Nonuniform-stretching

Fig. 3.2 Cases 1, 2, and 3 show steady, uniform extensional flows. Case 4 shows examples of more complex nonuniform stretching flows encountered in polymer processing operations. 

Here, the first term accounts for the nonuniform stretching of the coronal block, whereas the second term accounts for the excluded-volume interactions, the translational entropy of mobile ions, and (in the case of an annealing PE block) the free energy gain due to corona ionization, as specified by (66), (67). 

To specify the range of thermodynamic stability of micelles with morphology i, at a given salt concentration, Oion, we go beyond the boxlike model and incorporate polymer density gradients in the coronal domain and account for a nonuniform stretching of the blocks. We follow here the arguments of [20,22]. 

Weakly charged polyelectrolyte chains are nonuniformly stretched. The relation between the number of monomers N and chain size Re in salt-free solutions is given by eqn [11]. Thus, the overlap concentration c can be estimated as... 

Of course, this simple scaling picture does not account for the nonuniform stretching of a polyelectrolyte chain. In reality the chain is more strongly stretched in the middle than at the ends. Logarithmic corrections to the chain size may be obtained by allowing the blob size to vary along the chain. 

The uniaxial extensiometers described so far are suitable for use with viscous materials only. They cannot, for example, be used to measure the steady extensional viscosity of such commercially important polymers as nylons and polyesters used in the textile industry, and which may have shear viscosities as low as 100 Pa sec at processing temperatures. As a consequence, other techniques are needed but these invariably involve nonuniform stretching. Here one cannot require that the stress or the stretch rate be constant. Also, the material is usually not in a virgin (stress-free) state to begin with. One can therefore not obtain the extensional viscosity directly from these measurements. Nonetheless, data from properly designed non-uniform stretching experiments can be profitably analyzed with the help of rheological constitutive equations. In addition, such data provide a simple measure of resistance that polymeric fluids offer to extensional deformation. 

Liao Q, Dobrynin AV, Rubinstein M (2003) Molecular dynamics simulations of polyelectrolyte solutions nonuniform stretching of chains and scaling behavior. Macromolecules 36 (9) 3386-3398. doi 10.1021/ma025995f... 

It follows from previous discussion that the destabilizing electrostatic contribution grows in absolute value with x (with increasing A.). But the influence of the nonuniform electrical force is overwhelmed by the stabilizing bending and stretching contributions. As a result, the traditional smectic model cannot explain how a small transmembrane voltage can lead to membrane breakdown. The obvious solution is to abandon this approach and to develop an alternative, such as the pore formation model. However, as we noticed before, this approach postulates rather than proves the appearance of hydrophobic pores. 

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

Solvent polarization can also play a role in the stabilization of conformationally nonuniform particles. For example, 4-(l-phenylpiperidin-4-ylidene) cyclohexylidene propanedinitrile transforms into completely charge-separated species on photoirradiation. This species contains the C=C bond and bears two ion-radical centers N and C. As explained (Floogesteger et al. 2000), the species formed keeps a folded conformation in cyclohexane and a stretched conformation in benzene. 

The parison is inflated fast, within seconds or less, at a predetermined rate such that it does not burst while expanding. It is a complex process that involves expansion of a nonuniform membrane-like element. Because the extension ratio is high (above 10), it is difficult to calculate the final thickness distribution. Naturally, much of the recent theoretical research on parison stretching and inflation (as in the case with thermoforming) focuses on FEM methods and the selection of the appropriate rheological constitutive models to predict parison shape, thickness, and temperature distribution during the inflation. 

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