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The discussions until this point have been concerned with the polymerization of bifunctional monomers to form linear polymers. When one or more monomers with more than two functional groups per molecule are present the resulting polymer will be branched instead of linear. With certain monomers crosslinking will also take place with the formation of network structures in which a branch or branches from one polymer molecule become attached to other molecules. The structures of linear, branched, and crosslinked polymers are compared in Fig. 1-2. Consider the polymerization of an A reactant in the presence of a small amount of a B monomer Af containing f functional groups per molecule. The value of f is termed the functionality of the monomer. The product of this polymerization will be a branched polymer in which f chains are attached to a central branch point (i.e., an Af species). For the speci c case A j A of f 3, polymerization of A in the presence of B leads to the structure XVI. A A

careful consideration of this structure shows that there can be only one Af reactant molecule incorporated into each polymer molecule. Further, crosslinked species will not be formed. Reactions between two polymer molecules of the type described above cannot occur since all growing branches possess A functional groups at their ends. Branch chains from one molecule cannot react with those from another. (This would not be true if A groups were capable of reacting with each other. However, that is not the usual situation.) Polymerizations with extensive branching are considered elsewhere. Sections 2-10 and 2-16 discuss branching with and without crosslinking, respectively.

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Rotary switches are depicted showing, primarily, the common terminal (the contact common to all switch positions) and the variable number of outer terminals that the signal current applied to the common terminal will flow through when the switch s dial is turned to that outer terminal s position. Just as in the example of toggle switches, rotary switches are designed in various configurations. You ll find from few to many contacts on the switch diagram, depending upon the switch s design. (Some rotary switches use stacked platforms of contacts to achieve dozens of contact changes with one click of the dial.) Again, the letters SW label switches in schematic diagrams and refer you to the master parts list.

The molecular weight distribution in this type of nonlinear polymerization will be much narrower than for a linear polymerization. Molecules of sizes very much different from the average are less likely than in linear polymerization, since this would require having the statistically determined f branches making up a molecule all very long or all very short. The distribution functions for this polymerization have been derived statistically [Peebles, 1971; Schaefgen and Flory, 1948], and the results are given as

heavy responsibilities for the creditor These costs and responsibilities rarely change with interest rates Great-grandfather and -grandmother may have invested in mortgages on their neighbors farms If so, they drove around to collect the payments and watched their neighbors crops and their neighbors personal habits closely They earned a good part of their extra mortgage income It was always much easier to deposit money in a savings account than to own and service a portfolio of home mortgages hence the difference in rate, only in part, perhaps in small part, due to risk and in large part attributable to cost and convenience As the mortgage market came more nearly to resemble the bond market in recent decades, however, mortgage rates were no longer quite as stable relative to bond rates as they once were.

2-130 2-131

2-132

In the great advance and decline of rates between 1965 and 1989, the time pattern of mortgage rates was almost the same as that of bond yields CONSUMER CREDIT RATES AND LEGAL LIMITS Home mortgage rates, although above prime rates of interest, have been below the prevailing rates of interest on most forms of consumer credit with relatively short maturity This apparently has been true over the centuries During the Middle Ages, relatively modest rates often prevailed for long-term loans secured by farms or estates, but pawnshop rates were much higher, including even those charged by the publicly endowed pawnshops The reason for the higher rates was probably not primarily that the risk of making short loans on valuable security was greater than the risk of making long-term loans on homes, but rather that the cost of making many small short loans was very large.

Fig. 2-14 Weight fraction distribution plot for multichain polymerization. Plot 1, f 1; plot 2, f 2; plot 3 f 3; plot 4, f 4. After Howard [1961] (by permission of Iliffe Books and Elsevier, London).

Xw 1 1 f Xn 2-133

FIGURE C-18

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