Treatment of Polydispersity

In this section it will be outlined how the different molar masses contribute to the TDFRS signal. Of especial interest is the possibility of selective excitation and the preparation of different nonequilibrium states, which allows for a tuning of the relative statistical weights in the way a TDFRS experiment is conducted. Especially when compared to PCS, whose electric field autocorrelation function g t) strongly overestimates high molar mass contributions, a much more uniform contribution of the different molar masses to the heterodyne TDFRS diffraction efficiency t) is found. This will allow for the measurement of small 

The treatment of polydispersity becomes straight forward in the limit of low concentrations, where c0(l-c0) c0 = Ek cok.cokis the concentration of component k with molar mass Mk. Under this assumption, Eq. (18) can be solved for every component independently by 

DTk are the thermal and Dk = (Tk q2)1 the translational diffusion coefficient of the k- th component, respectively. The diffraction efficiency C,ket(t) is obtained according to Eq. (24) with a redefined memory function for f 0  

All potentially molar-mass-dependent quantities have been labelled with a summation index k in Eq. (30), which, in this form, also holds for dilute solutions of mixtures of chemically different species or copolymers with heterogeneity of both chemical composition and degree of polymerization. 

In analogy to Eq. (26), the response of a polydisperse sample to an excitation step from 0 to 1 at t = 0 is given by 

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The pedestrian approach to polydispersity that has been demonstrated up to here is an extension to three dimensions of the well-known rigorous treatment of polydispersity in one dimension by means of the Mellin convolution (Eq. (8.85), p. 168). 

Continuous thermodynamics provides a simple way for the thermodynamic treatment of polydisperse systems. Such systems consist of a very large number of similar species whose composition is described not by the mole fractions of the individual components but by continuous distribution functions. For copolymers, multivariate distribution functions have to be used for describing the dependence of thermodynamic behavior on molar mass, chemical composition, sequence length, branching, etc. 

Essentially all Industrial polymers are polydisperse. The effect of polymer polydispersity on phase equilibrium has been discussed previously by many authors, but the treatment of Tompa ( ) Is one of the most complete. For our purposes, the situation can be summarized as follows. Polydispersity has virtually no effect on vapor-liquid equilibria (as long as the polymer Is non-volatile). However, polymer polydispersity does have an Important Influence on liquid-liquid equilibria. 

Strictly speaking, monodisperse samples would be required for the determination of the Mark-Houwink coefficients. Since, however, the poly-dispersities of the nine individual fractions are only moderate (Mw/Mn 2) and since both Mw and [tj] are measured as weight averages with the same statistical weights, the error introduced by the incorrect treatment of the polydispersity could be neglected. 

The treatments of Kochendorfer, Porod, and Warren-Averbach identify superposition with the mathematical operation of a convolution. While this is true for translational superposition, for dilational superposition it is a coarse approximation that is only valid for small polydispersity. In the latter case the convolution must be replaced by the Mellin convolution (Eq. (8.85), p. 168) governed by a dilation factor distribution and the structure of the reference crystal, the structure of each observed crystal is generated by affine dilation of the reference crystal (Stribeck [2]). 

The formation of relatively ill-defined catalysts for epoxide/C02 copolymerization catalysts, arising from the treatment of ZnO with acid anhydrides or monoesters of dicarboxylic acids, has been described in a patent disclosure.968 Employing the perfluoroalkyl ester acid (342) renders the catalyst soluble in supercritical C02.969 At 110°C and 2,000 psi this catalyst mixture performs similarly to the zinc bisphenolates, producing a 96 4 ratio of polycarbonate polyether linkages, with a turnover of 440 g polymer/g [Zn] and a broad polydispersity (Mw/Mn>4). Related aluminum complexes have also been studied and (343) was found to be particularly active. However, selectivity is poor, with a ratio of 1 3.6 polycarbonate polyether.970... 

An excellent concise treatment of scattering—by molecules and particles, single and multiple—at an intermediate level is Chapter 14 of Stone (1963). Among the books devoted entirely to scattering by particles, that by Shifrin (1951) most closely resembles ours in that it discusses optical properties of bulk matter as well. Biit the two books that have influenced us most are those of van de Hulst (1957) and Kerker (1969) we are indebted to both authors. Another book on scattering, which emphasizes polydispersions, is by Deirmendjian (1969). 

Fitting the experimental distribution to distributions calculated as described above leads to the conclusion that the efficiency of reactions for the synthesis of the sample must be between 97 and 98% for each step [169]. Polydispersity indices derived from the calculated distributions include the trace low molecular weight species that would be present but undetectable at the experimental signal to noise ratio, giving a more realistic measure of polydispersity. Thus, this statistical treatment would place the polydispersity of the sample between 1.005 and 1.003. 

We shall not attempt to review and compare critically various theories of liquid crystallinity in this chapter. Inasmuch as theory based on a lattice model has proved most successful in the treatment of liquid crystallinity in polymeric systems, we shall present an abbreviated account of that theory confined to its essential aspects. The versatility of this theory has permitted its extension to polydisperse systems, to mixtures of rodlike polymers with random coils and to some of the many kinds of semirigid chains. These ramifications of the theory will be discussed in this chapter... 

In homogenous media, most of the transacylation reactions are reversible and as soon as the first polymer amide groups are formed, the same kind of reactions can occur both at the monomer and at the polymer amide groups. Unless the active species are steadily formed or consumed by some side reaction, a set of thermodynamically controlled equilibria is established between monomer, cyclic as well as linear oligomers and polydisperse linear polymer. The existence of these equilibria is a characteristic feature of lactam polymerizations and has to be taken into account in any kinetic treatment of the polymerization and analysis of polymerization products. The equilibrium fraction of each component depends on the size of the lactam ring, substitution and dilution, as well as on temperature and catalyst concentration. 

A complete theoretical treatment of z-average molecular weight has been developed by Lansing and Kraemer (1935), who introduced the concept. For a monodisperse system, Mn = Mw = Mz for a polydisperse system, this is not the case. The ratio MwIMn can be used as an indication of polydispersity (Stevenson, 1982). 

As mentioned above, measurements at finite concentrations lead to a non-vanishing influence of the structure factor S(q). For the overwhelming majority of the latex systems studied by SAS-experiments so far, colloid stability has been achieved by a screened Coulomb interaction [5,62,63]. The structure factor of such a system of particles interacting through a Yukawa-potential has been extensively studied theoretically by Klein and coworkers (see Ref. [63] and further citations given there) who extended the treatment to polydisperse systems. 

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