Distribution of molecular sizes

When surfactant molecules contain more than one distribution, for example, a distribution of chain lengths in the hydrophobic and hydrophilic portions, two-dimensional liquid chromatography (2DLC) is a very powerful method for complete analysis. One can get the full quantitation of the distribution of molecular size by using the 2DLC technique. For example, take a surfactant molecule like alcohol ethoxylates (AE s) having a general structure of... 

The explicit distribution of molecular sizes can now be obtained by solving recursively the Smoluchowski equation at Eq. (8). For ci we get... 

Spouge [ 21 ] has shown that for any kernel of this form the Smoluchowski equation can be solved analytically, i.e., the distribution of molecular size of polymer molecules at any extent of reaction prior to the gel point (if any) or at least moments of the distribution can be expressed explicitly. 

Equation (88) was used to demonstrate [44] the differences in distributions of molecular sizes in pre-gel stages of an RA3 polymerization with substitution effects as calculated according to a statistical and a kinetics model. The moments of distribution and the gel points were calculated by the numerical solution of a few ordinary differential equations that were derived from Eq. (88) and compared with analogous quantities calculated from a statistical model. 

Distribution of Molecular-Size Fractions (Dalton) of DOM Derived from Various Organic Wastes (% of Total DOM)... 

The distribution of molecular sizes in a polymer sample is usually expressed as the proportions of the sample with particular molecular weights. The mass of data contained in the distribution can be understood more readily by condensing the information into parameters descriptive of various aspects of the distribution. Such parameters evidently must contain less information than the original distribution, but they present a concise picture of the distribution and are indispensable for comparing different distributions. 

Size exclusion chromatography provides the distribution of molecular sizes from which average molecular weights can be calculated with the formulas summarized in Chapter 2. SEC is not a primary method as usually practiced it requires... 

Interchange reactions represent an approach to an equilibrium distribution of molecular sizes. Polymerizations that use very reactive monomers can proceed at room temperature [as in reaction (c) of Fig. 5-2]. They are kinetically controlled and will not produce random molecular weight distributions. Polymers from such syntheses will undergo interchange reactions, however, if they are subsequently heated to temperatures at which the equilibria such as that of reaction (5-14) can be reached in a reasonable time. 

ELquations (5-24) and (5-26) describe a random distribution of molecular sizes. This distribution is also known as a FlorySchulz distribution or a most probable distribution. Note that it is most probable only for linear step-growth polymers made under conditions which satisfy the assumptions in Section 5.4.1. If such polymers are made under other conditions (cf. Section 5.5), the molecular weight distributions of the products will not conform to the relations we have derived. However, when such polymers are melted and shaped, these processing conditions usually facilitate interchange reactions and randomization of the molecular weight distribution. 

The present refinement in linear siloxane polymerization is a monumental achievement resulting from the astute observations and ingenuity of many chemists over the past 120 years. The workers cited in this chapter are only some of the more recent contributors. Still, much work is yet to be done, and the critical reader should be left with many questions. For example, the equilibria 2 and 3 are traditionally the basis for explaining the distribution of molecular sizes and byproducts, but they exclude any role for the reactive chain ends. Yet, the accumulating evidence of the critical role of the counterions at the reactive ends in the mechanism of the process suggests that the equilibria may have to be rewritten to include the reactive ends. Definitive experiments are needed to settle the point. 

An extensive treatment of this subject has been ven very recently by Lichti el a . (1980), and a brief summary was given in an earlier paper (Lichti el ai, 1978). The model assumed for this treatment is a three-state model in which i is 0, I, or 2. An earlier paper (Lichti et al., 1977) applied a similar treatment to a two-state model in which i is 0 or 1. The treatment allows for the possibility that mutual termination may result in either combination or disproportionation. It also allows for the possibility of transfer to monomer. It has not, however, been possible to make allowance for branching and cross-linking. Prediction of the full distribution of molecular sizes, and not merely of particular moments of the distribution, has been achieved. The conclusion has been reached that compartmen-talization of the reaction leads to a broadening of the molecular-weight distribution. 

FIMS Fingerprint. Field ionization mass spectrometry of a mixture affords a spectrum of the molecular ions since fragmentation is minimal. Thus a distribution of molecular sizes and hydrocarbon classes can be obtained from a single analysis. This is illustrated in Figure 1 which compares the FIMS fingerprints for JP-5 from Shale-I and Shale-II refining. Distinct differences can be noted. The preponderance of alkanes (C... 

These seven samples were examined in 14 different laboratories by infrared, ultraviolet and mass spectrography. The mass spectrographic results are the most revealing. Table 16-4 gives the distribution of molecular size in the various samples according to the number of carbon atoms in the molecule. These results were obtained from parent-peak data. The occurrence of pronounced maxima in the distributions shows resolution into fraction groups centered on four or five adjacent molecular sizes. 

Above the gelatinization range (80°C, see Figure 6) the product distribution during reaction is considerably different than 60°C. Again, the distribution of molecular sizes is not random. 

Equation 3.6, together with Equation 3.4, describes a random distribution of molecular sizes this distribution is also known as the Flory-Schulz distribution or the most probable distribution [5]. Recently, Wutz and Kricheldorf [6] proposed a model describing the frequency distribution (/ ) and formulated the weight distribution (w,) of linear chains in step-growth polymerizations considering the cyclation reaction, which is one of the most important side reactions in step-growth polymerization. 

Finally, the parallel pore model was used to account for the actual distribution of molecular size and pore size. Eq. 3.S.d-9 for communicating pores, with an average value for tortuosity, was utilized ... 

Mildly condensed liquid resols, which are the more important of the two types of phenolic resins in the formulation of wood adhesives, have an average of fewer than two phenolic nuclei in the molecule. The solid resols average three to four phenolic nuclei but with a wider distribution of molecular size. Small amounts of simple phenol, phenolic alcohols, formaldehyde, and water are also present in resols. Heating or acidification of these resins causes cross-linking through uncondensed phenolic alcohol groups, and possibly also through reaction of formaldehyde liberated by the breakdown of the ether links. 

This material v/as again further depolymerised, under mild conditions with the aid of acetic acid, into four fractions with mean polymerisation degrees of 158, 100 54 and 31 respectively. The distribution of the molecular sizes of these four fractions is shown at the left of Fig. 22, leading to the result that ultimately important quantities of very small particles are also formed. Therefore in technical products derived from natural polymers the possibility of the isogel state must also be reckoned with even when Fig. 22. Distribution of molecular sizes (mean degree of the starting material is free from polymerisation p = 31, etc.) of specially prepared fractions low molecular products. of cellulose acetate... 

Weight-average molecular weight A polymer molar mass average which is the mean value of the weight distribution of molecular sizes, defined as M = LNiMf l = "LwiMirLwi, where Nj, Mi and Wi are the number of molecules, the molar mass, and the total with of the molecular species. 

For a system of homodisperse molecules, the value of a single decay constant can be determined from equation (13), a single exponential. However, scattering from a solution of polydisperse molecules results in a distribution of exponentials consequently, analysis of DLS data must be made using a probability function which accounts for the distribution of molecular sizes. 

It is shown in a subsequent section that in a water solution of oligomer salt containing a distribution of molecular sizes, addition of latex results in selective adsorption of smaller molecules at the particle-water interface. Use is made of this in refining crude fractions, as mentioned in the preceding section ... 

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