Session M31: Polymer Melts and Solutions

Uniaxial Extension of Entangled Polymer Melts close to T$_{\mathrm{g}}$

Transient (nonlinear) responses of entangled polymers to startup deformation indicate a transition from the initial elastic deformation to irreversible deformation (flow) [1-3]. This yielding behavior varies with the applied rate: at a higher rate the entanglement network can be strained to a higher degree before its breakdown. In this work, we subject entangled melts such as polystyrene to startup uniaxial extension to show how yielding takes place as a function of temperature. The objective is to explore whether there would be any mechanical signature of emergence of any secondary structure as the glass transition temperature T$_{\mathrm{g}}$ is approached from above. \\[4pt] [1] S. Q. Wang, S. Ravindranath, Y. Wang and P. Boukany, \textit{J. Chem. Phys}. \textbf{127}, 064903 (2007).\\[0pt] [2] Y. Y. Wang and S. Q. Wang, \textit{J. Rheol}. \textbf{53}, 1389 (2009).\\[0pt] [3] S. Q. Wang, S. Ravindranath and P. E. Boukany, \textit{Macromolecules} \textbf{44}, 183 (2011).   

Non-Gaussian chain stretching in simple shear of branched polystyrene solutions

Entangled polymers with long chain branching (LCB) exhibit a higher apparent viscosity than the zero-rate viscosity upon startup uniaxial extension whereas polymers either of linear chains or with LCB only show a lower transient viscosity than the zero-rate viscosity envelope. We report for the first time that simple shear of well-entangled polystyrene solutions with LCB produces a higher transient viscosity than the zero-shear envelope. In presence of sufficient LCB, non-Gaussian stretching can even show up in simple shear, which was previously observed only in uniaxial extension. Moreover, LCB resists against a structural breakdown of the entanglement network, postponing the stress overshoot to an unprecedented high shear strain of 30 units when the backbone of the PS would be nearly straightened without retraction and resulting elastic recovery as high as 20 strain units.   

Tube diameter of oriented polymer melts

The tube diameter is a key material parameter controlling the flow behavior of polymer melts. The Lin-Noolandi ansatz successfully accounts for the dependence of the tube diameter on polymer density, chain stiffness and diluent concentration. We extend the Lin-Noolandi ansatz to polymer melts under uniform tension. We find that the tube diameter $a$ decreases as $F^{-1/2}$ when the pulling force $F$ exceeds the thermal tension $k_BT/a$, and approaches a limiting value for typical flexible polymers near full extension of about half the unperturbed value. Our prediction is compatible with assumptions made in the GLaMM model [1] for polymer rheology. We have directly verified the predicted force-dependence of tube diameter by using isoconfigurational ensemble averaging [2] to measure the tube diameter in simulations of oriented polymer melts. In the simulations, the chains are oriented by pulling on the ends of the chains, and topologically equilibrated by allowing the chains to occasionally cross.\\[4pt] [1] R. Graham, A. Likhtman, T. McLeish, and S. Milner, {\textit J. Rheo.}, 47(2003):1171;\\[0pt] [2] W. Bisbee, J. Qin, and S. Milner, {\textit Macromolecules}, 44(2011):8972)   

An Intriguing Empirical Rule for Estimating the First Normal Stress Difference from Steady Shear Viscosity Data for Concentrated Polymer Solutions and Melts

The Cox-Merz rule and Laun's rule are two empirical relations that allow the estimation of steady shear viscosity and first normal stress difference, respectively, using small amplitude oscillatory shear measurements. The validity of the Cox-Merz rule and Laun's rule imply an agreement between the linear viscoelastic response measured in small amplitude oscillatory shear and the nonlinear response measured in steady shear flow measurements. We show that by using a lesser known relationship also proposed by Cox and Merz, in conjunction with Laun's rule, a relationship between the rate-dependent steady shear viscosity and the first normal stress difference can be deduced. The new empirical relation enables \textit{a priori} estimation of the first normal stress difference using only the steady shear viscosity vs shear rate data. Comparison of the estimated first normal stress difference with the measured values for six different polymer solutions and melts show that the empirical rule provides values that are in reasonable agreement with measurements over a wide range of shear rates; thus deepening the intriguing connection between linear and nonlinear viscoelastic response of entangled polymeric materials.   

Assumptions in Entanglement models and Their Effect on Non-Linear Rheology Predictions

While tube and slip-link theories are able to describe shear flow stresses qualitatively, and in some cases quantitatively, elongational flow prediction remains elusive. Both the GLaMM tube theory and primitive chain network simulations overpredict the magnitude of stress. As a result, several groups have suggested making the friction chain-conformation dependent, giving an enhancement to stress relaxation in elongational deformations when chains are highly oriented. Here we take a different tack, and examine the effect of typical assumptions and approximations made in these theories by use of the discrete slip-link model. Since the model exists on a relatively detailed level of description, it allows examination of assumptions without resorting to crude approximations. We find that while some of these approximations indeed fail in elongational flows at high strains, the theory is still unable to predict data. What's more, unlike other predictions, this model underpredicts the stress, and would therefore not be in agreement with the assumption of conformation-dependent friction as currently hypothesized.   

Microscopic Theory of Entangled Polymer Melt Dynamics: Flexible Chains as Primitive-Path Random Walks and Super Coarse-Grained Needles

We qualitatively extend our recent microscopic dynamical theory for the transverse confinement potential and diffusion of infinitely thin rigid rods (PRL, 107, 078102 (2011)) to construct a first-principles theory of topologically entangled melts of flexible polymer chains (PRL, 109, 168306 (2012)). Polymer coils are treated as ideal random walks of self-consistently determined primitive-path (PP) steps, and chain uncrossability is included exactly at the binary collision level. A strongly anharmonic (tube) confinement potential for a primitive path segment is derived and favorably compared with recent simulations. A fundamental basis is derived for the Lin-Noolandi conjecture that relates the tube diameter to the invariant packing length, along with the reptation scaling laws for the diffusion constant and terminal relaxation time, including numerical prefactors. The relationship of the PP-level theory to two simpler models, the melt as a disconnected fluid of primitive-path steps, and a super coarse-graining that replaces the entire chain by a needle corresponding to its end-to-end vector, is examined. Remarkable connections between the different levels of coarse graining are discovered.   

Entanglement elasticity in polymer chain melts: microscopic calculation of the rubbery plateau modulus via intermolecular correlations

Textbook models of stress relaxation in melts of entangled polymer chains are built on the assumption that intra-molecular or backbone stresses are the dominant contribution to the system's total stress. Numerous simulations over the last two decades have challenged this assumption, but calculating the intermolecular or non-bonded contribution to the stress has proven a daunting theoretical task. Building on our recent progress in microscopically constructing the transverse confinement field of entangled rods (PRL 107, 078102 (2011)) and ideal coils (PRL 109, 168306 (2012)), we explicitly separate stress correlations into intra- and inter-molecular terms, and calculate the contribution of intermolecular stress correlations in the ``plateau'' region of stress relaxation. We derive, with no adjustable parameters, the characteristic relation $G_e \sim k_BT/p^3$ (where $p$ is the packing length) with a prefactor that agrees within a factor of two with experiment and simulation. This theoretical advance has major implications for the effect of nonlinear deformation, confinement, and chain orientational ordering on entanglement elasticity.   

Influence of Reversibly Associating Side Group Bond Strength on Viscoelastic Properties of Polymer Melts

Reversible hydrogen-bonding between side-groups of linear polymers can sharply influence a material's dynamic mechanical behavior, giving rise to valuable shape memory and self-healing properties. Here, we investigate how bond-strength affects the bulk rheological behavior of functional poly(n-butyl acrylate) (PBA) melts. A series of random copolymers containing three different reversibly bonding groups (aminopyridine, carboxylic acid, and ureidopyrimidinone) were synthesized to systematically vary the side-group hydrogen bond strength ($\sim$26, 40, 70 kJ/mol). The materials' volumetric hydrogen-bond energy densities can be tuned by adjusting the side-group composition. By comparing the viscoelastic behavior of materials containing an equivalent bond energy density, with different bonding groups, the efficacy and cooperativity of reversible binding can be directly examined. Melt rheology results are interpreted using a state-of-ease model that assumes continuous mechanical equilibrium between applied stress and resistive stresses of entropic origin arising from a network of reversible bonds.   

Correction of Doi-Edwards' Green function in harmonic potential and its implication for stress-optical rule

We derive a corrected Green's function for a polymer chain trapped in a two-dimensional anisotropic harmonic potential with a fixed boundary condition. This Green's function is a modified version of what Doi and Edwards first derived to describe the polymer chain trapped in the tube-like domain of surrounding entangled polymers [J. Chem. Soc. Farad. Trans. II 74 (1978) 1802]. In contradiction to the results found by Ianniruberto and Marrucci using the incorrect Green's function [J. Non-Newtonian Fluid Mech. 79 (1998) 225], we find that the stress-optical rule is violated for any tube potential either circular or elliptic. The violation is due to the presence of the virtual springs to trap the chain in the tube rather than the anisotropy of the confinement potential.   

Explaining the absence of high-frequency relaxation modes of polymers in dilute solutions

Using multi-scale modeling, including Molecular Dynamics and Brownian dynamics (BD) simulations, we explain the long-mysterious absence of high frequency modes in the dynamics of isolated polymer chains in good solvents, reported years ago by Schrag and coworkers. The relaxation spectrum we obtain for a chain of 30 monomers at atomistic resolution is, remarkably, a single exponential while that of a chain of 100 monomers is fit by only two modes. This result is surprising in view of the many relaxation modes present in melts of such chains, but agrees perfectly with experimental observations (Peterson et al. J. Polym. Sci.: Part B 2001). We also performed BD simulations in which the explicit solvent molecules are replaced by a viscous continuum. Although the local dynamics is suppressed with the addition of bending, torsion, side groups and excluded volume interactions (as suggested in Jain and Larson, Macromolecules 2008), none of the BD simulations predict a single exponential relaxation for a short chain. Our results indicate that the chain dynamics at small length scales (down to a few Kuhn steps) is significantly different from the predictions of models based on a continuum solvent, and finally help explain the experimental results of Schrag and coworkers.   

Simultaneous determination of the interaction parameter and topological scaling features of polymers in dilute solutions

The RPA equation using the Debye polymer chain scattering function has been widely used to model polymer blends of linear chains in the melt where it is safe to assume a Gaussian conformation. When chains display more complicated topologies or when chains are in dilute solution Gaussian scaling no longer applies. In some cases the Zimm double extrapolation has been used to determine the second virial coefficient and the interaction parameter under the assumption that the deviation of chain scaling from a random walk is acceptable in the low qRg region such as when light scattering is used. If it is of interest to explicitly determine the nature of chain scaling, related to topology or solvent quality, as well as to quantify the thermodynamic interaction, such as in studies of cyclic and branched chains, networks, or polymers in good solvents, there is no analytically valid scattering model for data analysis. We propose the coupling of the unified scattering function with the RPA equation to analytically model these effects. Nevertheless, some issues remain to be resolved with star polymers in particular, such as scattering from highly branched high molecular weight symmetric stars in good solvents where it appears that the Daoud-Cotton model may be appropriate but a colloidal scattering model may be more appropriate.   

Anisotropic Thermal Conduction in Polymers and its Molecular Origins

The strong coupling of mechanical and thermal effects in polymer flows have a significant impact on both the processing and final properties of the material. Simple molecular arguments suggest that Fourier's law must be generalized to allow for anisotropic thermal conductivity in polymers subjected to deformation. In our laboratory we have developed a novel application of the optical technique known as Forced Rayleigh Scattering to obtain quantitative measurements of components of the thermal diffusivity (conductivity) tensor in polymers subjected to deformations. We report measurements of anisotropic thermal diffusivity and stress in molten, cross-linked and solid polymers subjected to several types of flows. The deformed samples have significant anisotropy in polymer chain orientation that results in significant anisotropy in thermal conductivity. Stress and thermal conductivity data support the validity of the stress-thermal rule, which is analogous to the well-known stress-optic rule. We also report measurements on solid polymers with isotropic polymer chain orientation that are under stress, which display rather unexpected behavior. These measurements are used to develop an understanding of the molecular origins of anisotropic thermal conduction in polymeric material   

Molecular modeling simulations in phase stability of polyethylene solutions at elevated pressures

Molecular dynamics (MD) simulations using the OPLS-AA force field are conducted to compute pressure, molecular weight dependence of Hildebrand's solubility parameters (SP) and density of hexane and high-density polyethylene (HDPE) at high pressures from 100 to 3000 bar. The electrostatic energy contribution to the cohesive energy and density leads to increases in the SP with pressure for molecular mechanical models (MMM) with and without electrostatic terms. The Flory-Huggins interaction parameter (IP) predicted from the pressure dependence of SPs and molar volumes decreases upon increasing pressure, indicating that miscibility improves by raising pressure. This is consistent with the solution polymerization process for producing PE, where pressure-induced phase separation (PIPS) is used to separate the polymer from solution. Exclusion of electrostatic potentials in the MMM results in larger IPs while the decreasing trend remains intact with and without electrostatic forces. There is a pressure limit beyond which the IP has less sensitivity to pressure indicating that PE miscibility is not further affected. It is shown that pressure increases the chemical potential factor of the phase stability condition, stabilizing the solution. These results contribute to the fundamental understanding of PIPS, an important demixing process poorly understood when compared to thermally-induced phase separation.   

Miscibility of Polymers in Supercritical Solvents

In this work we make use of our ability to correlate underlying thermodynamic behavior with trends in miscibility to study mixtures of polymers and supercritical carbon dioxide (scCO$_{2})$. scCO$_{2}$ has garnered significant interest as a ``green'' solvent, and supercritical solvent in general, for its highly accessible critical point. Experimental cloud point investigations have determined the miscibility for a range of polymers in scCO$_{2}$. We have used a simple equation of state (EOS) to study a series of poly-(acrylates) in scCO$_{2}$ solvent. Although polymer/scCO$_{2}$ mixtures have been modeled with some success in the past, the ability of an EOS to make accurate predictions has yet to be demonstrated. Our mixture modeling procedure yields parameters from pure component experimental data. Then, by pinning the mixed interaction parameter to the experimental critical temperature (T$_{\mathrm{c}})$ for one mixture from the series, we predict the T$_{\mathrm{c}}$ shifts for the remaining members. In addition to discussing miscibility we draw insight via the trends revealed from the parameterization of the pure component data, alone.   

The effect of topology on the conformations of cyclic polymers in melts

The bond fluctuation method is used to simulate both non-concatenated entangled and interpenetrating melts of cyclic polymers. We find that the swelling of interpenetrating rings upon dilution follows the same laws as for linear chains. The knotting probability of cyclic polymers decays exponentially as function of the number of blobs per chain. A power law dependence $f_{n}\sim\phi R^{2}\sim\phi^{0.77}N$ for the average number $f_{n}$ of linked rings per cyclic polymer at concentrations larger than the overlap volume fraction of rings $\phi^{*}$ is determined from the simulation data. The fraction of non-concatenated cyclic polymers displays an exponential decay $P_{OO}\sim\exp(-f_{n})$, which indicates $f_{n}$ to provide the entropic effort for not forming concatenated conformations. These results lead to four different regimes for the conformations of cyclic polymers in melts separated by critical lengths $N_{OO}$, $N_{C}$ and $N^{*}$ that describe the onset of concatenation, the cross-over between weak and strong compression, and the cross-over to an overlap dominated concatenation contribution. The four characteristic exponents describing ring size in these regimes are 1/2, 2/5, 3/8, and 4/9 as confirmed by simulation data for the first three regimes.