Affine deformation total

Cho and Kamal (2002) derived equations for the affine deformation of the dispersed phase, using a stratified, steady, simple shear flow model. It includes the effects of viscosity ratio and volume fraction. According to the equation, for viscosity ratio > 1, the deformation of the dispersed phase increases with the increase of the dispersed phase fraction. For compatibiUzed PE/PA-6 blends at high RPM (i.e., 100, 150, and 200 RPM) in the Haake mixer, the particle size decreases with concentration of the dispersed phase up to 20 wt%. This occurs because the total deformation of the dispersed phase before breakup increases as the volume fraction increases, and coalescence is suppressed. The increase of the particle sizes between 20 and 30 wt% results from the increase of coalescence due to the high dispersed phase fractions. The data for 1 wt% blends suggest that mixing in the Haake mixer follows the transient deformation and breakup mechanism, and that shear flow is dominant in the mixer. 

In fact, this derivation contains two main weaknesses. The first is the assumption that all chains between two crosslinks have the same end-to-md vector for the direction (parallel to the stretching axis) and modulus. This could be the case for a onedimensional material, but in an entangled or crosslinked melt the end-to-end vectors are expected to have all directions. Moreover, a distribution of the moduli is also also sensible. For the isotropic case, the end-to-end vector distribution is Gaussian, and integrating the total distribution leads back to the usual average = i — j b / 3. The simplest assumption for a deformed material is an affine deformation of this end-to-end distribution ... 

The first example is the totally affine deformation of the isotropic Gaussian conformation. In that case the deformation is the same on any scale, and thus For a uniaxial deformation X, ... 

It appears clearly that is a very slow function of X. if N is large (note the presence of N in the denominator of sinil/) while is still a fast one. This reflects the well-known fact that the orientation of a monomer is much smaller than the orientation obtained for a totally affine deformation. Thus, is much less anisotropic than in this example. 

Assuming the unaxial deformation of the polyethylene spherulite to follow an affine deformation hypothesis keeping its volume constant, as has been postulated by several authors,the number of crystallites orienting in a range from 6 to (0 -hd0 ) is given by equation (35), where Nq is the total number of crystallites within a spherulite and X is the extension ratio of the spherulite along the X3 axis. 

These end points are assumed to be a-junctions that deform affinely to the strain. The internal junctions are all regarded as r-junctions. The total number of chains in the network is... 

Other authors published interesting results on the radius of gyration for coextruded samples polystyrene of relatively high mass (M, p = M = 6 10 ) was used. A strip of polystyrene was inserted into a cylindrical billet of polyethylene, extruded at T = 127 °C at a pressure of 280 kg/cm and with a deformation rate of 0.1-0.2 cm/ mn. The deformation ratio, measured from marks on the sample, attained very large values k = 3, 4, 5, and 10. It is difficult to give a corresponding value of t for this deformation, but an indication is that the recovering was tested as total. The results, obtained only from a comparison to the affine behaviour, are presented in Sect. 5. 

A network in the liquid or amorphous state can be given a quantitative descrip-tion(4,9,10) by defining a chain as that portion of the molecule which traverses from one cross-linked unit to a succeeding one. It is convenient to characterize each chain by a vector r which connects the average position of its terminal units, namely, the cross-linked units. The number of chains v must be equal to the number of intermolecularly cross-linked units. If No is the total number of chain units in the network, then p is equal to v/No. The network can then be characterized by the number of chains and their vectorial distribution. When the network is deformed, a common assumption made is that the chain vector distribution is altered directly as the macroscopic dimensions. An affine transformation of the average position of the coordinates of the cross-links occurs. It is also usually assumed that the individual chains obey Gaussian statistics. 

To develop the tube theory of polymer motion, we consider the response of the melt to a step deformation. This is an idealized deformation that is so rapid that during the step no polymer relaxation can occur, and the polymer is forced to deform affinely, that is, to the same degree as the macroscopic sample is deformed. The total deformation, though rapid, is small, so that the chains deform only slightly this is called a small amplitude step strain. Because the deformation is very small, the distribution of chain configurations remains nearly Gaussian, and linear viscoelastic behavior is expected. In Chapter 4 we saw that the assumption of linear behavior makes it possible to use the response to a small step strain experiment to calculate the response to oscillatory shear or any other prescribed deformation. 

The different chain flexibility, or more precisely the flexibility of the soft part of the molecules, is responsible for the quite different fractions of the residual deformation, e, from the total one, as well as the limit of below which the change of L is more or less completely reversible (Figures 7 and 9). The flexibility of the soft parts strongly affects the character of microdeformation, making it to obey or disobey the affine model of deformation. The latter can only be applied for the homo PBT and PEE Arnitel EM400 (Figure 9). 

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