Stretching field

Figure 2 shows the increase in the rigidity (1 - f) of macromolecules induced by the field as a function of the parameter x = e/kT + Fl/kT. As soon as the flexibility decreases to f < 0.63, a system of molecules flexible in the state of rest will undergo a spontaneous transition into a nematic oriented state upon the action of the stretching field, just as it occurs for rigid molecules at rest. 

We studied the effect of the mechanical stretching field on the conformations of the macromolecules in the melt. It is known that for a freely jointed chain the Maxwell distribution of end-to-end distances holds50). 

When the chain undergoes extension, the stretching fields increases the number of trans isomers in the chain and, hence, decreases the effective chain flexibility which is related to by the equation (see Eq. (11))... 

Another objection concerns the orientation of the deformation field. It may happen that a lamella is perpendicular to the stretching field and in that case the striation thickness increases, which induces the reduction of the concentration gradient and slows the mixing rate. 

Figure 3-3 Calculating the stretching field and the Lyapunov exponent. In (a) a small material filament, represented by a vector /q, is convected by a flow. As a consequence, its length increases from the initial /q to / . The stretching (X) experienced by the material line after each period n is the ratio Zo//n. in (b) an array of small veetors is placed in the flow, and the stretching of each is measured and an average X can be calculated. In (c) a chaotic flow, X grows exponentially, and the exponent characterizing the growth rate (A, the Lyapunov exponent) can be calcnlated from the slope of the curve (In X> versus n.

Effective injection locations and fast-mixing areas can easily be identified. / The effect of diffusion is not measured by the stretching field. 

This measure can be used to characterize micromixing efficiency at all spatial locations. / p cannot be obtained directly in complex flows, but it is predictable from the stretching field. 

Since the sine flow is continuous everywhere, it is differentiable within the entire flow domain. This property leads to a piecewise analytical expression for the stretching field in the flow domain ... 

Figure 3-15 Intermaterial area density distribution for two cases of the sine flow at T = 1.6 (almost globally chaotic). The figure compares the distribution computed from the coarse-grained stretching field to the distribution computed from direct tracking of a continuous material filament. Although the latter method cannot be applied to most chaotic flows, the first case is a fairly straightforward computation in both model and real chaotic flows. The distributions of p are shown as a function of initial position (square), as a function of final position (circles), and as computed from direct filament tracking (triangles). All three curves collapse onto a single distribution.

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.

Figure 3-21 Contours of the stretching field in the standard Kenics mixer at Re = 10(10). The cross-sectional planes correspond to axial distances after (

Figure 3-23 shows the spatial distribution of intermaterial contact area, scaled by the overall length of the interface (p), at each axial position in the mixer. As discussed earlier in the chapter, the distribution of intermaterial area is related directly to the stretching field through the relationship X p. Qualitatively, this relationship can be confirmed if one compares the stretching field in Figure 3-21 to the field of intermaterial area densities in Figure 3-23. Quantitative proof was given in Figures 3-14 and 3-15, where the direct computation and prediction of p were compared for the sine flow. Once the intermaterial area in each of the 37 130 cells is normalized by the overall average, (p), the distribution and scale are identical for all four cross-sectional positions. In other words, the function p = p/(p) is invariant and describes the intermaterial area density at each period everywhere in the domain (i.e., each color represents the same range of p in...

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