Non-Newtonian behavior of polydisperse polymer melts as a consequence of the evolution of their relaxation spectra

The non-Newtonian flow of polydisperse polymer melts is shown to be described by a model according to which an increase in the shear rate leads to the suppression of the dissipative losses of the relaxation modes of each fraction. The higher the shear rate, the greater the suppression. The relaxation spectrum of each monodisperse fraction is represented by the Rouse distribution, and only this form of spectrum leads to a ??spurt?? effect at the critical shear stress. Hence, the physical content of the model that relates the non-Newtonian behavior of polymer melts to their molecular-mass distributions consists in the fact that the relaxation modes responsible for energy dissipation are gradually truncated from the side of high relaxation times. The higher the M of a given fraction, the greater the contribution of this part of the spectrum to the total viscous losses. In this case, the truncation of the spectrum from the side of high relaxation times is equivalent to the gradual ??elimination?? of high-molecular-mass fractions of the polydisperse polymer from the contribution to dissipation. The shear-rate-dependent evolution of the relaxation spectrum of the medium is the structural mechanism that causes the non-Newtonian flow of polymer melts. The efficiency of the proposed model is shown through calculation of the flow curves for polymers with known molecular-mass distributions. The calculation results are in agreement with the experimental data. The theoretical ideas developed with the use of the ?? function to describe molecular-mass distributions have made it possible to solve the inverse problem, i.e., to establish a quantitative relationship between the shape of the flow curve and the molecular-mass distribution and, thus, to calculate the molecular-mass distributions according to the shearrate dependence of the apparent viscosity.     

Nonlinear viscoelasticity of entangled polymer chains. I. Response of the slip-link model to elongation strains

The theoretical stress relaxation of polymer melts is derived from a new solution of the response of Doi's slip-link model in the plateau zone of the rubberlike part of the spectrum. The results of this treatment, based on the application of the statistical theory of elasticity to a virtual transient network of monodisperse entangled chains, are applied to the calculation of stress-strain relations for uniaxial extension performed at constant strain rate and to that of stress relaxation after such an elongation.     

Structure,conformation and dynamics of polymer chains at solid melt interfaces

We have performed molecular dynamics, and lattice Monte Carlo simulations of polymeric melts in the vicinity of solid surfaces. The structural features of the solid-melt interface were very simple. The interfacial width was comparable to the segment size. Inside this narrow interface the segment density profile was oscillatory. The density oscillations were much less pronounced than those present at solid-atomic liquid interfaces. On a scale much larger than the segment size, chain conformations were found to be identical with those of ideal chains next to a reflective barrier. In particular, the number of surface-segment contacts scaled like the square root of the molecular weight. Extensive molecular dynamics simulations showed that chain desorption times increase with molecular weight but at a rate much slower than the longest relaxation time of Rouse chains. Therefore, sufficiently long chains desorbed almost freely from the surface despite the presence of attractive surface-segment interactions. A study of chain relaxation dynamics confirmed that the Rouse modes constitute an appropriate set of normal coordinates for chains in the melt interacting with a solid surface. The effect of the surface on mode relaxation was significant. All relaxation processes of chains located within a couple of radii of gyration from the surface were slowed down considerably. This effect, however was approximately the same for fast and slow modes and independent of molecular weight for sufficiently long chains.     

Mechanical Spectral Hole Burning in polymer solutions

Mechanical Spectral Hole Burning (MSHB) was previously developed to investigate dynamic heterogeneity for polymeric materials, which exhibit relatively weak dielectric responses. In our previous work, MSHB was applied to a densely entangled block copolymer and successfully distinguishes the heterogeneous from the homogeneous states. Here, a series of polystyrene (PS) solutions was chosen to further investigate the effect of different types of heterogeneity on mechanical spectral hole burning. The three types of heterogeneity of interest include the entanglement spacing, the entanglement density (or number of entanglements per chain), and chain end density. The heterogeneity was varied by changing either solution concentration or molecular weight of the PS. Different types of dynamics from close to the Rouse regime into the terminal region were also examined. Our results are consistent with a heterogeneous dynamics over the time scales from close to Rouse regime into the rubbery plateau regime and the rubbery plateau‐to‐terminal flow transition regime. Terminal relaxation dynamics, on the other hand, were found to be homogeneous for the PS/diethyl phthalate solutions investigated. The results also indicate the hole properties are dominated by the type of dynamics rather than the length scale of heterogeneity. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2047–2062, 2009     

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.     

Hierarchical Modeling of Entangled Polymers

Summary : We demonstrate that it is possible to link multi-chain molecular dynamics simulations with the tube model using a single chain slip-links model as a bridge. This hierarchical approach allows significant speed up of simulations, permitting us to span the time scales relevant for a comparison with the tube theory. Fitting the mean-square displacement of individual monomers in molecular dynamics simulations with the slip-spring model, we show that it is possible to predict the stress relaxation. Then, we analyze the stress relaxation from slip-spring simulations in the framework of the tube theory. In the absence of constraint release, we establish that the relaxation modulus can be decomposed as the sum of contributions from fast and longitudinal Rouse modes, and tube survival. Finally, we discuss some open questions regarding possible future directions that could be profitable in rendering the tube model quantitative, even for mildly entangled polymers.     

The temperature dependence of the equilibrium and instantaneous compliances of low molecular weight polystyrene melts

The equilibrium compliance of three low molecular weight polystyrene melts has been determined over the range 15–70°K above T_g from dynamic viscoelastic measurements using alternating shear. Results are favorably compared with the previous results of Plazek and O'Rourke obtained from creep measurements, but agreement with predictions based on the Rouse theory is obtained only in the case of the highest molecular weight sample.     

Short‐time stretch relaxation of entangled polymer solutions investigated using full Rouse model predictions

We conduct a systematical investigation into the short‐time stretch relaxation behavior (i.e., shorter than the Rouse time but sufficiently longer than the glassy time) of entangled polymer liquid in single‐step strain flows, on the basis of theory/data comparisons for a broad series of type‐A entangled polymer solutions. First, within existing normal‐mode formulations, the Rouse model predictions on a full‐chain stretch relaxation in single‐step strain flows are derived for a popular 1‐D model proposed within the Doi–Edwards tube model, as well as for the original 3‐D model for nonentangled systems. In addition, an existing formula for the aforementioned 1‐D model that, however, rested upon a consistent‐averaging or the so‐called uniform‐chain‐stretch approximation is simultaneously examined. Subsequently, the previously derived formulas on chain stretch relaxation are directly incorporated into a reliable mean‐field tube model that utilizes the linear relaxation spectrum and the Rouse time constant consistently determined from linear viscoelastic data. It is found that the predictions of the 1‐D model differ substantially from that of the original 3‐D model at short times. Theory/data comparisons further indicate that the 1‐D model without approximations seems able to describe fairly well the nonlinear relaxation data under investigation. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1199–1211, 2006     

Nonlinear stress relaxation of an entangled linear polystyrene in single step‐strain flow: A quantitative theoretical investigation

The nonlinear stress relaxation of a nearly monodisperse, moderately entangled polystyrene solution (i.e., roughly seven entanglements per chain at equilibrium) in single step‐strain flow is investigated quantitatively by a detailed comparison of an existing set of experimental data with a simulation based on the tube model. The proposed simulation enables the effects of primary nonlinear relaxation mechanisms other than chain retraction to be identified more clearly and investigated individually. Two peculiar nonlinear relaxation behaviors are observed in this experiment. One is concerned with an apparent enhancement in the stress relaxation at short times, and the other is responsible for a seeming slowdown of the stress relaxation at long times. These findings are discussed within the tube model, in view of the effects of convective constraint release, partial strand extension, and nonaffine deformation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1281–1293, 2003     

Interchain coupled chain dynamics of poly(ethylene oxide) in blends with poly(methyl methacrylate): coupling model analysis

Quasielastic neutron scattering and molecular dynamics simulation data from poly(ethylene oxide) (PEO)/poly(methyl methacrylate) (PMMA) blends found that for short times the self-dynamics of PEO chain follows the Rouse model, but at longer times past t(c) = 1-2 ns it becomes slower and departs from the Rouse model in dependences on time, momentum transfer, and temperature. To explain the anomalies, others had proposed the random Rouse model (RRM) in which each monomer has different mobility taken from a broad log-normal distribution. Despite the success of the RRM, Diddens et al. [Eur. Phys. Lett. 95, 56003 (2011)] extracted the distribution of friction coefficients from the MD simulations of a PEO/PMMA blend and found that the distribution is much narrower than expected from the RRM. We propose a simpler alternative explanation of the data by utilizing alone the observed crossover of PEO chain dynamics at t(c). The present problem is just a special case of a general property of relaxation in interacting systems, which is the crossover from independent relaxation to coupled many-body relaxation at some t(c) determined by the interaction potential and intermolecular coupling/constraints. The generality is brought out vividly by pointing out that the crossover also had been observed by neutron scattering from entangled chains relaxation in monodisperse homopolymers, and from the segmental α-relaxation of PEO in blends with PMMA. The properties of all the relaxation processes in connection with the crossover are similar, despite the length scales of the relaxation in these systems are widely different.     

Molecular tube models of branched polymer melts

We apply the tube model to a specific problem in polymer melt dynamics - the rheology of star polymers as an additive to a monodisperse linear matrix. We find that the tube dilation picture of constraint release may be applied to the relaxation of the star fraction. There are four qualitatively different cases depending on the relative concentrations and relaxation times of the two fractions. Terminal relaxation times and relaxation spectra are calculated for each case and compared with available experimental data. The existence of a modified Rouse relaxation such that G(t)∼t^−3/2 is argued.     

Effects of chain length distribution on viscoelastic properties of entangled linear polymers: Blending laws for binary blends

The reptation idea of de Gennes and the tube model theory of Doi and Edwards are extended to explain the terminal viscoelastic properties of binary blends in the highly entangled state of two linear monodisperse polymers with different molecular weights M₁ and M₂. A modified tube model is proposed that considers the significance of the constraint release by local tube renewal in accounting for the relaxation process of the higher molecular weight chain. Its relaxation by both reptation and the constraint release is remodeled as the disengagement by pure reptation of an equivalent primitive chain. From knowledge of the longest relaxation times of the blend components, the stress equation is formulated from which blending laws of viscoelastic properties for the binary blends are derived. To force better agreement between theory and experiment at the pure monodisperse limits of the blends, a crude treatment to include the effect of contour-length fluctuation in the equivalent-chain model is also discussed. Theoretical predictions of the zero-shear viscosity and steady-state shear compliance are shown to be in good agreement with literature data on undiluted polystyrenes and polybutadienes over a wide range of the blend composition and M₂/M₁ ratio. The asymptotic of the laws for blends with M₂/M₁ → 1 and 0 are comparable to those from the relaxation spectrum proposed by others earlier on the basis of the tube model.     

Viscoelasticity of nanobubble‐inflated ultrathin polymer films: Justification by the coupling model

The nanobubble inflation method is the only experimental technique that can measure the viscoelastic creep compliance of unsupported ultrathin films of polymers over the glass–rubber transition zone as well as the dependence of the glass transition temperature (T_g) on film thickness. Sizeable reduction of T_g was observed in polystyrene (PS) and bisphenol A polycarbonate by the shift of the creep compliance to shorter times. The dependence of T_g on film thickness is consistent with the published data of free‐standing PS ultrathin films. However, accompanying the shift of the compliance to shorter times, a decrease in the rubbery plateau compliance is observed. The decrease becomes more dramatic in thinner films and at lower temperatures. This anomalous viscoelastic behavior was also observed in poly(vinyl acetate) and poly (n‐butyl methacrylate), but with large variation in the change of either the T_g or the plateau compliance. By now, well established in bulk polymers is the presence of three different viscoelastic mechanisms in the glass–rubber transition zone, namely, the Rouse modes, the sub‐Rouse modes, and the segmental α‐relaxation. Based on the thermorheological complexity of the three mechanisms, the viscoelastic anomaly observed in ultrathin polymer films and its dependence on chemical structure are explained in the framework of the Coupling Model. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013     

Analysis of orientational relaxation in binary blends of long and short polystyrene chains by fourier transform infrared dichroism and small-angle neutron scattering

Measurements of the local orientational order and average chain anisotropy in non-uniform polystyrene are reported. Fourier-transform infrared dichroism spectroscopy has been used to determine the effects of short deuterated chains (Mw = 500 to 188 000) on the orientational relaxation of long entangled chains (Mw = 2 000 000) in bidisperse melts uniaxially deformed above the glass transition temperature. While the long-chain relaxation is found to be dependent on the short-chain concentration, the local orientational order of the latter is molecular weight dependent consistent with the classical relaxation theories. The FTIR experiments are also combined with small-angle neutron scattering measurements which probe the deuterated-chain anisotropy in the defomed melts. There is evidence, from the combination of the two techniques, that although the short chains possess a negligible local orientational order, there exists an important anisotopy in the short chain distribution in space.     

Normal stress and shear stress in a viscoelastic liquid under steady shear flow: Effect of molecular weight heterogeneity

Under steady shear flow, the normal stress and the shear stress in both dilute and concentrated solutions of monodisperse poly-α-methylstyrenes and their blends were measured. It was confirmed that the molecular theories of Rouse and Zimm extended to concentrated solutions can explain the relation between the zero-shear normal stress coefficient and the zero-shear steady-flow viscosity for both monodisperse and polydisperse systems. Shear-rate dependence of steady-flow viscosity can be understood fairly well by the molecular entanglement concept proposed by Graessley so long as the polymer is monodisperse or the amount of the higher molecular weight component is high. However, zero-shear viscosity of blended systems cannot be explained quantitatively by the theory of Graessley. The shear-rate dependence of steady-state compliance of blended systems was also observed, and it can well be explained by the theory of Tanaka, Yamamoto, and Takano which interpreted the shear rate-dependent steady-state compliance in terms of the relaxation time spectrum and its variation with shear rate.     

Entanglement and network formation in polystyrene: Viscoelastic behaviour from pulsed NMR

A series of NMR proton spin-spin relaxation measurements in three monodisperse polystyrene samples over a range of temperatures shows the conditions required for the existence of a dynamic network in polystyrene due to entanglements. The density of entangled units, N_e, i.e. their number per unit mass, can be determined by this NMR technique and is found to decrease with increasing temperature, in a manner which indicates its dependence on free volume, as in viscosity. The T₂₁ relaxation time for low molecular weight-polystyrene and T_2S for high molecular weight polystyrene increase with increasing temperature approximately as does (viscosity)^−0.5. The NMR data provide a quantitative value for the elastic modulus, and its dependence on temperature, both being in agreement with the experimental data previously obtained from viscoelasticity. The model can be extended to analyse creep and subsequent recovery. The transition in viscosity from a M¹ to a M^3,4 can also be deduced from the NMR results.     

Cooperative polymer dynamics under nanoscopic pore confinements probed by field-cycling NMR relaxometry

Reptational dynamics of bulk polymer chains on a time scale between the Rouse mode relaxation time and the so-called disengagement time is not compatible with the basic thermodynamic law of fluctuations of the number of segments in a given volume. On the other hand, experimental field-cycling NMR relaxometry data of perfluoropolyether melts confined in Vycor, a porous silica glass of nominal pore dimension of 4 nm, closely display the predicted signatures for the molecular weight and frequency dependences of the spin-lattice relaxation time in this particular limit, namely T1 proportional M-1/2nu1/2. It is shown that this contradiction is an apparent one. In this paper a formalism is developed suggesting cooperative chain dynamics under nanoscopic pore confinements. The result is a cooperative reptational displacement phenomenon reducing the root-mean-squared displacement rate correspondingly but showing the same characteristic dependences as the ordinary reptation model. The tube diameter effective for cooperative reptation is estimated on this basis for the sample system under consideration and is found to be of the same order of magnitude as the nominal pore diameter of Vycor.     

Segmental orientation and chain relaxation of polymers by fourier transform infrared dichroism

Fourier transform infrared dichroism has been used to investigate molecular orientation in polymeric materials. It is first applied to characterize network behavior in some elastomeric systems such as model networks of poly(dimethylsiloxane). The strain dependence of segmental orientation is analyzed through networks of known degree of cross-linking and experimental results are compared with calculation predictions based on the rotational isomeric state formalism. Infrared dichroism spectroscopy has also been used to analyze orientational relaxation in binary blends of long and short polystyrene chains. The effect of short deuterated chains (M_w = 3000 to 72000) on the orientational relaxation of long entangled chains (Mw = 2 000 000) is examined in the bidisperse melts uniaxially deformed above the glass transition temperature. While the long chain relaxation is found to be dependent on the short-chain concentration, the local orientational order of the latter is molecular weight dependent in agreement with the classical relaxation theories.     

Influence of molecular stiffness on the dynamic structure factor

Characteristic features of the influence of molecular stiffness on the dynamic structure factor of macromolecules are briefly outlined. The relaxation times characterizing the internal dynamics of the macro‐molecules exhibit a crossover from Rouse‐Zimm to bending modes with increasing mode number. As a consequence the dynamic structure factor is strongly influenced by the molecular stiffness. In particular, a stretched exponential relaxation of the dynamic structure factor at scattering vectors larger than the inverse persistence length is predicted and confirmed by a comparison with experimental data. Moreover, the influence of polydispersity is discussed.