Molecular Stress Function Research Articles

Molecular stress function theory and analysis of branching structure in industrial polyolefins

Despite substantial progress in analytical techniques for polymer characterization, a realistic picture of branching structure in industrial polymers still remains at large. Using a number of assumptions, structure-based constitutive models can distinguish between linear and branched structures in a qualitative sense. More detail on branching architecture, such as the number and length of side chains, the sequence in which they exist on the backbone and their contribution to polymer chain relaxation is more or less unknown. In the current study, elongational behavior of four commercial polyolefins is compared using the predictions of the MSF (molecular stress function) theory. The results will then be used to analyze the branching in a group of strain-hardening polypropylenes synthesized using single site catalyst.

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a 2, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter $f_{\rm max}^2 $ describing maximum stretch of the polymer chains.

On Hadamard stability and dissipative stability of the molecular stress function model of non-linear viscoelasticity

We study the stability characteristics of the molecular stress function (MSF) model, i.e., a molecular constitutive theory for stress that extends the original Doi–Edwards model for linear polymers to the case of branched polymers, by repeating the assumption that the tension in the deformed chain is equal to its equilibrium value. We derive analytical, closed-form conditions for Hadamard stability under general 3-D high-frequency, short-amplitude wave disturbances in bi-quadratic form, which reduce to simple algebraic criteria for the cases of 1-D and 2-D disturbances. Application of the derived conditions in the case of general biaxial extension, which provides a simplified description of many processes encountered in industry and nature, shows that the MSF is Hadamard unstable for strains beyond 2. This casts doubts on its ability in predicting correct elastic response under rapid extensional deformations. The region of instability widens with the strengthening of network connectivity or the alignment strength of the flow. Dissipative stability of the MSF is examined using two necessary criteria: the first and less restrictive criterion requires the stress to be monotonically and unboundedly increasing function of strain in uniaxial elongation and simple shear. The second criterion requires the free energy and the rate of energy dissipation to be bounded functions of deformation. We find that while MSF satisfies the first stability criterion, violates the second, thus revealing thermodynamic inconsistencies in formulating the dissipative terms of the constitutive equation.

Elongational dynamics of multiarm polystyrene

The startup of uni-axial elongational flow followed by stress relaxation and reversed bi-axial flow has been measured for a branched polystyrene melt with narrow molar mass distribution using the filament stretching rheometer. The branched polystyrene melt was a multiarm Aq−C−C−Aq pom-pom polystyrene with an estimated average number of arms of q=2.5. The molar mass of each arm is about 28kg∕mole with an overall molar mass of Mw=280kg∕mole. An integral molecular stress function constitutive formulation within the “interchain pressure” concept agrees reasonably well with the experiments.

Verification of branch point withdrawal in elongational flow of pom-pom polystyrene melt

According to tube model ideas, chain stretch at deformation rates below the inverse Rouse time of the chain, is only possible for polymer topologies with two or more branch points. The basic topologies, which embody this idea, are the H molecule with two side chains, and the pom-pom molecule with q>2 side chains at each end of the backbone. According to the pom-pom hypothesis, maximum chain stretch of the backbone is limited by branch point withdrawal, i.e., the side chains are drawn into the tube of the backbone as soon as the relative tension in the backbone reaches a value of q. This hypothesis, which has never been verified before, can now be tested by considering recent elongational experiments by Nielsen et al. [Macromolecules 39, 8844–8853 (2006)] on a nearly monodisperse polystyrene pom-pom melt with q=2.5. The analysis presented is based on the original integral version of the pom-pom model, and on the molecular stress function (MSF) model with strain-dependent tube diameter. The material strain measure determined from the experiments is found to be consistent with a constant maximum stretch, independent of the elongation rate, which is, however, significantly larger than q. To achieve quantitative agreement between experiment and modeling, (1) dynamic dilution of the backbone, which increases the tube diameter of the backbone and reduces equilibrium tension in the backbone, (2) finite extensibility effects, (3) stretch relaxation causing a transition from chain stretch to tube squeeze at lower strain rates, and (4) the dynamics of branch point withdrawal need to be considered. Integrating all of these features in a MSF stretch evolution equation with multiple time scales, the fundamental pom-pom hypothesis is confirmed.

Reversed extension flow

A filament stretching rheometer (FSR) was used for measuring the start-up of uni-axial elongational flow followed by reversed bi-axial flow, both with a constant elongational rate. A narrow molecular mass distribution linear polystyrene with a molecular weight of 145 kg / mole was subjected to the start-up of elongation for three Hencky strain units and subsequently the reversed flow. The integral molecular stress function formulation within the ‘interchain pressure’ concept agrees with the experiments. In the experiments the Hencky strain at which the stress becomes zero (the recovery strain) in the reversed flow has been identified. The recovery strain is found to increase with elongational rate, and has a maximum value of approximately 1.45 . The Doi Edwards model using any stretch evolution equation is not able to predict the correct level of the recovery strain.

A constitutive analysis of transient and steady-state elongational viscosities of bidisperse polystyrene blends

The transient and steady-state elongational viscosity data of three bidisperse polystyrene blends were investigated recently by Nielsen et al. [J. Rheol. 50, 453–476 (2006)]. The blends contain a monodisperse high molar mass component (ML=390kg∕mol) in a matrix of a monodisperse small molar mass component (either MS=103kg∕mol or MS=52kg∕mol at two different weight fractions). The experimental data are analyzed in the framework of the molecular stress function model of Wagner et al. [J. Rheol. 49, 1317–1327 (2005)], which is based on the assumption of a strain-dependent tube diameter and the interchain pressure term of Marrucci and Ianniruberto [Macromolecules 37, 3934–3942 (2004)]. The dilution of the long chains in the matrix of the short chains is identified as the origin of a drastic increase in the tube-diameter relaxation time of the long chains, leading to a large stretching potential of the long-chain component and an increasing steady-state elongational viscosity with increasing strain rate. In ad...

Large amplitude oscillatory elongation flow

A filament stretching rheometer (FSR) was used for measuring the elongation flow with a large amplitude oscillative elongation imposed upon the flow. The large amplitude oscillation imposed upon the elongational flow as a function of the time t was defined as $\epsilon(t) = \dot{\epsilon}_0 t + \Lambda [1 - \cos( 2 \pi \Omega \dot{\epsilon}_0 t )]$ where e is the Hencky strain, $\dot{\epsilon}_0$ is a constant elongational rate for the base elongational flow, Λ the strain amplitude (Λ ≥ 0), and Ω the strain frequency. A narrow molecular mass distribution linear polystyrene with a molecular weight of 145 kg/mol was subjected to the oscillative flow. The onset of the steady periodic regime is reached at the same Hencky strain as the onset of the steady elongational viscosity ( Λ = 0). The integral molecular stress function formulation within the ‘interchain pressure’ concept agrees qualitatively with the experiments.

Gas displacement of polymer melts in a cylinder: Experiments and viscoelastic simulations

Series of isothermal gas-displacement of a polystyrene melt in a circular cylinder were performed. The experiments show an increase in the steady fractional coverage above a Newtonian level ( m = 0.6) at very low Deborah numbers, where the melt behaves like a diluted un-entangled system. At higher Deborah numbers, where the melt behaves as an entangled melt system, the steady fractional coverage decreases. Numerical flow calculations of the time dependent axisymmetric displacement in a circular cylinder were performed using the Lagrangian integral method. A molecular stress function (MSF) constitutive model was used to describe the fluid. The finite element simulations show quantitative agreement within experimental error for Deborah numbers 0.3 < De ≤ 1, based on a MSF model without molecular extension corresponding to a Doi–Edwards strain. At higher Deborah numbers molecular extension needs to be included in the MSF model to obtain the same agreement.

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-Nurnberg, 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 \( f^{2}_{{{\text{MAX}}}} \). 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 \( f^{2}_{{{\text{MAX}}}} \) 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.

Modeling non-Gaussian extensibility effects in elongation of nearly monodisperse polystyrene melts

Elongational viscosity and birefringence of two nearly monodisperse polystyrene melts with molar mass MW of 206000gmol−1 (PS206k) and 465000gmol−1 (PS465k), respectively, were measured simultaneously by Luap et al. [Rheol. Acta. 45, 83–91 (2005)]. The samples did not follow the stress optical rule (SOR). Elongational viscosity data can be modeled quantitatively by the molecular stress function (MSF) model of Wagner et al. [J. Rheol. 49, 1317–1327 (2005)], which is based on the assumption of a strain-dependent tube diameter and the interchain pressure term of Marrucci and Ianniruberto [Macromolecules 37, 3934–3942 (2004)], and which is modified here to account for non-Gaussian chain extension using the Padé approximation of the inverse Langevin function. The tube diameter relaxation time scales with MW2. While the transient elongational viscosity shows a small dependence on finite extensibility, the predicted steady-state elongational viscosity is not affected by non-Gaussian effects. The power-law exponent of the relation between steady-state elongational stress and Deborah number predicted is found to be 0.59 for PS465k, confirming a previous result of 0.6 for a similar molecular mass sample [Wagner et al. (2005)]. The power-law exponent is invariant with respect to temperature but slightly dependent on molar mass, thereby increasing with decreasing molar mass. Deviations from the SOR are described quantitatively by the MSF model by taking into account finite chain extensibility, and within the experimental window investigated, deviations from the SOR are predicted to be strain rate-, temperature-, and molar mass independent, in good agreement with experimental data.

The Rheology of Linear and Long‐chain Branched Polymer Melts

AbstractBy generalising the Doi‐Edwards tube model to the Molecular Stress Function theory, the non‐linear rheology of polymer melts can be described quantitatively. The strain‐hardening of linear polymer melts in extensional flows can be accounted for by a strain energy function, which reflects the increase of strain energy due to tube squeeze. In comparison to linear polymer melts, long‐chain branched polymer melts show enhanced strain‐hardening. This is due to the fact that while the backbone of the branched macromolecule is stretched by deformation, side chains are compressed. It is demonstrated that the experimentally observed slope of the elongational viscosity after inception of strain‐hardening depends on the ratio β of total molar mass to backbone molar mass as predicted by the model. The steady‐state (plateau) value of the elongational viscosity depends on the maximum relative stretch, $f_{MAX}^2$, which can be supported by chain segments and which represents the maximum elastic energy storable in the polymeric system.

A modification of the convective constraint release mechanism in the molecular stress function model giving enhanced vortex growth

The molecular stress function model with convective constraint release (MSF with CCR) constitutive model [M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387] is capable of fitting all viscometric data for IUPAC LDPE, with only two adjustable parameters (with difference found only on reported “steady-state” elongational viscosities). The full MSF with CCR model is implemented in a backwards particle-tracking implementation, using an adaptive method for the computation of relative stretch that reduces simulation time many-fold, with insignificant loss of accuracy. The model is shown to give improved results over earlier versions of the MSF (without CCR) when compared to well-known experimental data from White and Kondo [J.L. White, A. Kondo, Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation, J. Non-Newtonian Fluid Mech. 3 (1977) 41]; but still to under-predict contraction flow opening angles. The discrepancy is traced to the interaction between the rotational dissipative function and the large stretch levels caused by the contraction flow. A modified combination of dissipative functions in the constraint release mechanism is proposed, which aims to reduce this interaction to allow greater strain hardening in a mixed flow. The modified constraint release mechanism is shown to fit viscometric rheological data equally well, but to give opening angles in the complex contraction flow that are much closer to the experimental data from White and Kondo. It is shown (we believe for the first time) that a constitutive model demonstrates an accurate fit to all planar elongational, uniaxial elongational and shear viscometric data, with a simultaneous agreement with this well-known experimental opening angle data. The sensitivity of results to inaccuracies caused by representing the components of the deformation gradient tensor to finite precision is examined; results are found to be insensitive to even large reductions in the precision used for the representation of components. It is shown that two models that give identical response in elongational flow, and a very similar fit to available shear data, give significantly different results in flows containing a mix of deformation modes. The implication for constitutive models is that evaluation against mixed deformation mode flow data is desirable in addition to evaluation against viscometric measurements.

Quantitative prediction of transient and steady-state elongational viscosity of nearly monodisperse polystyrene melts

Elongational behavior of four narrow molar mass distribution polystyrene melts of masses 50 000, 100 000, 200 000, and 390 000, g∕mol, respectively was investigated up to Hencky strains of 5. All melts show strain hardening behavior. For the two highest molar mass polystyrenes, strain hardening starts at elongation rates larger than the inverse reptation time, and the steady-state elongational viscosities decrease with increasing elongation rate according to a power law with a power-law index of approximately −1∕2 instead of −1 as predicted by the original Doi–Edwards tube model. Marrucci and Ianniruberto [Macromolecules 37, 3934 (2004)] have introduced an interchain pressure term arising from lateral forces between the chain and the tube wall into the Doi–Edwards model to account for the latter effect. Based on the molecular stress function theory allowing for a strain-dependent tube diameter, we show that the transient and steady-state elongational viscosities of the nearly monodisperse polystyrene melts can be modeled quantitatively by assuming affine chain deformation balanced by the interchain pressure term of Marrucci and Ianniruberto. The interchain pressure is governed by a tube diameter relaxation time τa, which is found to be larger than the Rouse time τR of the chain, and which is the only parameter of the model. For monodisperse polystyrene melts of sufficient low molar mass, τa is larger than the reptation time, and a maximum in the steady-state elongational viscosity is predicted.

Numerical simulation of large amplitude oscillatory shear of a high-density polyethylene melt using the MSF model

We study the flow response in large amplitude oscillatory shear of the molecular stress function (MSF) model that has recently been proposed by Wagner et al. [M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387–1412]. The MSF model is derived from molecular theory and has only two parameters to describe the non-linear material response. The model predictions are analysed in both the frequency and time domain. It shows good agreement with experimental data for a linear high-density polyethylene melt. At low and medium strains, MSF model predictions are in excellent agreement with experimental data and predictions of a six-mode Giesekus model which has six parameters to describe the non-linear material response. At medium strains, the basic Doi–Edwards model, which has no non-linear parameters, already underpredicts the data. At high strains, the MSF model predictions agree slightly better with the experimental data than the Giesekus model. Surprisingly, however, it is the Doi–Edwards model that shows excellent agreement with experimental data at high strains. For the linear melt we consider, it outperforms the models that have non-linear parameters, both in the time and frequency domain.

Exponential shear flow of branched polyethylenes in rotational parallel-plate geometry

Exponential shear flow, as a strong flow with the potential to generate a high degree of molecular stretching, has attracted considerable interest in recent years. So far, exponential shear flow has been realized by either sliding-plate or cone-and-plate (CP) geometry. Both geometries guarantee homogeneous shear flow. Here, we present experimental data on exponential shear flow of several long-chain branched polyethylene melts with different degrees of strain hardening obtained by using parallel-plate (PP) geometry in a rotational rheometer. This type of geometry, which is standard in linear-viscoelastic characterization of polymer materials, produces inhomogeneous shear flow. A comparison of exponential shear flow data obtained by PP and CP geometry is made. Additionally, the experimental data are compared to predictions of the rubber-like liquid (RLL) and the molecular stress function (MSF) theories. For this purpose, the relaxation spectra of the polymer melts considered were obtained by standard linear-viscoelastic characterization. In addition, two irrotational parameters and one rotational parameter are required by the MSF theory. While the irrotational parameters were obtained from fitting to elongational viscosity data, the value of the rotational parameter was used as given in the literature. It can be concluded that viable experimental data in exponential shear flow can be obtained by PP geometry. For finite linear-viscoelasticity (RLL theory), predictions of reduced shear stress for CP and PP geometry coincide, but nonlinear material behavior (as modeled by the MSF theory) leads to small differences between both geometries. Furthermore, it is shown that the MSF predictions are in excellent agreement with the experimental data in exponential shear flow and that this type of flow leads to much less chain stretching than elongational flow.

The effects of polymer melt rheology on the replication of surface microstructures in isothermal moulding

Series of isothermal compression moulding experiments were performed with a polycarbonate and a polystyrene melt in a hot press. The bottom plate in the hot press was equipped with a microstructured nickel insert. The insert contained 10 parallel, rectangularly shaped microchannels with a depth of 9.4 μ m, a width of 22 μ m and a distance between the channels of 18 μ m. The channels were positioned parallel to the incoming molten plastic flow front. The polymer melt was frozen just before the flow-front of the melt reached the end of the inserts. The partly replicated microstructures were examined using a confocal laser scanning microscope. With increasing Deborah number, defined as D e = G ′ / G ″ , there is a considerable decrease in the (non-dimensional) length the flow-front has to move in order to fill the microchannels. Numerical flow calculations were performed using the Lagrangian Integral Method where the fluid is described by a molecular stress function (MSF) constitutive model. The numerical modelling of the flow was performed on two length scales, at a macro-level describing the flow between the mould plates and at a micro-level describing the flow into the structure. The information from the macro-level was passed to the micro- level as an applied local boundary condition.

On the bursting of linear polymer melts in inflation processes

Molten LLDPE and HDPE plates (thickness 2 mm) have been inflated into a circular cylinder (inner radius 31 mm) under isothermal conditions. Low deformation rates allow the plates to be inflated considerably into the cylinder, and at high inflation rates an early burst is observed. Axis-symmetric numerical simulation of the inflations have been performed, using a constitutive equation in the form of a separable memory integral where the strain dependence is described by the Linear Molecular Stress Function (L-MSF) model with dissipative convective constraint release. The material parameters in the constitutive model are obtained using liner viscoelastic (oscillatory shear) and uni-axial elongational measurements. The numerical simulations were performed for inflation of a flat plate and a perturbed plate, where a small circular cone was removed from the centre of the surface of the plate. This was done in order to investigate the stability of the inflations. It is shown that all of the inflations are hydrodynamically unstable, though the effect on the occurrence of the burst is limited. One exception is at slow inflation, where an unexpected burst may appear as a consequence of minute deviations from an ideal flat plate. All of the numerical calculations show quantitative agreement with the experiments for a wide range of experimental conditions. This strongly suggests that the initiation of the burst is a hydrodynamic phenomenon. The critical parameters in the inflation of molten linear polymers have been investigated using the Gel equation as a memory function (M(s)=Ans −(1+n)) and inflating the plate with a constant velocity for the top of the plate. The hydrodynamic burst in a linear polymer is mainly associated with the linear viscoelastic properties and only slightly with the non-linear strain dependence. Increased (linear) elasticity reduces the inflated volume, at the same inflation velocity, before the burst occurs. Furthermore, the critical parameter for the occurrence of the burst (whether or not the burst occurs) is related to the crossover point (G′=G″) in linear viscoelasticity.

A study of the quadratic molecular stress function constitutive model in simulation

Constitutive models that conform to separable KBKZ specification have been shown to fit steady-state strain hardening rheological data in planar and uniaxial elongational flows, but with inaccuracy in the rate of strain hardening. The single parameter molecular stress function model of Wagner [Rheol. Acta 39 (2000) 97–109] has been shown to accurately fit the rise-rate in experimental data for a number of strain hardening and strain-softening materials. We study this models accuracy against the well characterised IUPAC LDPE data, and present a method for full implementation of this model for flow solution which is suitable for incorporating into existing separable KBKZ software. A new method for particle tracking in arbitrarily aligned meshes, which is efficient and robust, is given. The quadratic molecular stress function (QMSF) model is compared to existing separable KBKZ based models, including one which is capable of giving planar strain hardening; the QMSF is shown to fit experimental rheological and contraction flow data more convincingly. The issue of ‘negative correction pressures’ notable in some Doi–Edwards based models is addressed. The cause is identified, and leads to a logical method of calculation which does not give these anomalous results.

Modeling strain hardening of polydisperse polystyrene melts by molecular stress function theory

The strain hardening of blends of polystyrene (PS) and ultra-high molecular weight polystyrene (UHMW-PS) in elongational flow is modeled by the molecular stress function (MSF) theory. Assuming that the ratios of strain energies stored in polydisperse and monodisperse polymers are identical for linear and nonlinear deformations, the value of the only non-linear parameter of the theory in extensional flows, the maximum molecular stress fmax, can be determined and is shown to be related to steady-state compliance Je0. Using only linear-viscoelastic data, the elongational viscosity of PS/UHMW-PS blends is consistently predicted by the MSF theory.