Flow-birefringence

To discuss flow birefringence, we have to make use of some results of the continuum theory developed in chapter 3. In analogy with (3.1.38) we write for the isotropic phase  

For an incompressible fluid = 0. All four tensors are symmetric and traceless. Further from Onsager s relations, p = ff. 

Now consider shear flow along x with a velocity gradient du/dz. The flow induces a birefringence proportional to the velocity gradient with the principal axes of the index ellipsoid inclined at 45° to the x, z axes. In the steady state = dr/dz. Therefore 

Therefore x yz represent the principal axes of the order parameter tensor. The difference between the dielectric constants (at optical frequencies) for polarizations along the x and z axes is 

The optical anisotropy of molten flexible-chain polymers is often very small in a quiescent (nonflowing) state owing to their nearly spherical chain configuration 

Birefringence setups can be designed to characterize molten materials undergoing isothermal homogeneous flow. The ranges of strains and strain rates also often coincide with those of rheometers, and consequently may be limited relative to those used in fabrication. Similarly, time-temperature superposition approaches may be used to expand the rate window. State-of-the-art setups suitable for rapid screening of new materials with research-scale quantities (5-20 g) are available for shear flow [72] and startup of uniaxial extensional flow [73,74]. 

Having obtained the basic formula, we can now study the flow birefringence of dilute polymer solutions. The contribution from the intrinsic birefringence is easily calculated. Comparing eqn (4.220) with eqn (4.134), we note that the intrinsic birefringence is proportional to the stress  

However, calculation of the form birefringence is tedious (see refe 66 and 70, and also Section 5.5). Here we will give a simple approximate treatment. Under weak shear flow eqn (4.115), the deformation of the structure factor g(ifc) is proportional to the dimensionless shear rate x/F, and will be written as 

Hence for large molecular weight, the form birefringence dominates. From eqns (4.240) and (4.246), it follows that 

In this model the polymer is made up of N beads connected by Af — 1 bonds, each having constant length b (see Fig. 4.16). In a small time interval At, each bead makes the following jump with probability w At. (i) For the internal beads (i.e., beads 2,3. N -1) 

To analyse this model, it is convenient to look at the bond vector v = H +i - R rather than R . The transition rule for v is 

---

Thomas and Rice [/. Appl. Mech., 40, 321-325 (1973)] applied the hydrogen-bubble technique for velocity measurements in thin hquid films. DureUi and Norgard [Exp. Mech., 12,169-177 (1972)] compare the flow birefringence and hydrogen-bubble techniques. 

B. H. Zimm. Dynamics of polymer molecules in dilute solutions Viscoelasticity, flow birefringence and dielectric loss. J Chem Phys 24 269-219, 1956. 

Janeschitz-Kriegl, H. Flow Birefringence of Elastico-Viscous Polymer Systems. Vol. 6, pp. 170-318. 

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... 

Isihara, A. and Guth, E. Theory of Dilute Macromolecular Solutions. Vol. 5, pp. 233-260. Janeschitz-Kriegl, H. Flow Birefringence of Elastico-Viscous Polymer Systems. Vol. 6, pp. 170-318. 

Zimm, BH, Dynamics of Polymer Molecules in Dilute Solution Viscoelasticity, Flow Birefringence and Dielectric Loss, Journal of Chemical Physics 24, 269, 1956. 

Lodge AS. A network theory of flow birefringence and stress in concentrated pol3mier solutions. Trans Faraday Soc 1956 52 120-130. 

This article builds upon an earlier review on the same subject by Porter and Johnson in 1966 (14), and on the recent treatise on viscoelasticity in polymers by Ferry (15). We have generally tried to maintain the same nomenclature as the latter. Recent reviews on the relation between the zero-shear viscosity and molecular structure (16), crosslinked networks (17), and flow birefringence (18) in this same journal cover portions of the subject. We have tried to minimise redundancy with these works while at the same time making the review reasonably self-contained. 

Fig. 5.12. Values of reduced compliance JcR for solutions of a narrow distribution polystyrene (Mw = 860000) according to several investigators and methods. Symbols Q from G (co) (175, 176), a from Nt (177), O- from N, (178), O from flow birefringence (179), - from flow birefringence (180), from G (o>) and Ns (181), and -O from creep...

Fig. 5.14. Reduced compliance vs molecular weight for undiluted polystyrenes of narrow molecular weight distributions. Symbols are O from creep recovery (163), Cr from G (w) (192), O- from flow birefringence (180), (X from (189), 9 from G (a>) (M>105 only) (124), jO extrapolated from steady state creep (191), -O from stress relaxation (165), and...

A final piece of evidence against both finite extensibility and internal viscosity is provided by flow birefringence studies. One would expect each to produce variations in the stress optical coefficient with shear rate, beginning near the onset of shear rate dependence in the viscosity (307). Experimentally, the stress-optical coefficient remains constant well beyond the onset of shear rate dependence in r for all ranges of polymer concentration (18,340). 

Janeschitz-Kriegl.H. Flow birefringence of elastico-viscous polymer systems. Advan. Polymer Sci. 6,170-318 (1969). 

Philippoff, W. Studies of flow birefringence of polystyrene solutions. Trans. Soc. Rheol. 7,45-59(1963). 

Philippoff, W., Gaskins,F.H., Brodnyan, J.G. Flow birefringence and stress. V. Correlation of recoverable shear strains with other rheological properties of polymer solutions. J. Appl. Phys. 28,1118-1123 (1957). 

Flow Birefringence of Elastico-Viscous Polymer Systems... 

---