Exponential shear

This shear flow is of interest, because it involves the exponential separation of material points, and thus seems likely to generate chain stretch, but is much easier to generate in the laboratory than extensional flows. However, whereas in extensional deformations the distance between material points on the same streamline increases exponentially with time, in exponential shear it is material points on different streamlines that separate in this way. In other words, uniform extension is an irrotational flow, while shear is an equal mixture of stretching and rotating flows. 

Exponential shear is therefore not a flow with constant stress history. The stress in this flow tends to grow without limit, even in an inelastic or linearly elastic fluid, and this makes the presentation of data an important issue. Doshi and Dealy [57] and Dealy [58] have argued that the results of an exponential shear experiment should be reported in terms of a time-dependent exponential viscosity rf that is defined in terms of the instantaneous shear rate ... 

Venerus, D. C. Exponential shear flow ofbranchedpolymer melts. RAeol Acte (2000) 39, pp. 71-79... 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Orthokinetic flocculation is induced by the motion of the Hquid obtained, for example, by paddle stirring or any other means that produces shear within the suspension. Orthokinetic flocculation leads to exponential growth which is a function of shear rate and particle concentration. Large-scale one-pass clarifiers used in water installations employ orthokinetic flocculators before introducing the suspension into the settling tank (see Water,... 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

In a detailed study on shear degradation of DNA, Adam and Zimm found a complex dependence of kc on solution viscosity. Considering that the macromolecules can rupture only after tumbling had brought them into the right configuration, these authors proposed to include solution viscosity into the pre-exponential factor A (l/r)s) [84],... 

Fig. 17. Response of CRL-8018 hybridoma cells to increasing levels of well-defined laminar shear in the concentric cylinder viscometer for 10 min. Spinner flask cultures were seeded with cells from routine T-flask cultures that were 3 days old. Cell samples were taken from the spinner flask cultures during late-exponential growth and sheared in the viscometer [17]...

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

Illustration Importance of reorientations. The stretching rate of long filaments in shear flow can be improved from being linear to exponential... 

Chain-growth polymerizations are diffusion controlled in bulk polymerizations. This is expected to occur rapidly, even prior to network development in step-growth mechanisms. Traditionally, rate constants are expressed in terms of viscosity. In dilute solutions, viscosity is proportional to molecular weight to a power that lies between 0.6 and 0.8 (22). Melt viscosity is more complex (23) Below a critical value for the number of atoms per chain, viscosity correlates to the 1.75 power. Above this critical value, the power is nearly 3 4 for a number of thermoplastics at low shear rates. In thermosets, as the extent of conversion reaches gellation, the viscosity asymptotically increases. However, if network formation is restricted to tightly crosslinked, localized regions, viscosity may not be appreciably affected. In the current study, an exponential function of degree of polymerization was selected as a first estimate of the rate dependency on viscosity. 

Plastic deformation is mediated at the atomic level by the motion of dislocations. These are not particles. They are lines. As they move, they lengthen (i.e., they are not conserved). Therefore their total length increases exponentially. This leads to heterogeneous shear bands and shear instability. 

For covalent crystals temperature has little effect on hardness (except for the relatively small effect of decreasing the elastic shear stiffness) until the Debye temperature is reached (Gilman, 1995). Then the hardness begins to decrease exponentially (Figure 5.14). Since the Debye temperature is related to the shear stiffness (Ledbetter, 1982) this softening temperature is proportional to C44 (Feltham and Banerjee, 1992). 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

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