Molecular stresses

Several attempts to relate the rate for bond scission (kc) with the molecular stress ( jr) have been reported over the years, most of them could be formally traced back to de Boer s model of a stressed bond [92] and Eyring s formulation of the transition state theory [94]. Yew and Davidson [99], in their shearing experiment with DNA, considered the hydrodynamic drag contribution to the tensile force exerted on the bond when the reactant molecule enters the activated state. If this force is exerted along the reaction coordinate over a distance 81, the activation energy for bond dissociation would be reduced by the amount ... 

The Stokes viscous drag equation predicts a proportionality between the molecular stress ( / ) with the product of solvent viscosity (qs) and fluid strain-rate... 

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)... 

In three directions, there will be nine stress components Ty. The viscous forces appear only when there are velocity gradients within the fluid. The forces per unit area (molecular stresses) acting on the body, it, both by the thermodynamic pressure and by the viscous stresses are given by... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

Thus, three mechanisms of alignment—one for frequencies and temperatures at which molecular stresses dominate, and the other two at conditions where stresses from the lamellar pattern dominate—can rationalize the alignment behavior of the low-molecular-weight PS-... 

Finally, a number of associated techniques have been used to sited light on the phenomenon of molecular fracture. These techniques do not estimate the inddence of fracture events but rather omcern phenomena or structures whidi bear strongly on the subject. Thus infra-red spectroscopy is employed to demonstrate the state of molecular stress prior to fracture, whilst low angle X-ray studies can be used both to assess the distribution of strain within a semi-crystalline polymer and to detect and measure the microvoids which are assodated with molecular fracture in such materials. 

Kausch and Becht proposed tlrat the local molecular stress could be written as ... 

Although numerous turbulence models are reported in the literature,1113 by far the most popular is the two-equation k-e model, first proposed by Jones and Launder.14 In this model, the turbulent stresses are recast in a form similar to the molecular stress tensor with mean velocity gradients, an assumption generally known as the Boussinesq hypothesis ... 

The macroscopic relationship between the total molecular stress tensor, which includes both the hydrostatic- or thermodynamic pressure and the viscous stresses is expressed as ... 

Wu D, Lenhardt JM, Black AL, Akhremitohev BB, Craig SL (2010) Molecular stress relief through a force-induced irreversible extension in polymer contoiff length. J Am Chem Soc 132 15936... 

The temperatures and the flexibility of polymeric molecules govern both the sizes of molecular segments that can be in motion and the frequencies at which that can occur. This in turn determines the rate at which polymer molecules respond to molecular stresses. In flexible polymers, if the thermal energy is sufficiently high, large segments can disengage and slip past each other quite rapidly in response to applied stress. This is a property of all elastomers. 

For statistically stationary flows of very dilute polymer solutions, the mean molecular stress and the mean strain rate are proportional. 

The set of equations for turbulent flows of an incompressible Maxwell fluid includes the continuity equation, the equation of motion, and Eq. (3). For short times and within the constant stress region, we neglect the convective contributions to Eq. (3) and assume that fluctuations in the molecular stress are governed by... 

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... 

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