Constitutive equations for polymer mole and rubbers: Lessons from the $20^{th}$ century

Wagner, Manfred H. The Korean Society of Rheology 1999 Korea-Australia rheology journal Vol.11 No.4

Refinements of classical theories for entangled or crosslinked polymeric systems have led to incommensurable models for rubber networks and polymer melts, contrary to experimental evidence, which suggests a great deal of similarity. Uniaxial elongation and compression data of linear and branched polymer melts as well as of crosslinked rubbers were analyzed with respect to their nonlinear strain measure. This was found to be the result of two contributions: (1) affine orientation of network strands, and (2) isotropic strand extension. Network strand extension is caused by an increasing restriction of lateral movement of polymer chains due to deformation, and is modelled by a molecular stress function which in the tube concept of Doi and Edwards is the inverse of the relative tube diameter. Up to moderate strains, $f^2$ is found to be linear in the average stretch for melts as well as for rubbers, which corresponds to a constant tube volume. At large strains, rubbers show maximum extensibility, while melts show maximum molecular tension. This maximum value of the molecular stress function governs the ultimate magnitude of the strain-hardening effect of linear and long-chain branched polymer melts in extensional flows.

Nonlinear rheology of linear polymer melts: Modeling chain stretch by interchain tube pressure and Rouse time

Wagner, Manfred H.,Rolon-Garrido, Victor H. The Korean Society of Rheology 2009 Korea-Australia rheology journal Vol.21 No.4

In flows with deformation rates larger than the inverse Rouse time of the polymer chain, chains are stretched and their confining tubes become increasingly anisotropic. The pressures exerted by a polymer chain on the walls of an anisotropic confinement are anisotropic and limit chain stretch. In the Molecular Stress Function (MSF) model, chain stretch is balanced by an interchain pressure term, which is inverse proportional to the $3^{rd}$ power of the tube diameter and is characterized by a tube diameter relaxation time. We show that the tube diameter relaxation time is equal to 3 times the Rouse time in the limit of small chain stretch. At larger deformations, we argue that chain stretch is balanced by two restoring tensions with weights of 1/3 in the longitudinal direction of the tube (due to a linear spring force) and 2/3 in the lateral direction (due to the nonlinear interchain pressure), both of which are characterized by the Rouse time. This approach is shown to be in quantitative agreement with transient and steady-state elongational viscosity data of two monodisperse polystyrene melts without using any nonlinear parameter, i.e. solely based on the linear-viscoelastic characterization of the melts. The same approach is extended to model experimental data of four styrene-butadiene random copolymer melts in shear flow. Thus for monodisperse linear polymer melts, for the first time a constitutive equation is presented which allows quantitative modeling of nonlinear extension and shear rheology on the basis of linear-viscoelastic data alone.

Nonlinear rheology of polymer melts: a new perspective on finite chain extensibility effects

Wagner Manfred H. The Korean Society of Rheology 2006 Korea-Australia rheology journal Vol.18 No.4

Measurements by Luap et al. (2005) of elongational viscosity and birefringence of two nearly monodisperse polystyrene melts with molar masses $M_{w}$ of $206,000g{\cdot}mol^{-1}$ (PS206k) and $465,000g{\cdot}mol^{-1}$ (PS465k) respectively are reconsidered. At higher elongational stresses, the samples showed clearly deviations from the stress optical rule (SOR). The elongational viscosity data of both melts can be modeled quantitatively by the MSF model of Wagner et al. (2005), which is based on the assumption of a strain-dependent tube diameter and the interchain pressure term of Marrucci and Ianniruberto (2004). The only nonlinear parameter of the model, the tube diameter relaxation time, scales with $M_{w}^{2}$. In order to get agreement with the birefringence data, finite chain extensibility effects are taken into account by use of the $Pad\'{e}$ approximation of the inverse Langevin function, and the interchain pressure term is modified accordingly. Due to a selfregulating limitation of chain stretch by the FENE interchain pressure term, the transient elongational viscosity shows a small dependence on finite extensibility only, while the predicted steady-state elongational viscosity is not affected by non-Gaussian effects in agreement with experimental evidence. However, 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 for the two nearly monodisperse polystyrene melts in good agreement with experimental data.

Nonlinear rheology of polymer melts: a new perspective on finite chain extensibility effects

Manfred H. Wagner 한국유변학회 2006 Korea-Australia rheology journal Vol.18 No.4

Measurements by Luap et al. (2005) of elongational viscosity and birefringence of two nearly monodisperse polystyrene melts with molar masses MW of 206,000gmol1 (PS206k) and 465,000gmol1 (PS465k) respectively are reconsidered. At higher elongational stresses, the samples showed clearly deviations from the stress optical rule (SOR). The elongational viscosity data of both melts can be modeled quantitatively by the MSF model of Wagner et al. (2005), which is based on the assumption of a strain-dependent tube diameter and the interchain pressure term of Marrucci and Ianniruberto (2004). The only nonlinear parameter of the model, the tube diameter relaxation time, scales with MW2. In order to get agreement with the birefringence data, finite chain extensibility effects are taken into account by use of the Pad approximation of the inverse Langevin function, and the interchain pressure term is modified accordingly. Due to a self-regulating limitation of chain stretch by the FENE interchain pressure term, the transient elongational viscosity shows a small dependence on finite extensibility only, while the predicted steady-state elongational viscosity is not affected by non-Gaussian effects in agreement with experimental evidence. However, 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 for the two nearly monodisperse polystyrene melts in good agreement with experimental data.

Nonlinear rheology of linear polymer melts: Modeling chain stretch by interchain tube pressure and Rouse time

Manfred H. Wagner,Víctor H. Rolón-Garrido 한국유변학회 2009 Korea-Australia rheology journal Vol.21 No.4

In flows with deformation rates larger than the inverse Rouse time of the polymer chain, chains are stretched and their confining tubes become increasingly anisotropic. The pressures exerted by a polymer chain on the walls of an anisotropic confinement are anisotropic and limit chain stretch. In the Molecular Stress Function (MSF) model, chain stretch is balanced by an interchain pressure term, which is inverse proportional to the 3rd power of the tube diameter and is characterized by a tube diameter relaxation time. We show that the tube diameter relaxation time is equal to 3 times the Rouse time in the limit of small chain stretch. At larger deformations, we argue that chain stretch is balanced by two restoring tensions with weights of 1/3 in the longitudinal direction of the tube (due to a linear spring force) and 2/3 in the lateral direction (due to the nonlinear interchain pressure), both of which are characterized by the Rouse time. This approach is shown to be in quantitative agreement with transient and steady-state elongational viscosity data of two monodisperse polystyrene melts without using any nonlinear parameter, i.e. solely based on the linear-viscoelastic characterization of the melts. The same approach is extended to model experimental data of four styrene-butadiene random copolymer melts in shear flow. Thus for monodisperse linear polymer melts, for the first time a constitutive equation is presented which allows quantitative modeling of nonlinear extension and shear rheology on the basis of linear-viscoelastic data alone.