Viscosity coefficients divergence

All physical parameters mentioned above are material specific and temperature dependent (for a detailed discussion of the material properties of nematics, see for instance [4]). Nevertheless, some general trends are characteristic for most nematics. With the increase of temperature the absolute values of the anisotropies usually decrease, until they drop to zero at the nematic-isotropic phase transition. The viscosity coefficients decrease with increasing temperature as well, while the electrical conductivities increase. If the substance has a smectic phase at lower temperatures, some pre-transitional effects may be expected already in the nematic phase. One example has already been mentioned when discussing the sign of Ua- Another example is the divergence of the elastic modulus K2 close to the nematic-smecticA transition since the incipient smectic structure with an orientation of the layers perpendicular to n impedes twist deformations. 

In eq 7.102, the difinsion coefficient, which controls the life time of the fluctuations, is assumed to follow eq 7.101 for a Brownian particle, where the radius R is replaced by the correlation length The shear viscosity also diverges at the critical point as but very weakly, with z 0.07. ... 

If there is a phase transition from a nematic to a smectic phase, pretransitional effects are observed in the neighbourhood of the transition [9, 44, 49-51]. Figure 10 [9] shows this behaviour for 4-/7-octyloxy-4 -cyanobiphenyl (80CBP) with a nematic-smectic phase transition at 340.3 K. Pretransitional effects cause a divergence of the shear viscosity coefficient 772 and the rotational viscosity /j. 

Theory and intuition show that at a transition from a nematic to a smectic A phase, where the mobility is restricted to two dimensions, the viscosity coefficient 772 > which is coupled with a movement in the forbidden dimension, will diverge. The divergence can be described by the expression [52-54]... 

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

As shown by Leutheusser, the above analysis can be extended to show that at glass transition the density fluctuations decay with a long-time power law (0 t with a = 0.395. As one approaches the transition, the viscosity is predicted to diverge as e- 1 and ebelow and above the transition, respectively. e = 1 - X/Xc, p = (1 + a)/2a, and p = p — 1, where p 1.7-2. It is shown by Kirkpatrick [30] that the diffusion coefficient near glass transition goes to zero as e1. ... 

The rather complex coefficient in Eq. (73) is given in Reference 9. It is evident that the series becomes divergent at the critical point, but in practice the coefficient in Eq. (73) is so small that it was estimated for the Reed and Taylor system that T — had to be about 0.2° or smaller for the dependence of the viscosity on the velocity gradient to be detectable. The effect is known to exist, and is quite striking in magnitude, but because the only experiment was done in a capillary viscometer, where d varies across the capillary, a quantitative interpretation is difficult. An attempt is being made to develop dtj as a function of d for large d (unpublished work by W. Botch and the author). 

It has already been noted in Section A5 that the temperature reduction factors in the transition and terminal zones may be somewhat different. An example of a very complete study by Plazek on a polystyrene with almost uniform molecular weight 46,900 is shown in Fig. 11-14. The creep compliance Jp t) reduced to a reference temperature of 100° by shift factors ar calculated from the viscosity—i.e., from equation 13—provides satisfactory superposition of data in the terminal zone, but in the transition zone the reduced data diverge. Alternatively, the recoverable compliance Jp(t) — t/rjo can be satisfactorily reduced in the transition zone with a slightly different set of reduction factors these, however, appear to give a slight divergence in the terminal zone (Fig. 11-15). Both sets of shift factors follow the WLF equation, equation 21, but with slightly different coefficients in Fig. 11-14,... 

Inserting (1.4.4) into (1.4.1) and differentiating with respect to time shows that rj is the effective coefficient of viscosity at large times. Note that J(t) may diverge for large /, in the case of a viscoelastic solid, as a< 1, for example. 

For the dependence of the Newtonian viscosity on the molar mass, again the two models (Rouse and reptation) diverge. As established in Chapter 6, the Rouse theory predicts that the viscosity of a fluid is the product of the friction coefficient (Iseg) times a factor F (see Chapter 6)... 

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