Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. II. Nonlinear polymer viscoelasticity

The nonlinear viscoelastic behavior of the Fraenkel-chain model is studied with respect to the constitutive equation of the Rouse model. Distinctly different from the results of the Rouse model, the Fraenkel-chain model gives the following characteristic nonlinear behavior: (a) The two distinct dynamic modes in the relaxation modulus GS(t,lambda)--as observed in the linear region reported in Paper I [Y.-H. Lin and A. K. Das, J. Chem. Phys. 126, 074902 (2007), preceding paper]--or in the first normal-stress difference function GPsi1(t,lambda) are shown to have different strain dependences: strain hardening for the fast mode and strain softening for the slow mode. (b) The Lodge-Meissner relation GS(t,lambda)=GPsi1(t,lambda) holds over the whole time range, which has been shown both analytically and by simulation. (c) The second normal-stress difference is nonzero, being positive in the fast-mode region and negative in the slow-mode region. The comparisons between orientation and stress for all tensor components consistently confirm the strong correlation of the slow mode as well as its entropic nature with the segmental-orientation anisotropy as shown in the linear region studied in Paper I. A consequence of this correlation is the applicability of the stress-optical rule in the slow-mode region. This also leads to the expectation that the damping function h(lambda)=G(S)(t,lambda)/G(S)(t,lambda-->0) and the ratio between the first and second normal-stress differences, N2(t,lambda)/N1(t,lambda), are described by the orientation tensor which has the same form as that given by Doi and Edwards [J. Chem. Soc. Faraday Trans. 2 74, 1789 (1978); 74, 1802 (1978)] with independent-alignment approximation for an entangled system. The similarity between the slow mode of an entanglement-free Fraenkel-chain system and the terminal mode of an entangled polymer system as observed in the comparison of theory, simulation, and experiment suggests that the close correlation of the entropic nature of the mode with the orientation anisotropy--as of the Fraenkel segment or the primitive step in the Doi-Edwards theory--is a generally valid physical concept in polymer viscoelasticity.     

Why does the rouse model fairly describe the dynamic characteristics of polymer melts at molecular masses below critical mass?

Generalization of the Rouse model without any use of the postulates concerning the Gaussian distribution of the vector connecting the ends of segments is advanced. In the initial (in general, nonlinear) Langevin equations, self-averaging over continuous fragments of a macromolecule naturally defines a linear term for the tagged chain, and this term differs from the entropy term of the classical Rouse model only by the numerical coefficient. According to the inertia-free approximation, the initial decay rates of correlation functions for the normal modes are described by the Rouse model independently of the character of fluctuations of the vector connecting the ends of the Kuhn segment. This statement is valid for any moment if the initial Langevin equations are treated in terms of the approximation of dynamic self-consistency. Simulation of the Fraenkel chains by the method of Brownian dynamics shows that decay of autocorrelation functions of shortwave normal modes is fairly described by the linearized equations for a given model of a chain and that the Rouse equation can be used for the long-wave modes. The results of this study make it possible to explain a marked difference between the lengths of the Kuhn and Rouse segments that is estimated from static and dynamic experiments.     

Segmental versus chain dynamics of linear polymers

Segmental dynamics of relatively short linear polymers are discussed in terms of two distinct contributions, one related to the local segmental motion (alpha relaxation) and the other to polymer-specific effects that reflect Brownian dynamics of the polymer under chain connectivity constraints (Rouse relaxation modes). These two aspects of polymer dynamics are reflected, though differently, in relaxation spectra of different experimental techniques. Two contrasting cases of the (collective) dipolar response (dielectric techniques) versus the individual segmental response (e.g., NMR spin-lattice relaxation spectroscopy) are considered. The second-rank orientational correlation function of an elementary (Kuhn) segment, directly related to NMR observables, is derived in terms of Rouse normal modes. The effect of alpha dynamics is estimated under the assumption of a separation of time scales which, as it is argued, is a necessary precondition of the Rouse approach. The relative magnitude of the polymer-related dynamics is expressed through the number of elementary Rouse units in the chain and the number of Kuhn segments in a Rouse unit. The results are discussed in the context of recent literature.     

Slow motion of long-chain molecules in solution with internal correlations

The Rouse equations for the dissipative motion of long-chain molecules are reformulated in terms of coupled Langevin equations. While in the conventional Rouse model the random forces acting on each bead are supposed to be uncorrelated, the calculation given here admits of any correlation between the forces. By virtue of the second fluctuation dissipation theorem this corresponds to an added friction factor, called the “internal viscosity” in agreement with earlier work of Kuhn and Kuhn, Cerf, and of Peterlin. In the mode decomposition, this added friction factor happens to be an even function of the mode frequency. From this general model it is possible to compute formally the mean-square displacement function. It takes a simple form if one assumes that forces are correlated only on adjacent beads and that their correlation functions are δ functions. The results of the calculation are best displayed in the Fourier transform of the meansquare displacement, which is directly related to the neutron incoherent scattering cross section. It is seen that the presence of internal viscosity modifies the behavior of the chain for a time scale of the order of the relaxation time of the elementary segment.     

Motion associated with a single rouse segment versus the alpha relaxation. 2

The dynamics in polystyrene melt and concentrated solution as probed by depolarized photon-correlation spectroscopy has been shown to reflect the motion associated with a single Rouse segment. In the concentrated solution case (entanglement-free), the analysis using the frictional factor K (=zetab2/kTpi2m2) extracted from the viscosity data in terms of the Rouse theory and aided by the Monte Carlo simulation based on the Langevin equation of the Rouse model confirms the conclusion in a precise manner. In the melt case (entangled), the Rouse-segmental motion as observed by depolarized photon-correlation spectroscopy is compared with the alpha relaxation and the highest Rouse-Mooney normal mode extracted from analyzing the creep compliance Jt of sample A reported in the companion paper. Another well-justified way of defining the structural (alpha-) relaxation time is shown basically to be physically equivalent to the one used previously. On the basis of the analysis, an optimum choice tauS = 18tauG (tauG being the average glassy-relaxation time) is made, reflecting both the temperature dependence of tauG and the effect on the bulk mechanical property by the glassy-relaxation process. In terms of thus defined tauS, two traditional ways of defining the alpha-relaxation time are compared and evaluated. It is shown that as the temperature approaches the calorimetric Tg, two modes of temperaturedependence are followed by the dynamic quantities concerning this study: One includes the time constant of the highest Rouse-Mooney normal mode, tauv; the temperature dependence of the viscosity corrected for the changes in density and temperature, eta/rhoT; and the average correlation time obtained by depolarized photon-correlation spectroscopy, tauc. The other, being steeper, is followed by the alpha-relaxation time tauS derived from the glassy-relaxation process and the temperature dependence of the recoverable compliance Jr(t) as obtained by Plazek. The comparison of the dynamic quantities clearly differentiates the motion associated with a single Rouse segment as characterized by tauv or tauc from the alpha-relaxation as characterized by tauS; due to the lack of clear definition of these two types of motion in the past and the proximity of one to the other in the time scale-actually the two crossing over each other-as the temperature is approaching Tg, the two modes could be easily confused. Below approximately 110 degrees C, the rate of tauc changing with temperature lags behind that of tauv is explained as due to the loss of effective ergodicity taking place in the system.     

Stress relaxation in polymer networks: equilibrium behavior and dynamics

The elastic and relaxational properties of a polymer network have been calculated using a stress based formulation based on the Rouse mode expansion [W. L. Vandoolaeghe and E. M. Terentjev, J. Chem. Phys. 123, 34902 (2005)]. In this article, we propose an improved Rouse mode expansion incorporating appropriate boundary conditions. In contrast to the previous work, this improved formulation provides a smooth crossover from the classical equilibrium result of rubber elasticity to the shorter-time-scale Rouse relaxation of a polymer melt. Our results are compared with the classical phantom network approach in equilibrium, as well as both equilibrium and dynamic elongation experiments. The model captures the qualitative features of the data well and some of the quantitative aspects, such as the exponents seen in the dynamic modulus G(omega).     

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     

Constrained Rouse model of rubber viscoelasticity

In this work we use a new approach to investigate the equilibrium and linear dynamic-mechanical response of a polymer network. The classical Rouse model is extended to incorporate quenched constraints on its end-boundary conditions; a microscopic stress tensor for the network system is then derived in the affine deformation limit. To test the model we calculate the macroscopic stress in equilibrium, corresponding to the long-time limit of relaxation. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. Although quite different in general, for small strains the model compares well with the classic equilibrium rubber-elasticity models. The dynamic shear modulus is obtained for a network relaxing after an instantaneous step strain by keeping track of relaxation of consecutive Rouse modes of constrained network strands. The results naturally cover the whole time range--from the dynamic glassy state down to the equilibrium incompressible rubber plateau.     

Chain contraction and nonlinear stress damping in primitive chain network simulations

Doi and Edwards (DE) proposed that the relaxation of entangled linear polymers under large deformation occurs in two steps: the fast chain contraction (via the longitudinal Rouse mode of the chain backbone) and the slow orientational relaxation (due to reptation). The DE model assumes these relaxation processes to be independent and decoupled. However, this decoupling is invalid for a generalized convective constraint release (CCR) mechanism that releases the entanglement on every occasion of the contraction of surrounding chains. Indeed, the decoupling does not occur in the sliplink models where the entanglement is represented by the binary interaction (hooking) of chains. Thus, we conducted primitive chain network simulations based on a multichain sliplink model to investigate the chain contraction under step shear. The simulation quantitatively reproduced experimental features of the nonlinear relaxation modulus G(t,γ). Namely, G(t,γ) was cast in the time-strain separable form, G(t,γ)=h(γ)G(t) with h(γ)=damping function and G(t)=linear modulus, but this rigorous separability was valid only at times t comparable to the terminal relaxation time, although a deviation from this form was rather small (within ±10%) at t>τ(R) (longest Rouse relaxation time). A molecular origin of this delicate failure of time-strain separability at t～τ(R) was examined for the chain contour length, subchain length, and subchain stretch. These quantities were found to relax in three steps, the fast, intermediate, and terminal steps, governed by the local force balance between the subchains, the longitudinal Rouse relaxation, and the reptation, respectively. The contributions of the terminal reptative mode to the chain length relaxation as well as the subchain length/stretch relaxation, not considered in the original DE model, emerged because the sliplinks (entanglement) were removed via the generalized CCR mechanism explained above and the reformation of the sliplinks was slow at around the chain center compared to the more rapidly fluctuating chain end. The number of monomers in the subchain were kept larger at the chain center than at the chain end because of the slow entanglement reformation at the center, thereby reducing the tension of the stretched subchain at the chain center compared to the DE prediction. This reduction of the tension at the chain center prevented completion of the length equilibration of subchains at t～τ(R) (which contradicts to the DE prediction), and it forces the equilibration to complete through the reptative mode at t?τ(R). The delicate failure of time-strain separability seen for G(t,γ) at t～τ(R) reflects this retarded length equilibration.     

Theory of relaxation properties of two‐dimensional polymer networks, 2. Local dynamic characteristics

Using normal mode transformation obtained in Part 1 of this series^[1], the exact analytical expressions for the mean‐square displacements of junctions and non‐junction beads, the autocorrelation functions of the end‐to‐end chain vectors between neighboring junctions, and those of subchain vectors of a two‐dimensional regular network consisting of "bead and spring" Rouse chains are obtained. Contributions of intra‐ and interchain relaxation processes to the local dynamic characteristics considered are compared. The time behavior of dynamic quantities obtained is estimated for different scales of motions. The possibility of describing long‐time relaxation of a two‐dimensional network by a simplified coarse‐grained network model is demonstrated. It is shown that the local relaxation properties of a two‐dimensional polymer network (as well as a three‐dimensional network) on scales smaller than the average distance between cross‐links are very close to those of a single Rouse chain. The large‐scale collective relaxation of the polymer networks having a two‐dimensional connectivity differs considerably from that of the three‐dimensional networks.     

Theory of relaxation properties of two‐dimensional polymer networks, 1. Normal modes and relaxation times

The local relaxation properties of polymer networks with a two‐dimensional connectivity are considered. We use the mesh‐like network model in which the average positions of junctions form the regular spatial structure consisting of square repeating units (network cells). The two‐dimensional polymer network consisting of “bead and spring” Rouse chains and the simplified coarse‐grained network model describing only the large‐scale collective relaxation of a network are studied. For both dynamic network models the set of relaxation times and the transformation from Cartesian coordinates of network elements to normal modes are obtained. Using the normal mode transformation obtained, in Part 2 of this series the exact analytical expressions for various local dynamic characteristics of the polymer network having a two‐dimensional connectivity will be calculated.     

Scaling of mesoscale simulations of polymer melts with the bare friction coefficient

Both the Rouse and reptation model predict that the dynamics of a polymer melt scale inversely proportional with the Langevin friction coefficient xi. Mesoscale Brownian dynamics simulations of polyethylene validate these scaling predictions, providing the reptational friction xi(R)=xi+xi(C) is used, where xi(C) reflects the fundamental difference between a deterministic and a stochastic propagator even in the limit of xi to zero. The simulations have been performed with Langevin background friction and with pairwise friction, as in dissipative particle dynamics. Both simulation methods lead to equal scaling behavior with xi(C) having almost the same value in both cases. The scaling is tested for the diffusion g(t), the shear relaxation modulus G(t), and the Rouse mode autocorrelations of melts of C(120)H(242), C(400)H(802), and C(1000)H(2002). The derived dynamical scaling procedure is very useful to reduce run-time in mesoscale computer simulations, especially if pairwise friction is applied.     

Diffusion Spectrum of Polymer Melt Measured by Varying Magnetic Field Gradient Pulse Width in PGSE NMR

The translational motion of polymers is a complex process and has a big impact on polymer structure and chemical reactivity. The process can be described by the segment velocity autocorrelation function or its diffusion spectrum, which exhibit several characteristic features depending on the observational time scale—from the Brownian delta function on a large time scale, to complex details in a very short range. Several stepwise, more-complex models of translational dynamics thus exist—from the Rouse regime over reptation motion to a combination of reptation and tube-Rouse motion. Accordingly, different methods of measurement are applicable, from neutron scattering for very short times to optical methods for very long times. In the intermediate regime, nuclear magnetic resonance (NMR) is applicable—for microseconds, relaxometry, and for milliseconds, diffusometry. We used a variation of the established diffusometric method of pulsed gradient spin-echo NMR to measure the diffusion spectrum of a linear polyethylene melt by varying the gradient pulse width. We were able to determine the characteristic relaxation time of the first mode of the tube-Rouse motion. This result is a deviation from a Rouse model of polymer chain displacement at the crossover from a square-root to linear time dependence, indicating a new long-term diffusion regime in which the dynamics of the tube are also described by the Rouse model.     

Influences of three kinds of springs on the retraction of a polymer ellipsoid in dissipative particle dynamics simulation

The impacts of Rouse spring, Fraenkel spring, and one kind of finitely extensible nonlinear elastic spring (FENE‐PM spring) on the surface tension‐induced retraction of a polymer ellipsoid in a matrix were compared using dissipative particle dynamics. Using the same spring constant, obvious differences among the three kinds of springs were found. A fast retraction process was observed from the hard Fraenkel spring, a slow process from the soft Rouse spring, and an intermediate process from the FENE‐PM spring. The effects of varying the spring constant and the chain length were also investigated. The results indicate that the influence of increasing spring hardness for a given spring type was significant; whereas, the influence of chain length was minor after five bonds were reached. The effects of varying the FENE‐PM r_m parameter were also studied to provide a reliable value for this study. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010     

Aging in a free-energy landscape model for glassy relaxation. II. Fluctuation-dissipation relations

Several fluctuation-dissipation relations are investigated for a simple free-energy landscape model designed to describe the primary relaxation in supercooled liquids. The calculations of the response and of the correlation functions are performed for a quench from a high temperature to a low temperature. In the model, all dynamical quantities reach equilibrium after long times, but for times shorter than the re-equilibration time they do not exhibit time-translational invariance and the fluctuation-dissipation theorem is violated. Two measures for these violations are considered. One such measure is given by the slope in a plot of the integrated response versus the correlation function and another one by the so-called fluctuation-dissipation ratio. It is found that these measures do not coincide and furthermore are not independent of the dynamical variable considered in the calculation. We propose to determine the fluctuation-dissipation ratio experimentally via measurements of the deuteron spin-lattice relaxation rate and the dielectric loss.     

Thermodynamic Limitations of the Reptation Model

Summary: The paper deals with the question whether the tube/reptation model of polymer chain dynamics is compatible with general laws of statistical physics. Based on a relation between the mean squared fluctuation of the number of segments in a given volume element and the isothermal compressibility of the polymer system, it follows straightforwardly that the tube/reptation model predicts fluctuations larger than permitted by thermodynamics on the time scale t ≳ τ_R, where τ_R is the Rouse relaxation time.     

A Monte Carlo study of a tethered polymer chain in a uniform field

We study the non‐uniform stretching and relaxation of a long flexible end‐anchored polymer chain of N monomers (32 ≤ N ≤ 1 024) in a uniform field B by means of an off‐lattice bead‐spring Monte Carlo model. Our simulational results for the case of a Rouse‐like polymer in the good solvent regime confirm the existence of “trumpet”‐ and “flower”‐type chain conformations, predicted recently by scaling analysis based on the notion of Pincus tensile blobs. The observed elongation of the chain and the critical fields, separating three different regimes of chain deformation, are found to obey the predicted scaling behavior. The segment density distribution matches that of a DNA molecule pulled from one end at constant velocity in a good solvent. As expected, the relaxation of the stretch to coil transition of the polymer of length N is determined by the typical Rouse time τ ∝ N^2ν+1.     

The single chain limit of structural relaxation in a polyolefin blend

The influence of composition on component dynamics and relevant static properties in a miscible polymer blend is investigated using molecular dynamics simulation. Emphasis is placed on dynamics in the single chain dilution limit, as this limit isolates the role of inherent component mobility in the polymer's dynamic behavior when placed in a blend. For our systems, a biased local concentration affecting dynamics must arise primarily from chain connectivity, which is quantified by the self-concentration, because concentration fluctuations are minimized due to restraints on chain lengths arising from simulation considerations. The polyolefins simulated [poly(ethylene-propylene) (PEP) and poly(ethylene-butene) (PEB)] have similar structures and glass transition temperatures, and all interactions are dispersive in nature. We find that the dependence of dynamics upon composition differs between the two materials. Specifically, PEB (slower component) is more influenced by the environment than PEP. This is linked to a smaller self-concentration for PEB than PEP. We examine the accuracy of the Lodge-McLeish model (which is based on chain connectivity acting over the Kuhn segment length) in predicting simulation results for effective concentration. The model predicts the simulation results with high accuracy when the model's single parameter, the self-concentration, is calculated from simulation data. However, when utilizing the theoretical prediction of the self-concentration the model is not quantitatively accurate. The ability of the model to link the simulated self-concentration with biased local compositions at the Kuhn segment length provides strong support for the claim that chain connectivity is the leading cause of distinct mobility in polymer blends. Additionally, the direct link between the willingness of a polymer to be influenced by the environment and the value of the self-concentration emphasizes the importance of the chain connectivity. Furthermore, these findings are evidence that the Kuhn segment length is the relevant length scale controlling segmental dynamics.