Semiempirical Constitutive Equations

The development of molecular constitutive equations for commercial melts is still a challenging unsolved problem in polymer rheology. Nevertheless, it has been found that for many melts, especially those without long-chain branching, the rheological behavior can be described by empirical or semiempirical constitutive equations, such as the separable K-BKZ equation, Eq. (3-72), discussed in Section 3.7.4.4 (Larson 1988). To use the separable K-BKZ equation, the memory function m(t) and the strain-energy function U, or its strain derivatives dU/dli and W jdh, must be obtained empirically from rheological data. 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

The stress tensor in this expression has been defined such that at equilibrium r = 0, rather than G8. Some of the proposed possible forms for the functions G( r, D) and H(a) are 

The notation tr stands for the trace of the tensor. These expressions contain parameters, such as and a, that must be obtained by fits to nonlinear rheological data. None of 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls 

---

Time constants are related to the relaxation times and can be found in equations based on mechanical models (phenomenological approaches), in constitutive equations (empirical or semiempirical) for viscoelastic fluids that are based on either molecular theories or continuum mechanics. Equations based on mechanical models are covered in later sections, particularly in the treatment of creep-compliance studies while the Bird-Leider relationship is an example of an empirical relationship for viscoelastic fluids. 

Owing to difficulties in deriving general constitutive equations for multiphase systems, rheologists had to resort to simplified theoretical or semiempirical dependencies derived for specific types of rheological tests and/or for specific multiphase systems. These, experimentally well-established relations, constitute the basic tools for the interpretation of rheological data for multiphase systems. They will be discussed in the following parts of the text. 

The above mathematical treatment constitutes the creation of the form of the semiempirical equations. To actually use these equations, they must be parameterized somehow (as stressed above, Dewar used experimental data). This is analogous to the situation in molecular mechanics (Chapter 3), where a force field, defined by the form of the functions used (e.g. a quadratic function of the amount by which a bond is stretched, for the bond-stretch energy term) is constructed, and must then be parameterized by inserting specific quantities for the parameters (e.g. values for the stretching force constants of various bonds). For each kind of atom A (a maximum of) six parameters is needed ... 

The computational requirements for calculating the electronic properties of large molecular systems from first principles equations are demanding and can become sometimes counter productive, particularly when the impractical first-principle steps of the calculation constitute an input for the overall computation. Semiempirical methods are built in with information of the atomic building blocks and, therefore, can render practical the ultimate goal of simulating coherent quantum dynamics processes in large scale molecular systems. 

---