Chemical sizing

When dealing with high molecular weight polymers it is convenient instead of 1 to introduce the composition vector Cwith components a=lall(a=l,...,m) which, in combination with the chemical size Z, is completely equivalent to 1. For such copolymers the variables Z and ta may be thought of as continuous and recourse can be made to a expression which relates SCD/W(1) to the size distribution (SD) /W(Z) and the composition distribution (CD) W( C l) of macromolecules of given size Z. 

Ra(l) concentration of a-th type radical with chemical size Z... 

The main statistical characteristic of the chemical structure of a heteropolymer among those pertaining to the first type is the distribution of molecules f( h, 12) for numbers l and h of their constituent monomeric units Mi and M2. In dealing with a high-molecular weight polymer, these numbers may be taken as continuous variables, uniquely specifying chemical size l=l + h and composition f = li/l of a macromolecule. Under such a consideration, it is more convenient instead of function /(Zi, l2) to use the equivalent function of Size-Composition Distribution (SCD) f(l, < ) This is possible to represent... 

When considering the composition inhomogeneity of Markovian copolymers, the finiteness of the chemical size of macromolecules cannot be ignored, because fractional composition distribution W(/ f) in the limit / -> oo turns out to be equal to the Dirac delta function 5(f - X). For macromolecules of finite size f2> 1 the function W(/ f) is the Gaussian distribution whose center and dispersion (Eq. 2) are described by relationships (Eq. 8) and the following one... 

The above phenomenon is due to the pronounced polydispersity of these products in their chemical size l described by the Flory exponential distribution. Because the composition of each macromolecule of the sample under investigation is unambiguously related to its degree of polymerization l, the Flory distribution for l in a polymer sample is responsible for its significant composition inhomogeneity. 

At low-conversion copolymerization in classical systems, the composition of macromolecules X whose value enters in expression (Eq. 69) does not depend on their length l, and thus the weight composition distribution / ( ) (Eq. 1) equals 5(f -X°) where X° = jt(x°). Hence, according to the theory, copolymers prepared in classical systems will be in asymptotic limit (/) -> oo monodisperse in composition. In the next approximation in small parameter 1/(1), where (/) denotes the average chemical size of macromolecules, the weight composition distribution will have a finite width. However, its dispersion specified by formula (Eq. 13) upon the replacement in it of l by (l) will be substantially less than the dispersion of distribution (Eq. 69)... 

For the calculation of the statistical characteristics of a copolymer by means of expressions (Eqs. 68-71), it is necessary to substitute into them the distribution function/w(D for the chemical size of macromolecules formed... 

An interesting approach has been employed in paper [74] to find the distribution f(li, l2) of copolymer chains for numbers l and h of monomeric units Mi and M2. This distribution is evidently equivalent to the SCD, because the pair of numbers k and I2 unambiguously characterizes chemical size (l = h + l2) and composition ( 1 = l] //, 2 = h/l) of a macromolecule. The essence of this approach consists of invoking the Superposition Principle [81] that enables the problem of finding the Laplace transform G(pi,p2) of distribution f(li,k) to be reduced to the solution of two subsidiary problems. The first implies the derivation of the expression for the generating function [/(z1",z 2n ZjX,z ) of distribution P(ti, M2 mt, m2), and the second is concerned with finding the Laplace transforms g (pi,p2) and (pi,p2) of distributions (Eq. 91). With these two problems solved, it is possible to obtain the characteristic function G(pi,p2) of distribution f(li,h) using the Superposition Principle formula... 

Using relationships Eqs. 92 and 94, it is an easy matter to find average values of chemical size l = l + and composition Xa = lafl of a copolymer obtained in regime 1 (Eq. 85). The expressions for them in terms of the parameters (Eq. 79) have a simple form... 

It is common during the dispersion process to use sodium hydroxide as this assists in ink removal and also in the more efficient dispersion of papers which have been chemically sized (see Chapter 7). Sodium hydroxide also has a swelling effect on cellulosic fibres and it is most probable that this too has a beneficial effect on strength. 

To test the applicability of statistical techniques for determination of the species contributions to the scattering coefficient, a one-year study was conducted in 1979 at China Lake, California. Filter samples of aerosol particles smaller than 2 ym aerodynamic diameter were analyzed for total fine mass, major chemical species, and the time average particle absorption coefficient, bg. At the same time and location, bgp was measured with a sensitive nephelometer. A total of 61 samples were analyzed. Multiple regression analysis was applied to the average particle scattering coefficient and mass concentrations for each filter sample to estimate aj and each species contribution to light scattering, bgn-j. Supplementary measurements of the chemical-size distribution were used for theoretical estimates of each b pj as a test of the effectiveness of the statistical approach. 

While the relations of chemical size and solubility are gratifying to recognize, we still notice that each compound class exhibits its own behavior (Fig. 5.2). Hence, we may wonder if there is any means to account for variations from compound class to compound class. Based on our visualizations of organic solute intermolecular interactions, it is not surprising to learn that parameters that quantify the importance of interactions like hydrogen bonding can be used to adjust for differences between compound classes. 

Fortunately, you are aware of the study by Aminabhavi and Naik (1998). In that work, the investigators deduced the molecular diffusivities of several alkanes in plastics including LLDPE. Plotting their data as a function of chemical size (here, molar volumes), you have their results as shown in the figure below. 

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