The molecular weight distributions in many living (e.g.
anionic, group transfer, cationic,
and radical) polymerizations strongly depend on the dynamics of various
equilibria between chain ends
of different reactivity. A very important special case is
equilibria between active and dormant centers.
Various mechanisms including uni- and bimolecular isomerization
(or activation/deactivation), aggregation,
and direct bimolecular activity exchange (“degenerative transfer”)
are discussed and compared. For all
these mechanisms (and for both fast and slow monomer addition), the
averages of the molecular weight
distribution and the polydispersity index (DPI),
P̄
w/P̄
n, are
derived in a unified way. Their dependencies
on three universal parameters are analyzed: (i) on the reactivity
ratio of the two species, λ =
k‘p/k*p,
(ii)
on the fraction of the more active species, α =
P*/I
0, which is determined by the initial
concentrations of
reagents, and (iii) on a generalized exchange rate parameter, β,
which quantifies the rate of exchange
relative to that of propagation. The dependence of β on the
initial concentrations of reagents is defined
by the mechanism of exchange and can be used as a mechanistic criterion
to distinguish between various
possible mechanisms. For the typical case β > 1, the PDI
decreases with monomer conversion, which is
a common observation in many living polymerizations where 10 ≤ β ≤
100 was determined. At full
conversion, a simple relation,
P̄
w/P̄
n ≈ 1
+ ϑ/β, is valid, where ϑ depends on α and λ. For the
common
case where one species is dormant this simplifies further to
P̄
w/P̄
n ≈ 1
+ 1/β. Generally, the molecular
weight distribution is narrower if monomer is added
slowly.