Viscous flow in polymers

Molten polymers and polymer solutions are, in general, non-Newtonian in their flow behaviour, with an apparent viscosity that decreases as the rate of shear is increased at constant temperature. 

Polymer viscosity 77 increases with relative molecular mass 

If the polymer solution or melt is placed in the space between two coaxial cylinders, one of which is kept fixed while the other is rotated, the liquid is not only subjected to a rate of shear, which generates a shear stress on the inner and outer cylinders, but, in addition, there is a tensile stress set up in the direction of the streamlines. In this case the stress is a hoop stress. This hoop stress produces an internal pressure in the Hquid. It is as if the liquid were surrounded by a stretched elastic membrane. Consequently, the liquid is forced up the wall of the inner cylinder to give what is known as the Weissenberg effect. 

Materials, and, in particular, polymers, that show viscoelastic behaviour, can be modelled by a combination of perfectly elastic Hookean springs and Newtonian viscous dashpots. For many polymers the behaviour at temperatures above when strains are small 1 per cent) is approximately represented by the so-called standard linear substance. This consists of a dashpot and spring in series (called a Maxwell element) and this combination in parallel with a second spring of different elastic modulus. In deformation, the strain of the Maxwell element and of the spring will be the same, say e. At a time t let Oi be the stress in the Maxwell element and 02 that in the spring. Then, if Ei is the Young s modulus of the Maxwell element and E2 that of the spring  

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In polymers, however, E levels off at a value independent of molecular mass. This means that in long chains the unit of flow is considerably smaller than the complete molecule. It seems that viscous flow in polymers takes place by successive jumps of segments until the whole chain has shifted. 

For a nematic polymer in a transition region from LC to isotropic state, maximal viscosity is observed at low shear rates j. For a smectic polymer in the same temperature range only a break in the curve is observed on a lgq — 1/T plot. This difference is apparently determined by the same reasons that control the difference in rheological behaviour of low-molecular nematics and smectics 126). A polymeric character of liquid crystals is revealed in higher values of the activation energy (Ef) of viscous flow in a mesophase, e.g., Ef for a smectic polymer is 103 kJ/mole, for a nematic polymer3 80-140kJ/mole. 

E. B. Bagley, The Separation of Elastic and Viscous Effects in Polymer Flow, Trans. Soc. Rheol., 5 355-368 (1961). 

In polymer processing, we frequently encounter creeping viscous flow in slowly tapering, relatively narrow, gaps as did the ancient Egyptians so depicted in Fig. 2.5. These flows are usually solved by the well-known lubrication approximation, which originates with the famous work by Osborne Reynolds, in which he laid the foundations of hydrodynamic lubrication.14 The theoretical analysis of lubrication deals with the hydrodynamic behavior of thin films from a fraction of a mil (10 in) to a few mils thick. High pressures of the... 

The FEM, which was originally developed for structural analysis of solids, has been very successfully applied in the past decades to viscous fluid flow as well. In fact, with the exponentially growing computer power, it has become a practical and indispensable tool for solving complex viscous and viscoelastic flows in polymer processing (20) and it is the core of the quickly developing discipline of computational fluid mechanics (cf. Section 7.5). 

Bagley EB (1961) The separation of elastic and viscous effects in polymer flow. Trans Soc Rheol 5 355-68. 

The Eyring model (16) was developed to describe the viscous flow in liquids. The fundamental ideas of this model can be applied to clarify some aspects of yield behavior of glassy polymers. 

Table 13.2. Observed activation energies for viscous flow of polymers at zero shear rate as T—>°o (E J [7], and the values of predicted by using the correlation developed for the molar viscosity-temperature function in this section. E is in units of 1()3 J/mole.

Pusher -700 and xanthan gum have larger molecular sizes in solution than Colloid and hydroxy ethyl cellulose. The dimensions in solution decrease with increasing salt concentration. Polyacrylamides are affected most severely by the presence of electrolytes. Polysaccharides are also affected by salt, but not to the same extent as polyacrylamides. Hydroxy ethyl cellulose is the most insensitive polymer to salt. Temprature can be inversely correlated with viscosity. Polyacrylamides have low activation energies for viscous flow. In order of decreasing temperature dependency are xanthan gum, Colloid , and hydroxy ethyl cellulose. 

It is evident from the above description that G (oj) and are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. The loss parameters G" oj) and on the other hand, reflect the nonrecoverable use of applied mechanical energy to cause viscous flow in the material. At a specified frequency and temperature, the dynamic response of a polymer in shear deformation can be summarized by any one of the following pairs of parameters G ( w) and G"( w), J ioj) and or absolute modulus G and tan 6. 

The ability to fluorescently tag individual DNA molecules and visualize their dynamics in these flows can considerably enhance the information we obtain from these flows. Numerous single molecule visualization experiments have already led to significant advancements in our understanding of polymer chain dynamics in viscous flows. In addition, these experiments have recently been combined with Brownian dynamics simulations, which simulate the motion of bead-rod chains in viscous flows and can quantitatively predict the chain conformation for given flow conditions. This powerful combination has allowed for the validation of detailed molecular scale physics, as well as the development of new physical insights. These conclusions are described in a recent review [10] we summarize a few of the salient conclusions here. 

Street [8,9] has carried out detailed studies of temperature and flow patterns for highly viscous non-Newtonian flows in polymer stirred tank reactors, particularly with respect to the effect of vessel geometry on reactor performance. His work is extended in this example to the case of a Rushton... 

Systems with vanishing gradients (a) catalyst peUet (b) viscous flow in a pipe (c) diffusion into capillary (d) polymer extruder. 

It is possible to think of yield and plastic deformation in polymers as a type of viscous flow, especially since glassy polymers.are basically frozen liquids that have failed to crystallize. Eyring developed a theory to describe viscous flow in liquids and it can be readily adapted to describe the behaviour of glassy polymers. The segments of the polymer chain can be thought of as being in a pseudo-lattice and for flow to occur a segment must move to an adjacent site. There will be a potential barrier to overcome,... 

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