Microscopic theory of orientational order, structure and thermodynamics in strained polymer liquids and networks

A microscopic integral equation theory of the segmental orientational order parameter, structural correlations and thermodynamics of strained polymer solutions, melts and networks has been developed. The nonclassical problem of the consequences of intermolecular excluded volume repulsions and chain connectivity is addressed. The theory makes several novel predictions, including effective power law dependences of the orientational order parameter on monomer concentration and chain degree of polymerization, and strain hardening of the bulk modulus. The predictions of a nearly classical strain dependence, and supralinear scaling with segment concentration, of the strain-induced nematic order parameter is in agreement with nuclear magnetic resonance experiments. The absolute magnitudes of the a priori calculated orientational order parameter agree with simulations and experiments to within a factor of 2. The possible complicating influence of "trapped entanglements" in crosslinked networks is discussed. Extensions of the theory are possible to treat the mechanical response of flexible polymer liquids and rubbers, and the structure, thermodynamics, and mechanical properties of strained liquid crystal forming polymers.     

Thermodynamics, orientational order and elasticity of strained liquid crystalline melts and elastomers

A microscopic polymer liquid-state theory has been developed for the structure, thermodynamics and mechanical properties of strained liquid crystalline elastomers. The theory captures the experimentally observed phenomenon of spontaneous distortion and establishes a direct correlation between it and the nematic order parameter. Strain induced softening of the elastic modulus is predicted to emerge due to coupling of the induced orientational order and anisotropic interchain excluded volume interactions. Comparison of our results with limited experiments shows good qualitative and sometimes quantitative agreement. The theory predicts that deformation in the liquid crystalline state results in an increase of the amplitude of density fluctuations (compressibility) which becomes more pronounced as chain degree of polymerization and/or segmental density are decreased.     

Experimental study of molecular entanglement in polymer networks

Elastomeric molecular networks have been prepared by endlinking polydimethyl siloxane molecules having functional chain ends, both in the presence of an unreactive polymeric diluent and in the undiluted state. Values of tensile (Young) modulus were found to be in good agreement with the simple molecular theory of rubberlike elasticity for networks prepared in a highly diluted state. For concentrated systems the modulus was anomalously high, however. The discrepancy can be attributed to chain entanglements. A second interpenetrating network was introduced into networks formed in the diluted state by replacing the diluent polymer by reactive polymer, which was then gelled in situ. The modulus of these combined networks was much higher than the sum of the moduli of the constituent networks, implying a large contribution from molecular entanglements. © 1994 John Wiley & Sons, Inc.     

Mechanistic model for deformation of polymer nanocomposite melts under large amplitude shear

We report the mechanical response of a model nanocomposite system of poly(styrene) (PS)-silica to large-amplitude oscillatory shear deformations. Nonlinear behavior of PS nanocomposites is discussed with the changes in particle dispersion upon deformation to provide a complete physical picture of their mechanical properties. The elastic stresses for the particle and polymer are resolved by decomposing the total stress into its purely elastic and viscous components for composites at different strain levels within a cycle of deformation. We propose a mechanistic model which captures the deformation of particles and polymer networks at small and large strains, respectively. We show, for the first time, that chain stretching in a polymer nanocomposite obtained in large amplitude oscillatory deformation is in good agreement with the nonlinear chain deformation theory of polymeric networks. © 2013 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013     

Linear rheology of water-soluble reversible neodymium(III) coordination polymers

The rheology of reversible coordination polymer networks in aqueous solution is studied. The polymers are formed by neodymium(III) ions and bifunctional ligands, consisting of two pyridine-2,6-dicarboxylate groups connected at the 4-positions by an ethylene oxide spacer. Neodymium(III) ions can bind three of these terdendate ligand groups. At high concentrations, the polymer networks yield viscoelastic materials, which can be described with the Maxwell model. The scaling of the elastic modulus, relaxation time, and zero-shear viscosity with concentration are in good agreement with the predictions of Cates' model that describes the dynamics of linear equilibrium polymers. This indicates that the networks have only few cross-links and can be described as linear equilibrium polymers. The gels are also thermo-reversible. At high temperatures, fast relaxation was found, resulting in liquidlike behavior. Upon cooling, the viscoelastic properties returned immediately. From the temperature dependence of the relaxation time, an activation energy of 49 kJ/mol was determined for the breaking and reptation of the polymers.     

A perturbed hard-sphere-chain equation of state for nematic liquid crystals and their mixtures with polymers

A perturbed hard-sphere-chain (PHSC) equation of state is presented to compute nematicisotropic equilibria for thermotropic liquid crystals, including mixtures. The equation of state consists of an isotropic term and an anisotropic term given by the Maier-Saupe theory whose contribution disappears in the isotropic phase. The isotropic contribution is the recently presented PHSC equation of state for normal fluids and polymers which uses a reference equation of state for athermal hard-sphere chains and a perturbation theory for the squarewell fluid of variable well width. The PHSC equation of state gives excellent correlations of pure-component pressure-volume-temperature data in the isotropic region and, combined with the Maier-Saupe theory, correlates the dependence of nematic-isotropic transition temperature on the pressure. Theory also predicts a nematic-isotropic biphasic region and liquid-liquid phase separation in a temperature-composition diagram of binary mixtures containing a nematic liquid crystal and a normal fluid or polymer. Theory and experiment show good agreement for pure fluids as well as for mixtures.     

Temperature dependence of relative modulus in filled polymer systems

The temperature dependence of relative modulus observed in filled thermoset, thermoplastic, and polyelectrolyte salt matrices is explained on the basis of induced stresses produced by the differences in the thermal expansion coefficients of the constituent materials. The analysis is based on the assumption that the modulus of the matrix in a filled polymer is less than that of the unfilled polymer. The temperature dependence of relative modulus is expressed as a function of the difference in thermal expansion coefficients, the volume fraction, the relative modulus in the unstressed state, and mechanical properties of the phases. Agreement is good between the analysis and experimental results for three systems: epoxy and glass, polyethylene and wollastonite, and a polyelectrolyte salt with mica and asbestos.     

Mesoscopic dynamics of inhomogeneous polymers based on variable cell shape dynamic self-consistent field theory

In this paper, we combine variable cell shape method with dynamic self-consistent field theory and extend to study structure and dynamics under shear for triblock copolymer melts. Due to shear, the calculation cell shape is variable and no longer orthogonal. Pseudospectral method is employed to solve the diffusion equation for chain propagator on the nonorthogonal coordinate and the shear periodical condition can be easily designed in terms of the variable cell shape method. By using this strategy, the shear induced morphology evolution is investigated for topologically complex polymeric systems such as linear and star triblock copolymers; the morphology of linear ABC triblock copolymers is more shear sensitive than that of star triblocks. In particular, once the chain propagator is obtained, the microscopic elastic stress and spatial stress distribution can be derived and thus the dynamic mechanical property can be calculated under shear. By imitating the dynamic storage modulus G' corresponding to any given morphology in the oscillatory shear measurements, we explore the relationship between the morphology and the storage modulus G' and extend to study the mechanism of phase separation dynamics as well as order-disorder transition (ODT) for linear and star triblock copolymers. The results show that the chain architecture can be easily distinguished by investigating the ODT, though the systems such as AB symmetric diblock and ABA triblock copolymers by coupling AB precursors almost exhibit similar microstructures. In addition, the storage modulus G' and loss modulus G" can be simultaneously determined in frequency sweeps of oscillatory shear measurements and the dependence of the moduli on phase separated patterns and the chain topology is investigated. The simulation findings are in qualitatively agreement with the experimental results.     

Elastic properties of crosslinked poly(vinyl alcohol) gels: Network topology

Current network theory exhibits inconsistencies which show up particularly clearly in deformation of networks prepared by crosslinking a polymer in solution. A check of theory can be obtained if one knows precisely the number of crosslinks in the network and if a range of deformations is applied to the network. In an effort to explore this problem we have examined the relation of shear modulus to crosslink density, primary molecular weight, and polymer concentration for a series of poly(vinyl alcohol) gels at low to intermediate concentrations. Aqueous poly(vinyl alcohol) solutions were crosslinked to form infinite networks using terephthalaldehyde. We find a large discrepancy with these poly(vinyl alcohol) gels between measured shear modulus and that calculated from classical elasticity theory assuming quantitative reaction of crosslinking. The ratio of measured to calculated modulus is independent of crosslink density for a given primary molecular weight and concentration. It shows linear dependence on polymer concentration prior to crosslinking and extrapolates to a critical concentration which is consistent with the effective sizes of the polymer molecules.     

Significance of cross correlations in the stress relaxation of polymer melts

According to linear response theory, all relaxation functions in the linear regime can be obtained using time correlation functions calculated under equilibrium. In this paper, we demonstrate that the cross correlations make a significant contribution to the partial stress relaxation functions in polymer melts. We present two illustrations in the context of polymer rheology using (1) Brownian dynamics simulations of a single chain model for entangled polymers, the slip-spring model, and (2) molecular dynamics simulations of a multichain model. Using the single chain model, we analyze the contribution of the confining potential to the stress relaxation and the plateau modulus. Although the idea is illustrated with a particular model, it applies to any single chain model that uses a potential to confine the motion of the chains. This leads us to question some of the assumptions behind the tube theory, especially the meaning of the entanglement molecular weight obtained from the plateau modulus. To shed some light on this issue, we study the contribution of the nonbonded excluded-volume interactions to the stress relaxation using the multichain model. The proportionality of the bonded/nonbonded contributions to the total stress relaxation (after a density dependent "colloidal" relaxation time) provides some insight into the success of the tube theory in spite of using questionable assumptions. The proportionality indicates that the shape of the relaxation spectrum can indeed be reproduced using the tube theory and the problem is reduced to that of finding the correct prefactor.     

Theory of nematic networks with finite extensibility

A statistical molecular model of polymer networks, which takes into account inter-molecular orientational interaction and constancy of the chain contour length, is proposed. A simple formula for the network free energy valid for any deformations is obtained. A system of equations which describes orientational-elastic interactions in polymer networks is derived. The influence of deformation and degree of cross linking on the conditions of network phase transition into the orientationally ordered state of the nematic type is studied. The deformation curves considerably deviate from those predicted by the classical elasticity theory. A comparison of the proposed theory with experimental data for natural rubbers is carried out.     

Molecular Dynamic Simulation of Side-Chain Liquid Crystalline Elastomer Under Load

Summary: We present a molecular dynamic simulation of a side chain liquid crystalline elastomer (LCE) under load. The LCE is composed of a flexible tetrafunctional diamond like network with rod-like mesogens attached to the network. As a precursor of the LC elastomer a flexible polymer network in a low molecular liquid-crystal (LC) solvent was used. The phase behavior of the LCE under uniaxial stretching up to the deformations of λ = 1.5 and 2.0 at different densities was studied. As in the non-stretched case upon density increase an isotropic to nematic phase transition occurs. However, in contrast to thermotropic side chain LC elastomers the stress induced shift transition is not observed. The stretching slightly increases the anisotropy of translational diffusion of mesogens in the nematic state. The stress-strain dependence for LCE both in the isotropic and the nematic states is obtained. Elastic modulus increases at high values of order parameter.     

Contribution of entanglements to the modulus of end-linked poly(ethylene oxide) networks

Poly(ethylene oxide) networks were made at 75 °C by end-linking of short linear chains with an excess of Tolonate HDT, a tri-functional aliphatic isocyanate. Molecular weights of poly(ethylene oxide)were 4000, 6000 and 12000. The observed elastic moduli were much higher than predicted from affine theory, indicating a large contribution from entanglements in contrast to the findings of previous studies. With increasing excess of cross-linker, the contribution from entanglements levels off at a value of approximately 5 MPa which is in good agreement with the rubber plateau modulus of pure poly(ethylene oxide).     

Liquid crystalline triblock copolymers. Mechanical behaviour and orientation under uniaxial strain

The stress‐strain and orientation behaviour of side‐chain liquid crystalline(SLCP) ABA triblock copolymers with a backbone of polystyrene‐block‐1,2‐polybutadiene‐block‐polystyrene and a cyanobiphenyl mesogen in the side chain was investigated in dependence of molecular weight. The polymer shows the behaviour of a thermoplastic liquid crystalline elastomer(TPLCE) in the nematic phase in a region between the glass transitions of the polystyrene block and the SLCP. The ultimate properties and E‐modulus is lower than for conventional thermoplastic elastomers. Under uniaxial strain liquid crystalline order perpendicular to the direction of strain is induced.     

Theory of glassy dynamics in conformationally anisotropic polymer systems

A mode coupling theory for the ideal glass transition temperature, or crossover temperature to highly activated dynamics in the deeply supercooled regime, T(c), has been developed for anisotropic polymer liquids. A generalization of a simplified mode coupling approach at the coarse-grained segment level is employed which utilizes structural and thermodynamic information from the anisotropic polymer reference interaction site model theory. Conformational alignment or/and coil deformation modifies equilibrium properties and constraining interchain forces thereby inducing anisotropic segmental dynamics. For liquid-crystalline polymers a small suppression of T(c) with increasing nematic or discotic orientational order is predicted. The underlying mechanism is reduction of the degree of coil interpenetration and intermolecular repulsive contacts due to segmental alignment. For rubber networks chain deformation results in an enhanced bulk modulus and a modest elevation of T(c) is predicted. The theory can also be qualitatively applied to systems that undergo nonuniversal local deformation and alignment, such as polymer thin films and grafted brush layers, and large elevations or depressions of T(c) are possible. Extension to treat directionally dependent collective barrier formation and activated hopping is possible.     

Second-harmonic generation in an optically poled azo-dye/polymer film

The power dependence of the optical poling process of a new azo chromophore in the polymethylmethacrylate matrix at room temperature has been investigated. The existing theory is found to be inconsistent with the experiment. A simple model based on the rate equation is proposed to describe the writing process of the chromophore/polymer system. The measured growth rate and the plateau second-harmonic generation intensity as induced by all optical poling are found to be in good agreement with the prediction of the simple model.     

Virtual isotropic‐nematic transitions in alkyl cyanobiphenyls

Pretransitional fluctuations in the isotropic phase of liquid crystalline and non‐liquid crystalline alkyl cyanobiphenyls have been investigated using light scattering and magnetic birefringence measurements. We find evidence for a virtual isotropic‐nematic phase transition in short‐chain alkyl cyanobiphenyls with no observable nematic phase. The measured temperature dependence of fluctuations is well‐described by mean‐field theory. Virtual phase transition temperatures extrapolated from separate light scattering and magnetic birefringence experiments are in good agreement. Landau–de Gennes model parameters for the compounds investigated are calculated from the experimental results.     

Molecular dynamics calculation to clarify the relationship between structure and mechanical properties of polymer crystals: the case of orthorhombic polyethylene

The molecular dynamics (MD) technique was used to calculate the temperature dependence of the structure, molecular motion, and mechanical property of the orthorhombic polyethylene (PE) crystal. The potential functional parameters reported by Karasawa et al. (J Phys Chem, 95 (1991) 2260) were refined further so that the vibrational frequencies of infrared and Raman bands, measured by us at ultra-low temperatures for the normal and fully deuterated PE, could be reproduced well. The flip-flop motion around the chain axis and the torsional motion of the skeletal chains were found to start above ca. 350 K and increase the amplitude of these motions progressively. Coupling these two types of chain motion resulted in a steep increase of the thermal vibration parameters or the mean-square-displacements of carbon and hydrogen atoms, corresponding well with the X-ray data. The lattice constants and the related linear thermal expansion coefficients were also found to be in good agreement with the observed data. The calculated Young's modulus along the chain axis decreased gradually with the increasing temperature: 330 GPa at 0 K to 280 GPa at room temperature. The latter was in good agreement with the value of 280–305 GPa evaluated from the Raman measurement of the longitudinal acoustic mode. Young's modulus was found to relate intimately with the chain contraction caused by the skeletal torsional motion. Only 0.3% contraction of the chain resulted in the reduction of the modulus by ca. 35%. A similar behavior was also seen in the trigonal polyoxymethylene and nylon 6 α forms.     

Microscopic theory of gelation and elasticity in polymer-particle suspensions

A simplified mode-coupling theory (MCT) of ergodic-nonergodic transitions, in conjunction with an accurate two-component polymer reference interaction site model (PRISM) theory for equilibrium structural correlations, has been systematically applied to investigate gelation, localization, and elasticity of flexible polymer-hard particle suspensions. The particle volume fraction at the fluid-gel transition is predicted to depend exponentially on reduced polymer concentration and size asymmetry ratio at relatively high colloid concentrations. In contrast, at lower particle volume fractions, a power-law dependence on polymer concentration is found with effective exponents and prefactors that depend systematically on the polymer/particle size ratio. Remarkable power-law and near universal scaling behavior is found for the localization length and elastic shear modulus. Multiple experiments for gel boundaries and shear moduli are in good agreement with the no adjustable parameter theory. The one exception is the absolute magnitude of the shear modulus which is strongly overpredicted, apparently due to nonequilibrium dense cluster formation. The simplified MCT-PRISM theory also captures the qualitative aspects of the weak depletion-driven "glass melting" phenomenon at high particle volume fractions. Calculations based on an effective one-component model of structure within a low particle volume fraction framework yield qualitatively different features than the two-component approach and are apparently all in disagreement with experiments. This suggests that volume fraction and size asymmetry dependent many-body screening of polymer-mediated depletion attractions at finite particle concentrations are important.     

Light scattering measurements of anisotropic viscoelastic coefficients of a main-chain polymer nematic liquid crystal

We report the first systematic measurements of elastic coefficients and viscosities that clearly demonstrate the distinction between rigid and semiflexible behaviour of a main-chain polymer nematic liquid crystal, a solution of poly-γ-benzyl glutamate (PBG) in mixed organic solvents. Quasi-elastic Rayleigh scattering studies show a crossover which occurs at a molecular chain length near the persistence length of PBG as the chain length increases. The results are in qualitatively good agreement with recent theoretical predictions for semiflexible chains. It is seen that bending distortions of an individual polymer play an important role in the fundamental nature of nematic elasticity and viscosity.