Molecular stress function

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. 

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 ... 

In the MSF theory, the function,/, in addition to simple reptation, is also related to both the elastic effects of tube diameter reduction, through the Helmholtz free energy, and to dissipative, convective molecular-constraint mechanisms. Wagner et al. arrive at two differential equations for the molecular stress function/ one for linear polymers and one for branched. Both require only two trial-and-error determined parameters. 

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). 

SWS7 Molecular Stress Function theory orientation tensor (3.4-10)... 

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],... 

Figure 4 shows the square of the molecular stress function, p, as a function of the average deformation ( ) for linear and long-chain branched polymer melts and for crosslinked rubbers (NR, PDMS). It is clear from Figure 4 that the amount of molecular stress which can be induced in linear macromolecules by deformation in the melt state, is... 

Figure 4 The square of the molecular stress function, f, as a function of average deformation -...

Keywords elongational viscosity, melt strength, extensibility, strain hardening, strain thinning, molecular stress function, flow induced crystallization, fiber spinning, blow molding, foam extrusion. 

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... 

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... 

Another strain measure that is closely related to the one defined by Eq. 10.14 was proposed by Wagner etal. [ 14]. This tensor involves a new scalar, which they call the molecular stress function. When used in an integral constitutive equation it was found to be able to describe the behavior of a high-density polyethylene in shear flow and several types of extensional deformation. 

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... 

Diveu C, Venereau E, Froger J, et al Molecular and functional characterization of a soluble form of oncostatin M/interleukin-31 shared receptor. J Biol Chem 2006 281 36673-36682. Stress C, Radtke S, Clahsen T, et al Oncostatin M receptor-mediated signal transduction is negatively regulated by SOCS3 through a receptor tyrosine-independent mechanism. J Biol Chem 2006 281 8458-8468. 

Beyond Cytotoxicity The Advent of Molecular and Functional Markers of Stress... 

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... 

We have to stress that these eonclusions will be valid independent of the approximations used to eompute the molecular wave functions. The reason is that they follow from the symmetry, which is identical for the exact and approximate wave functions. 

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. 

As discussed previously molecular stresses are a function of molecular strains which depend on sample morphology and chain orientation. Any real sample, therefore, contains a variety of differently stressed oscillators. The profile of the deformed band, D(p), representing such a system of oscillators can be expressed by a convolution integral ... 

IR spectroscopy can be used to study the effects of applied mechanical stress on highly oriented samples. The goal is to obtain the molecular stress distribution function, which is an important quantity for determining the stress relaxation moduli or creep compliances. Shifts in the peak frequencies are observed and an attempt is made to determine the molecular stress distribution by deconvolution [34]. The shifts as a function of stress for the 1168-cm band of oriented isotactic polypropylene [35] are shown in Fig. 4.37. 

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

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