Molecular stress function model

Fig. 3.7 (a) Uniaxial, (b) equibiaxial, and (c) planar extensional viscosities for a LDPE melt. [Data fromP. Hachmann, Ph.D. Dissertation, ETH, Zorich (1996).] Solid lines are predictions of the molecular stress function model constitutive equation by Wagner et al, (65,66) to be discussed in Section 3.4. 

M. H. Wagner, P. Rubio, and H. Bastian, The Molecular Stress Function Model for Polydisperse Polymer Melts with Dissipative Convective Constraint Release, J. Rheol., 45, 1387-1412 (2001). 

Wagner, M. H., Yamaguchi, M., Takahashi, M. Quantitative assessment of strain hardening of low-density polyethylene melts by the molecular stress function model./. Rheol. (2003) 47, pp. 779-793... 

It is the portion of the response curves around the maxima that are of primary interest in the characterization of nonlinear behavior, because this is where chain stretch has its most pronoimced effect. At longer times convective constraint release becomes dominant. Wagner etal. [23] used start-up of shear flow to evaluate the molecular stress function model for nonlinear behavior in which chain stretch and tube diameter are strain dependent. This theory was found to be suitable for describing an HDPE having a broad molecular weight distribution and an LDPE with random long-chain branching. 

Wagner, M. H., Rubio, P. Bastian, H. The molecular stress function model for polydisperse polynners melts with dissipative convective constraint release. /. Rheol. (2001) 45, pp. 1387-1412... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

The meaning of strain hardening and strain thinning is more clearly seen, when the effects of the linear-viscoelastic spectrum of relaxation times and the nonlinear strain measure Q on the elongational viscosity are separated. In the tube model, the strain measure can be represented by the second rank orientation tensor (describing the orientation of tube segments) and a molecular stress function f [6],... 

To describe the polymer stress in this equation, one can probe any of rheological constitutive models proposed for the long-chain branched polymers the partially extending convection (PEC) model of R. Larson, [117], the molecular stress function (MSP) theory of M. Wagner et al. [118,119], the modified extended pom-pom (mXPP) model of M.H. Verbeeten et al. [120], etc. Here, the PEC model has been chosen as it can be easily tuned to describe the overshoot position for a wide class of polymers by changing a value of the non-linear parameter Thus, = 1 in the case of linear polymer (Doi-Edwards limit), 0 < < 1 in the case of branched polymers, and = 0 in the... 

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

The requirements of theory both for solvation and transfer data of single ions are similar. A complete theory would require the knowledge of all molecular distribution functions and mean-force potentials between the ions and the solvent molecules. As already stressed in Section II such a theory is imavailable with the present state of knowledge. In the endeavour to represent solvation by models, the... 

Another, different way of utilizing the atomic hypothesis was realized in the Chemical Hamiltonian Approach [46] which exploits the LCAO model of quantum chemistry. It is important to stress that the LCAO (linear combination of atomic orbitals) expansion of the molecular wave functions is itself, in a certain sense, based on the concept of building blocks . In effect, the idea behind the LCAO method is the chemical experience that the basic building blocks of the molecules are the atoms. Thus the atomic orbitals obtained from atomic wave functions could be suitable for the expansion of molecular orbitals. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

The RNG model provides its own energy balance, which is based on the energy balance of the standard k-e model with similar changes as for the k and e balances. The RNG k-e model energy balance is defined as a transport equation for enthalpy. There are four contributions to the total change in enthalpy the temperature gradient, the total pressure differential, the internal stress, and the source term, including contributions from reaction, etc. In the traditional turbulent heat transfer model, the Prandtl number is fixed and user-defined the RNG model treats it as a variable dependent on the turbulent viscosity. It was found experimentally that the turbulent Prandtl number is indeed a function of the molecular Prandtl number and the viscosity (Kays, 1994). 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

The viscosity of some polymers at constant temperature is essentially Newtonian over a wide shear rate range. At low enough shear rates all polymers approach a Newtonian response that is, the shear stress is essentially proportional to the shear rate, and the linear slope is the viscosity. Generally, the deviation of the viscosity response to a pseudoplastic is a function of molecular weight, molecular weight distribution, polymer structure, and temperature. A model was developed by Adams and Campbell [18] that predicts the non-Newtonian shear viscosity behavior for linear polymers using four parameters. The Adams-Campbell model is as follows ... 

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