Nonlinear Step-Growth Polymerization

The reaction between similar /-functional molecules is the simplest case of nonlinear polymerization. An example is the etherification of pentaerythritol [27]  

Ignoring for the present the effects of any condensation products, we can schematically represent the polymerization of A/ (/ = 4) by 

Let the system react until some fraction p of the A groups have reacted p is given by 

E (W) ), the expected weight on any A looking into its parent molecule, will be the molecular weight of Ay plus the sum of the expected weights on each of the remaining / — 1 arms which is just E (W) ) for each arm. Thus, 

The weight-average molecular weight is obtained by picking an Ay or an A group at random and then finding the expected weight of the molecule of which it is a part that is, 

Let the system react until some fraction p of the A groups has reacted. Note that p = (Nao - Na)INa, where Na denotes the number of moles of A groups and the zero subscript indicates the value at zero time. Let us pick an A group at random, labeled as A in Eq. (5.105) and ask What is the weight, attached 

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Fig. 17. Extent of reaction at vitrification vs. reaction temperature for nonlinear step-growth polymerization (A + 2B2). All kinetic orders have the same p at vitrification. For mpdel parameters and system, see Fig. 16 caption. [Aronhime,...

During isothermal polymerization below Tg, the molecular weight and T, increase, and eventually T, will equal Tjure The main purpose of this section is to discuss the calculation of the time to vitrification, where vitrification is defined to occur when Tj, equals T ure- The concepts of vitrification and the TTT cure diagram are extended to linear systems for both step growth and chain reaction mechanisms, although most of the discussion will focus on the nonlinear step growth case, of which the cure of epoxy resins is an example. 

UnUke Flory s combinatorial approach, a Markovian analysis such as tot proposed by Macosko and Miller leads to easy derivations of expressions for M and for nonlinear polymers. Before generalizing to a reaction implying multivalent molecules having valence v, the case of a step-growth polymerization involving X4 tetravalent molecules and Ny Y- -Y molecules wiU be considered. 

Polymer growth J(c) showed nonlinear monomer concentration dependence in the presence of ATP (Carrier et al., 1984), while in the presence of ADP, the plot of J(c) versus monomer concentration for actin was a straight line, as expected for reversible polymerization. The data imply that newly incorporated subunits dissociate from the filament at a slower rate than internal ADP-subunits in other words, (a) the effect of nucleotide hydrolysis is to decrease the stability of the polymer by increasing k and (b) nucleotide hydrolysis is uncoupled from polymerization and occurs in a step that follows incorporation of a ATP-subunit in the polymer. Newly incorporated, slowly dissociating, terminal ATP-subunits form a stable cap at the ends of F-actin filaments. 

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