Conical deformation

In the discussion above we did not allow for the conical deformation of the helical structure, i.e., the polar angle 0 was considered to be equal to zero. In a more general case, the components of the director are taken in the form... 

Even in the thermodynamic equilibrium, 0 z) can form a periodic soliton-like structure with a period incommensurate with the helical pitch qo = 2itfPo [41]. When the field is applied the conical distortion can appear, especially for certain values of the twist elastic constants, K22 > Kzz [29, 42]. Such conical distortion can be accompanied by a compression of the hehcal pitch. The results of some earlier controversial observations were discussed in [6] and [1]. Probably, the conical deformation has been observed in a ternary cholesteric mixture by measuring the field-induced changes in the... 

The fact that the lower limit of the reflection band remains the same while the upper limit decreases indicate that the cholesteric structure and the helical pitch remains the same while molecules tilt to become oriented in the direction of the applied field. This mode of switching is associated with conical deformation and the equation describing the threshold voltage is given by... 

If the electric field is applied parallel to the helix, the situation is more complicated. For a short pitch, a shift of the reflection peak to shorter wavelengths (blue shift) can be observed [49]-[52]. It was assumed that above a threshold field, a conical deformation of the planar texture leads to a contraction of the pitch, and thus is shifted to shorter wavelengths [52]. However, it could be shown that the blue shift of the selective reflection... 

Let us assume a planar texture (helical axis at right angles to the boundaries) and a positive dielectric anisotropy. When an electric field is applied parallel to the helical axis the local optical axis experiences a dielectric torque which tries to tip it out of the cholesteric plane. The boundary forces, on the other hand, act in a direction to sustain the planar texture. Assuming that the deformation is uniform in the plane of the sample, Leslie [70] was able to show that below a certain threshold voltage Ug no deformation occurs. Above Ug the local optic axis tips out of the cholesteric plane conical deformation . Ug is given by... 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

Figure 9.12. Potential energy profile along (adapted from reference 10) near the fulvene conical intersection. The branching space consists of stretching and skeletal deformation of the five-membered ring.

Melt fracture results from a too large rubber-elastic deformation. This is governed by the polymer behaviour and by the rate of deformation. The strain rate must be lowered, which may be accomplished by a lower, and therefore unattractive, rate of production. The time scale of the elastic deformation can also be increased in another way, namely by making the conical channel to the die more pointed the same strain is then reached after a longer time. This goes at the cost of extra pressure and power, since the resistance of the chaimel is increased. 

Consider the vacuum forming of a polymer sheet into a conical mold as shown in Figure 7.84. We want to derive an expression for the thickness distribution of the final, conical-shaped product. The sheet has an initial uniform thickness of ho and is isothermal. It is assumed that the polymer is incompressible, and it deforms as an elastic solid (rather than a viscous liquid as in previous analyses) the free bubble is uniform in thickness and has a spherical shape the free bubble remains isothermal, but the sheet solidifies upon confacf wifh fhe mold wall fhere is no slip on fhe walls, and fhe bubble fhickness is very small compared fo ifs size. The presenf analysis holds for fhermoforming processes when fhe free bubble is less than hemispherical, since beyond this point the thickness cannot be assumed as constant. 

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