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A Molecular Theory for Fast Flows of Entangled Polymers
Abstract
The Doi−Edwards (DE) theory for the rheological properties of entangled polymer melts and solutions successfully predicts the response to large step-shear strains but fails to predict other nonlinear shear properties, such as the steady-state viscosity or the relaxation of stress after cessation of steady shearing. Many of these failures remain even in the extension of the theory by Marrucci and Grizzuti (Gazz. Chim. Ital. 1988, 118, 179)1 to allow deformation-induced “tube stretch”. Here, we find that a much more successful theory can be obtained by also accounting for “convective constraint release”, i.e., the loss of entanglement constraints caused by the retraction of surrounding chains in their tubes. (Marrucci, G. J. Non-Newtonian Fluid Mech. 1996, 62, 279 and Ianniruberto, G.; Marrucci, G. J. Non-Newtonian Fluid Mech. 1996, 65, 241).2,3 In the molecular model developed here, convective constraint release can both shorten the reptation tube and allow reorientation of interior tube segments. The revised model predicts many of the features of steady and transient shearing flows. These include a region of nearly constant steady-state shear stress at shear rates between the inverse zero-shear reptation time and the inverse Rouse time, similar to that seen in the experiments of Bercea et al. (Macromolecules 1993, 26, 7095)4 and also predicted by Marrucci and Ianniruberto (Macromol. Symp. 1997, 117, 233).5 The predictions of transient stresses after startup and cessation of shear are also in good agreement with experiments, as are predictions of nonmonotonicity in the extinction angle after stepup or stepdown in shear rate.
... The contour length of a polymer fluctuates within the tube on the Rouse time τ R of the entire chain [10]. Such fluctuations lead to retraction along the tube on the Rouse time [11,21,22]. The chain escapes the original confining tube on the longer reptation time τ d , forming a new set of entanglements. ...
... In the Doi-Edwards model, the reptation time is the time it takes a fixed-length chain to escape the tube by curvilinear diffusion [10]. In practice, the reptation time is reduced from Doi-Edwards prediction due to contour-length fluctuations [10,23] and CCR events [11,12,21,22] on the Rouse time. ...
... Mead et al. (MLD) developed a simple constitutive equation that accounts for reptation, retraction, and CCR, which predicts the stretch and orientation of polymers in flow [21]. While the MLD model successfully predicts many experimental observations in steady and transient shear flow, the proposed decoupling of stretch and orientation is less numerically stable than a single conformation tensor describing both stretch and orientation [24]. ...
We derive a thermodynamically consistent framework for incorporating entanglement dynamics into constitutive equations for flowing polymer melts. We use this to combine the convected constraint release (CCR) dynamics of Ianniruberto–Marriccui into a finitely extensible version of the Rolie–Poly model, and also include an anisotropic mobility as in the Giesekus model. The reversible dynamics are obtained from a free energy that describes both a finitely extensible conformation tensor and an ideal gas of entanglements along the chain. The dissipative dynamics give rise to coupled kinetic equations for the conformation tensor and entanglements, whose coupling terms describe shear-induced disentanglement. The relaxation dynamics of the conformation tensor follow the GLaMM and Rolie–Poly models, and account for reptation, retraction, and CCR. We propose that the relaxation time [Formula: see text] for entanglement recovery is proportional to the Rouse time [Formula: see text] which governs polymer stretch within the tube. This matches recent molecular dynamics simulations and corresponds to relaxing the entanglement number before the entire polymer anisotropy has relaxed on the longer reptation time [Formula: see text]. Our model suggests that claimed signatures of slow re-entanglement on the reptation time in step-strain experiments may be interpreted as arising from anisotropies in reptation dynamics.
... 10 Such fluctuations lead to retraction along the tube on the Rouse time. 11,20,21 The chain escapes the original confining tube on the longer reptation time τ d , forming a new set of entanglements. In the Doi-Edwards model the reptation time is the time it takes a fixed-length chain to escape the tube by curvilinear diffusion. ...
... In the Doi-Edwards model the reptation time is the time it takes a fixed-length chain to escape the tube by curvilinear diffusion. 10 In practice the reptation time is reduced from the Doi-Edwards prediction due to contour-length fluctuations 10,22 and CCR events 11,12,20,21 on the Rouse time. ...
... Mead, Larson, and Doi (MLD) developed a simple constitutive equation that accounts for reptation, retraction, and CCR, which predicts the stretch and orientation of polymers in flow 20 . While the MLD model successfully predicts many experimental observations in steady and transient shear flow, the proposed decoupling of stretch and orientation is less numerically stable than a single conformation tensor describing both stretch and orientation 23 . ...
We derive a thermodynamically consistent framework for incorporating entanglement dynamics into constitutive equations for flowing polymer melts. We use this to combine the convected constraint release (CCR) dynamics of Ianniruberto-Marriccui into a finitely-extensible version of the Rolie-Poly model, and also include an anisotropic mobility as in the Giesekus model. The reversible dynamics are obtained from a free energy that describes both a finitely-extensible conformation tensor and an ideal gas of entanglements along the chain. The dissipative dynamics give rise to coupled kinetic equations for the conformation tensor and entanglements, whose coupling terms describe shear-induced disentanglement. The relaxation dynamics of the conformation tensor follow the GLaMM and Rolie-Poly models, and account for reptation, retraction and CCR. We propose that the relaxation time $\tau_\nu$ for entanglement recovery is proportional to the Rouse time $\tau_R$ which governs polymer stretch within the tube. This which matches recent molecular dynamics simulations, and corresponds to relaxing the entanglement number before the entire polymer anisotropy has relaxed on the longer reptation time $\tau_d$.
... A first example is the work of Mead et al. [16], the so-called MLD model, which considers the effects of both CCR and stretch on the dynamics of the primitive path. The latter is defined as a coarse-grained chain that has the same topology as the tube itself. ...
... The latter depends on the stretch; at fast rates, it prescribes stretch relaxation, whereas at low and intermediate rates, it favors orientation relaxation. With these modifications, the MLD model predicts three regimes in the flow curve [16]: at _ γ , 1/τ d , the stress increases linearly with _ γ. At intermediate shear rates, the stress exhibits a plateau or a very weak overshoot, depending on the ratio τ d /τ R . ...
... Despite these promising features, however, a comprehensive comparison of model predictions and data, especially in the transient regime, is lacking. Due to the fact that the original model is computationally demanding, subsequent works focused on evaluating the performance of a simplified (toy) version of the full model [10,[16][17][18]. At fast rates, the toy model underpredicts the experimental stress overshoot and overpredicts its steady-state value [10,19]. ...
This work addresses the shear response of entangled linear polymers by assessing the current state-of-the-art model and proposing alternative directions. In particular, we examine the performance of the Graham–Likhtman–McLeish–Milner (GLaMM) nonlinear tube model in fast shear flows. Predictions are compared against experimental data for well-characterized, monodisperse, entangled linear polystyrene chains. Unlike previous works using the GLaMM model, finite extensibility is accounted for. Comparison of model predictions and data not only reveals an overall reasonable performance but also highlights limitations. For example, the predictions significantly depend on whether the contribution of contour length fluctuations to the retraction rate is accounted for or not. This specific sensitivity of the model is enhanced as the Rouse–Weissenberg number increases. Possible improvement of the model by modification of the existing mechanisms/model assumptions and/or by incorporation of overlooked mechanisms such as chain tumbling and CCR-driven disentanglement (CCR-D) is discussed. As a possible way to overcome the existing limitations, we propose an alternative approach based on the time marching algorithm (TMA) and focus on the description of the steady-state regime. Compared to the GLaMM model, this approach allows us to keep track of the relaxation state of each entanglement segment in a simple way, while ensuring consistency between chain stretch and constraint release mechanisms. Moreover, it accounts for CCR-D in an indirect manner. A key ingredient is the use of the recently advanced concept of the shear blob and its dependence on the shear rate. Our simple modification of TMA reproduces the experimental steady-state viscosities accurately for all samples and for all shear rates examined.
... However, this extraordinary tour de force had to be refined over the years to account for a better description of the experimental data, in particular to improve the molecular weight dependence of the reptation time which did not follow the predicted M 3 variation by de Gennes [3]. The model of reptation in a tube has been significantly improved over the years, by incorporating additional molecular mechanisms such as contour length fluctuation [36][37][38], constraint release [39][40][41][42][43], and chain stretching [44][45][46]. Doi and Edwards [47] proposed to account for the nonlinear rheological behavior by asserting that the external deformation acted on the tube, instead of the polymer chain [48]. ...
... Both studies concluded that the chains remained largely un-deformed under steady-state shear flow conditions for which extensive shear-thinning was present. These results represent a formidable challenge to the reptation model of melt deformation [36][37][38][39][40][41][42][43][44][45][46][47][48]. ...
Staudinger taught us that macromolecules were made up of the covalently bonded monomer repeat units chaining up as polymer chains. The chemical nature of the monomer directed the type of covalent bonds conferring most of the specific properties of the polymer. The more the number of repeat units the longer the chains and the more the possibility for the chains to assume a variety of shapes, from an extended elongated one to a more compact coiled one. Also, the chemical process that resulted in the synthesis of macromolecules produced many chains, often not with the same shape or size. The properties of the polymers improved when the chains became longer but it was more difficult to process them: their viscosity increased with molecular weight; viscosity was no longer an intensive property like it was for small liquids. The main question raised in polymer physics remains: how do these long chains interact and move as a group when submitted to shear deformation at high temperature when they are viscous liquids? This question is debated in a field of polymer physics called RHEOLOGY, whose purpose is to understand the viscoelastic aspects of polymer melts deformation. The current consensus is that we need to distinguish two cases: the deformation of “un-entangled chains” for macromolecules with molecular weight, M, smaller than Me, “the entanglement molecular weight”, and the deformation of “entangled” chains for M > Me. Several eminent scientists have extensively studied these 2 cases over the last 70 years. Paul J. Flory, in 1974, and Pierre-Gilles de Gennes, in 1991, have been awarded the Nobel Price in Chemistry and Physics, respectively, for their significant theoretical contribution to understand these challenging problems. For both of these authors the properties of polymers derive from the statistical characteristics of the macromolecule itself, the designated statistical system that defines the thermodynamic state of the polymer. Me, the molecular weight between entanglements, is defined from the rubber elasticity theory and is known to be equal to Mc/2 where Mc is the molecular weight for the entanglements when viscosity measurements are made. The current paradigm is that the viscoelasticity of un-entangled melts ( M < Mc) is well described by the Rouse model and that the entanglement issues raised by the impact of the increase of the length of the macromolecules on the melt viscoelasticity, when M > Mc, are well understood by the reptation model introduced by de Gennes and co-workers. Both models can be classified in the category of “chain dynamics statistics”. In this paper we examine in details the failures and the current challenges facing the current paradigm of polymer rheology: the Rouse model for M < Mc, the reptation model for M > Mc, the time-temperature superposition principle, the strain induced time dependence of viscosity, shear-refinement and sustained-orientation. The basic failure of the current paradigm and its inherent inability to fully describe the experimental reality is documented in this paper. In the discussion and conclusion of the paper we suggest that a different solution to explain the viscoelasticity of polymer chains and of their “entanglement” is needed. This requires a change of paradigm to describe the dynamics of the interactions within the chains and across the chains. A brief description of our currently proposed open dissipative statistical approach, “the Grain-Field Statistics”, is presented.
... Integral constitutive equations proposed more recently on the basis of molecular theories for linear and branched polymers have a more complicated mathematical structure (integrodifferential forms) and are beyond the scope of the present article. The interested reader is directed toward the developments made by Mead et al., 39 McLeish and Larson, 40 Ianniruberto and Marrucci, 41 Wagner et al., 42 Hassager et al., 43 and Hassager and Hansen, 44 to list a few. ...
The K–BKZ (Kaye–Bernstein, Kearsley, Zapas) rheological constitutive model is now 60 years old. The paper reviews the connections of the model and its variants with continuum mechanics and experimental evidence in polymer melt flow, presenting an up-to-date recap of research and major findings in the open literature. In the Introduction, an historical perspective is given on developments in the last 60 years of the K–BKZ model. Then, a section on mathematical modeling of polymer flows follows, including the governing equations of flow, the rheological constitutive equations (with emphasis on the viscoelastic integral constitutive equations of the K–BKZ type), dimensionless numbers controlling the flow, and relevant boundary conditions. The “Method of Solution” section reviews the major developments of numerical techniques for particle tracking and integral evaluation for the viscoelastic stresses. Finally, selected examples of successful application of the K–BKZ model in polymer flows are presented including considerations of wall slip and non-isothermal flows.
... We extend the model previously proposed by Costanzo et al. [5] for monodisperse polymers to the case of binary homopolymer blends. As suggested by several authors (see, e.g., [37,38]), the stress tensor σ is calculated by decoupling tube segment orientation and chain stretch: ...
... Shear banding in wormlike micelles is usually explained to result from a nonmonotonic relationship between shear stress and shear rate [3]. By contrast, since it is generally believed that entangled polymers exhibit monotonic constitutive behavior [4][5][6][7][8][9][10], the possibility of shear banding in entangled polymer solutions is hotly debated. ...
Shear banding in entangled polymer solutions is an elusive phenomenon in polymer rheology. One recently proposed mechanism for the existence of banded velocity profiles in entangled polymer solutions stems from a coupling of the flow to banded concentration profiles. Recent work [Burroughs et al., Phys. Rev. Lett . 126, 207801 (2021)] provided experimental evidence for the development of large gradients in concentration across the fluid. Here, a more systematic investigation is reported of the transient and steady-state banded velocity and concentration profiles of entangled polybutadiene in dioctyl phthalate solutions as a function of temperature [Formula: see text], number of entanglements ([Formula: see text]), and applied shear rate ([Formula: see text]), which control the susceptibility of the fluid to unstable flow-concentration coupling. The results are compared to a two-fluid model that accounts for coupling between elastic and osmotic polymer stresses, and a strong agreement is found between model predictions and measured concentration profiles. The interface locations and widths of the time-averaged, steady-state velocity profiles are quantified from high-order numerical derivatives of the data. At high levels of entanglement and large [Formula: see text], a significant wall slip is observed at both inner and outer surfaces of the flow geometry but is not a necessary criterion for a nonhomogeneous flow. Furthermore, the transient evolution of flow profiles for large Z indicate transitions from curved to “stair-stepped” and, ultimately, a banded steady state. These observed transitions provide detailed evidence for shear-induced demixing as a mechanism of shear banding in polymer solutions.
... More recent models incorporated other relaxation mechanisms, such as contour-length fluctuations, chain stretch, and convective constraint release (CCR) [3][4][5][6][7]. Some of these are the Doi-Edwards-Marrucci-Grizzuti (DEMG) model [8], which includes the effects of chain stretch on the simpler Doi-Edwards theory; the Mead-Larson-Doi (MLD) model [9], which incorporates CCR; and the Graham-Likhtman-Milner-McLeish (GLaMM) model [10], which starts from a stochastic differential equation describing the microscopic dynamics of a polymer chain that can be used to derive an equation for the stress. Another approach not based on tube theory is the slip-link model [11]. ...
Both entangled and unentangled polymer melts exhibit stress overshoots when subject to shearing flow. The size of the overshoot depends on the applied shear rate and is related to relaxation mechanisms such as reptation, chain stretch, and convective constraint release. Previous experimental work shows that melts subjected to interrupted shear flows exhibit a smaller overshoot when sheared after partial relaxation. This has been shown to be consistent with predictions by constitutive models. Here, we report molecular dynamics simulations of interrupted shear of polymer melts where the shear flow after the relaxation stage is orthogonal to the originally applied flow. We observe that, for a given relaxation time, the size of the stress overshoot under orthogonal interrupted shear is larger than observed during parallel interrupted shear, which is not captured by constitutive models. Differences in maxima are also observed for overshoots in the first normal stress and chain end-to-end distance. We also show that measurements of the average number of entanglements per chain and average orientation at different scales along the chain are affected by the change in shear direction, leading to nonmonotonic relaxation of the off-diagonal components of orientation and an appearance of a “double peak” in the average number of entanglements during the transient. We propose that such complex behavior of entanglements is responsible for the increase in the overshoots of stress components and that models of the dynamics of entanglements might be improved upon by considering a tensorial measurement of entanglements that can be coupled to orientation.
... melts under steady-state and startup of shear flow for several common variations of the tube model. Specifically, for the C 700 H 1402 melt, they compared the steady-state simulation results with predictions of Marrucci and Grizzuti's refinement of the original Doi-Edwards model (DEMG)[256] and the Mead-Larson-Doi model (MLD)[257] ...
Atomistic simulations of alkanes and polyethylenes have grown in utility and application over a 50-year period beginning at the earliest days of molecular dynamics research. This retrospective covers this period of time, aiming to present a coherent history of the development and implementation of nonequilibrium molecular dynamics simulations to one rather limited but immensely practical subject area, that of shear flows of linear, monodisperse alkane and polyethylene liquids. The development of accurate potential models to describe the energetic interactions between methyl and methylene groups is discussed at length from a historical perspective, as pertaining to the evolution of realistic united-atom models used in contemporary simulations. Molecular dynamics methodology is presented as relevant to the limited subject matter of the Review. Most importantly, the results of simulations tracing back 50 years are summarized for relevant published works known to the authors, building over time a coherent history of the subject and delineating the major impacts of the combined body of work on the field of polymer rheology, 50 years after its inception.
... Such more advanced approaches (Mead et al. 1998;Milner et al. 2001;Graham et al. 2003) consider the effects of both CCR and stretch on the dynamics of the primitive path, defined as a coarse-grained chain that has the same topology as the tube itself. Among these approaches, the most advanced one is considered to be the Graham-Likhtman-McLeish-Milner, GLaMM, model (Graham et al. 2003). ...
This work evaluates the performance of the Rolie-Poly (RP) model in uniaxial elongation and shear. Predictions of four RP versions are compared against literature data for monodisperse, entangled polystyrene (PS) chains. The examined RP versions differ in two respects, namely, the method by which they implement reptation and finite extensibility (FENE). In some cases, the reptation term leads to unrealistic strain hardening at transient shear. In contrast, predictions at relaxation after shear and uniaxial elongation are practically unaffected by variations in the reptation term. For all examined flows, the FENE treatment has a considerable influence on the model outcomes. Comparison of RP predictions and data reveals successes as well as limitations of the model. For example, some systematic disagreements are evident when the Rouse Weissenberg (WiR) number exceeds ten. Possible improvement of the model by incorporation of overlooked mechanisms like entanglement loss and chain tumbling is discussed.
... However, the interpretation of parallel superposition results is complicated by the coupling of the steady shear flow and oscillatory flow. The second study by Mead [25] uses the monodisperse and polydisperse MLD models to predict PSR and OSR moduli [26][27][28]. A potential weakness of the MLD model is that the stretch and orientation dynamics are treated using separate dynamical equations, which can lead to problems even in monodisperse rheology predictions [29]. ...
Understanding changes to microstructural dynamics under nonlinear deformations is critical for designing flow processes of entangled polymeric fluids, motivating the development of experimental methods to probe strain- and rate- dependent modifications to relaxation mechanisms. Although orthogonal superposition rheometry (OSR) holds promise as such a probe, the ability to interpret the superposition moduli accessible by OSR in the context of entangled polymer dynamics remains an open question. To fill this gap, we report model OSR predictions using detailed microstructural models for both monodisperse and polydisperse entangled polymers, i.e., the Rolie-Poly and the Rolie-Double-Poly models, respectively, which account for reptation, chain retraction, and convective constraint release. By combining numerical calculations with a perturbation analysis, we demonstrate that for polymers that can be described by a single-mode model, the OSR superposition moduli at different shear rates and frequencies can generally be collapsed onto a single master curve, with rate-dependent shift factors that depend on the nonlinear rate-dependent modification of polymer conformation and relaxation rates without changing the dominant relaxation mechanisms. We systematically study how the OSR moduli are sensitive to the shape and dispersity of the molecular weight distribution. We discuss the generality of our results for a broad class of constitutive models and suggest an analogy to Laun’s rule to relate OSR moduli to the first normal stress difference. Our results provide a foundation to guide the design and interpretation of future experiments and demonstrate that orthogonal superposition rheometry often probes features in nonlinear dynamics more directly than conventional rheometry techniques.
... Extensional rheology modeling: The modeling of the transient elongation behavior of branched polymers has been the subject of many studies. Among the models, the K-BKZ model is the best known, but it requires the determination of a damping function [12,13]. where m(t-t') is the memory function, C ¡1 is the Finger strain tensor, b is defined by the first and the second normal stress constant, and h is a damping function. ...
The objective of this work was to study the development of polypropylene (PP) foams with low density (< 100 kg m⁻³) using a CO2 batch process. To carry out this study, we used a PP-g-MA that was chemically modified by reactive extrusion to obtain polymer branched structures. Two reagent systems were selected on the basis of chemical reactions with maleic anhydride—the first one is based on the reaction between maleic anhydride and primary amines (MA/NH2, triamine, Jeffamine T-403), and the second one is based on ionic interactions from Zn neutralization. Both of these systems have shown that it is possible to obtain branched PP structures favorable to CO2 foaming and were compared to a commercial, high-melt-strength PP.
We have shown that the branched structure of polymer chains can be defined on the basis of a rheological criterion. It defines the notion of fractal behavior (coupling of relaxation modes), which polymer chains must have for nonlinear behavior (strain hardening). This criterion has been defined by analogy with the sol-gel transition, tan δ = 1.
Finally, in terms of modeling and simulation, the plasticizing effect of CO2 was modeled from the experimental rheological behavior of the PP plasticized with synthetic oil. A theoretical power law was derived and was then introduced in the mass balance equations to predict the cell growth during depressurization.
... Hence, it is accompanied by leveling in shear stress [63]. The plateaus are explained by a convective strain release and the further increase by polymer stretching [67][68][69]. In our case, we could not identify a plateau with a resumption of rising shear stress. ...
We study buffered aqueous solutions of deoxyribonucleic acid isolated from bacteriophage lambda (λ-DNA) at shear rates up to 10 5 s-1. The shear rates are accessed with a narrow-gap rheometer at gap widths down to 20 µm. At lower shear rates, our data merge with literature values. At high shear rates, the viscosity levels off into an infinite-shear viscosity plateau. Hence, the viscosity functions of buffered aqueous DNA solutions are now available for the entire shear-rate range from the first Newtonian plateau to that of infinite-shear viscosity. The latter hardly differs from the solvent viscosity. For the normal stress differences, we observe a power-law dependence on the shear rate close to previous findings up to shear rates of about 10 4 s-1. Beyond this shear-rate range, we observe a stepwise change with the shear rate. By means of agarose gel electrophoresis, we confirm that the λ-DNA is not fragmented during our rheometric study at high shear rates. Yet, at the highest shear rates studied, shear-induced changes of the DNA to structures not being able to travel through the gel appear.
... Numerous models have been proposed for improved predictions of nonlinear behaviors in fast flows or at high deformation rates [e.g. Mead et al. (1998;2015)]. A typical approach is to assume a hypothetical mechanism and add a related correction term to a current model. ...
This study focuses on comparing the individual polymer chain dynamics in an entangled polymeric liquid under different shear and extension rates. Polymer chains under various shear rates and extension rates were simulated using a stochastic-tube model [J. Rheol. 56: 1057 (2012)]. We developed a Matlab code to visualize and analyze the simulated configurations from the stochastic-tube model. We introduced new variables to determine how the extent of linearity changes with time for different shear rates, which is more useful than a typical end-to-end distance analysis. We identified whether the polymer chains undergo a tumbling rotation (slight elongation not accompanying contraction) or flipping rotation (elongation accompanying contraction). The simulation results indicate that the polymer chains exhibit a significant tendency to elongate at higher shear rates and occasionally experience flipping, while lower shear rates tend to exhibit very frequent tumbling. Furthermore, no rotations were observed under extensional flows. These results help clarifying uncertainty of previously proposed polymer deformation mechanisms of the convective constraint release and the configuration-dependent friction coefficient.
... The representative one is the tube model 6 , where the polymer motion is replaced by the single-chain dynamics confined in a tube-shaped constraint. Inspired by the remarkable success, several single-chain models have been proposed with the mean-field assumptions for the motion of surrounding chains [7][8][9][10][11][12][13][14][15][16] . Meanwhile, multi-chain models have also been developed to deal with the effects of neighboring chains explicitly [17][18][19][20][21][22][23][24][25] . ...
Although it has been established that the multi-chain slip-spring (MCSS) model can reproduce entangled polymer dynamics, the effects of model parameters have not been fully elucidated yet. In this study, we systematically investigated the effects of slip-spring density. For the diffusion and the linear viscoelasticity, the simulation results exhibited universality. Namely, the results from the simulations with various slip-spring densities can be superposed with each other by the conversion factors for the bead number per chain, unit of length, unit of time, and modulus. The diffusion and the viscoelasticity were in good agreement with the literature data for the standard bead-spring simulations, including the molecular weight dependence. The universality among the MCSS simulations with various slip-spring density also held under mild shear if the slip-spring density was not significantly high. The results imply that the level of coarse-graining for the MCSS model can be arbitrarily chosen as in the Rouse model.
... For quite a long time, this phenomenon has been experimentally observed only in worm-like micellar solutions [2] and attributed to the collapsing structure of micelles with increasing shear rate. In the case of polymer liquids, it is assumed that the formation of shear bands is impossible, even though it is theoretically predicted by a multitude of rheological models of polymer solutions and melts with non-monotonic flow curves [3,4]. ...
A Couette flow of liquid, described by a modified Vinogradov–Pokrovskii model with a non-monotonic flow curve is simulated. It is shown that the analytical solution of the stationary problem has an infinite set of solutions. The time-dependent problem is numerically simulated in the assumption that the components of the structural tensor take values corresponding to a current change in the velocity field. It is determined that the time it takes for the plate velocity to reach a given value significantly affects the velocity profile and the dependence of tangential stresses on an apparent shear rate. It is shown that, as this time decreases, the shear banding of the flow is observed not only for shear rates corresponding to the downward branch of the flow curve, but also in the entire domain of its ambiguity.
A variable-entanglement density constitutive model is developed for the description of the rheological properties of entangled polymer melts and concentrated polymer solutions using non-equilibrium thermodynamics (NET). It proposes two evolution equations: one for the average number of entanglements per chain and one for the orientation of entanglement strands. Direct comparison with non-equilibrium molecular dynamics simulation data shows that the model can accurately describe the loss of entanglements due to the applied flow for three molecular weights by using the same value for the convective constraint release (CCR) parameter. The CCR relaxation time depends on the trace of the inverse of the orientation tensor instead of an explicit dependency on the velocity gradient. Finally, the stress tensor contains an additional contribution inspired by the Curtiss-Bird or tumbling snake model. Overall, the model proposed here carefully derives via NET and builds upon the work of Ianniruberto-Marrucci when stretching is not considered.
Graphical abstract
Recent progress in creating micro and nano‐scale thermoset polymer fibers through extensional flow reveals remarkable mechanical properties. For instance, epoxy microfibers display a notable increase in stiffness, strength, and toughness as their diameter decreases. This size‐dependent behavior, well‐explored and explained in thermoplastic polymers, is far from being understood in thermoset polymers, as their densely cross‐linked network structure seems to restrain preferential directionality. Our theoretical analysis proposes that, during the pre‐gel curing phase, when the thermoset polymer begins clustering but remains in a liquid state, substantial cluster elongation is induced by the extensional flow. This elongated formation persists to some extent after curing completion, resulting in enhanced mechanical properties along the fiber's primary axis. Concurrently, the high extension reduces fiber diameter, leading to a power‐law diameter dependence of fiber stiffness. The model agrees well with experimental data from tensile tests on epoxy microfibers, highlighting the potential to fine‐tune mechanical properties by controlling the curing process, and laying the groundwork for future improvements.
Staudinger taught us that macromolecules were made up of covalently bonded monomer repeat units chaining up as polymer chains. This paradigm is not challenged in this paper. The main question raised in polymer physics remains: how do these long chains interact and move as a group when submitted to shear deformation at high temperature when they are viscous liquids? The current consensus is that we need to distinguish two cases: the deformation of “un-entangled chains” for macromolecules with molecular weight, M, smaller than Me, “the entanglement molecular weight”, and the deformation of “entangled” chains for M > Me. The current paradigm stipulates that the properties of polymers derive from the statistical characteristics of the macromolecule itself, the designated statistical system that defines the thermodynamic state of the polymer. The current paradigm claims that the viscoelasticity of un-entangled melts is well described by the Rouse model and that the entanglement issues raised when M > Me, are well understood by the reptation model introduced by de Gennes and colleagues. Both models can be classified in the category of “chain dynamics statistics”. In this paper, we examine in detail the failures and the current challenges facing the current paradigm of polymer rheology: the Rouse model for un-entangled melts, the reptation model for entangled melts, the time–temperature superposition principle, the strain-induced time dependence of viscosity, shear-refinement and sustained-orientation. The basic failure of the current paradigm and its inherent inability to fully describe the experimental reality is documented in this paper. In the discussion and conclusion sections of the paper, we suggest that a different solution to explain the viscoelasticity of polymer chains and of their “entanglement” is needed. This requires a change in paradigm to describe the dynamics of the interactions within the chains and across the chains. A brief description of our currently proposed open dissipative statistical approach, “the Grain-Field Statistics”, is presented.
The flow and transport of polymer solutions through porous media are ubiquitous in myriad scientific and engineering applications. With escalating interest in adaptive polymers, understanding the flow dynamics of their solutions is indispensable (yet lacking). Here, the hydrophobic-effect-driven reversible associations in a self-adaptive polymer (SAP) solution and its flow characteristics in a microfluidic-based "rock-on-a-chip" device have been analyzed. The hydrophobic aggregates were fluorescent labeled; this enabled a direct visualization of the in situ association/disassociation of the polymer supramolecular assemblies in pore spaces and throats. Furthermore, the influence of this adaptation on the macroscopic flow behavior of the SAP solution was analyzed by comparing its flow with that of two partially-hydrolyzed polyacrylamide (the molecular weight (MW)-equivalent HPAM-1 and ultrahigh-MW HPAM-2) solutions in the semi-dilute regime with similar initial viscosities. At low flow rates (with shear predominance), the SAP solution showed a low shear viscosity compared to HPAM-1, indicating a higher shear susceptibility for association than chain entanglement. Although the SAP exhibited the same elastic instability as the non-adaptive polymers above a threshold flow rate, the adaptable structure of the former advanced the onset of its viscoelastic-governed flow, providing a stronger flow resistance, possibly through an extension resistance. Furthermore, 3D-media analysis indicated that the reversible association/disassociation of SAP increased the accessible pore space during nonaqueous-liquid displacement, facilitating oil production.
To illustrate mechanisms of Payne effect in rubbers and their nanocomposites experiencing large amplitude oscillatory shear (LAOS), comparison studies were performed in styrene-isoprene-styrene (SIS) copolymers and their selectively crosslinked materials at temperatures below and above glass transition temperature of the polystyrene (PS) phase. It was found that under periodic dynamic shear, the strain softening is reversible when the polyisoprene (PI) phase, either crosslinked or not, is restricted by hard PS domains but it shows hysteresis once the PS domains disassociate. The strain softening can happen at the time scale of intrinsic Rouse relaxation of elastically active network strands. Critical stress of strain softening scales with number density of elastically active network strands, a simple relation being verified not only in the selectively crosslinked SIS copolymers but also in PI gum vulcanizates and carbon black filled PI compounds. Payne effect is traditionally used to term strain softening of highly filled rubber vulcanizates under LAOS deformation while evidenced herein is that the Payne effect of highly filled rubber vulcanizates shares the mechanism being common to the strain softening of SIS copolymers.
It is unsatisfactory that regarding the problem of entangled macromolecules driven out of equilibrium, experimentally based understanding is usually inferred from the ensemble average of polydisperse samples. Here, confronting with single-molecule imaging this common but poorly understood situation, over a wide range of shear rate we use single-molecule fluorescence imaging to track alignment and stretching of entangled aqueous filamentous actin filaments in a homebuilt rheo-microscope. With increasing shear rate, tube “softening” is followed by “hardening.” Physically, this means that dynamical localization first weakens from molecular alignment, then strengthens from filament stretching, even for semiflexible biopolymers shorter than their persistence length.
In this work, we mathematically derive the conditions for which empirical rheometric relations such as the Cox–Merz rule and Gleissle mirror relationship are satisfied. We consider the Wagner integral constitutive framework, which is a special limiting case of the Kaye–Bernstein Kearsley Zapas (K-BKZ) constitutive equation to derive analytical expressions for the complex viscosity, the steady shear viscosity, and the transient stress coefficient in the start-up of steady shear. We use a fractional Maxwell liquid model as the linear relaxation modulus or memory kernel within a non-linear integral constitutive framework. This formulation is especially well-suited for describing complex fluids that exhibit a broad relaxation spectrum and can be readily reduced to the canonical Maxwell model for describing viscoelastic liquids that exhibit a single dominant relaxation time. To incorporate the nonlinearities that always become important in real complex fluids at large strain amplitudes, we consider both an exponential damping function as well as a more general damping function. By evaluating analytical expressions for small amplitude oscillatory shear, steady shear, and the start-up of steady shear using these different damping functions, we show that neither the Cox–Merz rule nor the Gleissle mirror relation can be satisfied for materials with a single relaxation mode or narrow relaxation spectrum. We then evaluate the same expressions using asymptotic analysis and direct numerical integration for more representative complex fluids having a wide range of relaxation times and nonlinear responses characterized by damping functions of exponential or Soskey–Winter form. We show that for materials with broad relaxation spectra and sufficiently strong strain-dependent damping the empirical Cox–Merz rule and the Gleissle mirror relations are satisfied either exactly, or to within a constant numerical factor of order unity. By contrast, these relationships are not satisfied in other classes of complex viscoelastic materials that exhibit only weak strain-dependent damping or strain softening.
The molecular orientation in polymer fibers is investigated for the purpose of enhancing their optical properties through nanoscale control by nanowires mixed in electrospun solutions. A prototypical system, consisting of a conjugated polymer blended with polyvinylpyrrolidone, mixed with WO3 nanowires, is analyzed. A critical strain rate of the electrospinning jet is determined by theoretical modeling at which point the polymer network undergoes a stretch transition in the fiber direction, resulting in a high molecular orientation that is partially retained after solidification. Nearing a nanowire boundary, local adsorption of the polymer and hydrodynamic drag further enhance the molecular orientation. These theoretical predictions are supported by polarized scanning near-field optical microscopy experiments, where the dichroic ratio of the light transmitted by the fiber provides evidence of increased orientation nearby nanowires. The addition of nanowires to enhance molecular alignment in polymer fibers might consequently enhance properties such as photoluminescence quantum yield, polarized emission, and tailored energy migration, exploitable in light-emitting photonic and optoelectronic devices and for sensing applications.
This paper is concerned with simulating the viscoelastic flow of polystyrene melts in different contraction geometries based on single mode Rolie-Poly constitutive model. The 4:1 abrupt contraction flow of polystyrene melt is considered as a benchmark problem for validating the capabilities of Rolie-Poly model and numerical method and algorithm used in this work. Numerical results of the stress and velocity field illustrate that the present two-dimensional model and numerical method can reproduce predictions made by other published experimental and simulative results. To explore the rheological behaviors of the flowing polystyrene melt in different contraction geometries, the effect of die converging angle, contraction ratio and apparent shear rate on the flow field are discussed in detail. We compared some numerical results with the published experimental results, and reasonable matching between them was found. The present numerical results indicate that the die converging angle, contraction ratio and apparent shear rate are three important factors that are affecting the rheological behaviors of polystyrene melt in contraction flow. The magnitude of maximum PSD, first normal stress difference, and stretch ratio decrease with decreasing the converging angles, and largely reduced with a smooth entry shape. It shows that the shape of the convergence plays a major role in the foundation of extensional stress field, and a possible way to decrease the stress concentration at the contraction entrance of a die is to introduce a smooth entry shape. Numerical predictions of the distributions of principal stress difference (PSD), first normal stress difference, stretch ratio, shear stress, velocity, and so on in the flow field are also presented; the information contributes to a better understanding of the dynamic response of polymer melt forced into contraction geometries.
Graphical abstract
The current coarse-grained picture to represent polymer chain dynamics under uniaxial extensional flow (based on the Doi–Edwards model) fails to predict some scaling dependencies of material properties on deformation rate observed experimentally, specifically the monotonic thinning behavior of polymer melts. Recently, new mechanisms based on the concept of monomeric friction reduction have been proposed to explain this peculiar behavior; however, it is difficult to include them in the framework of the standard tube model. Therefore, in this work, we propose an alternative treatment which does not rule out friction reduction but uses a different approach. It considers that the chain can stretch up to a certain level that we determine based on the Pincus blob picture, in place of determining to which extend the chain stretch is reduced compared to its finite extensibility. To this end, we revisit the extensional rheological data of polystyrene melts and see how the specificities of chains under strong elongational flow can be integrated into a tube model. This requires accounting for possible flow-induced chain orientation, stretching, and disentanglement. In particular, we extend the picture of Pincus blobs and define different levels of stretch that a chain can reach as a function of the extensional rate by invoking a rate-dependent blob picture. While this approach requires introducing an additional parameter to describe the stretch relaxation time, the results are in good agreement with the experimental observations. This alternative but sound approach should contribute to the on-going discussion on the elongation of entangled polymers.
Both entangled and unentangled polymer melts exhibit stress overshoots when subject to shearing flow. The size of the overshoot depends on the applied shear rate and is related to relaxation mechanisms such as reptation, chain stretch and convective constraint release. Previous experimental work shows that melts subjected to interrupted shear flows exhibit a smaller overshoot when sheared after partial relaxation. This has been attributed to changes in the entanglement structure of the melt due to the applied flow, and the Rolie-Poly model has been used to show this behavior is consistent with tube theory. Here, we report molecular dynamics simulations of interrupted shear of polymer melts where the shear flow after the relaxation stage is orthogonal to the original applied flow. We observe that, for a given relaxation time, the size of the stress overshoot under orthogonal interrupted shear is larger than predictions of the Rolie-Poly model, and is larger than observed during parallel interrupted shear. Differences in maxima are also observed for overshoots in the first normal stress and chain end-to-end distance. We also show that measurements of the average number of entanglements per chain and average orientation at different scales along the chain are affected by the change in shear direction, leading to non-monotonic relaxation of the off-diagonal components of orientation and an appearance of a 'double peak' in the average number of entanglements during the transient.
In this work, the recently proposed frame-invariant Generalized Newtonian Fluid (GNF) constitutive equation [M. Zatloukal, “Frame-invariant formulation of novel generalized Newtonian fluid constitutive equation for polymer melts,” Phys. Fluids 32(9), 091705 (2020)] has been modified to provide uniaxial extensional viscosity at a high strain rate limit corresponding to molecular expression for a fully extended Fraenkel chain reported in Ianniruberto et al. [“Melts of linear polymers in fast flows,” Macromolecules 53(13), 5023–5033 (2020)]. It uses basic rheological and molecular parameters together with the ratio of monomeric friction coefficients for equilibrium and fully aligned chains. The modified GNF model was successfully tested by using steady-state uniaxial extensional viscosity data for well-characterized entangled polymer melts and solutions [namely, linear isotactic polypropylenes, poly(n-butyl acrylate), polyisoprenes, and polystyrenes] covering a wide range of strain rates, including those, at which the chain stretch occurs. Only two fitting parameters were sufficient to describe all uniaxial extensional viscosity data, one related to the Rouse stretch time and the other controlling the extensional thinning and thickening behavior at medium and high strain rates. The model was compared to five different advanced viscoelastic constitutive equations, which are based on Doi–Edwards theory and include chain stretch along with a number of important additions. The ability of the proposed GNF model to represent steady uniaxial extensional viscosities under fast flow conditions for entangled polymer fluids has been shown to be superior to the predictions of selected advanced viscoelastic constitutive equations. It is believed that the modified GNF model can be used in the stable modeling of non-Newtonian polymer liquids, especially in very fast steady-state flows where chain stretch begins to occur.
Rheology properties are sensitive indicators of molecular structure and dynamics. The relationship between rheology and polymer dynamics is captured in the constitutive model, which, if accurate and robust, would greatly aid molecular design and polymer processing. This dissertation is thus focused on building accurate and quantitative constitutive models that can help predict linear and non-linear viscoelasticity. In this work, we have used a multi-pronged approach based on the tube theory, coarse-grained slip-link simulations, and advanced polymeric synthetic and characterization techniques, to confront some of the outstanding problems in entangled polymer rheology. First, we modified simple tube based constitutive equations in extensional rheology and developed functional forms to test the effect of Kuhn segment alignment on a) tube diameter enlargement and b) monomeric friction reduction between subchains. We, then, used these functional forms to model extensional viscosity data for polystyrene (PS) melts and solutions. We demonstrated that the idea of reduction in segmental friction due to Kuhn alignment is successful in explaining the qualitative difference between melts and solutions in extension as revealed by recent experiments on PS. Second, we compiled literature data and used it to develop a universal tube model parameter set and prescribed their values and uncertainties for 1,4-PBd by comparing linear viscoelastic G’ and G” mastercurves for 1,4-PBds of various branching architectures. The high frequency transition region of the mastercurves superposed very well for all the 1,4-PBds irrespective of their molecular weight and architecture, indicating universality in high frequency behavior. Therefore, all three parameters of the tube model were extracted from this high frequency transition region alone. Third, we compared predictions of two versions of the tube model, Hierarchical model and BoB model against linear viscoelastic data of blends of 1,4-PBd star and linear melts. The star was carefully synthesized and characterized. We found massive failures of tube models to predict the terminal relaxation behavior of the star/linear blends. In addition, these blends were also tested against a coarse-grained slip-link model, the “Cluster Fixed Slip-link Model (CFSM)” of Schieber and coworkers. The CFSM with only two parameters gave excellent agreement with all experimental data for the blends.
Wormlike micelles (WLMs) are formed by reversible self-assembly of amphiphilic (e.g. surfactant) molecules usually with the aid of salt which acts to screen the electrostatic repulsion between the micelles to achieve giant structures. Their unique dynamics such as reversible scission give them interesting rheological properties which are sought in several industrial applications. It is therefore imperative to understand and predict these phenomena in WLMs using constitutive equations that incorporate reversible scission into the deformation dynamics of WLMs. We compare the predictions of the Vasquez-Cook-McKinley (VCM) (2007) model which treats WLMs as Hookean dumbbells that break at half-length to form two shorter dumbbells, to an analogous Brownian dynamics (BD) simulation of the same physical model. We find a discrepancy between their predictions and trace it to the absence in the VCM model of the internal position of the nascent breakage point in the long micelle, which is needed to satisfy microscopic reversibility of breakage and fusion. We correct this deficiency in the VCM model by extending an ensemble-averaged bead-spring phase space model of Wiest et al. (1989) to include reversible scission of two-spring chains. The revision tracks the conformations of the two halves of the long micelle and transmits this information to the short micelles upon breakage and thereby recovers complete agreement with the BD results. We extend the reversible scission kinetics to the slip-link tube model of Likhtman (2005) originally formulated for single polymer chains with entanglements. This facilitates the simulation of entangled WLMs and enables us to study the effects of entanglements on WLM rheology. We observe increased stresses for start-up shear flows in the entangled WLMs when breakage time was equal to reptation time. We propose that reversible scission acts as a means of constraint release which re-orientates chain segments in the velocity gradient direction and prevent retraction in the tube thereby causing increased stress. However, stress overshoot caused by the relaxation of the peak stress to a lower steady stress was observed in the fast-breaking regime. This suggests that reversible scission functions as both stress relaxation and constraint release mechanisms. We then investigate strain hardening, a nonlinear rheological property. We explore different kinds of WLMs for strain hardening and systematically study their strain hardening dependence on salt concentration and temperature. By measuring stress relaxation following a step strain, we observe that strain hardening is prevalent over a temperature range of 15 - 25 C for a solution of cetyl trimethyl ammonium bromide (CTAB) with the added hydrotrope, sodium salicylate at hydrotrope-to-surfactant concentration ratios between 0.5 - 3.0. The extent of strain hardening upon nonlinear step-strain deformation varies non-monotonically as a function of salt-to-surfactant ratio for different temperatures. A transition from strain hardening to softening or linear response is observed at strains that are dependent on temperature and concentration. Strain hardening was also observed in solutions of CTAB and hydrotrope, sodium 3-hydroxy-2-naphthoate. However, solutions of anionic sodium lauryl sulfate surfactants without hydrotrope but simple salt, sodium chloride strain softened, indicating that the hydrotrope is crucial to obtaining strain hardening in step strains. The results indicate a stress relaxation mechanism that is more complex than that of simple disentanglement and reversible scission, possibly involving strain-induced associations between micelles facilitated by hydrotropes that may act as physical crosslinkers.
Recent models have predicted entangled polymer solutions could shear band due to unstable flow-induced demixing. This work provides the first experimental probe of the in situ concentration profile of entangled polymer solutions under shear. At shear rates above a critical value, we show that the concentration and velocity profiles can develop bands, in quantitative agreement with steady-state model predictions. These findings highlight the critical importance of flow-concentration coupling in entangled polymer solutions.
Gum vulcanizates exhibit Payne effect and weak strain overshooting behavior, which is usually forgot in rubber industry. Herein the Payne effect of butadiene rubber (BR) vulcanizates is investigated for clarifying the effect of crosslinking density and sol content mediated by using coagent in conjugation with dicumyl peroxide. Toluene extraction and paraffin swelling of the vulcanizates are performed for clarifying the contribution of sol fraction and chains entanglement. The results show that the Payne effect and weak strain overshooting of gum vulcanizates are closely related to effective crosslinking density rather than network defects. This work is helpful for understanding the nonlinear rheological responses of the gum vulcanizates with defective network structure.
Filled vulcanizates exhibit the nonlinear Payne effect under dynamic deformations at high strain amplitudes, while the underlining mechanisms are still in dispute. Herein, polydimethylsiloxane (PDMS) networks are prepared via end-linking of α,ω-vinyl-terminated PDMS chains by the thiol-ene reaction, and the influences of silica on the dynamic rheological responses are investigated. The presence of silica tends to improve the crosslinking density alongside reinforcing the network, while silica itself does not contribute to the loss factor markedly. Furthermore, silica could promote the onset strain amplitude of the Payne effect and manifest the very weak overshoot of loss modulus near the onset amplitude. By creating master curves of the Payne effect of the filled vulcanizates referenced to the end-linking networks, it is evidenced that this effect and the accompanied weak overshoot are mainly rooted in the nonlinear Rouse dynamics of the network strands.
Multiscale simulation methods have been developed based on the local stress sampling strategy and applied to three flow problems with different difficulty levels: (a) general flow problems of simple fluids, (b) parallel (one-dimensional) flow problems of polymeric liquids, and (c) general (two- or three-dimensional) flow problems of polymeric liquids. In our multiscale methods, the local stress of each fluid element is calculated directly by performing microscopic or mesoscopic simulations according to the local flow quantities instead of using any constitutive relations. For general (two- or three-dimensional) flow problems of polymeric liquids (c), it is necessary to trace the history of microscopic information such as polymer-chain conformation, which carries the memories of past flow history, along the streamline of each fluid element. A Lagrangian-based CFD is thus implemented to correctly advect the polymer-chain conformation consistently with the flow. On each fluid element, coarse-grained polymer simulations are carried out to consider the dynamics of entangled polymer chains that show extremely slow relaxation compared to microscopic time scales. This method is successfully applied to simulate a flow around a cylindrical obstacle.
In some literature, entanglement molecular weight Me is discussed as a material property that characterizes magnitudes of entanglement for polymeric liquids. However, Me was introduced as a model parameter for theories that explain the plateau modulus, and the value is model dependent. A value of Me may induce misleading analysis if it is presented without noting the underlying theoretical model. Meanwhile, attempts in molecular simulations allow us to extract Me from a snapshot of entwining molecules via the topological analysis. However, the relation has not been fully clarified yet between the extracted network and the entangled polymer dynamics; the latter is the retarded molecular motion observed for dense polymeric liquids with high molecular weight. Analysis of the reduction of entanglement density under deformations is also a matter of discussion since some molecular theories can reproduce non-linear rheology with stable entanglement density.
Polymer melts with long-chain side branches and more than one junction point, such as commercial low density polyethylene (LDPE), have extensional rheology characterized by extreme strain hardening, while the shear rheology is very shear thinning, much like that of unbranched polymers. Working with the tube model for entangled polymer melts, we propose a molecular constitutive equation for an idealized polymer architecture, which, like LDPE, has multiple branch points per molecule. The idealized molecule, called a "pom-pom," has a single backbone with multiple branches emerging from each end. Because these branches are entangled with the surrounding molecules, the backbone can readily be stretched in an extensional flow, producing strain hardening. In start-up of shear, however, the backbone stretches only temporarily, and eventually collapses as the molecule is aligned, producing strain softening. Here we develop a differential/integral constitutive equation for this architecture, and show that it predicts rheology in both shear and extension that is qualitatively like that of LDPE, much more so than is possible with, for example, the K-BKZ integral constitutive equation.
Starting from the idea of equilibration, i.e., assuming that all molecular tensions are evenly distributed onto short and long chains in polydisperse polymer melts, we derive a general strain measure from a slip‐link model. By specifying disentanglement and slip of polymer chains, the strain measures of Lodge, Wagner, Doi, and Marrucci are shown to be special cases of this general strain measure. Predictions are compared to experimental data of uniaxial, planar, ellipsoidal, and equibiaxial extensions of a well‐characterized polydisperse polyisobutylene melt. The data do not support Doi’s assumption that the tube diameter remains unchanged by deformation. The relative tube diameter and its inverse, the molecular stress function, can be extracted directly from the data. The tension of the average polymer chain increases with increasing deformation, i.e., the polymer chain is stretched. At small strains, the relative cross section of the tube is inversely proportional to the average stretch of the tube.
Numerical predictions of the Doi-Edwards tube model with segmental stretch and a freely jointed chain spring model are presented for steady two-dimensional flows with a continuously varying degree of extensional and shear character. Our results are obtained from three sets of calculations by considering the effect of the flow-type parameter, the molecular weight and the number of entanglements per chain on the model predictions. The predicted degree of stretch and orientation, as well as specific rheological and optical properties that can be measured experimentally are presented. As anticipated, calculations reveal that the orientational dynamics are controlled by the reptative tube disengagement process, whereas the stretching process is controlled by the Rouse dynamics. Inclusion of segmental stretch fundamentally alters the character of the Doi-Edwards model. Calculations reveal that as the flow becomes increasingly extensional in character, significant steady state stretch is predicted with a commensurate modification of the material functions. According to calculated results, it is possible to have significant chain stretching without producing measurable changes in the stress optical coefficient.
The Doi-Edwards model with segmental stretch and a non-linear finitely extensible spring law is described and examined in simple flow situations where analytic results are derivable; namely oscillatory flow and steady state flow at high deformation rates. The model is shown to be consistent with the Bueche-Ferry hypothesis in fast large strain unidirectional flows but to violate this rule in small strain reversing flows. The discrepancy is identified with a preaveraging approximation used to describe the relative tube-chain velocity. Experimentally verifiable scaling rule for the birefringence as a universal function of a planar flow-type parameter and deformation rate are identified. Sensitivity to the extensional flow character, absent in the original tube model, manifests itself with the introduction of segmental stretch. Although the model generates a non-separable memory function kernel the deformation dependence of the memory function is quantitatively shown to have negligible impact on the predicted theological properties relative to the original Doi-Edwards model. With this simplification, relatively uncomplicated approximations to the segmental stretch model can be deduced.
Apparent slip within a few micrometers of a glass surface is visualized with a microscope using small glass spheres as tracers suspended in a highly entangled 20% solution of polystyrene of molecular weight 8.42 million, confined between glass plates in a plane Couette device. After step shears of 0.5–5 strain units, particles within a few micrometers of each surface move by 70–500 μm, enough to unload more than half of the imposed shear strain at intermediate strains. The slip displacement, when plotted logarithmically against time, show a peculiar ‘‘kink’’ in the vicinity of the ‘‘Rouse’’ time of 100 s, similar to the kink reported for the shear stress. These results show that the peculiar rheological properties reported for highly entangled solutions are produced by an effective slip phenomenon that occurs either at the wall or in a thin disentanglement layer near it.
Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 105 to 1.6 × 106 and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τw is proposed. τw is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/Me (where Me is the entanglement molecular weight). Limiting forms are obtained when M/Me ⩾ 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ1, which is identical for shear and normal stresses. τ1 can be simply related to the zero shear rate viscosity and the limiting elastic compliance.
Numerical and analytical methods are developed to invert the double reptation mixing rule to determine the molecular weight distribution (MWD). An analytic method involving Mellin transforms is developed for the case of a single exponential monodisperse relaxation function. A general analytic solution for the MWD is generated for a step‐function monodisperse relaxation function. Numerical methods are developed for more general multiple time constant monodisperse relaxation functions. Both analytical and numerical methods are simple, robust, and capable of appropriately handling error‐infected experimental data. The power‐law relaxation modulus associated with broad MWD commercial polymer is analytically inverted to generate the corresponding molecular weight distribution. MWDs calculated from rheological data for polybutadiene and polypropylene are in close agreement with the corresponding GPC data and are very sensitive to small amounts of high molecular weight material present. The fundamental origins of this sensitivity lie in the intrinsically nonlinear nature of the dependence of rheological properties on molecular weight. The quality of the results suggest that the ‘‘double reptation’’ mixing rule captures an essential feature of the physics of entangled polydisperse polymeric systems.
Stress relaxation behavior of concentrated solutions of high molecular weight polystyrene following a step shear strain is studied using both mechanical and optical birefringence techniques. Using the stress-optic law, which we find to be valid for our solutions, we obtain time-dependent shear stress and first normal stress difference values from birefringence measurement that are free of trans- ducer compliance effects. Similar to the previously reported experimental ob- servations of Fukuda, Osaki and Kurata, we obtain unusually low values for nonlinear shear moduli, much lower than the predictions of Do&Edwards model, for the sample with more than about 60 entanglements per molecule. Moreover, the shear stress and first normal stress difference data measured on this sample do not conform to the Lodge-Meissner relationship, especially at long times, suggesting the formation of regions in which the imposed strain is not homoge- neously distributed.
A study of the inelastic electron tunnelling spectra (IETS) of propynoic acid, propenoic acid and 3-methyl-but-2-enoic acid shows that these are chemisorbed from the vapour on to plasma-grown aluminium oxide as surface-bound carboxylate species. In the case of propynoic acid there is clear evidence for reaction of the triple bond at some of the adsorption sites. Similar considerations apply to the propenoic acid but not to the sterically more crowded 3-methyl-but-2-enoic acid. It is suggested that these surface reactions of the adsorbed carboxylate species may involve a Brønsted acid protonation of the π-electron systems present. Observed band intensities are held to be indicative of an adsorbate orientation to the surface which is close to the vertical.
The collection of trails blazed by Sam Edwards during half a century of fundamental research in theoretical physics is truly astonishing. He led theoretical physics into uncharted territories from his roots in quantum field theory — beginning with his seminal work on the transport properties of disordered metals, and continuing to the present day with his ground-breaking efforts to create a statistical mechanics of granular materials. Along the way, he and his collaborators developed the first modern theory of polymers in solution and in the rubbery state; created and explored the tube concept, which has had momentous implications for understanding the viscoelasticity of polymer melts; formulated the spin-glass problem and provided its first solutions using the method of replicas — work that has had profound implications in areas as diverse as combinatorial optimization, neural networks, as well as glassy systems; made important contributions to the still-unsolved problem of Navier-Stokes turbulence; and initiated the recent explosion of activity in the dynamics of growing interfaces. This book celebrates Sam's impact by collecting together and reprinting eleven of his papers, each of which played a seminal role and started a new field of study, each followed by one or more original articles by experts in the relevant fields demonstrating how the topics Sam started have developed to the modern day.
The apparent viscosities as a function of shear rate of five narrow-distribution polystyrenes were obtained using a CIL capillary rheometer. Relationships were derived between the shape of the flow curve and its dependence on molecular weight. Shift factors derived from these relationships were employed to reduce all the data (and some literature data) to a single master flow curve. It is shown that shift factors predicted by current theory do not satisfactorily reduce the data. A mechanism for non-Newtonian flow is postulated which ascribes the decrease in viscosity with increasing shear rate to an increase in the spacing of coupling entanglements.
The double‐reptation ansatz for stress relaxation in broadly polydisperse melts of linear chains relates the molecular weight distribution φ(n) and the dynamic viscosity η(ω), and has been used to infer φ(n) from rheological data. I show that the microscopic theory of constraint release implies the double‐reptation model is valid for practical distributions φ(n). I also provide a heuristic theoretical argument for the Cox–Merz empirical rule that the shapes of the functions η*(ω) and η(γ̇) are similar, which allows the shear‐thinning curve to be predicted.
The relationship between molecular weight distribution (MWD) and linear viscoelastic response of polymer melts was investigated with three commercial polymers and two blends (11 and 13 components, respectively) of nearly mono‐ disperse materials. Dynamic modulus master curves in the plateau and terminal zones were compared with predictions based on theoretically motivated combining rules, various descriptions of response for monodisperse melts, and the molecular weight distribution of each of the samples. Agreement with the Tsenoglou combining rule was excellent for the model mixtures, in which MWD was known by construction. Departures at low frequencies were found for the commercial samples, in which MWD was determined by size exclusion chromatography and light scattering. The discrepancy is attributed to uncertainties in the high molecular tail of distributions obtained by the dilute solution methods.
In the reported experiments, shear stress relaxation data were measured for a range of shear strains on several entangled polymer liquids in order to test time-strain factorability and the Doi-Edwards predictions about strain dependence. Undiluted samples and concentrated solutions of well-characterized linear and star polybutadienes with narrow molecular weight distributions were used. Some departures from time-strain factorability were found, but these appear to be nonsystematic, i. e. , they do not correlate in any obvious way with the polymer structure. The strain dependence, however, follows the pattern observed by Osaki and Kurata for polystyrene solutions. The Doi-Edwards prediction gives a good account of the results at intermediate entanglement densities (35,000 less than equivalent to cM less than equivalent to 150,000 for linear and star polybutadienes), but systematic departures are found at higher values. The causes of the latter are unknown.
The rheological constitutive equation of a condensed polymer system is presented based on the primitive chain model presented in Parts 1 and 2. The constitutive equation has the form of a BKZ equation and gives an explicit form for the memory kernel with the dependence of the rheological parameters on molecular weight and concentration. An interesting point is that it predicts a stress superposition law in the regime of non-linear viscoelasticity. The general features agree with experiments fairly well.
In this series of papers, the dynamics of polymers in melts and concentrated solutions are discussed with the eventual aim of constructing the rheological constitutive equation. The basic ideas are introduced in this paper. A mathematical model chain which describes the motion of the polymer in the fully entangled state is presented and its Brownian motion in equilibrium is studied. The model chain, called the primitive chain, shows much qualitatively different behaviour from that of the Rouse chain used in dilute solution theory.
Experiments measuring the birefringence following the inception of a steady shearing flow were conducted on a series of concentrated polyisoprene solutions. The steady‐state shear stress was also measured in a mechanical rheometer. The data obtained from the two experiments confirmed that the birefringence and stress were linearly related (stress‐optical law) over a range of shear rates that extended far into the non‐Newtonian region. Interesting nonlinear effects were also observed during the transient response at high shear rates. Both the shear stress and the first normal stress pass through maximums before reaching their steady‐state values. A maximum in the normal stress is not predicted by the original Doi‐Edwards model but a modification of their theory which includes chain stretching is in qualitative agreement with our results.
The molecular model used by Doi and Edwards in developing their rheological theory admits a simple form for the free energy of deformation, at least when the independent alignment approximation is used. A general expression is derived, as well as the specific form which applies to an instantaneous shear deformation. The latter is used to analyze the possibility that instabilities, or 'deformational phase separations,' arise in the course of stress relaxation experiments. The results of the analysis are compared with recent experimental results by Osaki and Kurata and by Vrentas and Graessley which show an anomalous relaxation behavior.
We present a theory of stress relaxation in star polymer melts with no adjustable parameters beyond those measurable in linear melts. We incorporate the effect of higher Rouse modes on star arm retraction and the Colby−Rubinstein scaling of entanglement length within “dynamic dilution”. Our results for G‘‘(ω) compare well with experimental data, with excellent agreement in shape and within a factor of 2 in time and modulus scales.
For concentrated polystyrene solutions, the effect of varying strain on the relaxation modulus can be described with the tube-model theory of Doi and Edwards at times longer than a certain characteristic time τk determined for each sample. The characteristic time is proportional to the second power of the number of entanglements per molecule and has been conjectured to define the complete equilibration of the fluctuation of the primitive path length. Here, we see that τk is about 4.5 times as large as the configurational relaxation time of an isolated (unentangled) polymer molecule in a hypothetical viscous medium that exerts the same frictional force to polymer segments as the polymer solution does. We also see that the same holds true for polystyrene solutions of relatively low concentrations if the entanglement molecular weight Me is assumed to be proportional to the -1.4 power of the concentration in this case.
Stresses and birefringences are measured for isotropic solutions of poly(γ-benzyl L-glutamate) as functions of shear rate and time at concentrations ranging from dilute to concentrated. The Doi theory and other Landau-de Gennes theories for rigid molecules predict a pretranstitional rise in the stress-optical ratio C - that is, the ratio of birefringence to stress - as the concentration approaches the value at which a first-order liquid-crystalline transition occurs. This predicted rise occurs because of critical slowing down as the concentration approaches the spinodal before being cutoff by the first-order transition. Our measurements of C for PBLG show a rise of this kind by a factor of 3.5. Kinetic rod-jamming effects in our data can be distinguished from the thermodynamic critical slowing down by combined use of stress and birefringence.
A method is applied for treating constraint release in the dynamics of branched polymers to an outstanding problem in the viscosity of star polymer melts. In the approximation of complete separation of time scales along a star arm, it is shown that cooperative effects dilute the effective entanglement by a factor 3, giving much improved quantitative agreement with rheological experiments. Polydispersity is found to affect the terminal time and viscosity only via the weight-average molecular weight.
Linear viscoelastic properties were examined on binary blends of monodisperse polystyrenes (PS) having extremely short 1-chains and relatively long 2-chains (respectively with weight-average molecular weights Mw1 and Mw2) to clarify the concept of entanglement. For dilute blends, in which 2-chains were not entangled with one another, the longest relaxation time was proportional to Mw13Mw22 for blends with Mw1 larger than the molecular weight between entanglements Me° and to Mw10Mw22 for blends with Mw1 < Me°. For concentrated blends, in which 2-chains were entangled with themselves, the 2-chains in blends with Mw1 < Me° behaved, at any frequency, as if they were in the solution, while for blends with Mw1 > Me° this solution-like behavior was observed only in the low-frequency terminal zone. These results indicate that 2-chains in dilute blends with Mw1 > Me° relax by tube renewal completely, while in concentrated blends they relax by tube renewal only partially. On the other hand, in either dilute or concentrated blends with Mw1 < Me° 1-chains do not constrain 2-chains at all. On the basis of these experimental results and the tube model, we conclude that at the onset of entanglement in monodisperse PS, the actual relaxation mode may change from the Rouse-like mode, resulting from a (virtual) very rapid tube renewal, to the reptation-like mode, presumably influenced by a partial tube renewal process.
The mechanism of constraint release for an entangled linear chain of polystyrene diffusing by reptation is emphasized, dissolving large N chains in a matrix of shorter Ns chains of the same species. Dynamic shear measurements allow us to define an average relaxation time τ̄(N,Ns) for the relaxation of the large chains, from which we derive another average relaxation time τ̄mod (N,Ns) for the tube modification induced by constraint release. We find that, according to the Klein and Graessley theories, the tube could be considered as a Rouse chain (τ̄mod(N,Ns) ∝N1.9∓0.1) but constraint release is not directly connected to the reptation of the Ns chains (τ̄mod(N,Ns)2.3±0.1). The process of constraint release included in the diffusion of chains in a binary blend at various component concentrations allows us to specify the idea of "cell" in the Graessley theory and to describe reasonably the variations of the limiting parameters η0 and Je0 with blend composition.
A theory is presented to describe the viscoelastic properties of star-shaped polymers. Disentanglement processes are considered to be dominated by diffusion of the ends of the arms in a (free energy) potential field. The existence of the potential function is supported by recent work of Helfand and Pearson on random walks entangled with a fixed obstacle net. When compared with new and existing experimental data on well-characterized polymeric stars, the theory predicts the correct molecular weight dependence of the viscosity and the steady-state shear compliance, as well as the frequency dependence of the dynamic moduli, G′(ω) and G″(ω). Useful variables for plotting time- or frequency-dependent relaxation data of star polymers are suggested by the theory and are considered.
One of the main goals of polymer science has been to relate the structure of macromolecular chains to their macroscopic properties. In particular, it has been hoped that one could relate the sizes of polymer coils to the degree to which they entangle with one another and thus to their viscoelasticity in the melt. In recent years, the availability of model polymers with nearly monodisperse molecular weight distributions and precisely controlled chemical structures has allowed for improved data both on rheology and on the dimensions of the chains. This has now allowed us to determine the correlations between such properties as chain dimensions, density, and plateau modulus and to show that some quite simple relations exist between them. The main body of these data is on polymers that can be considered to be models for polyolefins. These have been made by the hydrogeneration of polydienes synthesized by anionic polymerization techniques. In this way the molecular weight distribution can be made to be nearly monodisperse (M(w)/M(n) < 1.1) and the chemical structure is well controlled. For example, models of a wide range of ethylene-butene copolymers have been made by the saturation of polybutadienes with a range of vinyl content. Such polymers can be made at many molecular weights as well. The viscoelastic properties of these polymers have been measured very precisely, and their chain dimensions have been determined by small-angle neutron scattering. To a high degree of correlation, we find that the mean-square unperturbed end-to-end distance, 0, the density, rho, and molecular weight, M, are related to the plateau modulus, G(N)0, G(N)0 is-proportional-to { 0rho/M}3 a finding in accord with that of Ronca. This simple relationship gives us a deep understanding of what controls the rheology of these polyolefins and of how we might be able to predict the properties of as yet unsynthesized polymers.
For a series of cis-polyisoprene (PI) chains having almost identical molecular weights (M congruent-to 48K) but differently inverted type-A dipoles parallel along the chain contour, dielectric relaxation behavior was examined in homogeneous blends with a polybutadiene matrix (B-9; M = 9K). In the blends, the PI chains were dilute and entangled only with the shorter B-9 matrix. This matrix was dielectrically inert, and the dielectric loss (epsilon'') of the blends was attributed to global motion of the PI chains having type-A dipoles. Because of the differences of the location of the dipole inversion points, the PI chains of almost identical M exhibited remarkably different epsilon'' curves. As the inversion point shifted from the chain end to the center, the curve shifted to the higher frequency (omega) side and the relaxation mode distribution became first broader and then narrow again. Those PI chains relaxed locally as the shorter matrix B-9 chains rapidly diffused away and globally by accumulating such local processes. For this type of relaxation generally referred to as constraint release (CR) relaxation, the epsilon'' data enabled us to evaluate low-order eigenfunctions f(p)(n) (with the mode number p = 1-3) for a local correlation function, C(n,t;m) = (1/a2)[u(n,t).u(m,0)] (u(n,t) = nth bond vector at time t and a2 = [u2]). Those f(p)(n) were not largely but certainly different from sinusoidal eigenfunctions f(p)degrees(n) deduced from a conventional CR model assuming Rouse nature of CR processes. This Rouse CR model was further examined for an orientation function S(n,t) = (1/a2)[u(x)(n,t)u(y)(n,t)] that describes fundamental features of viscoelastic relaxation. For CR relaxation, S(n,t) can be generally expanded with respect to the dielectrically determined functions, [f(p)(n)]2. Storage moduli, G' and G-degrees', were thus calculated at low omega from the experimental f(p) and model f(p)degrees, respectively. Corresponding to the difference between f(p) and f(p)degrees, G' was not extremely but certainly larger than G-degrees' at intermediate to high omega. This result was in harmony with viscoelastic data for long and dilute probe chains entangled with much shorter matrices, meaning that the dielectric and viscoelastic relaxation processes of such probe chains are consistently described by a non-Rouse type CR mechanism. Differences between the actual CR behavior and the prediction of the Rouse CR model were discussed in relation to extra relaxation at the chain ends that was not considered in the model.
We consider the concepts of reptation and constraint release to model the dynamics of polydisperse linear polymers. The mechanisms of constraint release at work in concentrated polymer solutions can be divided into two categories. Tube dilation occurs when constraint release causes a widening of the effective tube confining the chain. Tube reorganization refers to relaxation of the tube due to motion of the surrounding chains without changing the effective tube diameter. By comparing the motion of the chain with the motion of the tube, we determine the effective tube diameter and conclude that the tube only dilates when some of the constraints are below the entanglement molecular weight. In binary mixtures of long and short chains that have entanglements between the long chains, Rouse motions of the tube are only allowed up to the length scale of entanglements between the long chains, a(L). We suggest that tube reorganization beyond those length scales occurs by reptation of the tube in the supertube of diameter a(L). Experimental tests are suggested to distinguish between the predictions of the newly proposed process of tube reptation and the older idea of tube dilation.
Strain-dependent relaxation moduli G(t,s) were measured for polystyrene solutions in diethyl phthalate with a relaxometer of the cone-and-plate type. Ranges of molecular weight M and concentration c were from 1.23 × 106 to 7.62 × 106 and 0.112 to 0.329 g/cm3. Measurements were performed at various magnitudes of shear s ranging from 0.055 to 27.2. The relaxation modulus G(t,s) always decreased with increasing s and the relative amount of decrease (i.e.,–log[G(t,s)/G(t,0)]) increased as t increased. However, the detailed strain dependences of G(t,s) could be classified into two types according to the M and c of the solution. When cM < 106, the plot of log G(t,s) versus log t varied from a convex curve to an S-shaped curve with increasing s. For solutions of cM > 106, the curves were still convex and S-shaped at very small and large s, respectively, but in a certain range of s (approximately 3 < s < 7) log G(t,s) decreased rapidly at short times and then very slowly; a peculiar inflection and a plateau appeared on the plot of log G(t,s) versus log t. The strain-dependent relaxation spectrum exhibited a trough at times corresponding to the plateau of log G(t,s). The longest relaxation time τ1(s) and the corresponding relaxation strength G1(s) were evaluated through the “Procedure X” of Tobolsky and Murakami. The relaxation time τ1(s) was independent of s for all the solutions studied while G1(s) decreased with s. The reduced relaxation strength G1(s)/G1(0) was a simple function of s (The plot of log G1(s)/G1(0) against log s was a convex curve) and was approximately independent of M and c in the range of cM <106. This behavior of G1(s)/G1(0) was in agreement with that observed for a polyisobutylene solution and seems to have wide applicability to many polymeric systems. On the other hand, log G1(s)/G1(0) as a function of log s decreased in two steps and decreased more rapidly when M or c was higher. It was suggested that in the range of cM < 106, a kind of geometrical factor might be responsible for a large part of the nonlinear behavior, while in the range of cM > 106, some “intrinsic” nonlinearity of the entanglement network system might be important.
We report measurements of the nonlinear relaxation moduli after a step-shear strain of polystyrene solutions with nearly monodisperse and with bidisperse distributions of molecular weight. We find, as have others, that for monodisperse solutions with M/Me > 60, there are anomalies, such as an unusually low nonlinear modulus and a kink in a plot of shear stress versus time after the step strain. Here M is the polymer molecular weight and Me is the entanglement molecular weight. We find that in the bidisperse solutions the anomalies persist as long as Mw/Me > 60, where Mw is the weight-averaged molecular weight of the bidisperse solution. The persistence of the anomalies in bidisperse solutions disagrees with a theory of Marrucci and Grizzuti that attributes the anomalies to strain inhomogeneities similar to shear banding. The Marrucci-Grizzuti theory predicts that as little as 10% short chains in the bidisperse mix should eliminate the anomalies, whereas in the experiments reported here at least 30% is required. Nevertheless the way in which the anomalies disappear at high strains when one increases the fraction of low-molecular-weight component is qualitatively similar to the theoretical predictions and supports the notion that strain inhomogeneities occur in these systems. © 1992 John Wiley & Sons, Inc.
A method for determining the molecular weight distribution (MWD) of a polymer melt has been developed using the dynamic elastic modulus (G'), plateau modulus (G), and zero shear complex viscosity (η). The cumulative MWD was found to be proportional to a plot of (G'/G)0.5vs. measurement frequency (ω). Frequency (ω) was found to be inversely proportional to (MW)3.4, as expected. Results were scaled to absolute values using the empirical relationship η ∝ (M̄w)3.4, where M̄w is the weight-average MW. M̄w, M̄n (number-average MW) and M̄w/M̄n calculated from melt measurements were found to agree with size exclusion chromatography usually well within 10 percent for broad and bimodal distribution samples. M̄w/M̄n tended to be approximately 20 percent higher for narrow distribution samples (M̄w/M̄n < 1.2) because we did not account for a finite distribution of relaxation times from a collection of monodisperse polymer chains. We also did not account for the plasticizing effect of short chains mixed with long ones which caused peak positions to be closer together for Theological vs, size exclusive chromatography (SEC) determinations of MW for bimodal distribution blends.
A theory for the relaxation of large strains in polymer melts is outlined. It is based on the Doi–Edwards slip-link model and the new concept of tube relaxation. Self-consistency makes this concept necessary when polymer melts are concerned. The discrepancy with previous non-self-consistent theories is not negligible and should be easy to observe experimentally. Special attention is given to the overall size and to the mean orientation of a labeled N-chain in a step-strained melt of P-chains. Using semiquantitative arguments, we predict different regimes depending on the respective values of N and P, and propose approximate evolution equations for each case. These equations may be used to design and interpret a variety of fluorescence, infrared, or NMR polarization measurements, as well as small-angle neutron scattering experiments.
A stiffened and automated Weissenberg rheogoniometer has been used to measure time-dependent nonlinear properties of several polymer solutions in simple shear flow. Departures from finite linear viscoelasticity are correlated with molecular structure. It is shown that certain features of behavior at high shear rates can be explained if the Doi–Edwards equilibration process is taken into account. A method is outlined for calculating the characteristic time τe for this process as a function of concentration, molecular weight, temperature, and branch length. This characteristic equilibration time is incorporated into a simplified BKZ constitutive equation. The proposed equation is found to adequately describe a wide range of experimental results both in step-strain and continuous shear flows.
A new model is presented for describing the time-dependent flow of entangled polymer liquids at high shear rates. The results were obtained by extending the Doi and Edwards theory to include the effect of chain stretching. This nonlinear phenomenon is predicted to occur when the product of the shear rate and longitudinal relaxation time of the polymer exceeds one. If a constant-shear-rate flow is started under these conditions, it is shown that the shear stress and the normal stress are considerably larger than that predicted by the original reptation model. We also find that both of these stresses can pass through maxima before reaching a steady state and that the times required to reach these maxima are constants independent of the shear rate. In general the new model requires the numerical solution of coupled partial differential equations. However, at the highest shear rates where reptative relaxation is no longer important, an analytical solution for the stresses is found. The results obtained here are shown to agree well with experimental data and to be an improvement over a simpler model recently proposed.
The stress relaxation behaviour of molten PMMA under large tensile deformations has been studied. Maximum values of the extension ratio were larger than 3.5. Similarly with previous results, the time dependence of the relaxing stress was found to be approximatively the same at all deformations. The deformation dependence was compared with the predictions of a theory byDoi andEdwards as well as with a modification of that theory which proves to agree better with the experimental results.Es wurde das Spannungsrelaxations-Verhalten von PMMA-Schmelzen unter groer Dehnbeanspruchung untersucht. Die maximalen Dehnungen waren dabei grer als 3,5. hnlich wie bei frheren Untersuchungen ergab sich die Zeitabhngigkeit der relaxierenden Spannung bei allen Deformationen als nahezu gleich. Der Deformationsverlauf wird mit den Voraussagen der Theorie vonDoi undEdwards, sowie denjenigen einer modifizierten Theorie verglichen. Es erweist sich, da letztere die experimentellen Ergebnisse noch besser zu beschreiben vermag.
The solution viscosity of narrow molecular weight distribution polystyrene samples dissolved in toluene and trans-decalin was investigated. The effect of polymer concentration, molecular weight and shear rate on viscosity was determined. The molecular weights lay between 5 [(g)\dot]\dot \gamma
. Different molecular weights were found to have the same viscosity in the non-Newtonian region of the flow curves and follow a straight line with a slope of – 0.83. A plot of log
0 versus logM
w
for 3 wt-% polystyrene in toluene showed a slope of approximately 3.4 in the high molecular weight regime. Increasing the shear rate resulted in a viscosity that was independent of molecular weight. The sloped (log)/d (logM
w
) was found to be zero for molecular weights at which the corresponding viscosities lay on the straight line in the power-law region.On the basis of a relation between
sp and the dimensionless productc [], simple three-term equations were developed for polystyrene in toluene andt-decalin to correlate the zero-shear viscosity with the concentration and molecular weight. These are valid over a wide concentration range, but they are restricted to molar masses greater than approximately 20000. In the limit of high molecular weights the exponent ofM
w
in the dominant term in the equations for both solvents is close to the value 3.4. That is, the correlation between
sp andc [] results in a sloped(log
sp)/d(logc []) of approximately 3.4/a at high values ofc [] wherea is the Mark-Houwink constant. This slope of 3.4/a is also the power ofc in the plot of
0 versusc at high concentrations.
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.
The flow of fluid polymers is often accompanied by flow instabilities. This phenomenon can be observed at the outlet of capillaries and is a characteristic phenomenon of polymer melts [1, 2]. In plastic processing this effect is called melt fracture. Because of the low Reynolds numbers at which melt fracture starts, it is assumed that these instabilities are caused by the viscoelastic properties of those melts. Up to now these assumptions could not be proved, because in most cases the elastic properties of the polymers are unknown at shear rates K where melt fracture begins.
Published data of the damping function of the shear relaxation modulus, h(), are reviewed. This is the ratio of the relaxation modulus measured at a finite magnitude of shear, , to that at the limit of = 0. Majority of the data are in accord with the universal function of the Doi-Edwards tube model theory, in which the damping or the decrease of h() is attributed to the contraction along the tube of extended polymer chains. The weaker damping seems to be attributed to 1) comb-branching such as in LDPE; 2) lack of entanglement in too short chains; 3) bimodal molecular weight distribution. However, a star-branching does not cause a deviation from the tube model theory and a broadness of molecular weight distribution is not a major origin of a weaker damping. A star-branched polystyrene with 15 arms exhibits no strain dependence: h() = 1. For highly entangled systems with more than 50 entanglement points per molecule, the strain dependence is stronger than that of the Doi-Edwards theory. This could be due to a slip or an instability of deformation in the material.
A recently suggested relaxation mechanism which is expected to be dominant in fast flows of polymer melts is combined here with the tube model of Doi and Edwards, and the corresponding constitutive equation is derived. The mechanism consists of the renewal of topology by convective displacement of entanglements. Detailed predictions for the flow curve in steady shear are obtained in order to test the agreement with the Cox-Merz rule. With respect to the basic tube model, significant progress is achieved.
It is argued that the flow directly contributes a mechanism by which the topological obstacles are renewed. As a consequence, the chain diffusivity is increased (or, equivalently, the friction coefficient is decreased) in fast flows. Such a nonlinear effect is incorporated in a simple dumbbell model so as to produce a constitutive equation in closed form. An extension to the case of multiple relaxation times is also briefly considered. The predictions compare favourably with the Cox-Merz rule as well as with the behaviour of polymer melts in elongational flows.
We consider the stochastic motions of a star-shaped flexible polymer trapped inside a fixed network. We predict : 1° a renewal time Tr for the star conformations which increases exponentially with the molecular mass M, and 2° a logarithmic relaxation of stress. We do not know whether these properties still hold for a homogeneous melt of stars, or not. On discute le mouvement brownien d'un polymère flexible en forme d'étoile, placé dans un réseau fixe (gel). On prédit : 1° un temps de renouvellement de la conformation globale Tr qui croit exponentiellement avec la masse M - donc beaucoup plus long que pour une chaine linéaire, 2° une relaxation logarithmique des tensions mécaniques. Nous ne savons pas, à l'heure actuelle, si ces propriétés doivent se retrouver pour des étoiles en phase fondue.
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