Peterlin function

The goieral form of the distribution function (123) can be particularized to yield the results of Debye, Peterlin and Stuart, and Frenkel. ... 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

The theory of non-Newtonian viscosity for ellipsoidal particles was first explicitly stated by Kuhn and Kuhn (1945), using Peterlin s distribution function (Peterlin, 1938) and Jeffery s hydrodynamic treatment (Jeffery, 1922-1923) [Eq. (10)]. More elegant treatments have recently been developed by Saito (1951), using the same ellipsoidal model, and also by Kirkwood and his co-workers (Kirkwood, 1949 Kirkwood and Auer, 1951 Kirkwood and Plock, 1956 Riseman and Kirkwood, 1956) for rodlike particles. The equivalence of the three theories has also been demonstrated by Saito and Sugita (1952). The general solution of Eq. (10) for the viscosity increment, v, can be expressed in the form... 

If the values of co and co from (33) and (34) are substituted in (32), a differential equation for the function q is obtained, for which a general solution by expansion in an infinite series has been obtained by Peterlin 96), 98), 99). The solution is best expressed with the aid of the parameter a = GjB, the ratio of velocity gradient to rotary diffusion constant. The general solution is very complex, and the terms converge slowly but for low values of [Pg.146]

It is possible to find an approximate expression for tj — rjs and 6 without finding the distribution function using a method similar to that of Peterlin (63). We start out with the Fraenkel dumbbell model, with the idea of letting H - ao later in order to get rigid dumbbell results. First Eq. (3.15) can be written down with Q successively chosen as XY,... 

Balta-Calleja and Peterlin [210] investigated annealing phenomena of drawn polypropylene. They found that the relaxation of tie molecules, shrinkage, and disorientation proceeded relatively fast, compared with the long period growth and the increase in density, which continued as a linear function of log time through 1000 min. 

For an unperturbed (Gaussian) spring-bead chain the mean square and mean reciprocal rij ) of the distance rij between its beads i and j are equal to a i — j and 6/na Y i —These are well-known facts and are cited here without proof (see Ref. [2]). How are these averages expressed when the chain is perturbed by intrachain excluded-volume interactions This problem was first brought us by Peterlin [56] in 1955. Solution of it needs information about Wij(R), but, as mentioned above, work on this function is as yet in the process of development. 

Miller et al. have recently reported infra-red dichroic data obtained for high density polyethylene crystallised under the orientation and pressure effects of a pressure capillary viscometer. Their data for a number of crystalline bands (including the 1894 cm" absorption) showed that the crystal c-axes were almost perfectly oriented (f 1) in the initial extrusion direction. The amorphous orientation functions were generally lower, but corresponded to an extension ratio between 2 and 7 when compared with the above results of Read and Stein and of Glenz and Peterlin. Further evidence was also obtained for the relatively high orientation of the amorphous component of the 2016 cm" band (U = 0-66-0-72). 

Non-linearity of the elastic force term in (4.1) can be formally eliminated by introducing Peterlin s approximation [14] which represents function E(h/Na) by the value for an average chain extension at any instant of time,... 

Consequence of the linearity introduced to (4.1) by the Peterlin s approximation is an affine evolution of the end-to-end vectors distribution function, W h,t), when assuming initial Gaussian distribution... 

An improvement in the agreement with experiment of predictions of crystal thickening rates on annealing as a function of long-period and temperature was suggested by Peterlin [29]. He proposed coUective mass transport by simultan-... 

Coppola et al. [142] calculated the dimensionless induction time, defined as the ratio of the quiescent nucleation rate over the total nucleation rate, as a function of the strain rate in continuous shear flow. They used AG according to different rheological models the Doi-Edwards model with the independent alignment assumption, DE-IAA [143], the linear elastic dumbbell model [154], and the finitely extensible nonlinear elastic dumbbell model with Peterlin s closure approximation, FENE-P [155]. The Doi-Edwards results showed the best agreement with experimental dimensionless induction times, defined as the time at which the viscosity suddenly starts to increase rapidly, normalized by the time at which this happens in quiescent crystallization [156-158]. 

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