Molecular expression for the shear stress

Because of the chemical decoration of each substrate, a confined fluid can be exposed to a shear strain by misaligning the substrates in the +x-direction according to 

Alternatively, we may derive a force expression for Txz following the derivation presented in Appendix E.3.2.2 for the stress tensor component r. It follows if we combine Eqs. (E.62) and (E.63) with Eq. (E.49) from which we obtain 

As before [see Eq. (5.65)] mechanical stability of the entire system requires 

Another interesting quantity is the shear modulus C44 where we use Voigt s notation ([12], see p. 14 in Ref. 196). We reemphasize the appropriateness of 

A microscopic definition of C44 can be derived directly from Eq. (E.49) that 

---

Here we derive a molecular expression for the shear stress parallel to the force expression for the compressional stress derived in Appendix E.3.1.2. How( ver, here wc employ a slightly iliffinimt definition of the auxiliary fiuic-tions. 9i, which we introduce via... 

At this point, we do not yet consider hydrodynamics in the example below where we shear a morphology, the velocity field is imposed from the outside. For most mesoscale polymer systems considered by the molecular modeling community, relaxation by internally driven hydrodynamics is relatively unimportant. Also, it is not easy to find a general-purpose model. At this point we do not have a good expression for the local stress if we do, we can extend the approach better to chemical engineering applications such as extrusion. 

Although the experiments described in the foregoing section are very helpful for developing our intuition about the behavior of viscoelastic fluids such as polymers, they are not suitable for obtaining and cataloging information about specific polymeric materials. For the characterization of polymers it is necessary to make careful measurements of stresses in systems where the velocity or displacement field is known within rather strict limits. These rheometric experiments provide information about one or more of the stress components as functions of shear rate, frequency, or of other controllable variables these functions are generally referred to as material functions , since they are different for each material. Once these material functions have been measured, they can be used to test various empirical or molecular expressions for the stress tensor (that is, the constitutive equation), or they can be used to establish the values of the parameters that appear in these stress-tensor expressions. 

This expression suffices to determine experimentally the shear stress. Having evaluated both To,/, and >o,/, we can readily obtain the viscosity function // (fg ). Figure E3.2b gives such data for low-density polyethylene. The data extend beyond the commonly accepted upper limit of shear rate for polymer melts, probably because of the low average molecular weight of the polymer. 

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 ... 

The influence of temperature on the relationship between the shear stress and the eurrent density was comprehensively addressed using the molecular sieve and permutite particles of a similar molecular stmeture dispersed into silicone oil [79]. The molecular stmeture of those two materials can be expressed as (M0) (Al203) t(Si02)v(IE0) where , x, y, z arc integral numbers and M represents the metallic atom. The particle was healed under at least 500 °C for several hours before it was mixed with silicone oil. The shear stress of the molecular sieve/silicone oil suspension versus the elevated and reduced temperature at an electric field 1.5 kV/mm... 

Assuming that the applied stress is transferred to the MWNT via a nanotube-matrix interfacial shear mechanism at the molecular level, a modified force-balance which accounts for pullout angle and is based on the expression for the nanotube-polymer interfacial shear strength, x, may be as follows ... 

The properties of i (0) for narrow distribution polymers have already been discussed in Section 5. The behavior of f jfy) at higher shear rates has only been determined for a few systems of well-characterized molecular structure. The experimental problems are more difficult than in the case of rj(y), so the conclusions here must be regarded as somewhat more tentative. Experimentally, t(y) and tj(y) depart from their zero shear values within the same range of shear rates (172). Shear rate sensitivity is much smaller when N is expressed as a function of shear stress (350). 

The main feature about molten high polymers (molecular weights higher than about 104) concerns the broadness of the relaxation spectrum that characterises the viscoelastic response of these systems. This broad two-dispersion spectrum may spread over a range of relaxation times going from about 10 9 up to several seconds [4]. It is well illustrated from the modulus of relaxation observed after applying a sudden stress to the polymer the resulting sudden deformation of the sample is then kept constant and the applied stress is released in order to avoid the flow of the polymer. For example, the release of the constraint oxy(t) is expressed as a function of the shear modulus of relaxation Gxy(t) ... 

---