Oscillatory birefringence

Now we refer to formula (10.13) for the relative permittivity tensor to determine the characteristic quantities in this case of strongly entangled linear polymers. We use expansions (7.32) and (7.43) for the internal variables to obtain the expression for the components of the tensor through velocity gradients 

Of course, these relations are trivial consequences of the stress-optical law (equation (10.12)). However, it is important that these relations would be tested to confirm whether or not there is any deviations in the low-frequency region for a polymer system with different lengths of macromolecules and to estimate the dependence of the largest relaxation time on the length of the macromolecule. In fact, this is the most important thing to understand the details of the slow relaxation behaviour of macromolecules in concentrated solutions and melts. 

One can turn to discussion of the dynamo-optical coefficient, defined by equation (10.22). The expression for the relative permittivity tensor (10.10) and equation (2.41) for the moments allow one to write 

The stress-optical coefficient C is defined by equation (10.27) and the relaxation times t,1 and t][ are defined by relations (2.30). One can see that the dynamo-optical coefficient of dilute polymer solutions depends on the non-dimensional frequency t w, the measure of internal viscosity ip and indices zv and 6 

For the components of dynamo-optical coefficient, one can find the equations, established by Thurston and Peterlin (1967), 

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The coupled phonon-polariton oscillations can be detected by measurement of oscillatory birefringence with a variably delayed probe pulse. (The transit time and the spectral content of the probe pulse also should show oscillatory time dependences.) As in forward ISRS, this pulse surfs along a crest or null of the polariton wave. Since the polariton radiates outward from the excitation beam, the probe pulse need not be overlapped spatially with the excitation pulse. By varying the spatial separation between the two parallel-propagating beams, the polariton group velocity and dispersion can be determined. Phonon-polariton dynamics in LiTaOj crystals were determined in this manner [36, 59]. An example of data is shown in Figure 9. 

The above description refers to a Lagrangian frame of reference in which the movement of the particle is followed along its trajectory. Instead of having a steady flow, it is possible to modulate the flow, for example sinusoidally as a function of time. At sufficiently high frequency, the molecular coil deformation will be dephased from the strain rate and the flow becomes transient even with a stagnant flow geometry. Oscillatory flow birefringence has been measured in simple shear and corresponds to some kind of frequency analysis of the flow... 

Fig. 2.5. Steady-state and dynamic oscillatory flow measurements on a 2 wt. per cent solution of polystyrene S 111 in Aroclor 1248 according to Philippoff (57). ( ) steady shear viscosity (a) dynamic viscosity tj, ( ) cot 1% from flow birefringence, (A) cot <5 from dynamic measurements, all at 25° C. (o) cot 8 from dynamic measurements at 5° C. Steady-state flow properties as functions of shear rate q, dynamic properties as functions of angular frequency m. Shift factor aT which is equal to unity for 25° C, is explained in the text, cot 2 % and cot 8 are expressed in terms of shear (see eqs. 2.11 and 2.22)...

Thurston and Schrag (187) have proposed to use the measurement of dynamic birefringence of solutions in oscillatory shear, in order to determine the influences of the above mentioned three parameters on the dynamic oscillatory behaviour at intermediate and high frequencies. 

In section 4.7, the Onuki-Doi theory for form birefringence and dichroism was developed and presented in equations (4.91) and (4.92). It is left to calculate the structure factor, S (q), as a function of flow. This was done in the limit of weak oscillatory flow for the... 

Formula (4.371) proves explicitly that the suspension birefringence is an even function of the applied field amplitude. For this reason, in response to the excitation of the frequency go the optical anisotropy Av oscillates with the basic frequency 2co. The higher-rank harmonics induced by the saturation behavior of Av((0) are the multiples of the basic one. It is also clear that besides the oscillatory contribution, the frequency spectrum of Av contains a constant component. 

Failure to use Eq. (7.9) has resulted in several errors in the literature Zimm s (79) calculation of flow birefringence, as pointed out by Williams (76) Kirkwood and Plock s (40) calculation of large amplitude oscillatory response, as pointed out by Paul (62) Williams and Bird s (78) calculation of oscillatory normal stresses, as pointed out by themselves in a later publication (77). 

This author does not know of any application where just the dichroism or the dichroism and mechanical properties are measured dynamically, similar to the dynamic X-ray or birefringence experiments. This is probably due to the experimental difficulty. Specifically one would need a fairly high speed detector which generally requires cryogenic cooling and is fairly costly. Secondly, the sensitivity of the dichroism method is not considered to be better than a few percent. Since only small dynamic strains are involved in such an oscillatory experiment, the sensitivity required must be better than this. This latter problem could probably be overcome by time averaging. 

The geometry used for flow birefringence measurements in an oscillatory shear (Miller and Schrag, 1975) consists of two flat, parallel surfaces, one of which is fixed the other is made to oscillate in the plane parallel to the fixed surface. The liquid is confined to the thin layer between the surfaces. The light beam propagates through the layer in a direction perpendicular to the flow direction and parallel to the surfaces. This experimental arrangement and its application are discussed in detail below. 

Cross-sectional view of the Miller-Schrag thin fluid layer (TFL) transducer, used to generate a precise sinusoidally time-varying shear flow for the oscillatory flow birefringence experiment (see Section 9.4.6). 

In this example (Mori et al., 1982), the experimental method is oscillatory electric birefringence (Morris and Lodge, 1986), in which... 

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