Alternative Relationships

In addition to the Langevin models and their Gaussian simplification there have been a number of other relationships derived to describe rubber elasticity. These may be roughly divided into three groups although in some studies the distinction is blurred. These are  

In 1940 Mooney proposed that the force-extension curve of a rubber in simple extension is more consistent with the semi-empirical formula  

This equation is based on the assumption that in rubber Hooke s Law is obeyed in simple shear. It will be noted that where C2 = 0, eqn 

It should however be noted that some variation from rubber to rubber has been demonstrated in a recent paper (Boyer and Miller 1977) and that this appears to be related to the interchain distance. A log-log plot of the ratio C2/C1 against the cross-sectional area per unit chain gave a unique linear negative slope for a range of polymers. 

In spite of its lack of success in this direction the Mooney equation has provided the basis for a number of phenomenological theories of rubber elasticity such as those by Rivlin and Saunders (1951). Discussion of such non-molecular theories is considered outside the scope of this book. 

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There are several possible alternative relationships for E (Lemhch, op. cit.). For simphcity, consider E = K C, where K is not necessarily the same as the equihbrium constant K. Substituting and integrating from the boundaiy condition of C = Cg at / = 0 yield... 

The present work demonstrates that (1) conclusions reached earlier for model chains also apply to realistic chains, (2) effects seen with finite chains may survive in extremely long chains, (3) the limit at large n for (o, 2 - 1) / (ar2 - 1) provides little information about the effect of excluded volume on the dipole moment of infinitely long chains, and (4) an alternative relationship between a 2 and ar2 may provide useful information on the relationship of and 0 for long chains. 

The majority of chemical companies offer a wide range of channel alternatives -e-channels, direct sales, customer service, third party distribution, technical service. This multi-channel approach has greatly increased customer flexibility and the choice of alternative relationship models, but has done little to reduce the investment in direct customer interface capabilities or to differentiate players service offerings. Primarily, it has added complexity and costs to the delivery system. 

An alternative relationship (9), based on absolute reaction-rate theory, has been derived by Marcus (1956, 1960, 1964) and Hush (1961, 1967, 1968, 1975). Scandola and Balzani (1979) found that the use of this equation gave... 

It turns out that there is an alternating relationship between the number of electron pairs (double bonds) undei going bond leorganization and the stereochemistry of ring opening or closure. Polyenes with an even number of electron pairs undergo thermal electrocyclic reactions in a conrotatory sense, whereas polyenes with an odd number of electron pairs undergo the same reactions In a disrotatory sense. 

An alternative relationship is a curve such that, when used with an approximation of global U distribution, it is in agreement with the global mean estimate derived from the technique (Wanninkhof, 1992 henceforth W92). Although this is a sensible approach, the main difficulty is in deciding the appropriate shape of the curve. Wanninkhof used a dependence of on based on results from wind tunnel studies and n was assumed to be 0.5. 

Substituting the value of Gg obtained from this equation into Eq. (6.56) gives the following alternative relationship between G (oi) and G"(o>) ... 

This criticism can be formalized in terms of a simple physical model that furnishes an alternative relationship between o (x) and / for a structural parameter x with equilibrium value Xq in the absence of perturbing forces. The energy increase for small deformations x Xq is assumed to be quadratic and the crystal environment is supposed to exert some perturbing force a on the system, so that the linearly perturbed energy becomes ... 

Here the mass velocity G is used in place of its equal Vp. Dividing both sides of Eq. (12.27) by the product DGjpi) CpPlk) gives an alternate relationship... 

Laura and Susemihl [52] gave several values of the parameter As for several values of At The alternate relationship [91] is... 

An alternative relationship between the concentration and the reflected intensity is now widely used in near-infrared diffuse reflectance spectroscopy. The relationship is analogous to the Beer-Lambert law, which is discussed in Chapter 5 ... 

Note In passing, it should be noted that recent work " has indicated that N70 = constant, is probably not applicable when gassing. However, it is only used here to illustrate the approach and an alternative relationship could be selected as appropriate in order to produce definitive scale-up... 

This equation yields an alternative relationship for the partition constant in HS in which K is found to be inversely proportional to a product of the partial pressure of component i if it were pure and its activity coefficient according to... 

An interesting critical analysis of the validity of empirical relationships for the assessment of size-dependent growth has been made by Rojkowski (1978) who proposes alternative relationships which merit consideration if a more rigorous assessment of the problem is considered necessary. 

In the case of Rco < 0.01, the simpler function log l/Rdiffuse reflectance. Since such small R,o value is usually foimd in the near-IR region, therefore, an alternative relationship between concentration and reflected intensity in near-IR is defined as ... 

An alternate relationship has also been proposed by the same authors,... 

Both expressions have been criticized, and an alternative relationship has been used with some success [27] ... 

Structural Abstraction - supporting generalised n-ary, optional and alternative relationships between GSN elements... 

As shown in the last section, R W>ult s and Henry s laws are permissible solutions of the differential expression (7 40) which governs vapour-liquid equilibria at constant temperature and constant (or effectively constant) total pressure. However, these laws are not the only functional relationships connecting and Xi which are compatible with (7 40), and in the present section we shall discuss some alternative relationships for the special case of a binary system. 

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