Orientation tensors

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

The written relations define the relative permittivity tensor for the system, which is formulated below to within second-order terms in the orientation tensor... 

In conformity with the significance of the terms employed by investigators of anisotropy (Tsvetkov et al. 1964), the effects associated with the first-order terms in equation (10.6) may be called the effects of intrinsic anisotropy, while the second-order effects may be referred to as the effects of mutual interaction. In the second approximation, the principal axes of the relative permittivity tensor do not coincide, generally speaking, with the principal axes of the orientation tensor. It is readily seen that interesting situations may arise when Aa < 0 in this case, the coefficients of the first- and second-order terms have different signs. 

Let us note that the contribution from anisotropy due to the difference in the isotropic part of the polarisability between segments and solvents molecules, a — ao, ought to be added to expression (10.6). This is a first-order term in the orientational tensor (Tsvetkov et al. 1964). We shall not consider this contribution to the anisotropy, as it is not so important for the very concentrated solutions under consideration. 

If another average orientation state besides uniaxial alignment is of interest for an anisotropic filler, orientation tensors are used to perform orientation averaging [7]. 

Doraiswamy and Metzner noted that use of the LCF approach is permissible at concentrations above that which would correspond to the transition from isotropic to aligned morphology, ( ) > 8/p. The theory provided fair description of the stress-strain dependence for systems containing 10 wt% GF and excellent agreement for those with 40 wt% GF. Also, the approach gave good predictions of the diagonal terms of the second-order orientation tensor. 

A far more compact method involves the use of orientation tensors. Orientation tensors are defined as even-ordered tensors by relationships such as... 

The starting point for using an orientation tensor to predict fiber orientation is the evolution equation of Jeffery for the motion of an isolated fiber in a Newtonian fluid. Jeffery s equation is valid for dilute fiber suspensions where there are no fiber-fiber interactions. 

Jeffery s equation was extended to concentrated solutions by Folgar and Tucker who added a diffusion term to account for the fiber-fiber interaction. In terms of the orientation tensor, the Tucker-Folgar equation has the form... 

In order to achieve these goals, we have adopted a multi-scale approach that comprises molecular and mesoscopic models for the liquid crystal. The molecular description is carried out in terms of Monte Carlo simulations of repulsive ellipsoids (truncated and shifted Gay-Berne particles), while the mesoscopic description is based on a dynamic field theory[5] for the orientational tensor order parameter, Q. ... 

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],... 

Although the distribution function provides a general description of the orientation state in the suspension, the numerical solution of the Fokker-Planck equation is computationally expensive. One needs a more compact and efficient description of fiber orientation for use in modeling of process. A proper approach is to use orientation tensors (Advani and Tucker 1987). Orientation tensors are defined in term of the ensemble average of the dyadic products of the unit vector p, i.e.. 

Fig. 5.1 Some fiber orientation distributions and corresponding second-order orientation tensor components a fully aligned in the 1-direction b random in the 1-2 plane c random in 3-D space...

Since xj/ is an even function, the average of products of odd number of components of p is zero. Also, since the distribution function is normalized (Eq. 5.2) and p is a unit vector, one has akk= 1- From the definition, one can also find the symmetry = Uji. Therefore, the second-order orientation Uy has only five independent components. Figure 5.1 shows some extreme cases of fiber orientation distributions and corresponding values of the second-order orientation tensor components. 

Multiplying Eq. 5.6 by tensor pt pj and then integrating it over the orientation space, one obtains the following evolution equation for the orientation tensor ... 

A family of closure approximations called orthotropic closures was developed by Cintra and Tucker (1995). Since the fourth-order orientation tensor aiju must take... 

The IBOF closure proposed by Chung and Kwon (2002) uses the same idea of the natural closure, approximating the fourth-order orientation tensor in terms of the second-order orientation tensor and its invariants. They showed that there are only three independent among the six p s that appear in Eq. 5.22. They chose P4 and / 6 as the independent parameters and the rest can be expressed in terms... 

Phelps and Tucker (2009) further assumed that C could also be a function of the orientation tensor, and they proposed the following expression ... 

Instead of solving the evolution equation in terms of the orientation tensor, one can simulate the stochastic equation such as Eq. 5.7 for the orientation vector p without the need of closure approximations, using the numerical technique for the simulation of stochastic processes (Ottinger 1996) known as the Brownian dynamics simulation. Once trajectories for aU fibers are obtained, the orientation tensor can be calculated in terms of the ensemble average of the discrete form ... 

Fig. 5.4 Brownian dynamics simulation results of orientation tensor components distributed along the radial distance of a center-gated disk (From Zheng et al. (2000), with permission from Society of Plastics Engineers Inc.)...

In reality, the fibers in a short-fiber reinforced composite are rarely completely aligned, so that a further step of calculations is required to account for the effect of the distribution of fiber orientation on the actual properties of the composite. The procedure of averaging properties over all directions by writing with the orientation distribution is termed orientation averaging. The orientation averaging can be formulated in terms of the orientation tensor a and... 

The orientation of fibers can be represented by the second order orientation tensor defined as, ... 

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