Striation thickness distribution

For miscible polymer systems with low values of diffusion coefficient, the polymers form lamellar structures with crisp material surfaces, that is, sheets of one polymer intimately intertwined in the other in a nested three-dimensional structure. The lamellar structures appear as alternating striations on a cross-sectional plane as in Figure 8.5a, with a characteristic striation thickness, s, which follows a distribution, defined as the striation thickness distribution [141]. A usual definition of striation thickness s is ... 

In distributive mixing, the flow field produces deformation in the dispersed phase, without inducing breakup. As a consequence, a lamellar morphology of alternating striations of dispersed phase and the matrix is obtained, which can be characterized by the striation thickness distribution. Note that the striation thickness (5) is a function of the initial length scale (la), viscosity of the dispersed phase r]i), total shear strain (y) (product of shear rate and time), volume fraction of the dispersed phase (4>a), and viscosity of the matrix (rim) [59] (Eq. (3.1)) ... 

Some methodologies of dynamical systems theory developed from model flows that can be extended to real 3D flows are introduced next. These new tools—stretching calculations, striation thickness distributions, and so on—are essential if one is to obtain detailed statistical information regarding the microstructure emerging in a mixing process and the dynamics of a mixing operation. 

Figure 3-16 Striation thickness distributions after nine time units in the sine flow, T = 1.2. The light- and dark-color curve is the distribution formed by the light and the dark mixture component. The horizontal shift in the mean is due to different amounts of light and dark material present in the mixture.

In the preceding two sections, it was shown how to measure local micromixing intensity using intermaterial area distribution [H(log p)] and length scale distributions [H(logs)]. These tools are not directly applicable to 3D flows, because the resolution of the smallest length scales in 3D mixture structures is, to date, computationally prohibitive. In this section we present a predictive method for striation thickness distributions applicable to either 2D or 3D chaotic flows. 

The time evolntion, shape, and scaling of stretching distributions in globally chaotic flows is almost one-to-one with the same properties of striation thickness distributions. The proportionality is the intermaterial area density (p) that links the stretching field with the number of striations influenced by each value of X. If we consider that the number of stretching values is constant at each flow period, while the number of striations increases (i.e., Nj (p) (X)), the striation thickness distribution as predicted from the stretching field is... 

Figure 3-17 Comparison of the striation thickness distribution calculated directly from simulating the evolution of material interfaces (as in Figure 3-12) and as predicted from the stretching field.

The real power of using stretching computations to characterize chaotic flows lies in the fact that stretching is the link between the macro- and micromixing intensities in laminar mixing flows. In this section we describe the method for computing striation thickness distribution in our 3D example, the Kenics mixer. 

Prediction of Striation Thickness Distributions in Realistic 3D Systems... 

Another important component of mixing processes, diffusion, remains to be included in the discussion. The ability to measure or predict striation thickness distributions in chaotic flows provides the link between mixing on the macroscopic and microscopic scales. Diffusion and stretching are intimately coupled because as material filaments are stretched by the flow, the rate of diffusion is... 

---