Viscoelastic Properties Of Polymer Melts Research Articles

Equilibrium and non-equilibrium molecular dynamics approaches for the linear viscoelasticity of polymer melts

Viscoelastic properties of polymer melts are particularly challenging to compute due to the intrinsic stress fluctuations in molecular dynamics (MD). We compared equilibrium and non-equilibrium MD approaches for extracting the storage (G′) and loss moduli (G″) over a wide frequency range from a bead-spring chain model in both unentangled and entangled regimes. We found that, with properly chosen data processing and noise reduction procedures, different methods render quantitatively equivalent results. In equilibrium MD (EMD), applying the Green−Kubo relation with a multi-tau correlator method for noise filtering generates smooth stress relaxation modulus profiles from which accurate G′ and G″ can be obtained. For unentangled chains, combining the Rouse model with a short-time correction provides a convenient option that circumvents the stress fluctuation challenge altogether. For non-equilibrium MD (NEMD), we found that combining a stress pre-averaging treatment with discrete Fourier transform analysis reliably computes G′ and G″ with a much shorter simulation length than previously reported. Comparing the efficiency and statistical accuracy of these methods, we concluded that EMD is both reliable and efficient, and is suitable when the whole spectrum of linear viscoelastic properties is desired, whereas NEMD offers flexibility only when some frequency ranges are of interest.

Modeling and simulation of weld line location and properties during injection molding based on viscoelastic constitutive equation

Weld lines are one of the typical quality issues of injection molded parts. They are unavoidable even for moderate complex products. To explore the formation of weld lines in injection molding, a flow model based on viscoelastic constitutive equation is presented in this article to predict and evaluate of weld lines location and properties, in which Rolie-Poly constitutive equation is used to describe the viscoelastic properties of polymer melts and level set method is used to capture melt front. Furthermore, the formation and evolution processes of weld lines are investigated by co-injection molding technique in simulation, since the processes are determined by the flow front and rheological history of the melt streams. Numerical results show that weld lines can be predicted accurately, and the formation and evolution processes of weld line can be shown more clearly by co-injection molding technique. In addition, properties of polymer melt such as velocity, pressure, principal stress difference, and molecular orientation in the weld line regions are investigated and presented.

Rheological properties of crystallizing polylactide: Detection of induction time and modeling the evolving structure and properties

AbstractLinear viscoelastic properties of polymer melts are highly sensitive to any structural changes, including molecular weight changes and the formation and growth of crystallites. Here, we make use of this sensitivity to study the homogeneous crystallization of polylactide. As this polymer is rather quickly susceptible to thermal degradation even at moderate temperatures, it is essentially impossible to study homogeneous crystallization in the absence of degradation. Thus, the evolution of complex viscosity of a polylactide due to thermal degradation in the absence of crystallization was studied. A simple empirical model is used to characterize the variation of complex viscosity due to thermal degradation and to determine the induction time of homogeneous crystallization at wide range of degrees of supercooling. Next, the evolution of complex viscosity due to crystallization was measured at several temperatures. Based on the results, a phenomenological model describing the viscosity evolution during homogeneous crystallization is proposed and validated. Finally, the linear viscoselastic data in the early stages of crystallization are shown to be consistent with gelation due to the formation of a network of tie molecules between spherulites. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 812–822, 2010

Multiscale simulation of polymer melt viscoelasticity: Expanded-ensemble Monte Carlo coupled with atomistic nonequilibrium molecular dynamics

We present a powerful framework for computing the viscoelastic properties of polymer melts based on an efficient coupling of two different atomistic models: the first is represented by the nonequilibrium molecular dynamics method and is considered as the microscale model. The second is represented by a Monte Carlo (MC) method in an expanded statistical ensemble and is free from any long time scale constraints. Guided by recent developments in nonequilibrium thermodynamics, the expanded ensemble incorporates appropriately defined ``field'' variables driving the corresponding structural variables to beyond equilibrium steady states. The expanded MC is considered as the macroscale solver for the family of all viscoelastic models built on the given structural variable(s). The explicit form of the macroscopic model is not needed; only its structure in the context of the general equation for the nonequilibrium reversible irreversible coupling or generalized bracket formalisms of nonequilibrium thermodynamics is required. We illustrate the method here for the case of unentangled linear polymer melts, for which the appropriate structural variable to consider is the conformation tensor $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathbf{c}}$. The corresponding Lagrange multiplier is a tensorial field $\mathbf{\ensuremath{\alpha}}$. We have been able to compute model-independent values of the tensor $\mathbf{\ensuremath{\alpha}}$, which for a wide range of strain rates (covering both the linear and the nonlinear viscoelastic regimes) bring results for the overall polymer conformation from the two models (microscale and macroscale) on top of each other. In a second step, by comparing the computed values of $\mathbf{\ensuremath{\alpha}}$ with those suggested by the macroscopic model addressed by the chosen structural variable(s), we can identify shortcomings in the building blocks of the model. How to modify the macroscopic model in order to be consistent with the results of the coupled micro-macro simulations is also discussed. From a theoretical point of view, the present multiscale modeling approach provides a solid framework for the design of improved, more accurate macroscopic models for polymer melts.

Development of a Sliding Plate Rheometer to Measure the High Frequency Viscoelastic Properties of Polymer Melts

Abstract A newly designed and constructed sliding plate rheometer is used to measure the high frequency (210 Hz) linear viscoelastic properties of two model polymers: polybutene (PB) and polydimethylsiloxane (PDMS). Using well-known rheological models, extrapolations of the viscoelastic measurements obtained on a rotational parallel plate rheometer to a frequency of 210 Hz are used to assess the performance of the high frequency sliding plate rheometer. Good agreement between the extrapolated and measured data demonstrates the ability of the sliding plate rheometer to measure the high frequency rheological properties of both Newtonian and shear-thinning materials.

The role of intermolecular interactions in the viscoelastic properties of polymer melts

AbstractThe exact microscopic expression for the stress tensor in polymer liquids contains a tensor product of the the segment position vector with the total, intra‐ plus inter‐chain, force acting on the segment. On the other hand, the widely accepted theory of viscoelasticity of polymer melts1) is based on the assumption, that contributions from interchain interactions to the viscosity of polymer melts is negligible relative to the effectively intrachain entropic interactions. Starting from the exact Green‐Kubo formula for the viscosity, the Rouse dynamic correlation functions, and Newton's second law, we show that the intrachain assumption is inadequate. Rather, the intrachain and interchain forces acting on polymer segments cancel each other largely. The intrachain contribution therefore cannot be dominant as anticipated in the usual treatment1), or, in other words, the interchain contribution cannot be ignored. The main contribution to viscoelastic properties of polymer melts can only arise from a part of the total stress tensor as already suggested by M. Fixman based on a different argument2). It is concluded that the viscosity is of a purely interchain nature, and is determined by the tensor product of the vector connecting the centers‐of‐mass of neighboring macromolecules on the one hand, and the total force by which macromolecules interact, on the other, just in the case of simple liquids.

Prediction of linear viscoelastic response for entangled polyolefin melts from molecular weight distribution

AbstractThe linear viscoelastic properties of polymer melts depend strongly and systematically on the molecular weight distribution. A molecular theory relating dynamic modulus and molecular weight distribution for linear polymers, developed and confirmed earlier with data for three other polymer species, is applied here to commercial samples of isotactic polypropylene and high density polyethylene. Experimental master curves are compared with predictions based on only the fundamental rheological parameters of the species and molecular weight distributions as obtained by the methods of size exclusion chromatography. Agreement is fairly good for the two polypropylene samples, about the same as had been found earlier for the other species, but it is highly variable for the ten polyethylene samples. We attribute this variability to differences among high density polyethylenes in the frequency, length, and type of long‐chain branching. However, we could find surprisingly little supporting evidence for this from such supposed signatures of long branches in polyethylene as the flow activation energy Ea. Measured values of Ea agreed well with the literature results for linear polyethylene; none showed the elevation in Ea that would be expected for polyethylene with long branches.

A simulation system for viscoelastic properties of polydisperse polymer melts

A simulation system is developed for evaluating viscoelastic properties of polymer melts with arbitrary polydispersity in molecular weight, starting from the known chemical structure of the polymers. The critical molecular weight for entanglement and molecular weight dependence of zero shear viscosity are evaluated based on theories of van Krevelen and Berry‐Fox by optimizing parameters associated with chain flexibility, chain length and effective molar weight per repeating unit. In the evaluation of viscoelastic properties of polymer melts, simple and reasonable shape of relaxation spectrum is assumed for polydisperse systems. Using molecular parameters evaluated above, frequency dependence of viscoelastic functions is calculated from the relaxation spectrum. This method is shown to be superior to the usual way of calculation of viscoelasticity based on blending rules: The time needed for the calculation is very short and does not depend on the polydispersity in molecular weight. Combination of the present method with an appropriate constitutive equation gives a system for evaluating nonlinear viscoelastic properties of polydisperse polymers.

A Simulation System for Viscoelastic Properties of Polydisperse Polymer Melts

A simulation system is developed for evaluating viscoelastic properties of polymer melts with arbitrary polydispersity in molecular weight, starting from the known chemical structure of the polymers. The critical molecular weight for entanglement and molecular weight dependence of zero shear viscosity are evaluated based on theories of van Krevelen and Berry-Fox by optimizing parameters associated with chain flexibility, chain length and effective molar weight per repeating unit. In the evaluation of viscoelastic properties of polymer melts, simple and reasonable shape of relaxation spectrum is assumed for polydisperse systems. Using molecular parameters evaluated above, frequency dependence of viscoleastic functions is calculated from the relaxation spectrum. This method is shown to be superior to the usual way of calculation of viscoelasticity based on blending rules: The time needed for the calculation is very short and does not depend on the polydispersity in molecular weight. Combination of the present method with an appropriate constitutive equation gives a system for evaluating nonlinear viscoelastic properties of polydisperse polymers.

A note on estimating mastercurves

There are several quantities in theology which show a scaling behavior. One well known example is the time-temperature superposition principle of material functions characterizing the linear viscoelastic properties of polymer melts. We propose a mathematical shift procedure for the calculation of mastercurves and the corresponding scaling factors from experimental data which show such a scaling behavior. In order to demonstrate the applicability of the shift procedure mastercurves and scaling factors are determined for material functions of several polystyrene melts and for the specific viscosity of polyisobutylene in cyclohexane.

Weak ergodicity breaking and aging in disordered systems

We present a phenomenological model for the dynamics of disordered (complex) systems. We postulate that the lifetimes of the many metastable states are distributed according to a broad, power law probability distribution. We show that aging occurs in this model when the average lifetime is infmite. A simple hypothesis leads to a new functional fornl for the relaxation which is in remarkable agreement with spin-glass experiments over nearly five decades in time. In spite of fifteen years of dispute, the theory of equilibrium spin-glasses is not yet settled (1, 2, 3). Dynamical effects are, however, likely to be the dominant aspect in experiments. One of the most striking aspects of the dynamics of spin-glasses in their low temperature phase is the aging phenomenon- a rather peculiar and awkward feature from the thermodynamics point of view : the relaxation of a system depends on its history. More precisely, if a system is field-cooled below its spin-glass temperature, the magnetization relaxation depends on the Waiting time t~ between the quench and the switch off of the magnetic field (4, 5, 6). Similar effects are observed on the viscoelastic properties of polymer melts (7), magnetic properties of HTC superconductors (8) and more recently on the relaxation after a heat pulse in Charge Density Wave systems (9). Analytical fits of the magnetization relaxation as a function of time have been proposed. In (6), it is proposed that the initial «stationary» part of the relaxation is a power-law With a small (negative) exponent. For times longer than the Waiting time, relaxation is Well fitted by a « stretched » exponential decay, provided that an effective time is introduced (6). In (4, 5), however, a stretched exponential of the «real» time was found for relatively short times. Many phenomenological theories have been devised to account for the stretched exponential decay lo-14), and for the slow part and aging (6, 15-18). We however feel that the basic mechanism underlying aging has not been fully appreciated (see however (17)) although it is, in our opinion, one of the constitutive properties of spin-glasses. The aim of this note is to suggest that aging is related to ergodicity breaking which has, in these systems, a peculiar and perhaps unexpected meaning. We find on some simple models that- according to our definition, see below « Weak » ergodicity breaking occurs in the spin-glass phase, defined as the phase Where the Edwards-Anderson parameter is non-zero. We see however no reason Why the two should be linked in general, and suggest that the appearance of aging could be

Phenomenological theory of the dynamics of polymer melts. II. Viscoelastic properties

A phenomenological theory of the nonmechanical and viscoelastic properties of polymer melts is developed. Consistent with computer simulation results [A. Kolinski, J. Skolnick, and R. Yaris, J. Chem. Phys. 86, 1567, 7164, 7174 (1987)], that fail to find evidence for reptation as the dominant mechanism of long distance motion in a melt, we assume that the long-time behavior of a chain is that of a Rouse-like chain having a number of slow moving points, each with a friction constant proportional to the degree of polymerization n. Coupled with the assumption of rubber like behavior at short times made previously by Doi and Edwards [J. Chem. Soc., Faraday Trans. 2 74, 1802 (1978)], the theory predicts that over a broad molecular weight range the shear viscosity η scales with n as approximately the 3.4 power of the molecular weight, and that η∼n3 in the infinite molecular weight limit. Furthermore, the theory rationalizes the origin of the different crossover molecular weights for the shear viscosity and the self-diffusion coefficient, D. It also accounts for the origin of the intermediate time coupling of the center-of-mass motion into the internal coordinates and for the time dependence of the single bead positional autocorrelation functions seen in previous simulations. Proceeding by analogy to Graessley [J. Poly. Sci. Poly. Phys. Ed. 18, 27 (1980)], in the large molecular weight limit, phenomenological expressions for D and η are derived and comparison is made with experiment.

A newly designed coaxial cylinder-type dynamic rheometer

A newly designed dynamic rheometer is described. This apparatus is designed for the purpose of the investigation of the physical ageing process of gels and for the determination of the viscoelastic properties of polymer melts. With this instrument the dynamic moduli can be determined very accurately in a range of about 8 decades (− 2 < logG′ < 5.3 and − 3 < logG″ < 5.4;G′ andG″ in N/m2) and in a frequency range of more than 6 decades (5 ⋅ 10−5 − 100 Hz) with an accuracy and stability of 0.025%. Results are shown of measurements on a silicon oil (with loss angles deviating only little from 90°), on a gel of polyvinyl chloride in Reomol (with loss angles deviating only little from 0°) and on a polystyrene melt for which, at low frequencies, the dependencies of the storage and loss moduli on the frequency are quadratic and linear, respectively.

Measurement of dynamic viscoelastic properties of polymer melts and liquids by the modified rheovibron

A new technique is presented which permits the quantitative characterization of the dynamic viscoelastic properties of polymer melts and liquids. A new sample holding system with oscillatory shear platen and modification of the amplifier and oscillating unit made it possible to measure rheological properties of the viscous liquids using the Rheovibron. The dynamic shear modulus, viscosity, and internal friction of acrylic dope, and silicone fluids are obtained by using the new procedure and developed mathematical expressions. This technique will be useful in studies on the rheological properties characterization of polymer melts and liquids in conjunction with process parameters.

Computer simulation of the dynamics of highly entangled polymers. Part 1.—Equilibrium dynamics

We present a computer simulation of the stochastic motion of a highly entangled polymer chain in three dimensions. Very good agreement is found between the results of these simulations and the predictions of the theory of reptation, first propounded by de Gennes and developed by Doi and Edwards to account for the viscoelastic properties of polymer melts. It is shown that a polymer diffusing through a network of entanglements obeys the laws TR∝M3; DR∝N–2; 〈r2(t)〉∝t1/4 where DR is the centre-of-mass diffusion constant, TR the chain reaction time and 〈r2(t)〉 the mean-square displacement of a monomer at short times (≪TR).

Computer simulation of the dynamics of highly entangled polymers. Part 2.—Static properties of the primitive chain

In this paper we examine the static properties of the primitive chain developed by Doi and Edwards to model the viscoelastic properties of polymer melts. An analytical calculation is presented to show that the primitive chain segment length, a, depends on the density of monomers, ρ, as a∝ρ–1/2. A computer simulation is described which models the primitive path. Verification is found for the above result. Also, it is found that the number of segments on the primitive chain, Npp is proportional to N, the number of segments on the original chain. The probability distribution for the number of segments between entanglements is given by P(n)∝ exp (–n/Ne) where Ne is the average number of segments between entanglements.

An Improved Rheometer Design Used to Measure Viscoelastic Properties of Polymer Melts

A fixture consisting of a bob and a special cup was developed to ease the charging of polymer melt to the gap of a concentric‐cylinder rheometer. Dynamic measurements with the improved rheometer for two polyolefins below melt temperatures were compared with results obtained on the Rheovibron and the torsion pendulum. Shear moduli were generated for polypropylene, polyethylene,and poly(vinyl chloride) melts by oscillating the bob in the axial mode at amplitudes effecting relatively large strain. Bob oscillations in the rotational mode were also investigated. The phase angles and complex shear moduli differed slightly for different modes of oscillation. At strains of 9%, the time‐temperature shift factor αT remained constant throughout the test‐frequency range for an isotactic polypropylene. However, αT increased with increasing frequency for strains of 98%. A time‐strain shift factor, αε, was found to satisfactorily superimpose polypropylene moduli data generated at high and low strain levels.

The effect of molecular weight and weight distribution upon polymer melt rheology

AbstractA major objective in polymer rheology is to predict a fluid's response to a general deformation from molecular information. A method has been developed which allows one to predict the viscoelastic properties of polymer melts from a limited amount of rheological and molecular data for the polymer. The input parameters are: (a) zero‐shear viscosity; (b) molecular weight distribution; (c) temperature and density; and (d) constants relating Graessley's relaxation time to the Rouse relaxation time. The technique then “simulates” a discrete relaxation spectrum using G′ and G″ data from the Rouse theory and finally requires that a continuum model of polymer viscoelasticity be fit to shear viscosity data predicted by Graessley's theory. Examples of the utility of the procedure are given to illustrate the role of molecular weight and weight distribution in determining rheological behavior.

A phenomenological description of the non-linear viscoelastic properties of polymer melts and concentrated solutions

The non-linear stress response of molten polymers and concentrated solutions to an applied strain is formulated in terms of a material response function. The response function is considered to depend explicitly on the resulting stress history and it is shown how many of the non-linear features of these systems can be described both qualitatively and quantitatively. The particular equations used involve one non-linear term in addition to the usual linear response terms. Finally the non-linear equation is interpreted in terms of a plausible model process.

Blending Laws for High-Molecular-Weight Polymer Melts

Blending laws have been previously proposed to describe the effects of molecular weight distribution (MWD) on the viscoelastic properties of polymer melts. The blending laws represent the relaxation spectrum of a blend HB(τ) as a weighted sum of the relaxation spectra of the individual components, each shifted in time by an amount characteristic of the interactions between the molecular species. In general, the binary blending law is ( Remark: Graphics omitted. ) The weighing factor Vij, and shift factors λij may be obtained directly from the calculated relaxation spectra of blends of narrow MWD fractions. These parameters are used to test the applicability of two types of blending laws: (1) The quadratic blending law, Vij=wiwj, where wk is the weight fraction of the k-th component; (2) The simple blending law, Vij=viδij. The experimental results give qualified support to the quadratic blending law but place strong restrictions on the magnitude and shape of the cross relaxation spectra term Hij. These require that Hij≠Hji, Hij/Hii=constant, and λij=λii. In addition, the restrictions imply that the simple and the quadratic blending laws are equivalent representations of HB(τ).