Coil-stretch deformations

Rao et al. (1981) reported the primary structure of guar gum and CMC to be quite stable after heating at 210-260°F for 5-20 min, but there was a loss of T fl. The results of this heating study suggested possible coil-stretch deformation of an equilibrium structure in the gums beyond their elastic limit and an infinitely long t1 after cooling. Coil-stretch deformations can... 

Chauveteau and co-workers 24, 48) examined the flow of PEO and HPAA through the extensional flow produced in severe constrictions. They concluded that a coil-stretch transition was responsible for the dilatant behavior observed, and that the critical shear rate required was of the order of 10 times the reciprocal of the Rouse relaxation time. Perhaps the most extensive studies have been those of Haas and co-workers (25, 26, 49). They have explored the critical dilatant behavior on flow through porous media and pursued the hypothesis that the phenomenon is primarily due to a coil-stretch transition beyond a critical deformation rate. They attempted a semiquantitative description based upon the dependence of the lowest order relaxation time of the random coil upon polymer type, molecular weight, solvent quality, and ionic environment. 

The flow timescale can be defined in terms of an inverse shear rate or a residence time, for example. When De is small, polymer chains have ample time to relax toward equilibrium, and the fluid behavior is viscous. When De is large, the polymer chains elongate in the flow and do not have time to relax. In this hmit, the fluid behavior is predominantly elastic. The coil-stretch transition occurs at De = above which a majority of polymer chains elongate with the flow. Below this hmit, a majority of polymer chains do not deform significantly away from the equihb-rium coil size. 

Chilcott and Rallison (1988) simulated the zero Reynolds number flow of a dumbbell model with finite extensibility across a cylinder. In particular, they performed time-dependent calculations and reported the regions of high deformations near stagnation points, especially downstream from stagnation points. Likewise, Ambari et al. (1984) have considered the underlying phenomena of coil-stretch transition of macromolecules in laminar flow about cylindrical obstacles. 

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. 

Small deformations of the polymers will not cause undue stretching of the randomly coiled chains between crosslinks. Therefore, the established theory of rubber elasticity [8, 23, 24, 25] is applicable if the strands are freely fluctuating. At temperatures well above their glass transition, the molecular strands are usually quite mobile. Under these premises the Young s modulus of the rubberlike polymer in thermal equilibrium is given by ... 

Usually, the molecular strands are coiled in the glassy polymer. They become stretched when a crack arrives and starts to build up the deformation zone. Presumably, strain softened polymer molecules from the bulk material are drawn into the deformation zone. This microscopic surface drawing mechanism may be considered to be analogous to that observed in lateral craze growth or in necking of thermoplastics. Chan, Donald and Kramer [87] observed by transmission electron microscopy how polymer chains were drawn into the fibrils at the craze-matrix-interface in PS films [92]. One explanation, the hypothesis of devitrification by Gent and Thomas [89] was set forth as early as 1972. 

Fig. 11.9 Types of linear continuous-flow reactors (LCFRs). (a) Continuous plug flow reactor (CPFR) resembling a batch reactor (BR) with the axial distance z being equivalent to time spent in a BR. (b) A tabular flow reactor (TFR) with (tq) miscible thin disk of reactive component deformed and distributed (somewhat) by the shear field over the volume, and (b2) immiscible thin disk is deformed and stretched and broken up into droplets in a region of sufficiently high shear stresses, (c) SSE reactor with (cj) showing laminar distributive mixing of a miscible reactive component initially placed at z = 0 as a thin slab, stretched into a flat coiled strip at z L, and (c2) showing dispersive mixing of an immiscible reactive component initially placed at z — 0 as a thin slab, stretched and broken up into droplets at z — L.

One of the prominent features of polymeric liquids is the property to recover partially the pre-deformation state. Such behaviour is analogous to a rubber band snapping back when released after stretching. This is a consequence of the relaxation of macromolecular coils in the system every deformed macro-molecular coil tends to recover its pre-deformed equilibrium form. In the considered theory, the form and dimensions of the deformed macromolecular coil are connected with the internal variables which have to be considered when the tensor of recoverable strain is to be calculated. Further on, we shall consider the simplest case, when the form and dimensions of macromolecular coils are determined by the only internal tensor. In this case, the behaviour of the polymer liquid is considered to describe by one of the constitutive equations (9.48)-(9.49) or (9.58). 

When describing dilatant behavior, the maximum stretch rate, e, in the converging flow at the contraction is a better parameter, but more difficult to be calculated. Instead of the term stretch rate, other authors also used deformation rate (e.g., Chauveteau, 1981) or elongational rate (e.g.. Sorbic, 1991). The shear-thickening viscosity is also called elongational viscosity (often referred to as the Trouton viscosity Sorbie, 1991) or extensional viscosity in the literature. James and McLaren (1975) reported that for a solution of polyethylene oxide (a flexible coil, water-soluble polymer physically similar to HPAM), the onset of elastic behavior at maximum stretch rates was of the order of 100 s and shear rates of the order of 1000 s. In this instance, the stretch rate is about 10 times lower than the shear rate. However, some authors use shear rate instead of stretch rate in defining the Deborah number—for example, Delshad et al. (2008). 

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