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.     

Determining polymer molecular weight distributions from rheological properties using the dual-constraint model

Although multiple models now exist for predicting the linear viscoelasticity of a polydisperse linear polymer from its molecular weight distribution (MWD) and for inverting this process by predicting the MWD from the linear rheology, such inverse predictions do not yet exist for long-chain branched polymers. Here, we develop and test a method of inverting the dual-constraint model (Pattamaprom et al., Rheol Acta 39:517–531, 2000; Pattamaprom and Larson, Macromolecules 34:5229–5237, 2001), a model that is able to predict the linear rheology of polydisperse linear and star-branched polymers. As a first step, we apply this method only to polydisperse linear polymers, by comparing the inverse predictions of the dual-constraint model to experimental GPC traces. We show that these predictions are usually at least as good, or better than, the inverse predictions obtained from the Doi–Edwards double-reptation model (Tsenoglou, ACS Polym Prepr 28:185–186, 1987; des Cloizeaux, J Europhys Lett 5:437–442, 1988; Mead, J Rheol 38:1797–1827, 1994), which we take as a “benchmark”—an acceptable invertible model for polydisperse linear polymers. By changing the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, the predictions of the dual-constraint model are slightly improved. These results suggest that models that are complex enough to predict branched polymer rheology can be inverted, at least for linear polymers, to obtain molecular weight distribution. Further work will be required to invert such models to allow prediction of the molecular weight distribution of branched polymers.     

A rheological model with constant approximate volume for immiscible blends of ellipsoidal droplets

We develop a set of evolution equations describing the effects of a general deformation field on the shape, size, and orientation of constant-volume droplets suspended in a Newtonian fluid. The rheological characteristic functions of this incompressible and immiscible polymer blend model are also derived in tandem with the abovementioned set. The constant-volume constraint is implemented using a recent methodology (Edwards BJ, Dressler M, Grmela M, Ait-Kadi A (2002) Rheological models with microstructural constraints. Rheo Acta (in press)) and discussed relative to the similarly volume-constrained model of Almusallam et al. (Almusallam AS, Larson RG, Solomon MJ (2000) A constitutive model for the prediction of ellipsoidal droplet shapes and stresses in immiscible blends. J Rheol 44:1055–1083). Sample results are reported for step-strain and shear relaxation profiles.     

Uniaxial deformation and relaxation of polymer blends: relationship between flow and morphology development

Uniaxial elongational flow followed by stress relaxation of a dilute mixture of polystyrene/polymethylmethacrylate) PS/PMMA with PS (5 wt%) as a dispersed phase was investigated. The behavior of the blend was found to be dominated by the PMMA matrix during elongation and by the interface during the relaxation at long time. Such a behavior was related to drop deformation and shape recovery during the relaxation process as was confirmed by morphological analyses on samples quenched within the rheometer just after elongation and at various times during the relaxation process. The morphology and the rheological material functions variation were compared to the Yu model (Yu W, Bousmina M, Grmela M, Palierne JF, Zhou C (2002) Quantitative relationship between rheology and morphology in emulsions. J Rheol 46(6):1381–1399).     

A method for calculating rheological and morphological properties of constant-volume polymer blend models in inhomogeneous shear fields

We show how to formulate two-point boundary-value problems in order to compute fully-developed laminar channel and tube flow profiles for viscoelastic fluid models. The formulation is applied to Couette and pressure-driven flows separately, or a combination of both. The application of this methodology is illustrated analytically for the Upper-Convected Maxwell Model, and it is applied computationally for the Phan-Thien/Tanner and Giesekus Models. Numerical solutions exist for the last two models [J.Y. Yoo, H.C. Choi, On the steady simple shear flows of the one-mode Giesekus fluid, Rheol. Acta 28 (1989) 13–24; P.J. Oliveira, F.T. Pinho, Analytical solution for fully developed channel and pipe flow of Phan-Thien–Tanner fluids, J. Fluid Mech. 387 (1999) 271–280; M.A. Alves, F.T. Pinho, P.J. Oliveira, Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid, J. Non-Newtonian Fluid Mech. 101 (2001) 55–76], allowing verification of the computational technique. Subsequently, the computational algorithm is applied to the constant-volume polymer blend models of Maffettone and Minale [P.L. Maffettone, M. Minale, Equation of change for ellipsoidal drops in viscous flow, J. Non-Newtonian Fluid Mech. 84 (1999) 105–106 (Erratum), J. Non-Newtonian Fluid Mech. 78 (1998) 227–241] and Dressler and Edwards [M. Dressler, B.J. Edwards, The influence of matrix viscoelasticity on the rheology of polymer blends, Rheol. Acta 43 (2004) 257–282; M. Dressler, B.J. Edwards, Rheology of polymer blends with matrix-phase viscoelasticity and a narrow droplet size distribution, J. Non-Newtonian Fluid Mech. 120 (2004) 189–205]. Rheological and morphological properties of the model blends are thus obtained as functions of the spatial position within the channel, applied pressure drop, and shear rate at the wall.     

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.     

Linear viscoelastic models: Part I. Relaxation modulus and melt calibration

Constitutive models for the linear viscoelasticity of polymers are presented for the relation between the relaxation modulus and the molecular weight distribution (MWD). We also compute the MWD from a simulated relaxation modulus curve by first obtaining the rheologically effective distribution (RED) as a function of time, and converting this into the MWD by melt calibration; that is, the relation between timescale and the molecular weight. This procedure has similarities with the widely used universal calibration with solved polymers. The main principles of our technique are applied here to familiar relaxation modulus data, for which we present two models: (1) an analytical model derived from control theory, which is known capable of modelling partially observed system and (2) a practical characteristic model for obtaining usable results. No relaxation time or spectrum procedures are used to model the process of linear viscoelastic relaxation. The use of relative calculations and melt calibration dispenses with the need to know the real chain structures such as branching and entangled chain dynamics, and the model remains useful for future investigations of polymer chain structures and dynamics, such as using tube theory.     

The transition between undiluted and oligomer-diluted states of nearly monodisperse polystyrenes in extensional flow

We have measured the startup and steady extensional viscosity of two narrow molar mass distributed (NMMD) polystyrenes, a 910 kg/mole and a 545 kg/mole, diluted in a NMMD 4.29 kg/mole styrene oligomer, with a wide concentration range from 90 down to 17%. The constant interchain pressure model, proposed by Rasmussen and Huang (Rheol Acta 53(3):199–208 (2014a)), predicts the extensional viscosity well for the dilutions with lower concentrations. However, for the 70 and 90% 545 kg/mole samples which represent the transition between the diluted and undiluted states, the model predictions are less satisfactory. Another concept based on interchain pressure, proposed by Wagner (Rheol Acta 53(10):765–777 (2014)), also shows agreement with the measured data.     

Predictions of linear viscoelastic properties for polydisperse entangled polymers

Different blending laws have been proposed in the literature to describe the polydispersity effect on the rheological behavior of polymer melts. In this paper predictions of linear viscoelastic properties of entangled polydisperse polymers have been derived from the double reptation mixing rule. The results in terms of the relaxation modulus, the zero shear-rate viscosity, η₀, and the steady-state compliance, J _e ⁰, have been obtained using three different relaxation functions for the monodisperse fractions, namely the Tuminello step function, the single exponential function and the BSW function. Both discrete and continuous molecular weight distributions (MWDs) have been investigated. The Generalized Exponential Function (GEX) has been considered in the continuous case. The results showed that, in systems with a large number of components, the predictions of linear viscoelastic properties mainly depend on the double reptation mixing rule assumption, while the choice of the relaxation function is not crucial. In particular, the mathematical simplicity of the Tuminello step relaxation function has allowed analytical computation of the linear viscoelastic properties in closed form. Indeed, the analytical results indicated a dependence of η₀ on the MWD that could be expressed in terms of (M _z/M _w)^0.8, in agreement with experimental results reported in the literature. In the case of J _e ⁰, the analytical model defines a dependence on (M _z/M _w)^5.5, i.e. as expected a strong dependence on the MWD is predicted for the steady-state compliance. Finally, dynamic moduli have been computed from the relaxation modulus and their predictions have been favorably compared with experimental results from the literature. Received: 19 July 1999/Accepted: 24 November 1999     

A regularization-free method for the calculation of molecular weight distributions from dynamic moduli data

There are several models for the determination of molecular weight distributions (MWDs) of linear, entangled, polymer melts via rheometry. Typically, however, models require a priori knowledge of the critical molecular weight, the plateau modulus, and parameters relating relaxation time and molecular weight (e.g., k and in =kM). Also, in an effort to obtain the most general MWD or to describe certain polymer relaxation mechanisms, models often rely on the inversion of integral equations via regularization. Here, the inversion of integral equations is avoided by using a simple double-reptation model and assuming that the MWD can be described by an analytic function. Moreover, by taking advantage of dimensionless variables and explicit analytic relations, we have developed an unambiguous and virtually parameter-free methodology for the determination of MWDs via rheometry. Unimodal MWDs have been determined using only a priori knowledge of the exponent and dynamic moduli data. In addition, the uncertainty in rheological MWD determinations has been quantified, and it is shown that the reliability of the predictions is greater for the high-molecular-weight portion of the distribution.     

A 2D model for tube orientation and tube squeezing in fast flows of polymer melts

Motivated by recent data of Hassager and coworkers [A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Elongational viscosity of narrow molar mass distribution polystyrene, Macromolecules, 36 (2003) 5174–5179], we develop a new tube model describing the non-linear behaviour of entangled monodisperse linear polymers. Within the context of well established tube theories, the model accounts for the effect of flow on tube diameter, thus somehow picking up a long standing suggestion by Wagner and Schaeffer [M.H. Wagner, J. Schaeffer, Non-linear strain measures for general biaxial extension of polymer melts, J. Rheol., 32 (1992) 1–26] among others. Since tubes with a deformation dependent cross section are rather difficult to deal with, we here limit model development to an artificial two-dimensional situation. The simple 2D model has the advantage of illustrating the new physical assumptions more transparently, and it already proves to correctly predict most qualitative features typically shown by shear and elongational data in the non-linear range.     

Interchain tube pressure effect in extensional flows of oligomer diluted nearly monodisperse polystyrene melts

We have derived a constitutive equation to explain the extensional dynamics of oligomer-diluted monodisperse polymers, if the length of the diluent has at least two Kuhn steps. These polymer systems have a flow dynamics which distinguish from pure monodisperse melts and solutions thereof, if the solvent has less than two Kuhn steps, e.g. is not a chain. The constitutive equation is based on a phenomenological tube-based model within the methodology of the molecular stress function approach. The nonlinear dynamics have been explained as a consequence of a constant thermal interchain pressure originating from the short polymer chains (e.g. the oligomers) on the wall of the tube containing the long chains. The nonlinear dynamics are uniquely defined by the Rouse time and the maximal extensibility of the long polymer chains. Both are linked to the entanglement length. The relation between the Rouse times and entanglements have been established based on published extensional experiments on nearly monodisperse polystyrene melts. The constitutive equation has shown agreement with the experimental startup of and steady extension data from Huang et al. (Macromolecules 46:5026–5035, 2013a) based on 285 and 545 kg/mol polystyrenes diluted in styrene oligomers containing 3.3 (1.92 kg/mol) and 7.3 (4.29 kg/mol) Kuhn steps.     

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.     

Recoil from elongation using general network models

In this paper, we use two new models and the irreversible KBKZ model of Wagner (Rheol Acta 18:681–692, 1979) to describe the famous experiments of Meissner (Rheol Acta 10:230–242, 1971) on the recoil of polyethylene. The new models are based both on network and reptation-type ideas. One of the new models (PTT-X) is a member of the PTT family and shows good agreement with polyethylene data in shearing, elongation, and recoil from elongation.     

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.     

Thermodynamic admissibility of the extended Pom-Pom model for branched polymers

The thermodynamic consistency of the eXtended Pom-Pom (XPP) model for branched polymers of Verbeeten et al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model, J. Rheol. 45 (4) (2001) 823–843; W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model (vol 45, pg 823–843, 2001), J. Rheol. 45 (6) (2001) 1489] as well as its modified version [J. van Meerveld, Note on the thermodynamic consistency of the integral pom-pom model, J. Non-Newtonian Fluid Mech. 108 (1–3) (2002) 291–299] is investigated from the perspective of non-equilibrium thermodynamics, namely the General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) framework. The thermodynamic admissibility of the XPP model is shown for both its original and modified form. According to the GENERIC formalism, the parameter α introduced by Verbeeten et al. to predict non-zero second normal stress in shear flows must fulfill the condition 0 ≤ α ≤ 1.     

Rheological behavior of poly(methyl methacrylate)/polystyrene (PMMA/PS) blends with the addition of PMMA-<Emphasis Type="BoldItalic">ran</Emphasis>-PS

In this work, the dynamic behavior of poly(methyl methacrylate)/polystyrene blend to which P(S_0.5-ran-MMA_0.5) was added was studied. Several blend (ranging from 5 to 20 wt% of dispersed phase) and copolymer (up to 20 wt% with respect to dispersed phase) concentrations were studied. The rheological behavior of the blends was compared to Bousmina’s (Rheol Acta 38:73–83, 1999) and Palierne’s (Rheol Acta 29:204–214, 1990) generalized models. The relaxation spectra of the blends were also inferred, and the results were analyzed in light of the analysis of Jacobs et al. [J Rheol 43:1495–1509, 1999]. The relaxation spectra of the blends with smaller dispersed phase (below 10 wt%) and larger copolymer concentrations (above 0.4 wt%) showed the presence of four relaxation times, two corresponding to the blend phases, τ _F , corresponding to the relaxation of the shape of the dispersed phase of the blend and that can be attributed to the relaxation of Marangoni stresses tangential to the interface between the dispersed phase and matrix. The experimental values of and were used to infer the interfacial tension (Γ) and the interfacial complex shear modulus (β) for the different blends, Γ decreased with increasing copolymer concentration. β decreased with increasing blend dispersed phase concentration and decreasing copolymer concentration. The predictions of Palierne’s generalized model were found to corroborate the experimental data once the values of Γ and β, found analyzing the relaxation spectra, were used in the calculations. Bousmina’s model was found to corroborate the data only for larger dispersed phase concentration. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

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.     

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.