Mass independent fractionation theory

FIGURE 2.3 Heavy oxygen isotope enriched mass independent fractionation, experiment [59] and theory [13]. 

Theory of Mass Independent Isotope Fractionation of Ozone... 

The principal focus of the present article is on the "mass-independent isotope fractional effect" (MIF) found in atmospheric and laboratory produced ozone. When this MIF occurs, a plot of the positive or negative "enrichment" in samples versus that of in those same samples has a slope of approximately unity, rather than its typical value of about 0.52. The 0.52 is the value expected using conventional transition state theory when nuclear tunneling effects are absent. For an isotope Q, 5Q is defined in per mil as 1000 [(Q/0)/(Q/0)std - 1]/ where Q/O is the ratio of Q to in the sample and std refers to its value in some standard sample, standard mean ocean water. An example of a three-isotope plot showing a slope of 0.52 is given in Figure 2.1. 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

The mean-field theory correctly predicts that the osmotic pressure is independent of molar mass in semidilute solution. However, the mean-field theory does not take into account the correlations between monomers along the chain, but instead assumes that they are distributed uniformly as in a solution of monomers. This is the reason why the two-body interac-Tioiis [Eq. (5.41)] become important at volume fraction --------------------... 

In Eq. (29.23) W 0) is the distribution of constraints among different points along the network chain and 0 = i/n is the position of the /th segment of the chain as a fraction of the contour length between two crosslinks. If the distribution is uniform, then W 0) = 1 inside the integrand of Eq. (29.23). In the case when constraints are assumed to affect only fluctuations of junctions (as in the constrained-junction theory), 0 is limited to 0 = 0 or 0 = 1 only. [95] It is important to note that this theory does not reduce identically to the constrained-chain theory, because the latter characterizes the deformation-dependent fluctuations of the centers of mass of the chains and not the deformation-independent fluctuations of the midpoints [95]. 

It has been reported (5) that the elastic modulus of ABS resins prepared by either mass or emulsion polymerization can be represented by a single relationship with the dispersed phase volume fraction. This is in agreement with the theory that the modulus of a blend with dispersed spherical particles depends only on the volume fraction and the modulus ratio of particles to matrix phase. Since the modulus of rubber is almost 1000 times smaller than the modulus of the matrix SAN, the rubber particle volume fraction alone is the most important parameter controlling modulus values of ABS resins. Even for rubber particles containing a high occlusion level, as in ABS produced by mass polymerization, the modulus of the composite particle still remains imchanged from pure rubber, suggesting a unique relationship between modulus and dispersed phase volume fraction. Also, the modulus of a material is a small strain elastic property and is independent of particle size in ABS. The effects of rubber content on modulus and on tensile... 

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