both the original GSE and the corrected hydrodynamic Stokes–Einstein (HSE) equations. These results are compared to small amplitude oscillatory non-equilibrium molecular dynamics (NEMD) simulations, which show that the HSE analysis correctly predicts the G∗ values at low frequencies but breaks down when either fluid inertia is important or the bead size is too small to “see” a continuum. For small R/L, the GSE and HSE results are similar, although a small correction is still required for the finite box size and computational cost is significantly larger. The HSE equation allows the use of smaller box sizes, reducing computational costs by more than an order of magnitude.

The discrete slip-link theory is a hierarchy of strongly connected models that have great success predicting the linear and nonlinear rheology of high-molecular-weight polymers. Three of the four parameters of the most-detailed model, which can be extracted from primitive-path analysis, give quantitative agreement with experimental data for all examined chemistries (polystyrene, polyisoprene, polybutadiene, and polyethylene). Here, we attempt to extract the remaining friction parameter from atomistic simulations. In particular, an available quantum chemistry-based force field for polyethylene oxide (PEO) was used to perform molecular-dynamics simulations of a 12 kDa melt. The Kuhn friction is obtained from the mean-squared displacement of the center-of-mass of the chains (MSD of COM) in the melt. The result is also corroborated using the relaxation modulus calculated through the Green–Kubo formula. Once the four parameters are determined for any chemistry, all parameters for all members of the slip-link hierarchy are determined. Then, using a coarser member of the hierarchy, the dynamic modulus of a 256 kDa PEO melt was predicted. The predictions are compared to experimental measurements performed at the same temperature. Unfortunately, the extracted friction is about 30% larger than the one observed in the experiment. However, two fundamentally different methods, one utilizing the MSD of COM and the other the relaxation modulus, gave consistent results for the extracted Kuhn friction. Therefore, the discrepancy presumably arises from insufficient accuracy in the force field. Nonetheless, the work demonstrates that theory predictions without adjustable parameters should be possible.

A simple mean-field microswimmer model is presented. The model is inspired by the nonequilibrium thermodynamics of multi-component fluids that undergo chemical reactions. These thermodynamics can be rigorously described in the context of the GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework. More specifically, this approach was recently applied to non-ideal polymer solutions [T. Indei and J. D. Schieber, J. Chem. Phys. 146, 184902 (2017)]. One of the species of the solution is an unreactive polymer chain represented by the bead-spring model. Using this detailed description as inspiration, we then make several simplifying assumptions to obtain a mean-field model for a Janus microswimmer. The swimmer model considered here consists of a polymer dumbbell in a sea of reactants. One of the beads of the dumbbell is allowed to act as a catalyst for a chemical reaction between the reactants. We show that the mean-squared displacement (MSD) of the center of mass of this Janus dumbbell exhibits ballistic behavior at time scales at which the concentration of the reactant is large. The time scales at which the ballistic behavior is observed in the MSD coincide with the time scales at which the cross-correlation between the swimmer's orientation and the direction of its displacement exhibits a maximum. Since the swimmer model was inspired by the GENERIC framework, it is possible to ensure that the entropy generation is always positive, and therefore, the second law of thermodynamics is obeyed.

Collagen is heavily hydroxylated. Experiments show that proline hydroxylation is important to triple helix (monomer) stability, fibril assembly, and interaction of fibrils with other molecules. Nevertheless, experiments also show that even without hydroxylation, type I collagen does assemble into its native D-banded fibrillar structure. This raises two questions. Firstly, even though hydroxylation removal marginally affects macroscopic structure, how does such an extensive chemical change, which is expected to substantially reduce hydrogen bonding capacity, affect local structure? Secondly, how does such a chemical perturbation, which is expected to substantially decrease electrostatic attraction between monomers, affect collagen’s mechanical properties? To address these issues, we conduct a benchmarked molecular dynamics study of rat type I fibrils in the presence and absence of hydroxylation. Our simulations reproduce the experimental observation that hydroxylation removal has a minimal effect on collagen’s D-band length. We also find that the gap-overlap ratio, monomer width and monomer length are minimally affected. Surprisingly, we find that de-hydroxylation also has a minor effect on the fibril’s Young’s modulus, and elastic stress build up is also accompanied by tightening of triple-helix windings. In terms of local structure, de-hydroxylation does result in a substantial drop (23%) in inter-monomer hydrogen bonding. However, at the same time, the local structures and inter-monomer hydrogen bonding networks of non-hydroxylated amino acids are also affected. It seems that it is this intrinsic plasticity in inter-monomer interactions that preclude fibrils from undergoing any large changes in macroscopic properties. Nevertheless, changes in local structure can be expected to directly impact collagen’s interaction with extra-cellular matrix proteins. In general, this study highlights a key challenge in tissue engineering and medicine related to mapping collagen chemistry to macroscopic properties but suggests a path forward to address it using molecular dynamics simulations.

Investigations on thermal transport in cross-linked elastomers subjected to elongational deformations are reviewed and discussed. The focus is on experimental research, in which the deformation-induced anisotropy of the thermal conductivity tensor in several common elastomeric materials is measured using novel optical techniques developed in our laboratory. These sensitive and noninvasive techniques allow for the reliable measurement of thermal conductivity (diffusivity) tensor components on samples in a deformed state. When combined with measurements of the stress in deformed samples, we are able to examine the validity of the stress–thermal rule, which predicts a linear relationship between the thermal conductivity and stress tensor in deformed polymeric materials. These results are used to shed light on possible underlying mechanisms for anisotropic thermal transport in elastomers. We also present results from a novel experimental technique that show evidence of a deformation dependence of the heat capacity, which implies that, in addition to the usual entropic contribution, there is an energetic contribution to the stress in deformed elastomers.

We evaluate the thermodynamic consistency of the anisotropic mobile slip-link model for entangled flexible polymers. The level of description is that of a single chain, whose interactions with other chains are coarse grained to discrete entanglements. The dynamics of the model consist of the motion of entanglements through space and of the chain through the entanglements, as well as the creation and destruction of entanglements, which are implemented in a mean-field way. Entanglements are modeled as discrete slip links, whose spatial positions are confined by quadratic potentials. The confinement potentials move with the macroscopic velocity field, hence the entanglements fluctuate around purely affine motion. We allow for anisotropy of these fluctuations, described by a set of shape tensors. By casting the model in the form of the general equation for the nonequilibrium reversible-irreversible coupling from nonequilibrium thermodynamics, we show that (i) since the confinement potentials contribute to the chain free energy, they must also contribute to the stress tensor, (ii) these stress contributions are of two kinds: one related to the virtual springs connecting the slip links to the centers of the confinement potentials and the other related to the shape tensors, and (iii) these two kinds of stress contributions cancel each other if the confinement potentials become anisotropic in flow, according to a lower-convected evolution of the confinement strength or, equivalently, an upper-convected evolution of the shape tensors of the entanglement spatial fluctuations. In previous publications, we have shown that this cancellation is necessary for the model to obey the stress-optical rule and the Green-Kubo relation, and simultaneously to agree with plateau modulus predictions of multichain models and simulations.

Background: Atrioesophageal fistula (AEF) is a severe complication of left atrial radiofrequency (RF) ablation for the treatment of atrial fibrillation. Proactive cooling with a novel cooling device has been shown to significantly reduce thermal injury, with no AEFs yet encountered after thousands of treatments worldwide. Computer modeling suggested that lethal temperatures could be avoided with moderate-power and moderate-duration RF ablation. We aimed to evaluate the effects of proactive cooling during high-power short-duration (HPSD) ablation, and compare the computed results to clinical data and recently published experimental data. Methods: A computer model accounting for the left atrium and esophagus including the active cooling device was created. We used both the Arrhenius equation and 50oC isotherm to estimate the esophageal thermal damage during 50 W, 10 second and 90 W, 4 second RF ablations. Results: Under control conditions, temperatures across the esophageal wall exceeded the lethal isotherm, in agreement with recent experimental data. With proactive esophageal cooling in place, temperatures in the esophageal tissue were significantly reduced, with the resulting fraction of esophageal damage reduced by 74% under 50 W and 10 seconds of ablation, and by 82% under 90 W and 4 seconds of ablation. Esophageal damage was restricted to the epi-esophageal region, sparing the remainder of the esophageal tissue, including the mucosal surface. Conclusions: Proactive esophageal cooling significantly reduced temperatures and the resulting fraction of damage in the esophagus during HPSD ablation. These findings offer a mechanistic rationale explaining the absence of AEF encountered to date using proactive esophageal cooling.

Engineered materials exhibit amazing and useful out-of-equilibrium properties. Some are soft but tough; others can harvest waste heat to produce electricity. Their properties often depend on how the materials are processed; during processing they can exhibit complex flow behavior unlike that of simple fluids. Classical descriptions like the Navier–Stokes equation or Hookean elasticity do not capture the mechanics of such materials. Instead, modeling emergent complex behavior requires simultaneous dynamical descriptions on both macroscopic and microscopic length scales. Such multiscale modeling relies on physical insight; the examples discussed here, which use minimal mathematics, show that the growing field is ripe for contributions from physicists, mathematicians, materials scientists, and engineers.

We present a more efficient way to predict the dynamic modulus of a broadly disperse, entangled polymer melt that uses the discrete slip-link model. Polydisperse blends provide a computational challenge because of the large number of molecular weights that contribute to the modulus. Here, we simulate the order 10 probes to capture the sliding dynamics of chains, but a continuous background to capture constraint dynamics. Contributions of other molecular weights can be interpolated from the simulated probes. Since we use the continuous background, additional probes can be added as needed without re-doing the prior simulations. We show that the method is capable of predicting the modulus well for two different chemistries. Comparison with a simplified double reptation model for a third chemistry shows agreement between the two approaches for frequencies near and below the crossover point of the dynamic modulus even though the distribution is rather narrow.We present a more efficient way to predict the dynamic modulus of a broadly disperse, entangled polymer melt that uses the discrete slip-link model. Polydisperse blends provide a computational challenge because of the large number of molecular weights that contribute to the modulus. Here, we simulate the order 10 probes to capture the sliding dynamics of chains, but a continuous background to capture constraint dynamics. Contributions of other molecular weights can be interpolated from the simulated probes. Since we use the continuous background, additional probes can be added as needed without re-doing the prior simulations. We show that the method is capable of predicting the modulus well for two different chemistries. Comparison with a simplified double reptation model for a third chemistry shows agreement between the two approaches for frequencies near and below the crossover point of the dynamic modulus even though the distribution is rather narrow.

The text considers the ideal gas and van der Waals equations of state in some detail. In addition to these, we summarize here several more-accurate equations of state. This appendix presents but a small fraction of such equations. A more comprehensive discussion of the quality of such equations is given in [118]. Note that all fluid equations of state reduce to the general ideal gas at low densities. For small deviations from ideal behavior, the virial expansion is the most reliable. However, it is not appropriate for predicting vapor–liquid equilibria. The cubic equations of state are straightforward to use and computationally simple. However, some sacrifice must be made for accuracy. Of these, the Peng–Robinson, Soave–Redlich–Kwong, and Schmidt–Wenzel equations are usually superior. However, all cubic equations are suspect near the critical region. For an excellent review of many such equations, see [5]. If computational ease should be sacrificed for accuracy, the Benedict–Webb–Rubin and the Anderko–Pitzer equations are usually more accurate. The parameters in the cubic PVT equations of state are usually determined from the critical properties of a fluid, and these equations are given. Critical values for a few substances are given in Table D.3 in Appendix D. More values can be found from the NIST web page. If the critical values of a substance are not known, they may be estimated from group methods on the basis of the chemical structure of the substance. These methods are reviewed in [102].

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.

We present a novel infrared thermography technique to investigate the dependence of heat capacity on deformation in cross-linked polymers. This phenomenon is directly related with the longstanding question of whether or not there is an energetic contribution to the stress in deformed polymers and, in general, to the nonisothermal, viscoelastic behavior of polymer materials. By tracking the temperature evolution of samples heated by a laser, we are able to measure heat capacity changes relative to the equilibrium value in an elastomer subjected to uniaxial extension. We find that the heat capacity increases with elongation in lightly cross-linked cis-1,4-polyisoprene. Remarkably, the onset of heat capacity dependence on deformation is observed at strains similar to those required to achieve finite extensibility. The deviation from the equilibrium value of heat capacity is consistent with an independent set of experiments comparing anisotropy in thermal diffusivity from forced Rayleigh scattering and therma...

We use straightforward energy and entropy balances to test the thermodynamic consistency of microstructural rheological models. The method utilizes the same mathematical methods as classical transport phenomena, so it is much simpler to use than the much more rigorous GENERIC formalism. The cost of this simplicity is that fewer restrictions are actually tested than those in either the single-generator or the two-generator formalisms. The proposed test does provide, however, a separation of energy and entropy, leading to an interesting internal energy balance. More importantly, it leads to two requirements for non-negative entropy production: one closely related to a virtual work argument, important during flow, and a second that guarantees adherence to the second law of thermodynamics during microstructural relaxation. These criteria do not appear to be in conflict with the requirements of the more rigorous formulations and are much simpler to implement. Several illustrative examples are given with models using the conformation tensor level of description. As expected, the models that use a relaxation function that is proportional to the free energy gradient are straightforward to check. These include the Hookean dumbbell, the FENE-P, and the Giesekus models, which are shown to satisfy the first and second laws. With a little more work, models with relaxation functions not driven by the free energy gradient only can also be checked for thermodynamic compliance with the proposed formalism. Examples of these are the GLaMM and Rolie–Poly models; they violate the first and second laws of thermodynamics, respectively. A recently proposed FENE-mode model is also checked, which superficially satisfies both laws, but fails to have an analytic free energy. Finally, the application of the formalism to a nonlinear dumbbell model that uses the probability density for chain conformations is also illustrated. In that case, satisfaction of the fluctuation-dissipation theorem and a positive mobility guarantee compliance with the first and second laws.

We analyze the effects of polymer concentration, Cp, and physical crosslinking ratio, rc, on the linear viscoelasticity of lightly entangled, linear, associating polymers with multiple crosslinkable groups along the chain backbone. To accomplish this goal, we utilize three novel tools: a robust tunable chemistry based on electrophilic methacryl-succinimidyl modified poly(N-isopropylacrylamide); a modified state-of-the-art entanglement theory, called the discrete slip-link molecular model; and a novel experimental technique, surface fluctuation specular reflection. This experimental technique allows the extraction of the complex viscoelastic modulus covering up to six decades of frequency, from the sample's surface fluctuations at constant temperature. We demonstrate that our theoretical model with a consistent set of parameters can qualitatively reproduce the observed features of the complex viscoelastic modulus over the entire probed frequency range. Furthermore, the effects of Cp and rc on the viscoelas...

We compare predictions of two of the most advanced versions of the tube model, namely the “Hierarchical model” by Wang et al. [J. Rheol. 2010, 54, 223] and the BoB (branch-on-branch) model by Das et al. [J. Rheol. 2006, 50, 207], against linear viscoelastic G′ and G″ data of binary blends of nearly monodisperse 1,4-polybutadiene 4-arm star polymer of arm molar mass 24 000 g/mol with a monodisperse linear 1,4-polybutadiene of molar mass 58 000 g/mol. The star was carefully synthesized and characterized by temperature gradient interaction chromatography and by linear rheology over a wide frequency region through time–temperature superposition. We found large failures of both the Hierarchical and BoB models to predict the terminal relaxation behavior of the star/linear blends, despite their success in predicting the rheology of the pure star and pure linear polymers. This failure occurred regardless of the choices made concerning constraint release, such as assuming arm retraction in “fat” or “skinny” tubes....

Pelvic organ prolapse (POP) is characterized by weakening of the connective tissues and loss of support for the pelvic organs. Collagen is the predominant, load-bearing protein within pelvic floor connective tissues. In this study, we examined the nanoscopic structures and biomechanics of native collagen fibrils in surgical, vaginal wall connective tissues from healthy women and POP patients. Compared to controls, collagen fibrils in POP samples were bulkier, more uneven in width and stiffer with aberrant D-period. Additionally, the ratio of collagen I (COLI) and collagen III (COLIII) is doubled in POP with a concomitant reduction of the amount of total collagen. Thus, POP is characterized by abnormal biochemical composition and biophysical characteristics of collagen fibrils that form a loose and fragile fiber network accountable for the weak load-bearing capability. The study identifies nanoscale alterations in collagen as diagnostic markers that could enable pre-symptomatic or early diagnosis of POP. From the Clinical Editor Pelvic organ prolapse (POP) occurs due to abnormalities of the supporting connective tissues. The underlying alterations of collagen fibers in the connective tissues have not been studied extensively. In this article, the authors showed that collagen fibrils in POP patients were much different from normal controls. The findings may provide a framework for the diagnosis of other connective diseases.

Summary Accurately characterizing the forces acting on particles in fluids is of fundamental importance for understanding particle dynamics and binding kinetics. Conventional asymptotic solutions may lead to poor accuracy for neighboring particles. In this paper, we develop an accurate boundary integral method to calculate forces exerted on particles for a given velocity field. We focus our study on the fundamental two-bead oscillating problem in an axisymmetric frame. The idea is to exploit a correspondence principle between the unsteady Stokes and linear viscoelasticity in the Fourier domain such that a unifying boundary integral formulation can be established for the resulting Brinkman equation. In addition to the dimension reduction vested in a boundary integral method, our formulation only requires the evaluation of single-layer integrals, which can be carried out efficiently and accurately by a hybrid numerical integration scheme based on kernel decompositions. Comparison with known analytic solutions and existing asymptotic solutions confirms the uniform third-order accuracy in space of our numerical scheme. Copyright © 2016 John Wiley & Sons, Ltd.

We demonstrate the ability to simulate complex flows of entangled polymer melts using a high-fidelity slip-link model. Given the strong connections of the underlying molecular model to an atomistic description, nearly ab initio predictions of complex processing are feasible. Moreover, the method retains sufficient information which might allow extraction of detailed polymer chain conformations imposed by the flow. The macroscopic transport equations are solved using smoothed-particle hydrodynamics consistently with the stresses calculated using stochastic simulation of an ensemble of polymer chains in each particle. The polymer model uses only a single adjustable parameter whose value is determined from equilibrium stress relaxation. All other parameters are determined from atomistic simulation. Thereafter, nonlinear rheology predictions are made without any parameter adjustment. As a demonstration, we simulate journal-bearing flow of a moderately entangled polymer. Although the flows considered here are two dimensional, the required computational resources demonstrate that three dimensional flow calculations are accessible.

We investigate the ability of a coarse-grained slip-link model and a simple double reptation model to describe the linear rheology of polydisperse linear polymer melts. Our slip-link model is a well-defined mathematical object that can describe the equilibrium dynamics and non-linear rheology of flexible polymer melts with arbitrary polydispersity and architecture with a minimum of inputs: the molecular weight of a Kuhn step, the entanglement activity, and Kuhn step friction. However, this detailed model is computationally expensive, so we also examine predictions of the cheaper double reptation model, which is restricted to only linear rheology near the terminal zone. We report the storage and loss moduli for polydisperse polymer melts and compare with experimental measurements from small amplitude oscillatory shear. We examine three chemistries: polybutadiene, polypropylene and polyethylene. We also use a simple double reptation model to estimate parameters for the slip-link model and examine under which circumstances this simplified model works. The computational implementation of the slip-link model is accelerated with the help of graphics processing units, which allow us to simulate in parallel large ensembles made of up to 50,000 chains. We show that our simulation can predict the dynamic moduli for highly entangled polymer melts over nine decades of frequency. Although the double reptation model performs well only near the terminal zone, it does provide a convenient and inexpensive way to estimate the entanglement parameter for the slip-link model from polydisperse data.

Traditional application of thermodynamics to engineering problems involves processes that are flowing. For example, an engineer might design a refrigerator in which a refrigerant is pumped in a continuous cycle through a coil of tubing, or a generator where steam is pumped in a power cycle through several pieces of equipment. Hence, engineering has placed a lot of emphasis on balances in flowing systems. Such flow systems involve time as an independent variable. However, thermodynamics applies only to equilibrium states, and the introduction of time is strictly forbidden. The study of time-dependent processes actually falls within the domains of transport phenomena and non-equilibrium thermodynamics . Whereas the field of transport phenomena is relatively well advanced and well understood, non-equilibrium thermodynamics is a developing field of research, and the fundamental postulates are by no means agreed upon [11, 94]. We will restrict ourselves here to the simplest of such time-dependent systems. Namely, we will assume that our system is in a local state of equilibrium . Such an assumption allows us to use the quantities derived for equilibrium systems as local variables that depend upon position and time. This simplification is usually applicable whenever the local response time of a system is much smaller than the time scale of the whole process. In this way, we can simplify many engineering flow problems to equivalent equilibrium thermodynamics problems.

The word polymer comes from the Greek polymeres , meaning many units. It is used to designate molecules with large molecular weights. There is no exact cutoff to determine when a large molecule is actually polymeric, but chained molecules of molecular weight greater than a few thousand are typically considered to be polymers. Scientists and engineers are concerned both with naturally occurring and with synthetic polymers. Naturally occurring polymers are important in biotechnological and biomedical fields, for example as DNA, RNA, microtubules, or actin filament, and in other fields, such as in the form of cellulosics in high-strength fibers. Synthetic polymers play important roles in advanced materials such as plastics, liquid crystals, and fiber-reinforced composites. The manufacture of these materials, as well as numerous applications, can often require that the polymer be dissolved in solvents, for example in spin-coating processes. Alternatively, material scientists often wish to combine the desirable properties of two different polymers in a plastic alloy by blending different mixtures of the two. In this chapter, we consider the phase behavior of polymer solutions and the miscibility of polymer blends. The starting point for describing these properties is the Flory–Huggins equation for the Gibbs free energy of mixing. In Section 11.1 we compare the experimental data for the solubility of various polymer–solvent mixtures with the Flory–Huggins equation of state. We see that the theory is insufficient to describe all systems. Hence, in Section 11.2 we also consider generalizations to the Flory–Huggins approach.

A microscopic molecular model of polymeric molecules that captures the effects of topological constraints is used to consider how microphase segregation can alter the distribution of entanglements both in space and along chain contours. Such topological constraints are obtained by using the Z1 algorithm, and it is found that for diblock copolymers in the lamellar morphology they are not homogeneously distributed, but instead exhibit a spatial dependence as a consequence of the self-organization of the polymer blocks. The specific shape of the inhomogeneous distribution is affected by the molecular weight of the copolymer. The microscopic information obtained by these calculations is then compared with the corresponding results generated from a coarser description of entangled block copolymers that includes soft intermolecular interactions and slip-springs, whose role is to incorporate the effects of entanglements that are lost during coarse-graining. This comparison is helpful for improving coarse-grained...

Molecular dynamics simulations and experimental measurements were used to investigate the thermal and mechanical properties of cross-linked phenolic resins as a function of the degree of cross-linking, the chain motif (ortho–ortho versus ortho–para), and the chain length. The chain motif influenced the type (interchain or intrachain) as well as the amount of hydrogen bonding. Ortho–ortho chains favored internal hydrogen bonding whereas ortho–para favored hydrogen bonding between chains. Un-cross-linked ortho–para systems formed percolating 3D networks of hydrogen bonds, behaving effectively as “hydrogen gels”. This resulted in differing thermal and mechanical properties for these systems. As cross-linking increased, the chain motif, chain length, and hydrogen bonding networks became less important. Elastic moduli, thermal conductivity, and glass transition temperatures were characterized as a function of cross-linking and temperature. Both our own experimental data and literature values were used to valid...

A signature feature of collagen is its axial periodicity visible in TEM as alternating dark and light bands. In mature, type I collagen, this repeating unit, D, is 67 nm long. This periodicity reflects an underlying packing of constituent triple-helix polypeptide monomers wherein the dark bands represent gaps between axially adjacent monomers. This organization is visible distinctly in the microfibrillar model of collagen obtained from fiber diffraction. However, to date, no atomistic simulations of this diffraction model under zero-stress conditions have reported a preservation of this structural feature. Such a demonstration is important as it provides the baseline to infer response functions of physiological stimuli. In contrast, simulations predict a considerable shrinkage of the D-band (11-19%). Here we evaluate systemically the effect of several factors on D-band shrinkage. Using force fields employed in previous studies we find that irrespective of the temperature/pressure coupling algorithms, assumed salt concentration or hydration level, and whether or not the monomers are cross-linked, the D-band shrinks considerably. This shrinkage is associated with the bending and widening of individual monomers, but employing a force field whose backbone dihedral energy landscape matches more closely with our computed CCSD(T) values produces a small D-band shrinkage of < 3%. Since this force field also performs better against other experimental data, it appears that the large shrinkage observed in earlier simulations is a force-field artifact. The residual shrinkage could be due to the absence of certain atomic-level details, such as glycosylation sites, for which we do not yet have suitable data.

Despite strong industrial interest, only recently have quantitative predictions for entangled polymer rheology become possible. Major advances are achieved with an alternative approach to entanglement modeling—slip-links. For example, the discrete slip-link model (DSM) allows for equilibrium and high-flow-rate predictions with a strong connection to a molecular basis. Slip-link models are applicable to arbitrary composition and chain architectures, but usually incur high numerical cost. Recent advances in coarse-graining allowed for a speedup of order one hundred, but further improvements are required for the ultimate goal—simulations of nonhomogeneous flows. On the other hand, it is possible to rely on advanced computational devices to obtain additional speed-up. GPUs are one of the most rapidly developing kind of computational devices, and provide immense computational power, but also have some critical restrictions on numerical operations. Since slip-link models have a unique level of description, our ...

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.

Instability and structural transitions arise in many important problems involving dynamics at molecular length scales. Buckling of an elastic rod under a compressive load offers a useful general picture of such a transition. However, the existing theoretical description of buckling is applicable in the load response of macroscopic structures, only when fluctuations can be neglected, whereas membranes, polymer brushes, filaments, and macromolecular chains undergo considerable Brownian fluctuations. We analyze here the buckling of a fluctuating semiflexible polymer experiencing a compressive load. Previous works rely on approximations to the polymer statistics, resulting in a range of predictions for the buckling transition that disagree on whether fluctuations elevate or depress the critical buckling force. In contrast, our theory exploits exact results for the statistical behavior of the worm-like chain model yielding unambiguous predictions about the buckling conditions and nature of the buckling transition. We find that a fluctuating polymer under compressive load requires a larger force to buckle than an elastic rod in the absence of fluctuations. The nature of the buckling transition exhibits a marked change from being distinctly second order in the absence of fluctuations to being a more gradual, compliant transition in the presence of fluctuations. We analyze the thermodynamic contributions throughout the buckling transition to demonstrate that the chain entropy favors the extended state over the buckled state, providing a thermodynamic justification of the elevated buckling force.

The idea that the dynamics of concentrated, high–molecular weight polymers are largely governed by entanglements is now widely accepted and typically understood through the tube model. Here we review alternative approaches, slip-link models, that share some similarities to and offer some advantages over tube models. Although slip links were proposed at the same time as tubes, only recently have detailed, quantitative mathematical models arisen based on this picture. In this review, we focus on these models, with most discussion limited to mathematically well-defined objects that conform to state-of-the-art beyond-equilibrium thermodynamics. These models are connected to each other through successive coarse graining, using nonequilibrium thermodynamics along the way, and with a minimal parameter set. In particular, the most detailed level of description has four parameters, three of which can be determined directly from atomistic simulations. Once the remaining parameter is determined for any system, all parameters for all members of the hierarchy are determined. We show how, using this hierarchy of slip-link models combined with atomistic simulations, we can make predictions about the nonlinear rheology of monodisperse homopolymer melts, polydisperse melts, or blends of different architectures. Mathematical details are given elsewhere, so these are limited here, and physical ideas are emphasized. We conclude with an outlook on remaining challenges that might be tackled successfully using this approach, including complex flow fields and polymer blends.

The discrete slip-link model (DSM) was developed to describe the dynamics of flexible entangled polymer melts. With just three molecular-weight- and chain-architecture-independent parameters—the molecular weight of a Kuhn step MK; entanglement activity β; and Kuhn step shuffling characteristic time τK—DSM is able to predict simultaneously the linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. Without any adjustment, DSM shows excellent agreement with shear flow experiments and elongational flows with stretch up to ∼10–20. Universality observed between entangled melts with the notable exception of high-strain elongations suggests that the average number of entanglements per chain is the primary characteristic of the system. Therefore, theoretical predictions for systems with differing numbers of Kuhn steps per chain but roughly the same number of entanglements should be equivalent when rescaled. In this work, we present a scaling of the DSM parameters, which has no si...

The mechanical properties of a cross-linked polydimethylsiloxane (PDMS) network swollen with nonreactive entangled PDMS solvent was previously studied experimentally. In this article, we use the discrete slip-link model to predict its linear and nonlinear rheology. Model parameters are obtained from the dynamic modulus data of pure solvent. Network rheology predictions also require an estimate of the fraction and architecture of dangling or inactive strands in the network, which is not directly measurable. The active strand fraction is estimated from dynamic modulus measurements, and the molecular weight is adjusted to fit the dynamic modulus data. Then, the nonlinear rheology can be predicted without adjustments. These successful predictions strongly suggest that the observed rheological modification in the swollen blend arises from the constraint dynamics between the network chains and the dangling ends.

We consider four criteria of acceptability for single-chain mean-field entangled polymer models: consistency with a multi-chain level of description, consistency with nonequilibrium thermodynamics, consistency with the stress-optic rule, and self-consistency between Green–Kubo predictions and linear viscoelastic predictions for infinitesimally driven systems. Each of these topics has been considered independently elsewhere. However, we are aware of no molecular entanglement model that satisfies all four criteria simultaneously. Here we show that an idea from Ronca and Allegra, generalized to arbitrary flows, can be implemented in a slip-link model to create a model that does satisfy all four criteria. Aside from the direct benefits of agreement, the result modifies the relation between the initial relaxation modulus G(0) and the entanglement molecular weight Me. If this implementation is correct, current estimates for Me would require modification that brings their values more in line with estimates based on topological analysis of molecular dynamics simulations.

The discrete slip-link model (DSM) was developed to describe the dynamics of flexible polymer melts. The model is able to predict linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. The model also shows good agreement with dielectric relaxation experiments, except for the single data set available for bidisperse linear systems with a small volume fraction of long chains. In this work, both shear and elongational flow predictions obtained using the DSM without parameter adjustment are shown. Model predictions for shear flow agree very well with experimental results. The DSM is able to capture the transient response as well as the steady-state viscosity. However, for elongational flow, agreement is unsatisfactory at large strains. The DSM captures the onset of strain hardening, but after a Hencky strain between 2 and 3, it predicts transient strain softening, whereas experiments show only monotonic growth. We explore a number of assumptions and approximations of the model and their effect on flow predictions. The approximations are related to the neglect of these phenomena, which are expected to be more sensitive in elongational flow: finite extensibility, convective constraint release, and deformation of dangling ends. We indeed find that shear flow predictions are insensitive to these approximations, but elongational flow is affected. However, none of these effects is able to bring prediction in line with experiments. We conclude that the currently accepted view of entanglement dynamics is incomplete.

Anisotropic thermal conduction was measured in two amorphous polymers that were quenched immediately after being subjected to uniaxial elongation in the molten state. The quenching is performed so that the flow-induced orientation is retained and the samples are essentially in a stress-free state. A novel optical technique based on forced Rayleigh scattering is used to measure the two independent components of the thermal diffusivity tensor as a function of strain and strain rate. The thermal diffusivity is found to increase in the direction parallel, and decrease in the direction perpendicular, to the direction of elongation. Thermal diffusivity data along with measurements of the tensile stress at the point of quenching were used to evaluate the stress-thermal rule, which is analogous to the well-known stress-optic rule. The stress-thermal rule was found to be valid for both polymers over a range of strains and strain rates. Since the quenched samples have orientation only, it appears that the primary source of anisotropy in thermal conductivity is the anisotropy of polymer chain orientation.

Type I collagen is the predominant collagen in mature tendons and ligaments, where it gives them their load-bearing mechanical properties. Fibrils of type I collagen are formed by the packing of polypeptide triple helices. Higher-order structures like fibril bundles and fibers are assembled from fibrils in the presence of other collagenous molecules and noncollagenous molecules. Curiously, however, experiments show that fibrils/fibril bundles are less resistant to axial stress compared to their constituent triple helices-the Young's moduli of fibrils/fibril bundles are an order-of-magnitude smaller than the Young's moduli of triple helices. Given the sensitivity of the Young's moduli of triple helices to solvation environment, a plausible explanation is that the packing of triple helices into fibrils perhaps reduces the Young's modulus of an individual triple helix, which results in fibrils having smaller Young's moduli. We find, however, from molecular dynamics and accelerated conformational sampling simulations that the Young's modulus of the buried core of the fibril is of the same order as that of a triple helix in aqueous phase. These simulations, therefore, suggest that the lower Young's moduli of fibrils/fibril bundles cannot be attributed to the specific packing of triple helices in the fibril core. It is not the fibril core that yields initially to axial stress. Rather, it must be the portion of the fibril exposed to the solvent and/or the fibril-fibril interface that bears the initial strain. Overall, this work provides estimates of Young's moduli and persistence lengths at two levels of collagen's structural assembly, which are necessary to quantitatively investigate the response of various biological factors on collagen mechanics, including congenital mutations, posttranslational modifications and ligand binding, and also engineer new collagen-based materials.

Particle rheology is used to extract the linear viscoelastic properties of an entangled polymer melt from molecular dynamics simulations. The motion of a stiff, approximately spherical particle is tracked in both passive and active modes. We demonstrate that the dynamic modulus of the melt can be extracted under certain limitations using this technique. As shown before for unentangled chains [Karim et al., Phys. Rev. E 86, 051501 (2012)PLEEE81539-375510.1103/PhysRevE.86.051501], the frequency range of applicability is substantially expanded when both particle and medium inertia are properly accounted for by using our inertial version of the generalized Stokes-Einstein relation (IGSER). The system used here introduces an entanglement length d_{T}, in addition to those length scales already relevant: monomer bead size d, probe size R, polymer radius of gyration R_{g}, simulation box size L, shear wave penetration length Δ, and wave period Λ. Previously, we demonstrated a number of restrictions necessary to obtain the relevant fluid properties: continuum approximation breaks down when d≳Λ; medium inertia is important and IGSER is required when R≳Λ; and the probe should not experience hydrodynamic interaction with its periodic images, L≳Δ. These restrictions are also observed here. A simple scaling argument for entangled polymers shows that the simulation box size must scale with polymer molecular weight as M_{w}^{3}. Continuum analysis requires the existence of an added mass to the probe particle from the entrained medium but was not observed in the earlier work for unentangled chains. We confirm here that this added mass is necessary only when the thickness L_{S} of the shell around the particle that contains the added mass, L_{S}>d. We also demonstrate that the IGSER can be used to predict particle displacement over a given timescale from knowledge of medium viscoelasticity; such ability will be of interest for designing nanoparticle-based drug delivery.

A theoretically informed entangled polymer simulation approach is presented for description of the linear and non-linear rheology of entangled polymer melts. The approach relies on a many-chain representation and introduces the topological effects that arise from the non-crossability of molecules through effective fluctuating interactions, mediated by slip-springs, between neighboring pairs of macromolecules. The total number of slip-springs is not preserved but, instead, it is controlled through a chemical potential that determines the average molecular weight between entanglements. The behavior of the model is discussed in the context of a recent theory for description of homogeneous materials, and its relevance is established by comparing its predictions to experimental linear and non-linear rheology data for a series of well-characterized linear polyisoprene melts. The results are shown to be in quantitative agreement with experiment and suggest that the proposed formalism may also be used to describe the dynamics of inhomogeneous systems, such as composites and copolymers. Importantly, the fundamental connection made here between our many-chain model and the well-established, thermodynamically consistent single-chain mean-field models provides a path to systematic coarse-graining for prediction of polymer rheology in structurally homogeneous and heterogeneous materials.

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.

Anisotropic thermal transport in a crosslinked poly- isoprene (natural rubber) subjected to uniaxial elongation is investigated experimentally. Using a novel optical technique based on forced Rayleigh scattering, two components of the ther- mal diffusivity tensor are measured as a function of stretch ratio. The thermal diffusivity is found to increase in the direction par- allel, and decrease in the direction perpendicular, to the stretch direction. The level of anisotropy for the natural rubber is substan- tially lower than that reported by Tautz 50 years ago but compa- rable to that found in our previous studies on molten polymers, quenched thermoplastics, and other crosslinked elastomers. Thermal diffusivity data along with measurements of the tensile stress were used to evaluate the stress-thermal rule, which was found to be valid over the entire range of stretch ratios. In con- trast, failure of the stress-optic rule was observed at stretch ratios well below the largest value at which the stress-thermal rule was valid. This suggests that the degree of anisotropy in thermal con- ductivity depends on both orientation and stretch of polymer chain segments. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 00: 000-000, 2012

Dielectric spectroscopy is often used as a tool complementary to rheology for investigation of relaxation processes of a viscoelastic medium. In particular, dielectric relaxation of type-A polymers reveals the autocorrelation function of the end-to-end vector of a chain, whereas the relaxation modulus is the autocorrelation function for stress, which depends on more-local structural moments of the chain conformation. Here we examine published data for monodisperse linear, bidisperse linear and star-branched polyisoprene using the discrete slip-link model (DSM). Although the DSM makes predictions very similar to tube theory for linear viscoelasticity, there are some noticeable differences in the predicted contributions of different processes. Also, the DSM uses a single mathematical object for blends and varying chain architectures, meaning that the two adjustable parameters should be independent of molecular weight, blending, architecture or flow field. Here we use one set of parameters (aside from the te...

Objectives Mild therapeutic hypothermia has been shown to improve outcomes after adult cardiac arrest but remains underused. Development of easier methods than currently exist to induce therapeutic hypothermia may help increase use of this treatment. We developed a mathematical model to evaluate the potential to induce mild therapeutic hypothermia through the esophagus. Methods Using a finite element mathematical modeling software package incorporating Pennes Bioheat equation, we predicted the changes in body temperature that would occur with placement of an esophageal cooling device containing recirculating chilled water at 10°C. Results Patient temperature under the simulated conditions decreased from 37°C to 33°C in approximately 40 minutes. Distribution of body temperature was not uniform in our model, with the skin surface and extremities showing a greater temperature decrease than in the patient's core. Conclusions Our computer simulations suggest that inducing mild therapeutic hypothermia via an esophageal route is feasible.

SynopsisViscoelastic properties of condensed soft matter can be estimated by following the trajectory of an embedded micron-sized particle in a method called passive microbead rheology. Data analysis of passive microbead rheology is usually based on formulas that relate bead displacement statistics to the dynamic modulus of the material in the frequency-domain. Therefore, methods of analysis require conversion of the data to the frequency-domain using numerical Fourier transform routines. These methods are known to introduce errors associated with frequency discretization and finite window size. Time-domain data analysis methods based on a single bead trajectory were introduced by Fricks et al. [SIAM J. Appl. Math. 69, 1277–1308 (2009)] as an alternative to the frequency-domain formulas. We have expanded these ideas with the aim of performing Monte Carlo simulations on synthetic data to evaluate and compare analysis algorithms for systems in which particles are trapped in linear or nonlinear traps. Brownian dynamics simulations were used to generate trajectories of beads embedded in viscoelastic materials having a discrete relaxation spectrum, with multiple relaxation times. We show that by including a small purely dissipative element in the memory function of the generalized Langevin equation (GLE), we can eliminate inertia-related fast variables directly from the GLE to find an inertia-less GLE, avoiding the singularity reported by McKinley et al. [J. Rheol. 53, 1487–1506 (2009)]. Using the inertia-less GLE, the computational cost of the simulations is reduced by nearly 5 orders of magnitude. We also show that, in real systems, this purely dissipative element can arise from fluid inertia, since for the bead the Basset force acts dissipative at high frequencies.SynopsisViscoelastic properties of condensed soft matter can be estimated by following the trajectory of an embedded micron-sized particle in a method called passive microbead rheology. Data analysis of passive microbead rheology is usually based on formulas that relate bead displacement statistics to the dynamic modulus of the material in the frequency-domain. Therefore, methods of analysis require conversion of the data to the frequency-domain using numerical Fourier transform routines. These methods are known to introduce errors associated with frequency discretization and finite window size. Time-domain data analysis methods based on a single bead trajectory were introduced by Fricks et al. [SIAM J. Appl. Math. 69, 1277–1308 (2009)] as an alternative to the frequency-domain formulas. We have expanded these ideas with the aim of performing Monte Carlo simulations on synthetic data to evaluate and compare analysis algorithms for systems in which particles are trapped in linear or nonlinear traps. Browni...

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.

Although the LVE predictions of monodisperse systems by the discrete slip-link model (DSM) are at least as good as those made by tube models, there are significant differences in contributions to r...