Nonlinear dynamics and conformational changes of linear entangled polymers using the Rolie-Poly model with an “effective” maximum contour length

The extended Rouse-CCR tube model for linear entangled polymers recently proposed by the authors (Kabanemi and Hétu, J Non Newtonian-Fluid Mech 160:113–121, 2009), designed to capture the progressive changes in the average internal structure (kinked state) of polymer chains, is here used to analyze, by means of a time-dependent three-dimensional finite element method, chain segment dynamics, pressure drop, and stability of flow through a 4:1:4 constriction in a tube. The model predicts an enhancement of the pressure drop in the stretch-dominated flow regime, which is also observed experimentally. This excess pressure drop was not associated with the onset of flow instability. The model also predicts kinked configurations within chain segments in the entry section to the constriction tube, at the inception of flow, and prior to the development of upstream vortices. It is also shown how these kinked configurations within chain segments influence pressure drop transients.     

A mesoscopic tube model of polymer/layered silicate nanocomposites

Rheology of isothermal suspensions of completely exfoliated silicate lamellae in polymer melts is investigated. In order to express more faithfully the physics involved in low shear rates and low frequencies, we model the polymer molecules composing the melt as chains whose motion is confined to a tube formed by surrounding chains and lamellae. In the absence of lamellae, the model reduces to the mesoscopic model of reptating chains developed in Eslami and Grmela (Rheol Acta, 2008). If the chains are seen only as FENE-P dumbbells, the model reduces to the model developed in Eslami et al. (J Rheol 51:1189–1222, 2007). Responses to oscillatory, transient, and steady shear flows are calculated and compared with available experimental data. Particular attention is payed to the region of low shear rate and low frequency.     

The interchain pressure effect in shear rheology

Recently, the tube diameter relaxation time in the evolution equation of the molecular stress function (MSF) model (Wagner et al., J Rheol 49: 1317–1327, 2005) with the interchain pressure effect (Marrucci and Ianniruberto, Macromolecules 37:3934–3942, 2004) included was shown to be equal to three times the Rouse time in the limit of small chain stretch. From this result, an advanced version of the MSF model was proposed, allowing modeling of the transient and steady-state elongational viscosity data of monodisperse polystyrene melts without using any nonlinear parameter, i.e., solely based on the linear viscoelastic characterization of the melts (Wagner and Rolón-Garrido 2009a, b). In this work, the same approach is extended to model experimental data in shear flow. The shear viscosity of two polybutadiene solutions (Ravindranath and Wang, J Rheol 52(3):681–695, 2008), of four styrene-butadiene random copolymer melts (Boukany et al., J Rheol 53(3):617–629, 2009), and of four polyisoprene melts (Auhl et al., J Rheol 52(3):801–835, 2008) as well as the shear viscosity and the first and second normal stress differences of a polystyrene melt (Schweizer et al., J Rheol 48(6):1345–1363, 2004), are analyzed. The capability of the MSF model with the interchain pressure effect included in the evolution equation of the chain stretch to model shear rheology on the basis of linear viscoelastic data alone is confirmed.     

Elongational viscosity of LDPE with various structures: employing a new evolution equation in MSF theory

Molecular stress function theory with new strain energy function is used to analyze transient extensional viscosity data of seven low-density polyethylene (LDPE) melts with various molecular structures as published by Stadler et al. (Rheol Acta 48:479–490, 2009) Pivokonsky et al. (J Non Newton Fluid Mech 135:58–67, 2006) and Wagner et al. (J Rheol 47(3):779–793, 2003). The new strain energy function has three nonlinear viscoelastic material parameters and assumes that the total stored energy of a branched molecule is given by different backbone and side chains stretching. The model parameters have been fitted for each LDPE in order to correlate with the supposed macromolecular structure expected from the type of synthesis. Most probable molecular structures for these LDPEs are comb and Cayley tree structures for respectively low- and high-molecular weight parts.     

The MSF model: relation of nonlinear parameters to molecular structure of long-chain branched polymer melts

The elongational viscosity data of model PS combs (Hepperle J, Einfluss der Molekularen Struktur auf Rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen. Doctoral Thesis, University Erlangen-Nürnberg, 2003) are reconsidered by including the interchain pressure term of Marrucci and Ianniruberto [Macromolecules 37:3934–3942, 2004] in the Molecular Stress Function model [Wagner et al., J Rheol 47(3):779–793, 2003, Wagner et al., J Rheol 49:1317–1327, 2005d]. Two nonlinear model parameters are needed to describe elongational flow, β and . The parameterβ determines the slope of the elongational viscosity after the inception of strain hardening. It is directly related to the molecular structure of the polymer and represents the ratio of the molar mass of the (branched) polymer to the molar mass of the backbone alone. β follows from the hypothesis of Wagner et al. [J Rheol 47(3):779–793, 2003] that side chains are compressed onto the backbone. We consider also the case that side chains are oriented by deformation, but not stretched, and found little difference in the model predictions. The parameter represents the maximum strain energy stored in the polymeric system and determines the steady-state value of the viscosity in extensional flows. The relation of this energy parameter to the molecular structure is discussed. Good correlations between the energy parameter and different coil contraction ratios, as determined either experimentally or calculated theoretically by considering the topology of the macromolecule, are found. The smaller the relative size of the polymer coil, the larger is the energy parameter and the more strain energy can be stored in the polymeric system. Presented at the 3rd Annual European Rheology Conference, AERC2006, Crete, Greece.     

Elastic yielding after step shear and during LAOS in the absence of meniscus failure

This work examines the possibility that the previously observed elastic yielding, i.e., nonquiescent relaxation after a large step shear (Ravindranath and Wang, Macromolecules 40:8031–8039, 2007) is due to an intrinsic experimental difficulty technically known as edge fracture. By redesigning the rheometric apparatus to eliminate edge failure, we show by an example of a well-entangled polymer solution that elastic yielding still occurs in the absence of any edge failure. We are also able to confirm that shear banding during large amplitude oscillatory shear (Ravindranath and Wang, J Rheol 52:341–358, 2008a) is an inherent rheological characteristic related to internal yielding of the entanglement network.     

X-ray scattering measurements of particle orientation in a sheared polymer/clay dispersion

We report steady and transient measurements of particle orientation in a clay dispersion subjected to shear flow. An organically modified clay is dispersed in a Newtonian polymer matrix at a volume fraction of 0.02, using methods previously reported by Mobuchon et al. (Rheol Acta 46: 1045, 2007). In accord with prior studies, mechanical rheometry shows yield stress-like behavior in steady shear, while time dependent growth of modulus is observed following flow cessation. Measurements of flow-induced orientation in the flow-gradient plane of simple shear flow using small-angle and wide-angle X-ray scattering (SAXS and WAXS) are reported. Both SAXS and WAXS reveal increasing particle orientation as shear rate is increased. Partial relaxation of nanoparticle orientation upon flow cessation is well correlated with time-dependent changes in complex modulus. SAXS and WAXS data provide qualitatively similar results; however, some quantitative differences are attributed to differences in the length scales probed by these techniques.     

Transient rheological responses in sheared biaxial liquid crystals

We study the rheological response of monodomain ellipsoidal biaxial liquid crystal polymers (BLCP) as well as bent-core or V-shaped liquid crystal polymers (VLCP) subject to steady and time-dependent small amplitude oscillatory shear in selected regions of the model as well as flow parameter space. We adopt the two newly developed hydrodynamical kinetic theories for ellipsoidal BLCPs and VLCPs, respectively (Sircar and Wang, PRE 78:061702, 2008, J Rheol 53:819–858, 2009; Sircar et al., Comm Math Sci (in press), 2010), in which a generalized Straley’s potential is used to represent the pairwise mean-field interaction of the mesoscopic system in biaxial phases. Transient shear stresses and normal stress differences corresponding to steady and small amplitude oscillatory shear are investigated; their variations with respect to the strength of the intermolecular potential, types of biaxial interaction, and changes in the aspect ratios for ellipsoidal BLCPs and the bent angle for VLCPs are explored.     

Numerical simulation of a drop undergoing large amplitude oscillatory shear

Direct numerical simulations are conducted for a Newtonian drop in a Newtonian matrix subjected to large amplitude oscillatory shear flows. In the experimental study of Guido et al. (in Rheol Acta 43:575–583, 2004), the drop shape is found to oscillate at higher harmonics of the forcing frequency when the capillary number is increased. Their phenomenological model requires a much smaller capillary number for predicting the harmonic nature of the experimental data. In this paper, computational results on the evolution of drop length and inclination angle are obtained at the same fluid and flow properties as the experiments, and are shown to reasonably reproduce the experimental data. In particular, the computed velocity fields around the drop are shown to elucidate the over-rotation, which is a mechanism for the experimentally observed harmonics.     

On the use of the model proposed by Leonov for the explanation of a secondary plateau of the loss modulus in heterogeneous polymer–filler systems with agglomerates

The purpose of the presented work was to test the capability of the model proposed by Leonov (J Rheol 34:1039–1068, 1990) for the prediction of secondary plateaus on the storage and loss moduli during small-amplitude oscillatory shear flow experiments on filled or heterogeneous polymer melts. Though the occurrence of a plateau on the storage modulus can be well explained in the frame of a filler network, a plateau on the loss modulus can hardly be described with the classical models. In the Leonov model, the continuum of dissipative processes is attributed to the rupture of flocs of particles. Experiments with polyolefins filled with magnesium hydroxide show that there is a clear connection between the amount of agglomerates and the occurrence of a plateau on the loss modulus. However, the value of the critical strain for floc rupture that can be calculated from the experiment shows that the processes responsible for the low-frequency dissipation are rather changes of configuration within the agglomerates than floc rupture. These processes are not described by the Leonov model, and the predicted strain dependence of the plateau is not observed experimentally.     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Rheological aspects of dense lignite–water suspensions; structure development on consecutive flow loops

Aspects of dense lignite–water slurries (LWS) rheology were investigated using controlled stress and controlled strain rheometers with parallel disks and Couette geometries. During the preparation of the slurries, the achieved solids volume fractions were up to 0.425 and the particle size distributions were polydispersed with sizes up to 300 μm. In the ascending parts of consecutive flow loops, a slope transition of the flow curve was observed and studied in relation to the solids volume fraction. The obtained results with the different geometries and rheometers were qualitatively the same. By following the model proposed by Cheng (Rheol Acta 42:372–382, 2003) for thixotropic fluids, and taking into account the yield stress appearance, a suitable correlation for LWS is proposed, which is consistent with the experimental flow curves.     

In situ observations of unusual drop deformation and wobbling in simple shear flow

Deformation and wobbling of a liquid drop immersed in a liquid matrix were studied under mild shear conditions for various viscosity ratios. In situ visualization experiments were conducted on a homemade transparent Couette cell incorporated to the Paar Physica MCR500 shear rheometer. The effect of drop or matrix elasticity was examined and was found to play a major role in both deformation and wobbling processes. Experimental results were compared to Jackson and Tucker (J Rheol 47:659–682, 2003), Maffettone and Minale (J Non-Newton Fluid Mech 78:227–241, 1998) and Yu and Bousmina (J Rheol 47:1011–1039, 2003) ellipsoidal models. It was found that the agreement between the Newtonian models and the experimental results required an increase in the drop viscosity. Such increment in viscosity was found to scale with the first normal stress difference.     

A classical problem revisited: rheology of nematic polymer monodomains in small amplitude oscillatory shear

We revisit the classical problem of the viscoelastic response of nematic (liquid crystal) polymers to small amplitude oscillatory shear. A multiple time scale perturbation analysis is applied to the Doi–Hess mesoscopic orientation tensor model to describe key features observed of longtime experiments, both physical (Moldenaers and Mewis, J Rheol, 30:567–584, 1986; Larson and Mead, J Rheol, 33:1251–1281, 1989b) and numerical (herein). First, there is a very slow time scale drift in the envelope of oscillations of the major director; we characterize the mean director angle and the envelope of oscillation. Second, there are bistable asymptotic orientational states, distinguished in that they are precisely the zero-stress orientational distributions noted in Larson and Mead (J Rheol, 33:185–206, 1989a). Third, the drift dynamics and asymptotic mean director angle are determined by the initial orientation of the director, not by material properties; we characterize the domain of attraction of each bistable state. Finally, the director drift leads to a predicted longtime decrease in the storage and loss moduli, consistent with experimental observations.

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Creep dynamics of non-entangled miscible polymer blends and block copolymers

The bead-spring model, as a fundamental model of polymer physics, has been widely utilized so far for polymer chains under the strain-controlled conditions. Nevertheless, full analysis of conformational dynamics of the polymer chains during the creep (stress-controlled) process has not been given until recently by Watanabe and Inoue (Rheol. Acta 43: 634–644, 2004a). In this paper, this analysis is extended to disordered block copolymers and miscible polymer blends for which an effect of frictional distribution/heterogeneity manifests. Due to the requirement of constant stress, segments of different blocks of the block copolymers as well as those of different components of the polymer blends exhibit correlated anisotropic change during the creep process. Furthermore, this change contains two stages with a growth of the orientational correlation among the segments, that is, the first segmental stage where the anisotropic change reflects the intrinsic mobility of segments, and the second global stage where the anisotropy approaches the steady-state defined with respect to the specific position of a chain.     

Direct comparison of equibiaxial elongational viscosity measurements from lubricated squeezing flow and the MultiAxiales Dehnrheometer

We present the first direct comparisons of rheological data from the lubricated squeezing flow (LSF) technique and the MultiAxiales Dehnrheometer (MAD) instrument developed by Meissner and coworkers (J Rheol 47:989–1010, 2003). Comparisons of transient equibiaxial elongational viscosity are carried out at strain rates well into the nonlinear regime on low-density polyethylene and polystyrene melts. We find data obtained using LSF deviate from the MAD data when the Hencky strain reaches a value of approximately 1, which we interpret as a failure of the LSF technique. The strain at which the LSF technique fails is relatively insensitive to experimental parameters including strain rate. For Hencky strains larger than 1, LSF data display behavior that could easily be mistaken for the phenomenon of strain hardening.     

RETRACTED ARTICLE: Kuhn–Grün analysis of polarized Rayleigh and Raman scattering experiments to deduce segmental orientation in polymeric systems

The intensity of Rayleigh and Raman scattered light from molecular structural units is proportional to the quadratic polarizability tensor and the derived polarizability tensor, respectively. The orientation of the polymer skeletal backbone is directly related to the orientation of the scattering structural units comprising it. The mathematical structure of the quadratic scattering tensors for a single Kuhn bond are deduced in terms of the unit vector along a Kuhn bond from symmetry considerations alone (Boehler 1987). Subsequent application of the Kuhn–Grün conditional probability analysis (Kuhn and Grün, Kolloid Z 101:248–271, 1942), which uses a freely jointed chain model, yields a general expression for the quadratic Raman and polarizability tensors for a single chain segment with five independent terms. Each term is multiplied by a spectroscopic parameter that is a complex function of the intrinsic spectroscopic tensors and the orientation distribution of monomers within an elementary Kuhn bond. A small stretch analysis of the Kuhn–Grün representation of the quadratic polarizability reveals that independent fourth moments of the segmental orientation distribution function can only be determined experimentally when the deformation or stretch of the flexible polymer is large and finite, thus severely restricting a primary advantage of the Raman and Rayleigh scattering methods. A general segmental additivity theorem is rigorously proven which demonstrates that polarized scattering experiments physically reflect the average orientation and stretch of flexible polymer skeletal backbone segments, or sub-segments, independent of chain architecture or molecular weight. Constitutive equations are fundamentally constructed to determine Kuhn bond orientation and are intrinsically related to the Kuhn–Grün analysis. The decoupling approximation, which is always invoked in Doi–Edwards type models of entangled polymeric liquids, is examined in light of the Kuhn–Grün analysis.     

Measurements and model predictions of transient elongational rheology of polymeric nanocomposites

Transient elongational rheology of two commercial-grade polypropylene (PP) and the organoclay thermoplastic nanocomposites is investigated. A specifically designed fixture consisting of two drums (SER Universal Testing Platform) mounted on a TA Instruments ARES rotational rheometer was used to measure the transient uniaxial extensional viscosity of both polypropylene and nanoclay/PP melts. The Hencky strain rate was varied from 0.001 to 2 s^− 1, and the temperature was fixed at 180°C. The measurements show that the steady-state elongational viscosity was reached at the measured Hencky strains for the polymer and for the nanocomposites. The addition of nanoclay particles to the polymer melt was found to increase the elongation viscosity principally at low strain rates. For example, at a deformation rate of 0.3 s^− 1, the steady-state elongation viscosity for polypropylene was 1.4 × 10⁴ Pa s which was raised to 2.8 × 10⁴ and 4.5 × 10⁴ Pa s after addition of 0.5 and 1.5 vol.% nanoclay, respectively. A mesoscopic rheological model originally developed to predict the motion of ellipsoid particles in viscoelastic media was modified based on the recent developments by Eslami and Grmela (Rheol Acta 47:399–415, 2008) to take into account the polymer chain reptation. We show that the orientation states of the particles and the rheological behavior of the layered particles/thermoplastic hybrids can be quantitatively explained by the proposed model.     

Nematic polymer mechanics: flow-induced anisotropy

In this paper, we model and compute flow-induced mechanical properties of nematic polymer nano-composites, consisting of transversely isotropic rigid spheroids in an isotropic matrix. Our goal is to fill a gap in the theoretical literature between random and perfectly aligned spheroidal composites (Odegard et al. in Compos. Sci. Technol. 63, 1671–1687, 2003; Gusev et al. in Adv. Eng. Mater. 4(12), 927–931 2002; Torquato in Random heterogeneous materials. Springer, Berlin Heidelberg New York, 2002; Milton in The Theory of Composites. Cambridge University Press, Cambridge, 2002) by modeling the influence of nano-particle volume fraction, flow type and flow rate on nano-composite elasticity tensors. As these influences vary, we predict the degree of elastic anisotropy, determining the number of independent moduli, and compute their values relative to the nano-particle and matrix moduli. We restrict here to monodomains, addressing features associated with orientational configurations of the rod or platelet ensemble. The key modeling advance is the transfer of symmetries (Forest et al. in Phys. Fluids 12(3), 490–498, 2000) and numerical databases (Forest et al. in Rheol. Acta 43(1), 17–37, 2004a, Rheol. Acta 44(1), 80–93, 2004b) for the orientational probability distribution function of the nematic polymer ensemble into the classical Mori–Tanaka effective elasticity tensor formalism. Isotropic, transversely isotropic, orthotropic, monoclinic, and maximally anisotropic elasticity tensors are realized as volume fraction, imposed flow type and flow strength are varied, with 2, 5, 9, 13 or 21 independent moduli for the various symmetries.     

Relaxation time spectrum molecular weight distribution relationships

Single exponential decay relationships, which define the molecular weight distribution (MWD) of a polymer as a function of the polymer’s relaxation time spectrum (RTS), have been derived by Wu (Polym Eng Sci 28:538–543, 1988) and Thimm et al. (J Rheol 43:1663–1672, 1999). Experimental validation studies with monodisperse polymers, with quite precisely known MWDs, have been used to test their reliability. It has been established that neither formula is always able to accurately recover the MWDs of monodisperse polymers from their experimentally determined RTS. In this paper, different and more general relationships, based on theoretical results of Anderssen and Loy (Bull Aust Math Soc 65:449–460, 2002a) for decays of the form , where the derivative of θ(t) is a completely monotone function, are derived, analyzed, and applied. It is shown how to transform these general relationships to equivalent single exponential decay relationships for which Laplace transform solutions are derived. In order to illustrate the interrelationship between an RTS and its corresponding MWD, an explicit analytic solution is given. The paper concludes with a discussion of the rheological implications for the BSW model.