Alfrey approximation

Figure 4.18 The Alfrey approximation the stress relaxation of a Maxwell unit A is replaced by a step function B. The curve C represents relaxation of a typical viscoelastic polymer...

Figure 4.19 The Alfrey approximation for the relaxation time spectrum H ty. (a) from the stress relaxation modulus G(t) (b) from the real and imaginary parts G and G2, respectively, of the complex modulus G(co)...

The reader may use the Alfrey approximation (see Section 4.3.2) to derive relaxation and retardation time spectra from the data of Figure 6.1. These spectra can be approximated by a wedge and box distribution [3], shown by the dotted lines in Figure 6.2. 

Both physical and technological properties of copolymers are influenced by the sequence distribution in the macromolecular chains. The mathematical relationships governing the distribution, first developed by Alfrey and Goldfinger (7), are based upon kinetic and statistical considerations implying three fundamental assumptions a) steady state copolymerization, b) terminal effect only (i.e. influence of the last, but not of the penultimate unit of a growing chain on the addition of the next monomeric unit), and c) constant monomer feed. Under these assumptions, which may be defined as a first order approximation, the copolymers are described by two quantities, the ratio / of the molar fractions of the two monomers and the product of reactivity ratios... 

All the above factors controlling monomer and radical reactivities contribute to the rate of polymerization, but in a manner which makes it difficult to distinguish the magnitude of each effect. Attempts to correlate copolymerization tendencies based on these factors are thus mainly of a semiempirical nature and can, at best, be treated as useful approximations rather than rigorous relations. However, a generally useful scheme was proposed by Alfrey and Price [23] to provide a quantitative description of the behavior of diferent monomers in radical polymerization, with the aid of two parameters, for each monomer rather than for a monomer pair. These parameters are denoted by Q and e and the method has been called the Q — e scheme. It allows calculation of monomer reactivity ratios r and T2 from properties of monomers irrespective of which pair is used. The scheme assumes that each radical or monomer can be classified according to its reactivity or resonance effect and its polarity so that the rate constant... 

Attempts to correlate copolymerization tendencies are thus mainly on a semi-empirical footing and must be treated as useful approximations rather than rigorous relations. A genraally useful scheme was proposed by Alfrey and Price, who denoted the reactivities or resonance effects of monomers by a quantity Q and radicals by F, whereas the polar properties were assigned a factor e, which is assumed to be the same for both a monomer and its radical. 

Alternatively, one can use various approximations (see Ferry, 1980) to determine the relaxation spectra directly from modulus data or a mathematical function fit to the data. A method that works well is known as Alfrey s rule, in which the exponential function in the integral in Eq, 7,42 is 0 at small x s and 1 at large x s and is thus replaced by a step function H(t-T). With this simplification, Eq, 7,42 can be differentiated to obtain... 

The exact formal relationships between the various viscoelastic functions are conveniently expressed using Fourier or Laplace transform methods (cf. Section 5.4.2). However, it is often adequate to use simple approximations due to Alfrey in which the exponential term for a single Kelvin or Maxwell unit is replaced by a step function, as shown schematically in Figure 5.18. 

There are several other cases which are commonly encountered. Eor thin annular dies Eqs. 7.54 through 7.59 can be adopted directly by making appropriate replacements for W and h. Eor multilayer extrusion through an annulus in which the gap is too thick to apply the thin slit approximation, the appropriate equations are derived in Problem 7B.8. For more than two layers the equations for flow through parallel plates can be generalized as follows (Schrenk and Alfrey, 1976). Referring to Figure 7.51 we can obtain expressions... 

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