First-shell substitution effect

With the method applied it is possible to take into account substitution effects both in the A and C monomers, as explained elsewhere (12). The substitution effect factor Kjj indicates the factor by wKTch the reaction rate between monomer I and any other monomer L is multiplied for each previous bond formed between monomers I and J (i.e. first-shell substitution effects (7)). For positive substitution effects Kjj is larger than 1, for negative effects it is smaller than 1. 

Fig. 59. a First shell substitution effect the probability for a reaction of functionality x depends on whether y or z or both had already reacted, b Second shell substitution effect the probability of reaction for a unit Xi and x2 depends on how many of the functional groups in y and z had reacted... 

We have now to take care that the various generations are properly connected. This is achieved as usual by introducing labels Si, S2, and s3 to the auxiliary variables. Furthermore, there may be reasons to assume that there will be a certain hindrance in the reaction if adjacent functionalities are already occupied. (First shell substitution effect.) Taking all these requirements together, Gordon and Parker were led to set up the generating functions as follows ... 

For simplicity, we start with a single tri-functional monomer RA3, the functional groups of which react with each other with the first shell substitution effect. It is now convenient to write down six types of reactions [44], one for each pair of reacting units of substitution degree 0,1, and 2. They are presented in Table 6 together with the appropriate rate constants. Similarly as in previous sec-... 

Table 6. The types of reactions in model polymerization of RA3 monomer with functional groups reacting with the first shell substitution effect. The subscripts at the symbols of reagents correspond to the codes of molecules (i,j-mers, see the text) and those at the rate constants indicate the degrees of substitution of reacting units. The left and right column corre-...

It is not difficult to extend the reasoning to the general case of RAf polymerization with the first shell substitution effect approximation. By defining the function... 

The monomer units (building blocks from which the molecules are built-up) differ in the number of reacted functional groups, i.e., in the number of bonds in which they take part, or, in other words, in the number of bonds they issue — the term which will be used later on. For a single-component system, this distribution is sufficient (for multicomponent systems the types of bonds are to be specified, too) for the build-up of trees, if the so-called first-shell substitution effect (fsse) is operative. Fsse means that the reactivity of a group in a unit is independent of the state of the groups in the neighbouring units. 

Finally, an entirely different approach to simulating gelation is the Dynamic Monte Carlo (DMC) method, in which chemical reactions are modeled by stochastic integration of phenomenological kinetic rate laws [23]. This has been used successfully to understand the onset of gel formation, first-shell substitution effects, and the influence of cyclization in silicon alkoxide systems [24—26]. However, this approach has not so far been extended to include the instantaneous positions and diffusion of each oUgomer, which would be necessary in order for the calculation to generate an actual model of an aerogel that could be used in subsequent simulations. 

Kasehagen LJ, Rankin SE, McCormick AV, Macosko CW (1997) Modeling of first shell substitution effects and preferred cyclization in sol-gel polymerization. Macromolecules 30 3921-3939. 

A first-shell substitution effect (ESSE) is the simplest kind of departure from ideal behavior. It is the situation where reactivity is affected only by the reaction of the functional groups attached to the same monomer unit. FSSEs are encountered in many polycondensations, as reactivities of functional groups in monomers are often different from the reactivities of end groups in polymers because of mutual steric, resonance, or electrostatic interactions. 

To account for first-shell substitution effects (FSSEs), a more general expression [Eq. (65)] can be written, using stoichiometric coefficients v ... 

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