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The phenomenological Gardiner-Collett approach not only assumes that the radiation field inside the cavity is given by the same expressions as for a perfect cavity, but also assumes that the quantization formalism is the same. In other words, it assumes that the an obey the usual commutation rules, (9.29)

2-5

and also with diacid monomer:

It is possible to show (Problem 9.5) that these commutation rules follow from (9.8)

2-6



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of the perfect cavity modes orthonormality properties and the canonical commutation relations for the radiation fields (Problem 9.6). So far so good, but to carry on and find the coupling, we must describe the modes of the outside. We proceed in the same way as for the intracavity fields, expanding the outside fields in terms of free-space modes with a perfect mirror at x = O. Then

Dimer reacts with itself to form tetramer:

Eout(x) =

2-7

-i21 O dk Evac(k) {b(k) - bt(k)} sinkx,





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The tetramer and trimer proceed to react with themselves, with each other, and with monomer and dimer. The polymerization proceeds in this stepwise manner with the molecular weight of the polymer continuously increasing with time (conversion). Step polymerizations are characterized by the disappearance of monomer early in the reaction far before the production of any polymer of suf ciently high molecular weight (approximately >5000 10,000) to be of practical utility. Thus for most step polymerizations there is less than 1% of the original monomer remaining at a point where the average polymer chain contains only $10 monomer units. As will be seen in Chap. 3, the situation is quite different in the case of chain polymerization. The rate of a step polymerization is, therefore, the sum of the rates of reaction between molecules of various sizes, that is, the sum of the rates for reactions such as

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(9.30)

Monomer Dimer + Dimer + Trimer + Trimer + Trimer + Tetramer Tetramer Tetramer Tetramer Pentamer Pentamer + monomer monomer dimer monomer dimer trimer + monomer + dimer + trimer + tetramer + trimer + tetramer dimer trimer tetramer tetramer pentamer hexamer pentamer hexamer heptamer octamer octamer nonamer

(9.31) These expressions define the operators b(k) and bt(k). With this definition, b(k) t and b (k) automatically satisfy the usual commutation relations of annihilation and creation operators of the continuum (Problem 9.7):

The reaction mixture at any instance consists of various-sized diol, diacid, and hydroxy acid molecules. Any HO-containing molecule can react with any COOH-containing molecule. This is a general characteristic of step polymerization. The kinetics of such a situation with innumerable separate reactions would normally be dif cult to analyze. However, kinetic analysis is greatly simpli ed if one assumes that the reactivities of both functional groups of a bifunctional monomer (e.g., both hydroxyls of a diol) are the same, the reactivity of one functional group of a bifunctional reactant is the same irrespective of whether the other functional group has reacted, and the reactivity of a functional group is independent of the size of the molecule to which it is attached (i.e., independent of the values of n and m). These simplifying assumptions, often referred to as the concept of equal reactivity of functional groups, make the kinetics of step polymerization identical to those for the analogous small molecule reaction. As will be seen shortly, the kinetics of a polyesteri cation, for example, become essentially the same as that for the esteri cation of acetic acid with ethanol. 2-1b Experimental Evidence

[b(k), bt (k')) = 8(k - k')

These simplifying assumptions are justi ed on the basis that many step polymerizations have reaction rate constants that are independent of the reaction time or polymer molecular

TABLE 2-1 Rate Constants for Esteri cation (25 C) in Homologous Compoundsa,b Molecular Size (x) 1 2 3 4 5 6 8 9 11 13 15 17

(9.32)

k 104 for (CH2)x(CO2H)2 6.0 8.7 8.4 7.8 7.3

Substituting (9.4) and (9.5) in (9.30) and (9.31), mUltiplying by the respective mode function and integrating over x outside the cavity, we find that

Rate constants are in units of L mol 1 s 1. Data from Bhide and Sudborough [1925].

b(k) - bt(k)

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