The Doi-Edwards model without independent alignment

The well-known constitutive equation of Doi and Edwards incorporates an approximation called “independent alignment”. Although the approximation is acceptable in most cases, there are instances where it generates predictions which may be off by an order of magnitude or even opposite in sign. By completing an analysis initiated in a previous paper and there limited to slow flows, the approximation is here removed for the general case. The constitutive equation which is obtained resembles an integro-differential equation previously obtained by Doi. Differences and similarities among the various approximations are thoroughly discussed both in their physical significance and for what regards the simplicity of use of the corresponding constitutive equations.     

The nonlinear strain measure of polyisobutylene melt in general biaxial flow and its comparison to the Doi-Edwards model

The nonlinear strain measure of a polyisobutylene (PIB) melt as determined by analysis of uniaxial, planar, ellipsoidal, and equibiaxial extensions is compared to the predictions of the molecular model of Doi and Edwards. It is found that the universal strain function of the Doi-Edwards model is unable to predict the nonlinear behavior of this polymer melt in general extensional flow. The qualitative agreement between predictions and experimental data for the strain dependence of shear stress and first normal stress difference in shear flow that was considered as powerful evidence for the correctness of the Doi-Edwards model seems to be accidental. The exaggerated strain dependence of the model suggests a need to reconsider the assumptions concerning the chain retraction process.Presented at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.Dedicated to Professor F.R. Schwarzl on the occasion of his 65th birthday     

Nonlinear viscoelasticity of PP/PS/SEBS blends

The nonlinear viscoelastic behavior of polypropylene/polystyrene (PP/PS) blends compatibilized or not with the linear triblock copolymer (styrene-ethylene-/butylene-styrene, SEBS) was investigated. Start-up of steady-shear at rates from 0.1 to 10 s^–1 was carried out using a controlled strain rotational rheometer and a sliding plate rheometer for strain histories involving one or several shear rates. The shear stress and first normal shear stress difference were measured as functions of time, and the morphologies of the samples before and after shearing were determined. For each strain history except that involving a single shear rate of 0.1 s^–1 the blends showed typical non-linear viscoelastic behavior: a shear stress overshoot/undershoot, depending on the history, followed by a steady state for each step. The first normal stress difference increased monotonically to a steady-state value. The values of the stresses increased with the addition of SEBS. The shear stress overshoot and undershoot and the times at which they occurred depended strongly on the strain history, decreasing for a subsequent shear rate step performed in the same direction as the former, and the time at which stress undershoot occurred increased for a subsequent shear rate step performed in the opposite direction, irrespective of the magnitude of the shear rate. This behavior was observed for all the blends studied. The time of overshoot in a single-step shear rate experiment is inversely proportional to the shear rate, and the steady-state value of N₁ scaled linearly with shear rate, whereas the steady-state shear stress did not. The average diameter of the dispersed phase decreased for all strain histories when the blend was not compatibilized. When the blend was compatibilized, the average diameter of the dispersed phase changed only during the stronger flows. Experimental data were compared with the predictions of a model formulated using ideas of Doi and Ohta (1991), Lacroix et al. (1998) and Bousmina et al. (2001). The model correctly predicted the behavior of the uncompatibilized blends for single-step shear rates but not that of the compatibilized blends, nor did it predict morphologies after shearing.     

Polymeric liquids in extension: fluid mechanics or rheometry?

We use a transient 3D free surface finite element method to simulate flow of entangled polymer fluids in the dual cylinder wind-up extensional rheometer. The constitutive equations are K-BKZ integral representations of the Doi–Edwards models with and without the independent alignment approximation (IA). It is demonstrated that the actual kinematics in this rheometer is a mixture of planar and uniaxial extension. Moreover, the ratio of planar to uniaxial deformation is highly dependent upon whether IA is invoked. Without IA, the flow has a tendency toward planar extension, while it tends to be more uniaxial with IA invoked. As a second illustration of the techniques, we simulate the phenomenon of delayed rupture after rapid extension of entangled polymer systems. It is demonstrated that this phenomenon can be explained on the basis of the Doi–Edwards model in terms of a Considere-type instability after chain stretch relaxation.     

Quantitative predictions of linear viscoelastic rheological properties of entangled polymers

The “dual constraint” model developed by Mead, Van Dyke et al. is here extended by inclusion of “early-time” contour-length fluctuations and constraint-release Rouse relaxation, and then evaluated by comparing its predictions with literature data for over 50 different linear and star polymers. By combining the reptation model of Doi and Edwards with contour-length fluctuations and constraint release, the model provides a systematic approach to prediction of the rheological properties of polymers. The parameters are taken from the literature and used consistently for linear polymers, star polymers, and their mixtures having the same chemical compositions. In most cases, the predictions of the model appears to agree well with data for monodisperse, bidisperse, and polydisperse linear and star polymers, except at low molecular weights. Received: 23 December 1999 Accepted: 28 March 2000     

Mesoscopic tube model of fluids composed of worm-like micelles

Rheological model of fluids involving Brownian relaxation, reptation, diffusion, and scission–recombination processes as relaxation mechanisms is formulated. Numerical solution of a particular example of the model displays the S-shape form of the shear rate versus shear stress curves observed in worm-like micellar solutions.     

Constitutive equations for the Doi–Edwards model without independent alignment

We present two representations of the Doi–Edwards model without Independent Alignment explicitly expressed in terms of the Finger strain tensor, its inverse and its invariants. The two representations provide explicit expressions for the stress prior to and after Rouse relaxation of chain stretch, respectively. The maximum deviations from the exact representations in simple shear, biaxial extension and uniaxial extension are of order 2%. Based on these two representations, we propose a framework for Doi–Edwards models including chain stretch in the memory integral form.     

Coupling between flow and diffusion at polymer/polymer interfaces: large amplitude oscillatory shear experiments

 Coupling between flow and diffusion at symmetric polymer/polymer interfaces has been investigated. Polystyrene/polystyrene sandwich assemblies were subjected to large-amplitude oscillatory shear (LAOS) using a sliding-plate rheometer (SPR) and the stress was monitored as a function of time. The results were treated using a new model combining Wagner's model with the theory of Doi and Edwards. The model explicitly expresses the influence of the strain and stress amplitudes frequency and time on the self-diffusion process. The apparent self-diffusion coefficient was found to increase with welding time, in agreement with our previous results obtained using small-amplitude oscillatory shear measurements. However, it was found in the present case that the self-diffusion coefficient depends strongly on the strain and stress amplitudes and frequency, and its steady state value was found to be larger than that determined from small-amplitude oscillatory shear measurements. It appears that the large strain oscillatory shear field continuously increases the density of chain ends at the interface and thus increases the flux of mass transport. Received: 30 January 2001 Accepted: 12 June 2001     

Viscoelasticity of low molecular weight polymers and the transition to the entangled regime

The viscoelasticity of unentangled polystyrene melts has been investigated in terms of terminals parameters: zero-shear viscosity, steady-state compliance and relaxation spectrum. The Rouse model applies well for molecular weights lower than the average molecular weight between entanglements, providing that one takes into account the proper variations of the radius of gyration. Moreover, local motions at the scale of Kuhn segments have to be considered in order to describe correctly the relaxation modes intermediate between the terminal zone and the glassy plateau. On the other hand, reptation models are commonly used for describing the entangled regime. We propose an expression of the shear modulus which accounts not only for the terminal modes (reptation, tube length fluctuations and tube renewal), but also for the relaxation modes responsible for the plateau zone and the transition of the glassy plateau. A crossover region between the unentangled and untangled regimes is located around . When the molecular weight increases, a shift transfer of Rouse modes towards reptation modes occurs. That leads to a continuity of the expression of the shear modulus over the entire range of molecular weights. Received: 29 December 1997 Accepted: 27 July 1998     

A modified tube dilatation theory for the relaxation of entangled linear polymer melts

In this paper, the “tube dilatation” or “tube Enlargement” concept introduced by Marrucci, is revisited in the case of broad entangled linear polymer melts. Using the tube fluctuation relaxation function of Doi and a linear mixing rule, the model implicitly contains both the features of double reptation and time modification by tube renewal. It has been shown that theoretical arguments of both double reptation with tube renewal and tube dilatation can be used to take into account the effect of polydispersity on the distribution of relaxation times. The model has been tested for some polymers with various N/Ne values. However, experiments indicate that the loss of only one entanglement does not systematically induce a change of the relaxation times through a constraint release and tube renewal process. The freeing of a critical volume, larger than the volume of a tube segment, is required to induce an efficient dilution effect on relaxation times.     

Coarse-grained model of entangled polymer melts in non-equilibrium

A coarse-grained model developed for entangled polymeric systems and calibrated to represent melts in equilibrium (Rakshit, Picu, J Chem Phys 125:164907(1)–(10), 2006) is used to model shear flows. The model is a hybrid between multimode and mean-field representations: chain inner blobs are constrained to move along the chain backbone and the end blobs are free to move in 3D and continuously redefine the diffusion path for the inner blobs. Therefore, contour length fluctuations and reptation are captured. Constraint release is implemented by tracing the position of chain ends and performing a local relaxation of the chain backbones once end retraction is detected. This algorithm takes advantage of the multi-body nature of the model and requires no phenomenological parameters other than the length of an entanglement segment. The model is used to study start-up and step strain shear flows and reproduces features observed experimentally such as the overshoot during start-up shear flow, the Lodge–Meissner law, the monotonicity of the steady state shear stress with the strain rate, and shear thinning at large . These simulations are performed in conditions in which using a fully refined model of the same system would have been extremely computationally demanding or simply impossible with the current methods.     

A network model with free-strand dynamics for polymer melts

 A network model for polymer melts is presented in which disentangled strands relax under flow conditions and may rejoin the network before complete relaxation. For simplicity, we study Gaussian strands that move affinely when incorporated in the network. Network strands are created and lost according to a time constant λ. Free strands have their dynamics given by the Bird-DeAguiar model as a crude representation of reptation and the hindered rotation experienced by polymer strands in melts. The model yields a shear-thinning viscosity with overshoot in the start-up viscosity η⁺ (t). The double-step strain results compare well with available experimental data. Received: 10 July 2000 Accepted: 10 July 2001     

The weak shear kinetic phase diagram for nematic polymers

We study the shear problem for nematic polymers as modeled by the molecular kinetic theory of Doi (1981), focusing on the anomalous slow flow regime. We provide the kinetic phase diagram of monodomain (MD) attractors and phase transitions vs normalized nematic concentration (N) and weak normalized shear rate (Peclet number, Pe). We then overlay all rheological features typically reported in experiments: alignment properties, normal stress differences and shear stress. These features play a critical role in the synthesis between theory and experiment for nematic polymers (Larson 1999; Doi and Edwards 1986). MD type is routinely used for rheological shear characterization: cf., flow-aligning 5CB (Mather et al. 1996a), tumbling PBT (Srinivasarao and Berry 1991), and 8CB (Mather et al. 1996b), evidence for a wagging regime (Mewis et al. 1997), out-of-plane kayaking modes (Larson and Ottinger 1991), and evidence for chaotic major director dynamics (Bandyopadhyay et al. 2000). MD transitions correlate with sign changes in normal stresses (Larson and Ottinger 1991; Magda et al. 1991; Kiss and Porter 1978, 1980). Furthermore, structure formation in shear devices appears to be correlated with monodomain precursor dynamics (Tan and Berry 2003; Forest et al. 2002a). In this paper we combine seminal kinetic theory results (Kuzuu and Doi 1983, 1984; Larson 1990; Larson and Ottinger 1991; Faraoni et al. 1999; Grosso et al. 2001), symmetry observations (Forest et al. 2002b), and mesoscopic results on the fate of orientational degeneracy in weak shear (Forest and Wang 2003; Forest et al. 2003a), together with our resolved numerical simulations, to provide the kinetic flow-phase diagram of Doi theory in the weak shear regime, 0<Pe<1, for infinitely thin rods. We report the "birth" of key rheological features at the onset of flow: sign changes and local maxima and minima in normal stress differences (N₁ and N₂) associated with MD transitions. These results serve as the basis for continuation of the kinetic phase diagram to Pe>1 ; as the definitive benchmark for any mesoscopic or continuum model; and experimental data can be compared in order to determine accuracy and limitations of the Doi theory in weak shear.     

Numerical study of chain conformation on shear banding using diffusive Rolie-Poly model

Shear-banding phenomenon in the entangled polymer systems was investigated in a planar Couette cell with the diffusive Rolie-Poly (ROuse LInear Entangled POLYmers) model, a single-mode constitutive model derived from a tube-based molecular theory. The steady-state shear stress ?? _s was constant in the shear gradient direction while the local shear rate changed abruptly, i.e., split into the bands. We focused on the molecular conformation (also calculated from the Rolie-Poly model) around the band boundary. A band was found also for the conformation, but its boundary was much broader than that for the shear rate. Correspondingly, the first normal stress difference (N ₁) gradually changed in this diffuse boundary of the conformational bands (this change of N ₁ was compensated by a change of the local pressure). For both shear rate and conformation, the boundary widths were quite insensitive to the macroscopic shear rate but changed with various parameters such as the diffusion constant and the relaxation times (the reptation and the Rouse times). The broadness of the conformational banding, associated by the gradual change of N ₁, was attributed to competition between the molecular diffusion (in the shear gradient direction) and the conformational relaxation under a constraint of constant ?? _s.     

Rheological characterization and modeling of end-functionalized polybutadienes

The rheological characterization and modeling of a series of polybutadienes obtained by anionic solution polymerization is presented in this work. The polybutadienes are synthesized using two different initiators: R,R′,R′′-silyloxyalkyllithium (F1) and R,R′,R′′-silylalkyllithium (F3). In addition, a polybutadiene obtained with a conventional alkyllithium initiator (n-butyllithium) is used as a reference. The rheological characterization is carried out under small amplitude oscillatory shear in the stress-controlled mode. Microstructure, molecular weight, and molecular weight distribution are determined by FTIR and GPC. The vinyl content of the polybutadienes synthesized using the functionalized initiators is similar to that obtained with n-butyllithium (8–11%). Materials obtained with F1 show a relatively low polydispersity within a narrow molecular weight range (250,000–300,000 g/mol), while samples obtained with F3 cover a wider range of molecular weights (65,000–670,000 g/mol) and display higher values of polydispersity. In all cases, a parallel reaction using propylene oxide in the termination step is done to place a functional group at the chain ends. The effect of this group on the rheological behavior appears to be negligible. Three rheological models are used and their predictions of the experimental data are compared. The models include the Doi and Edwards reptation model, expressions using a discrete spectrum of relaxation times based in the rubber-like liquid constitutive equation and the fractional Maxwell equation in which a given analytical relaxation-spectrum is used. Relevant relations are obtained between the models' parameters and the molecular properties of these systems, which in turn are related to the presence of functional groups at the polymer chain ends.     

Analytic derivation of the Cox�CMerz rule using the MLD ��toy�� model for polydisperse linear polymers

A general constitutive formalism, the ??na?ve?? polydisperse MLD model, has been developed by Mead et al. (Macromolecules 31:7895?C7914, 1998) and Mead (Rheol Acta 46:369?C395, 2007) at both the tube coordinate level and the mathematically simplified ??toy?? level independent of the tube coordinate. The model includes constraint release generated by convection-driven chain retraction (which is equivalent to ??convective constraint release?? (CCR)), reptation, and tube contour length fluctuations. The properties of the mathematically simplified na?ve polydisperse ??toy?? MLD model are explored in linear and nonlinear steady shear flows where we analytically derive the Cox?CMerz rule relating the steady shear viscosity to the modulus of the linear viscoelastic dynamic viscosity. The Cox?CMerz rule relating the linear viscoelastic material properties and the nonlinear material properties is shown to be a direct consequence of convective constraint release. The specific feature of CCR that leads to this result is that the relaxation rate due to convective constraint release is proportional to the shear rate, $\dot{{\gamma }}$ , independent of molecular weight. The viability of this well-known empirical relationship is a direct consequence of a coincidence in the mathematical structure of the linear viscoelastic material properties and convective constraint release. There is no physical analogy or relationship between the molecular relaxation mechanisms operative in linear (diffusive relaxation) and nonlinear (convective relaxation) flow regimes. The polydisperse MLD model predictions of the individual molecular weight component contributions to the flow curve, and interpretations thereof, are effectively identical to those first postulated by Bersted (J Appl Polym Sci 19:2167?C2177, 1975, J Appl Polym Sci 20:2705?C2714, 1976). Following the theoretical developments, a limited experimental study is executed with a commercial polydisperse polystyrene melt. Nearly quantitative agreement between the polydisperse MLD theory and experimental measurements of steady-shear viscosity and dynamic moduli is achieved over a wide range of shear rates.     

Stress relaxation as a microstructural probe for immiscible polymer blends

Relaxation has been investigated in immiscible blends that consist of slightly viscoelastic components. Both the shear and normal stresses have been measured after cessation of steady shear flow as well as after transient shear histories. The latter can generate a fibrillar structure which can relax by either retraction or break-up via end-pinching or Rayleigh instabilities. Each of these three relaxation mechanisms is reflected in the shape of the stress curves, from which also the corresponding structural time scales can be deduced. The experimental results have been used to evaluate the Doi-Ohta and Lee-Park models for immiscible blends. The scaling relations by Doi-Ohta are confirmed by the experimental results, but none of the existing models can correctly predict the complex relaxation behaviour observed for a highly deformed droplet phase. In the present study an alternative approach has been proposed. The stress relaxation due to fibril break-up via Rayleigh instabilities has been predicted successfully by combining physical models for the structural changes with the basic approach of the Doi-Ohta model.     

An evaluation of the Larson-Doi model for liquid crystalline polymers using recoil

The mesoscopic models for the rheological properties of liquid crystalline polymers proposed by Larson and Doi in 1991 and Kawaguchi and Denn in 1999 are based on phenomenological expressions that describe the evolution of the defect density and the contribution of the “texture” to the stress. In the present work, we attempt to assess some of these assumptions by monitoring how the energy stored in the texture of liquid crystalline materials evolves during shear flows. For that purpose, strain recovery is measured as a function of the applied strain for flow reversal and intermittent flow. Solutions of poly-benzylglutamate in m-cresol, hydroxypropylcellulose in water and a nematic surfactant solution are used as model systems. Although the behaviour is described qualitatively by the model, discrepancies between the predictions and the experiments are observed, especially when the shear history includes rest periods. Received: 14 July 1999 /Accepted: 30 August 1999     

Shear and elongational flow behavior of acrylic thickener solutions. Part II: effect of gel content

We have investigated the effect of crosslink density on shear and elongational flow properties of alkali-swellable acrylic thickener solutions using a mixing series of the two commercial thickeners Sterocoll FD and Sterocoll D as model system. Linear viscoelastic moduli show a smooth transition from weakly elastic to gel-like behavior. Steady shear data are very well described by a single mode Giesekus model at all mixing ratios. Extensional flow behavior has been characterized using the CaBER technique. Corresponding decay of filament diameter is also well fitted by the Giesekus model, except for the highest crosslink densities, when filament deformation is highly non-uniform, but the non-linearity parameter α, which is independent of the mixing ratio, is two orders of magnitude higher in shear compared to elongational flow. Shear relaxation times increase by orders of magnitude, but the characteristic elongational relaxation time decreases weakly, as gel content increases. Accordingly, variation of gel content is a valuable tool to adjust the low shear viscosity in a wide range while keeping extensional flow resistance essentially constant.     

Prediction of multiple overshoots in shear stress during fast flows of bidisperse polymer melts

We present a differential constitutive model of stress relaxation in polydisperse linear polymer melts and solutions that contains contributions from reptation, contour-length fluctuations, and chain stretching. The predictions of the model during fast start-up and steady shear flows of polymer melts are in accord with experimental observations. Moreover, in accordance with reported experimental literature (Osaki et al. in J Polym Sci B Polym Phys 38:2043–2050, 2000), the model predicts, for a range of shear rates, two overshoots in shear stress during start-up of steady shear flows of bidisperse polymer melts having components with widely separated molar masses. Two overshoots result only when the stretch or Rouse relaxation time of the higher molar mass component is longer than the terminal relaxation time of the lower molar mass component. The “first overshoot” is the first to appear with increasing shear rate and occurs as a result of the stretching of longer chains. Transient stretching of the short chains is responsible for the early time second overshoot. The model predictions in steady and transitional extensional flows are also remarkable for both monodisperse and bidisperse polymer solutions. The computationally efficient differential model can be used to predict rheology of commercial polydisperse polymer melts and solutions.