Copolymerization

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COPOLYMERIZATION

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COPOLYMERIZATION

Introduction

This article is primarily concerned with chain copolymerization. Chain copolymer-
ization is important as it allows a wide array of functional molecules to be designed
and incorporated into polymeric materials. In addition, fundamental studies on
copolymerization allow the mapping out of monomer structure–reactivity rela-
tionships that underpin our knowledge of polymerization reaction mechanisms.

In recent years it has become evident that the reaction mechanism involved

in free-radical copolymerization propagation is appreciably more complicated than
originally postulated in the terminal model developed back in the 1940s. There is
still some reluctance to acknowledge this fact as it is often more convenient to sim-
plify the mechanism. In some instances this may be appropriate; however, in many
cases where speciﬁc information about radical reactivity is garnered from the data
reduction, signiﬁcant problems may arise if the interpretation is oversimpliﬁed. In
an attempt to clarify the position, we would like to draw an analogy with the devel-
opment of understanding in the study of a human cell. As experimental techniques
(particularly the electron microscope) have become more sophisticated for study-
ing cells, our recognition of the complexity of cellular processes has been enhanced
and new theories and ideas have been advanced to accompany the increased ap-
preciation of cellular processes, leading to an understanding of the importance and
role of apoptosis in designing new cancer treatments. In a similar way as experi-
mental methods have improved to study free-radical copolymerization (most no-
tably, size-exclusion chromatography, pulsed-laser polymerization, soft ionization
mass spectrometry and ab initio calculations) our understanding of the complex-
ity of radical copolymerization has improved and it would be a mistake if all these
new experimental data were ignored, leaving theory restricted to the basic termi-
nal model developed in the 1940s on the basis of fairly rudimentary experimental
data. The complexity of radical processes should be recognized while allowing
for the fact that by using carefully designed synthetic methods where some ki-
netic pathways are suppressed (ie, controlled/living radical polymerization) we
can largely overcome many of the side reactions to generate bespoke polymer
structures. In 1985–1987 the implicit copolymerization model was developed by
Professor Fukuda, when his team discovered that the terminal model failed in the
prediction of average propagation rate coefﬁcients. In subsequent years it has be-
come clear, mainly through advanced quantum calculations, that the cause of fail-
ure of the terminal model can be ascribed to a combination of radical stabilization,
entropic interactions, and partial charge transfer in the transition state (so-called
polarity effects), which are all signiﬁcantly inﬂuenced by penultimate units. This
leads to a conclusion that the physical chemistry underpinning copolymerization
can be best described by an explicit modelling approach. This complexity causes
some problems in the data reduction of copolymerization experiments. However,
for the simple prediction of composition the terminal model, although fairly crude,
provides an adequate method in many instances. The problems only arise when it
is used to study absolute rate coefﬁcients (cross-propagation rate coefﬁcients) or
when it is used to predict other quantities such as average propagation rates or se-
quence distribution. Therefore it is imperative that the limitations of the terminal

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395

model are recognized particularly when adding layers of experimental complexity
(controlled/living radical copolymerization). Very similar arguments can be made
for the area of termination in free-radical copolymerization with the recent devel-
opment of SP-laser experiments, enabling the study of the reaction with greater
precision and accuracy than was previously possible. A continuous development
of knowledge via copolymerization research should result from a simultaneous
progression in both experimental techniques and modiﬁed theory.

This article progresses to discuss the newly emerging ﬁeld of controlled/living

radical copolymerization, and the novel materials that can be made using these
highly versatile synthetic methods. Finally we cover in brief ionic chain copoly-
merization and cross-linking copolymerization.

Propagation Kinetics and Copolymerization Models

Copolymerization propagation kinetics have perhaps the most signiﬁcant impact
on copolymer properties. The rates of the individual propagation reactions in a
free-radical copolymerization not only affect the overall rate of polymerization and
the molecular weight distribution of the resulting polymer, but also directly govern
the copolymer composition and microstructure (ie, whether the monomers form
block, alternating, or random copolymers), and the chain-end composition of the
living radicals (which in turn affects the chain-end composition of the dead poly-
mer). Furthermore, it is the geometry of the transition structure in the propagation
reaction that governs the stereochemistry of the resulting polymer. The ability to
understand the fundamental inﬂuences on the propagation reaction (both from a
kinetic and mechanistic point of view) thus enables these important properties to
be modeled, and hence controlled, simply by manipulating the reaction conditions
(such as solvent type, monomer feed composition, and temperature).

There are an enormous number of different copolymerization models avail-

able in the literature. The simplest kinetic model, the terminal model, was derived
in the 1940s and, while it continues to receive widespread use, 60 years of exper-
imental and theoretical studies have revealed serious deﬁciencies in this model
for a wide range of systems. In response to these problems, numerous alternative
copolymerization models have been derived. These models differ primarily in their
assumptions as to what factors affect the rate of the propagation step. For example,
in the terminal model it is assumed that the composition of the terminal unit of
the propagating radical (ie, the

α- and β-substituents) and the monomer affect the

propagation rate, while in the penultimate model it is assumed that the penulti-
mate unit of the propagating radical (ie, the

γ - and possibly δ-substituents) is also

important. Other models incorporate possible side reactions that might affect the
overall propagation rate, such as depropagation or various types of solvent effect.

In what follows we brieﬂy outline the assumptions and equations correspond-

ing to the most important copolymerization models. To keep this as a compact and
accessible summary, we do not present derivations for the equations. Neither do
we present the lengthy equations needed to predict the various triad and pen-
tad fractions under speciﬁc models. For more detailed information of this nature,
the reader is referred to the original references. In addition, a general account
of the main steps in deriving a copolymerization model is found in Reference 1.

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For the sake of simplicity, we present equations for each of the types of models in
their low-conversion, “instantaneous” forms, but conclude with a brief discussion
of the extension of the models to their conversion-dependent “integrated” form.
Having outlined all of the models, we then discuss the main experimental evi-
dence for or against them, and make some general suggestions concerning their
applicability to speciﬁc types of system. We then conclude with a short discussion
of reactivity in free-radical copolymerization. More fundamental aspects of this
topic are addressed in a separate article (see C

OMPUTATIONAL

UANTUM

HEMISTRY

FOR

REE

-R

ADICAL

OLYMERIZATION

Terminal Model

In the terminal model it is assumed that the terminal unit of a propagating poly-
mer radical is the only factor inﬂuencing its reactivity, and that side reactions are
not signiﬁcant. As a result, there are only four types of propagation reaction in
the free-radical copolymerization of any two given monomers (M

and M

From this assumption, Jenkel (2), Mayo and Lewis (3), and Alfrey and Goldﬁnger
(4) all independently derived an expression for the copolymer composition (F

)

as a function of the monomer feed fractions (f

and f

) and the reactivity ratios

and r

) of the monomers:

+ f

where

= j and i, j = 1 or 2

(1)

Alfrey and Goldﬁnger (4) also derived expressions for the sequence distribution
and the number-average degree of polymerization expected for a copolymerization
obeying the terminal model. They later extended the terminal model to describe
polymerizations involving three or more monomers (5,6). Fukuda and co-workers
(7) derived the following expression for the copolymerization propagation rate
constant

under the terminal model:

+ 2f

+ r

]

+ [r

]

(2)

This equation follows from the kinetic analysis of copolymerization by Melville
and co-workers (8) and Walling (9), who arrived at an expression for the overall
rate of copolymerization assuming a terminal model for both propagation and
termination.

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397

Explicit and Implicit Penultimate Models

In the explicit penultimate model, it is assumed that both the terminal and penul-
timate units of a polymer radical may affect the rate of the propagation reaction.
As in the terminal model, side reactions are considered to be insigniﬁcant. The
explicit penultimate model was ﬁrst suggested in 1946 by Merz and co-workers
(10), who derived equations for predicting the composition and sequence distri-
bution under this model. A full description of the model—including an expression
for

—has since been provided by Fukuda and co-workers (11), and it is their

notation that is used below. In addition, penultimate model equations for the case
of terpolymerization have been published by Coote and Davis (12).

In the presence of a penultimate unit effect, there are eight different types

of propagation reactions:

From their eight different propagation rate constants, four different monomer
reactivity ratios (r

and r

) and two radical reactivity ratios (s

) can be deﬁned as

follows.

iii

iij

jii

jij

jii

iii

where

= j and i, j = 1 or 2

These are used to calculate the adjusted parameters ¯r

and ¯k

¯r

= r

+ f

where

, j = 1 or 2 and i = j

(3)

¯k

= k

iii

+ f

where

, j = 1 or 2 and i = j

(4)

These are used in place of r

and k

, in the terminal model expressions for compo-

sition and

(ie, eqs. 1 and 2 above).

The implicit penultimate model was ﬁrst suggested by Fukuda and co-

workers (7) in 1985, in order to describe their observation that the terminal model
could be ﬁtted to the composition data for the copolymerization of styrene with
methyl methacrylate, though it could not simultaneously describe the propagation
rate coefﬁcients. In this model, the following restriction is placed on the explicit
penultimate model.

iii

iij

= r

jii

jij

where

= j and i, j = 1 or 2

(5)

The penultimate unit effect is thus assumed to be absent from the monomer re-
activity ratios, which are equivalent to their terminal model forms, and only to

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COPOLYMERIZATION

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exist in the radical reactivity ratios (ie, through values of s

= 1). This amounts

to assuming that the magnitude of the penultimate unit effect on reactivity is
independent of the type of monomer with which it is reacting (since the equality
k

jii

iii

= k

jij

iij

follows directly from the assumption that r

= r

). In other words,

it assumed that there is a penultimate unit effect on reactivity but not selectivity.
On the basis of this assumption (ie, eq. 5), equation 3 collapses into

¯r

= r

jii

jij

= r

iii

iij

where

= j and i, j = 1 or 2

(6)

Thus the adjusted monomer reactivity ratios of the penultimate model are re-
placed simply by their corresponding terminal model values. However, since the
penultimate unit effect can remain in the radical reactivity ratios (ie, through val-
ues of s

= 1), equation 4 does not collapse to its equivalent terminal model form

(ie, ¯k

= k

iii

). Since the composition and triad/pentad fraction equations contain

only ¯r

terms, they collapse to the corresponding terminal model equations. How-

ever, since it contains both ¯r

and ¯k

terms, the propagation rate equation, though

simpler than that of the explicit penultimate model, continues to differ from that
of the terminal model. There is thus an implicit penultimate unit effect—that
is, a penultimate unit effect on the propagation rate but not the composition or
sequence distribution.

Solvent Effects

Solvent Polarity Effects.

When polar interactions are important in the

transition structure of the propagation reaction, the polarity of the solvent may
affect the propagation rate. This may be explained as follows. Polar interactions
are said to occur when the transition structure is stabilized by charge transfer
between the reacting species (13). The amount of charge transfer, and hence the
amount of stabilization, is inversely proportional to the energy difference between
the charge transfer conﬁguration and the product and reactant conﬁgurations that
combine to make up the ground-state wave function of the transition structure.
Now it is known that polar solvents can stabilize charged species, as seen in
the favorable effect of polar solvents on both the thermodynamics and kinetics
of reactions in which charge is generated (13). Thus, when charge transfer in
the transition structure is important, the relative stability of the charge-transfer
conﬁguration, and thus of the transition structure, will be affected by the polarity
of the solvent. Hence, when polar interactions are important in a propagation
reaction, a polar solvent can stabilize the transition structure and hence lower
the reaction barrier.

When such effects are important, the polarity of the solvent affects radical

selectivity as well as radical reactivity. This is because the extent to which charge-
transfer stabilization can occur, and hence the extent to which polar solvents can
further enhance these effects, depends on both reacting species. For instance,
in a free-radical copolymerization, it is likely that polar interactions would be
more important in the cross-propagations (when the monomer and radical bear
different substituents and thus have different electronic properties) than in the

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399

homopropagations (when the monomer and radical bear the same substituents).
Hence the effect of solvent polarity on the stability of the transition structure
(and thus the propagation rate) would generally be expected to be greater in the
cross- than in the homopropagation reactions. Thus, there would be a net effect
of solvent polarity on the reactivity ratios of the copolymerization (regardless of
whether or not these polar interactions are inﬂuenced by the penultimate unit, or
merely by the terminal unit).

There are two cases to consider when predicting the effect of solvent polarity

on copolymerization propagation kinetics:

(1) the solvent polarity is constant as a function of the monomer feed compo-

sition for the given copolymerization system, as would be the case if the
solvent polarity is dominated by an added diluent, or if the comonomers
have similar dielectric constants, such as in the bulk copolymerization of
styrene with methyl methacrylate

(2) the solvent polarity varies with the comonomer feed mixture for the given

system, as would be the case in a bulk copolymerization of monomers with
signiﬁcantly different dielectric constants, such as styrene and acrylonitrile

In case (1), the effect on copolymerization kinetics is simple. The various

reactivity ratios would vary from solvent to solvent, but, for a given copolymer-
ization system, they would be constant as a function of the monomer feed ratios.
The copolymerization kinetics would simply follow the appropriate base model for
that system (according to which side reactions were assumed to be important, and
which units were assumed to affect radical reactivity).

In case (2), the effect of the solvent on the copolymerization kinetics is more

complicated since the various reactivity ratios would not be constant as a function
of the monomer feed. To model such behavior, it would ﬁrst be necessary to select an
appropriate base model for the copolymerization (based on the usual assumptions).
It would then be necessary to replace the various reactivity ratios (included in the
base model as constants) by functions of the composition of the comonomer feed
mixture. These functions would need to relate the reactivity ratios to the solvent
polarity, and then relate the solvent polarity to the comonomer feed composition,
and would thus vary depending upon the particular chemical properties of the
monomers. It is therefore difﬁcult to suggest a general kinetic model to describe
these systems, but it is clear that such effects would result in deviations from the
behavior predicted by their appropriate base model.

Radical Complexes.

Solvents can also interfere with the propagation

step via the formation of radical–solvent complexes. When complexation occurs,
the complexed radicals are generally more stable than their corresponding un-
complexed radicals as it is this stabilization that drives the complexation reaction.
Thus, in general, one might expect complexed radicals to propagate more slowly
than their corresponding uncomplexed radicals, if indeed they propagate at all.
However, in the special case that one of the comonomers is the complexing agent,
the propagation rate of the complexed radical may instead be enhanced if propa-
gation through the complex offers an alternative less-energetic reaction pathway.
In any case, the complexed radicals would be expected to propagate at a rate

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COPOLYMERIZATION

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different from their corresponding free radicals; thus, the formation of radical–
solvent complexes would affect the copolymerization propagation kinetics.

A terminal radical–complex model for copolymerization was formulated by

Kamachi (14). He proposed that a complex is formed between the propagating
radical and the solvent (which may be the monomer) and that this complexed rad-
ical propagates at a different rate compared with the corresponding uncomplexed
radical. Under these conditions (and assuming that the terminal unit is the only
unit affecting radical reactivity), there are eight different propagation reactions
to characterize in a binary copolymerization. These are as follows:

There are also two equilibrium reactions for the formation of the complex (as-
suming that there is only one complexing agent which can complex with either
radical):

On the basis of these reactions, Kamachi (14) derived expressions for the copoly-
mer composition under this model. Fukuda and co-workers (15) later provided
expressions for the composition, triad/pentad fractions, and propagation rate by
deriving expressions for ¯r

and ¯k

, which could be used in place of r

and k

in the

terminal model equations (ie, eqs. 1 and 2 above). These are as follows:

¯k

= k

+ ¯s

[S]

+ K

[S]

(7)

¯r

= r

+ ¯s

[S]

+ (r

/¯r

)¯s

[S]

(8)

where

¯r

cii

cij

¯s

cii

, i, j = 1 or 2 and i = j

Variants of this model may be derived by assuming an alternative basis model
(such as the implicit or explicit penultimate model) or by making further assump-
tions as to the nature of the complexation reaction. For instance, in the special
case that the complexed radicals do not propagate (ie, ¯s

= 0 for all i), the reac-

tivity ratios are not affected (ie, ¯r

= r

for all i), and the complex formation serves

only to remove radicals (and monomer, if monomer is the complexing agent) from
the reaction, resulting in a solvent effect that is analogous to a bootstrap effect
(see below).

Monomer Complexes.

A solvent may also interfere in the propagation

step via complexation with the monomer. As was the case with radical–solvent
complexes, complexed monomer might be expected to propagate at a rate different
from that of free monomer, since complexation might stabilize the monomer, alter
its steric properties, and/or provide an alternative pathway for propagation. In

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401

examining the effect of such complexation on copolymerization kinetics, there are
a number of different mechanisms to consider. In the case that the complex is
formed between the comonomers, there are three alternatives:

(1) The monomer–monomer complex propagates as a single unit, competing

with the propagation of free monomer.

(2) The monomer–monomer complex propagates as a single unit, competing

with the propagation of free monomer, but the complex dissociates during
the propagation step and only one of the monomers is incorporated into the
growing polymer radical.

(3) The monomer–monomer complex does not propagate, and complexation

serves only to alter the free monomer concentrations.

In the case that the complex is formed between one of the monomers and an
added solvent, there are two further mechanisms to consider:

(4) The complexed monomer propagates, but at a different rate to that of the

free monomer.

(5) The complexed monomer does not propagate. Models based on mechanisms

(1) and (2) are known respectively as the monomer–monomer complex par-
ticipation (MCP) and dissociation (MCD) models. Mechanisms (3) and (5)
would result in a solvent effect analogous to a bootstrap effect (see below),
while mechanism (4) would result in a model similar to the MCD model,
though it would be based on a slightly different equilibrium expression.
The MCP and MCD models are outlined in the following paragraphs.

The Monomer–Monomer Complex Participation (MCP) Model was ﬁrst sug-

gested by Bartlett and Nozaki (16), later developed by Seiner and Litt (17), and
reﬁned by Cais and co-workers (18). In this model, it is assumed that the two
monomers can form a 1:1 donor–acceptor complex and add to the propagating
chain as a single unit in either direction. Assuming that the terminal unit is the
only unit affecting radical reactivity, eight addition reactions and an equilibrium
constant are required to describe the system.

The composition and propagation rate can be expressed in terms of the following
parameters (15).

◦

+ (A

◦

+ (A

◦

(9)

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COPOLYMERIZATION

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◦

)

+ (A

◦

)

+ (A

+ A

◦

)

+ (A

◦

)

(10)

where

= 1 + r

Q f

◦

and

= 1 + r

Q f

◦

= 1 + s

+ r

− 1

Q f

◦

and

= 1 + s

+ r

− 1

Q f

◦

= 1 + r

+ r

− 1

Q f

◦

and

= 1 + r

+ r

− 1

Q f

◦

2Q f

◦

[Q( f

− f

)

+ 1]

+ 4Q f

− [Q( f

− f

)

+ 1] and Q = K[M]

= feed composition of M

and f

= [M

]/[M]

where [M] is the total monomer feed and [M

◦

] the uncomplexed M

, r

iij

iji

, s

iij

where

, j = 1 or 2 and i = j

As with all of the models, variations are possible by making a different assumption
as to which units can affect radical reactivity. For instance, Brown and Fujimori
(19) have derived expressions for composition and sequence distribution for the
“Comppen” model, a complex-participation model that is based on the penultimate
model.

In the Monomer–Complex Dissociation (MCD) Model it is assumed that the

monomer–monomer complex described in the MCP model dissociates upon addi-
tion to the chain, with only one unit adding (20). A model based on this mechanism
was ﬁrst formulated by Karad and Schneider (21), and later generalized by Hill
and co-workers (22). Assuming that only the terminal unit of the radical affects the
propagation step, eight rate constants and two equilibrium constants are required
to describe the system.

As with the penultimate and radical–complex models, it is possible to express the
model in terms of equations for ¯r

and ¯k

, which could be used in place of r

and

in the terminal model equations 1 and 2. These are as follows (15):

¯k

= k

+ s

[C]

+ K

[C]

(11)

¯r

= r

+ s

[C]

+ (r

[C]

(12)

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COPOLYMERIZATION

403

where

, r

iic

ijc

, s

iic

, i, j = 1 or 2 and i = j

This form of the model is for monomer–solvent complexes generally. Under the
MCD model, the complexing agent C is the other monomer (M

). As with all of the

solvent effects models, variations are possible by making a different assumption
as to which units can affect radical reactivity.

Monomer Partitioning: The Bootstrap Effect.

In the bootstrap model,

solvent effects on the propagation rate are attributed to solvent partitioning and
the resulting difference between bulk and local monomer concentrations. In this
way, a solvent could affect the measured propagation rate coefﬁcient without
changing the reactivity of the propagation step. Bootstrap effects may arise from
a number of different causes. As noted above, when radical–solvent or monomer–
solvent complexes form and the complexes do not propagate, the effect of complex-
ation is to alter the effective radical or monomer concentrations, thereby causing a
bootstrap effect. Alternatively, a bootstrap effect may arise from some bulk prefer-
ential sorption of one of the comonomers around the growing (and dead) polymer
chains. This might be expected to occur if one of the monomers is a poor solvent for
its resulting polymer. A bootstrap effect may also arise from a more localized form
of preferential sorption in which one of the comonomers preferentially solvates
the active chain end rather than the entire polymer chain. In all cases, the result
is the same: the effective free-monomer and/or radical concentrations differ from
those calculated from the monomer feed ratios, leading to a discrepancy between
the predicted and actual propagation rates.

Copolymerization models based upon a bootstrap effect were ﬁrst proposed

by Harwood (23) and Semchikov (24). Harwood suggested that the terminal model
could be extended by the incorporation of an additional equilibrium constant re-
lating the effective and “bulk” monomer feed ratios. Different versions of this
so-called bootstrap model may be derived depending upon the baseline model
assumed (such as the terminal model or the implicit or explicit penultimate mod-
els) and the form of equilibrium expression used to represent the bootstrap ef-
fect. In the simplest case, it is assumed that the magnitude of the bootstrap ef-
fect is independent of the comonomer feed ratios. Hence in a bulk copolymeriza-
tion, the monomer partitioning may be represented by the following equilibrium
expression:

= K

1 bulk

2 bulk

(13)

The equilibrium constant K may be considered as a measure of the bootstrap effect.
Using equation 13 to eliminate the effective monomer fractions ( f

and f

) from

the terminal model (ie, eqs. 1 and 2 above), replacing them with the measurable
“bulk” fractions ( f

bulk and f

bulk), the following equations for composition (25)

and

may be derived (26).

K f

1 bulk

2 bulk

K f

1 bulk

+ f

2 bulk

+ K f

1 bulk

(14)

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COPOLYMERIZATION

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2 bulk

+ K f

1 bulk

+ 2K f

1 bulk

2 bulk

+ r

2 bulk

K f

1 bulk

+ r

2 bulk

(15)

It should be noted that the above equations are applicable to a bulk copoly-

merization. When modeling solution copolymerizations under the same condi-
tions, the same equations may be used for predicting copolymer composition since
it is only the relationship between bulk and local monomer feed ratios that deter-
mines the effect on the composition and microstructure of the resulting polymer.
However, some additional information about the net partitioning of monomer and
solvent between the bulk and local phases is required before the propagation rate
can be modeled. Modeling of the propagation rate in a solution copolymerization
could be achieved by rewriting the above equilibrium expression in terms of mo-
lar concentrations (rather than comonomer feed ratios), and including the solvent
concentration in this expression.

The bootstrap model may also be extended by assuming an alternative model

(such as the explicit penultimate model) as the baseline model, and also by allow-
ing the bootstrap effect to vary as a function of monomer feed ratios. Closed ex-
pressions for composition and sequence distribution under some of these extended
bootstrap models may be found in papers by Klumperman and co-workers (25,27).

Depropagation

All of the models discussed thus far have assumed that the propagation steps
are irreversible. However, while this is usually a reasonable assumption, at high
temperatures (and, for some bulky or exceptionally stable monomers, even at
low temperatures) the depropagation reaction is signiﬁcant. Consideration of the
depropagation reactions leads to kinetic models that are quite different from the
terminal model and thus some deviations from the terminal model may be caused
by the reversibility of one or more of the propagation steps. A large number of
depropagation models are possible, depending upon both the basis model assumed
and the assumptions that are made as to the nature of the depropagation reaction
(such as which units can depropagate, and which units of the polymer radical can
affect the depropagation rate). Early depropagation models were derived by Barb
(28) (who incorporated depropagation into a simpliﬁed terminal MCP model),
and Walling (29) (who incorporated depropagation into a simpliﬁed penultimate
model). Lowry (30) and later Wittmer (31,32) derived composition equations for
a number of different sets of assumptions concerning the depropagation reaction,
assuming the terminal model for propagation. Howell and co-workers (33) gener-
alized these equations, and provided corresponding expressions for the sequence
distribution. Expressions for the propagation rate coefﬁcient for some of these
cases have recently been published by Martinet and Guillot (34) and by Kukulj
and co-workers (35). Given the large number of possible depropagation models,
the various equations will not be reproduced here but can be found in the original
papers.

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COPOLYMERIZATION

405

Conversion-Dependent Models

The equations for copolymer composition presented thus far are “instantaneous”
or low-conversion equations, relating the copolymer composition to the exact
monomer feed present in the reactor. However, during the course of copolymer-
ization, monomers with differing reactivities tend to be incorporated into copoly-
mers at different rates. As a result, in a batch reaction, this leads to a drift in
the comonomer feed composition, which in turn leads to a drift in the copolymer
composition. To model this behavior it is necessary to integrate the copolymer
composition equation. This is achieved by noting that the copolymer composition,
given as F

in the instantaneous equations, is in its more general form actually

the derivative d[M

]/d[M

] which can be integrated. Let us consider, as an exam-

ple, the following integrated equation for the terminal model, published by Mayo
and Lewis (3):

log

]

0
2

− r

log

]

0
2

]

0
1

−

− r

)

− r

)(1

− r

)

log








− r

+ (r

− 1)

]

− r

+ (r

− 1)

0
1

0
2








(16)

In this equation, [M

] and [M

] refer to the instantaneous monomer concentrations

and [M

] and [M

] to the initial concentrations. An alternative form (expressed

in terms of total molar conversion of monomer) was later published by Meyer and
Lowry (36):

[M]

]

− δ

where

α =

− r

, β =

− r

, γ =

− r

)(1

− r

)

, δ =

− r

(17)

Using either form of the equation, one can calculate the instantaneous comonomer
feed composition as a function of the conversion, knowing the initial feed composi-
tion. This can then be used to calculate the copolymer composition, sequence dis-
tribution, and propagation rate using the corresponding instantaneous equations.

Model Discrimination: Which Model to Use for a Given System?

The choice of copolymerization model for any given system is often one of great
debate in the polymer literature. This, to a large extent, is caused by the fact that
all of the alternative models contain a number of characteristic constants (such
as monomer and radical reactivity ratios). Since these are difﬁcult to measure
directly, they are usually (if not always) estimated by treating them as adjustable
parameters and ﬁtting the copolymerization model to the available data. As a
result, many different models can often be made to ﬁt the same experimental data.
If the sole purpose of the model is merely to provide a shorthand version of the
existing experimental data, provided the model provides a good ﬁt to that data, any
of the ﬁtted models are appropriate to use. If however the estimated parameters
are then to be used to make other model predictions or to deduce something about

406

COPOLYMERIZATION

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the reaction mechanism, it is necessary to select the copolymerization model with
the correct assumptions.

In what follows we examine which assumptions are likely to be valid for

different types of copolymerization systems. In doing this, we make a distinction
between what we refer to as a “basis model,” a model containing the set of assump-
tions likely to be applicable to most of the copolymerization systems, and models
for “exceptional systems,” in which additional system-speciﬁc inﬂuences are likely
to be operating in conjunction with those accounted for by the basis model. We
have recently reviewed and assessed the experimental evidence for the different
copolymerization models both for basis systems (37) and exceptional systems (38)
and will not attempt to reproduce these reviews here. Instead we merely outline
the main conclusions of this work and make some general comments concerning
the applicability of the alternative models.

Basis Model.

For many years, people believed that the terminal model

was the basis of copolymerization propagation kinetics because it could be ﬁtted
to the composition data for most systems tested. However, in 1985 Fukuda and co-
workers (7) demonstrated that the terminal model failed to predict the propagation
rate coefﬁcients for the copolymerization of styrene with methyl methacrylate—
a system for which the composition data had been widely ﬁtted by the terminal
model (see Fig. 1). These results were later conﬁrmed by several independent
groups, both for the styrene–methyl methacrylate system (under a wide range
of different conditions), and also several other copolymerizations—indeed for al-
most all systems so far tested (37). It now appears likely that the failure of the
terminal model to describe simultaneously the composition and propagation rate
coefﬁcients of ordinary free-radical copolymerization systems is general, with the
terminal model being applicable only to those exceptional systems in which the
comonomers have very similar reactivities.

Although there is strong evidence against the terminal model, there has

been much reluctance to abandon this model since this would undermine the 50

Fig. 1.

Shown here is Fukuda’s 40

◦

data (a) and composition data (b) (7). The solid

line on each graph is the terminal model predictions using the reactivity ratios that provide
the best ﬁt to the composition data (r

STY

= 0.523 and r

MMA

= 0.460) while the dotted line

in each graph is the terminal model predictions using the reactivity ratios that provide the
best ﬁt to the

data (r

STY

= 2.5724 and r

MMA

= 0.7973). Clearly the terminal model can

ﬁt the composition or the

data but not both simultaneously.

Vol. 9

COPOLYMERIZATION

407

years of terminal model reactivity ratios that have been and continue to be pub-
lished (see, eg, the large listing of terminal model reactivity ratios in the Polymer
Handbook (39), and the various empirical schemes, such as the Patterns (40) and
Q–e schemes (41), that have been developed for predicting these parameters). The
implicit penultimate model was proposed as a solution to this problem. As seen
above, by assuming that the penultimate unit affected only the reactivity but not
the selectivity of the propagating radical, it is possible to obtain a copolymeriza-
tion model that retains the terminal model composition equation but includes a
non-terminal model

equation. This thus enabled the failure of the terminal

model

equation to be accounted for without abandoning the terminal model

composition equation and the accompanying large database of reactivity ratios.

However, recent experimental and theoretical work has shown that the as-

sumptions of the implicit penultimate model are unlikely to be applicable to the
majority of copolymerization systems. We have recently published a review of this
evidence (37,42), which draws on direct experimental and theoretical measures of
reactivity ratios, model testing in a range of copolymerization systems, and other
tests of the mechanism of the propagation step via, for example, the examina-
tion of solvent effects on reactivity ratios. These studies provide strong evidence
for penultimate unit effects but, in all cases where penultimate unit effects have
been measured directly, effects on radical selectivity have been shown to be sig-
niﬁcant. In other words all available evidence contradicts the assumption of the
implicit penultimate model that the penultimate unit affects reactivity but not
selectivity.

Furthermore, recent theoretical studies have provided a rationalization for

this result that suggests it is likely to be a general feature of free-radical poly-
merization. As noted earlier, for a penultimate unit effect in the reaction barrier
to be implicit, it is necessary for this penultimate unit effect to occur in the ab-
sence of polar, steric, or other forms of direct interaction that, by their very nature,
are dependent on and will thus vary with the chemical structure of the reacting
monomer. Fukuda proposed (11) that an implicit effect could occur if the penulti-
mate unit affected only the radical stability. A radical stabilization penultimate
unit effect would affect the reaction enthalpy, and the reaction barrier (via the
Evans–Polanyi rule). Provided there are no other types of penultimate unit effect
operating (eg, steric or polar effects), such a penultimate unit effect would be im-
plicit. However, high-level ab initio molecular orbital calculations of a representa-
tive range of model propagation reactions have shown that, although penultimate
unit effects in radical stability are signiﬁcant, these effects largely cancel from
the reaction barrier—owing to the early transition structure in these exothermic
reactions (43). As a result, for such a penultimate unit effect in the barrier to be
signiﬁcant, the corresponding effect on radical stability has to be enormous and
thus requires substituents with very strong electron-withdrawing or -donating
properties. The theoretical calculations demonstrated that large penultimate unit
effects of this nature simultaneously affected the charge-transfer conﬁgurations
of transition structure, leading to explicit polar penultimate unit effects. Given
that the majority of copolymerization systems have highly exothermic propaga-
tion steps and thus early-transition structures, it would seem that, in general,
implicit radical stabilization penultimate unit effects are unlikely to occur in the
absence of explicit polar effects.

408

COPOLYMERIZATION

Vol. 9

In addition, steric penultimate unit effects should also lead to explicit (rather

than implicit) penultimate unit effects, as they would obviously be sensitive to
the steric properties of the monomer with which they were reacting. Recently,
experimental evidence for signiﬁcant entropic penultimate unit effects has been
published (44), conﬁrming predictions made in an earlier theoretical study (45).
These studies further conﬁrm that, if there are penultimate unit effects in free-
radical copolymerization, then they are most likely to be explicit effects.

To summarize, we know ﬁrstly from simple model-testing studies spanning

the last 20 years that for almost all systems tested, the terminal model can be
ﬁtted to

or composition data for a copolymerization system, but not both si-

multaneously. Secondly, more recent experimental and theoretical studies have
demonstrated that the assumption of the implicit penultimate model—that the
penultimate unit affects radical reactivity but not selectivity—cannot be justiﬁed.
Therefore, on the basis of existing evidence, the explicit penultimate model should
replace the terminal model as the basis of free-radical copolymerization propaga-
tion kinetics, and hence the failure of the terminal model

equation must be

taken as a failure of the terminal model and hence of the terminal model compo-
sition equation. This means that the terminal model composition equation is not
physically valid for the majority of systems to which it has been applied.

Exceptional Systems.

It is likely that side reactions are also important

in a large number of copolymerization systems. As seen above, their presence fur-
ther complicates the copolymerization kinetics by adding extra parameters to the
copolymerization models, thereby making model discrimination even more dif-
ﬁcult. Indeed (with the possible exception of those few systems that have been
shown to obey the terminal model in all respects) it is very difﬁcult to ﬁnd any
copolymerization system for which there is consensus about the type of copoly-
merization model that should be applied. There is, however, strong independent
evidence for the various types of side reactions in speciﬁc copolymerization sys-
tems, and hence for these systems such side reactions need to be incorporated into
copolymerization models and taken into account when studying the propagation
reaction. Indeed, for some systems, the presence of side reactions such as com-
plexation may provide a means of controlling the composition and microstructure
of the copolymer and, in some cases, even its stereochemistry. In what follows, we
make a few general comments about the types of systems in which the various
side reactions are likely to be occurring.

Depropagation is known to become signiﬁcant with increasing polymeriza-

tion temperature, and most polymerizations are known to have a ceiling temper-
ature beyond which polymerization will not proceed. This arises because of the
exothermic and exentropic nature of the propagation step for most polymerization
reactions (ie, since

G = H − T S and both H and S are negative, there

will exist a temperature T

such that for all T

> T

G > 0). For most poly-

merization systems, these ceiling temperatures are well above ordinary operating
conditions; however, for polymerizations involving exceptionally stable or bulky
monomers such as CO, SO

, and

α-methylstyrene, depropagation is signiﬁcant

at much lower temperatures and thus needs to be incorporated into copolymer-
ization models. The importance of depropagation in a particular system can be
determined directly from measurements of the entropy and enthalpy of the prop-
agation reaction, or indirectly by measuring the ceiling temperature of each of

Vol. 9

COPOLYMERIZATION

409

the comonomers. The use of monomers such as

α-methylstyrene to assist in the

control of molecular weight by exploiting their slow propagation kinetics is often
used in industry as a facile method of reaction control (46,47).

Monomer–monomer complexes are known to form in solutions containing an

electron donor monomer with an electron acceptor monomer. Many such com-
plexes have been detected spectroscopically (48) and one of the most well-known
examples is that between maleic anhydride and styrene. There has been much
dispute over the role these complexes play in the reaction mechanism, concern-
ing, for example, whether they can propagate via an MCP or MCD mechanism,
or whether they are unreactive and merely result in a bootstrap effect. For some
complexes, the heat of formation is less than the heat of propagation and hence
it seems unlikely that the MCP model could be appropriate for these systems, as
the complex should be disrupted during the propagation step. In other systems,
however, this is not the case and there is reasonable circumstantial evidence that
these complexes could be propagating as a single unit. For instance, it has been ob-
served that in the copolymerization of N-phenylmaleimide with chloroethyl vinyl
ether, the stereochemistry at succinimide units is predominately cis (as it would
be in the complex) and random elsewhere, and further that the proportion of
cis linkages was correlated with the variables with which the concentration of
the complex was also correlated (49). It seems likely that, depending on factors
such as the strength of the complex, the role of monomer–monomer complexes in
free-radical copolymerization will vary and model discrimination should be on a
case-by-case basis. However, where there is (spectroscopic) evidence for their ex-
istence, they must always be taken into account when deriving copolymerization
models.

Radical–solvent complexes are more difﬁcult to detect spectroscopically; how-

ever, they do provide a plausible explanation for many of the solvent effects ob-
served in free-radical homopolymerization—particularly those involving unsta-
ble radical intermediates (such as vinyl acetate) where complexation can lead to
stabilization. For instance, Kamachi (50) observed that the homopropagation rate
of vinyl acetate in a variety of aromatic solvents was correlated with the calculated
delocalization stabilization energy for complexes between the radical and solvent.
If such solvent effects are detected in the homopolymerization of one or both of the
comonomers, then they are likely to be present in the copolymerization systems as
well. Indeed, radical–complex models have been invoked to explain solvent effects
in the copolymerization of vinyl acetate with acrylic acid (51). Radical–solvent
complexes are probably not restricted merely to systems with highly unstable
propagating radicals. In fact, radical–solvent complexes have even been proposed
to explain the effects of some solvents (such as benzyl alcohol, N,N-dimethyl for-
mamide, and acetonitrile) on the homo- and/or copolymerizations of styrene and
methyl methacrylate (52–54). Certainly, radical–solvent complexes should be con-
sidered in systems where there is a demonstrable solvent effect in the copolymer-
izations and/or in the respective homopolymerizations.

Polarity effects can also result in solvent effects in free-radical copolymer-

ization. Recent theoretical studies (43,55–57) of small-radical addition reactions
suggest that in a wide range of cross-propagation reactions, the transition struc-
ture is stabilized by the contribution of charge-transfer conﬁgurations. When this
is the case, the extent of stabilization (and hence the propagation rate) will be

410

COPOLYMERIZATION

Vol. 9

inﬂuenced by the dielectric constant of the reaction medium. Solvent effects on
reactivity ratios have been known for a long time and at least some of these
[for instance those for styrene-methyl methacrylate copolymerization (58–60) and
styrene–acrylonitrile copolymerization (61,62) in a variety of solvents] have been
shown to correlate with the dielectric constant of the reaction medium. Further-
more, it is possible that solvent effects in other systems may in part be attributable
to polarity effects, although additional effects (such as radical–solvent complexes
or monomer partitioning) obscure a direct correlation. As noted above, polarity
effects need only be explicitly considered in a copolymerization model if the po-
larity of the reaction medium varies signiﬁcantly as a function of the monomer
feed fraction (as it would in a bulk copolymerization of two monomers with widely
differing dielectric constants). However, it is worth noting that polarity effects are
likely to be operating at least to some extent in copolymerizations of monomers
with even moderately different electronegativities and, where such effects do oc-
cur, the reactivity ratios (and hence composition and sequence distribution) can
be manipulated by varying the solvent polarity.

Bootstrap effects arise from a number of different causes. For instance, a boot-

strap effect can be expected if there is preferential sorption of one of the monomers
around the growing polymer chain—as might be expected if one of the monomers is
a poor or nonsolvent for the resulting polymer. This can sometimes be established
by examining the appropriate polymer–solvent interaction parameters. Nonideal
mixing of the comonomers can also be independently established via measure-
ments of solution thermodynamics. Indirect evidence for bootstrap effects arising
from preferential solvation of the polymer chain can be obtained by examining
the copolymer composition as a function of molecular weight, since it would be
expected that this type of bootstrap effect would lead to chain length effects on
copolymer composition, even for nonoligomeric systems that would normally be
expected to satisfy the long-chain assumption. On the basis of this method and
measurements of solution thermodynamics, Semchikov (24,63) has shown that
such bootstrap effects do indeed operate in a number of bulk copolymerization
systems including AN–STY, STY–MA, VAC–STY and VAc–NVP, but not in the
system STY–MMA (at 298–343K). Bootstrap effects can also be expected if unreac-
tive monomer–monomer, monomer–solvent, radical–solvent, or radical–monomer
complexes form. As noted above, there is evidence for many of these complexes
in certain types of system, though there is frequently dispute as to the role these
complexes play in the reaction mechanism. Bootstrap effects should certainly be
considered when attempting to model systems for which there is demonstrable
evidence for the existence of such complexes.

Which Model to Use and When: Practical Recommendations

It is clear from the above discussion that any physically realistic model for free-
radical copolymerization kinetics should be based on the penultimate model and,
depending on the system, should also incorporate various types of solvent effects
and/or depropagation. The resulting model would not only be very complex, but
would also contain numerous characteristic constants (ie, reactivity ratios, equi-
librium constants). This poses a major problem as such parameters are difﬁcult to

Vol. 9

COPOLYMERIZATION

411

measure directly, and are therefore usually treated as adjustable parameters, and
estimated as part of the model-ﬁtting process. As we have argued elsewhere (1),
even in oversimpliﬁed models there are usually more adjustable parameters than
are mathematically required to ﬁt data. This results in large and highly correlated
uncertainties in the parameter estimates, with multiple parameter sets provid-
ing excellent ﬁts to the experimental data. Hence, the use of physically realistic
models may not be practical for most copolymerization systems, unless they are
accompanied by independent measurements of the model-parameters.

As a solution to this problem, we recommend that, provided they can be ﬁt-

ted to the data, simpliﬁed models such as the implicit penultimate model (and
hence the terminal model composition equation) continue to be used for descrip-
tive purposes. However, it is important to recognize that the resulting parameter
estimates will at best convey qualitative information about radical reactivity and
should not be used to make quantitative predictions of quantities other than those
to which they were ﬁtted. When physically meaningful information is required,
more direct approaches should be adopted. For example, substituent effects on
radical reactivity and selectivity in small-radical models can be studied directly
via a host of experimental approaches, and also via high-level ab initio molecular
orbital calculations. Various spectroscopic techniques can be used to study radi-
cal and monomer complexes, and indeed ab initio molecular orbital calculations
could also be used to tackle this problem. Information on possible monomer parti-
tioning can be accessed from polymer–solvent interaction parameters, and other
measurements of solution thermodynamics. Likewise, thermodynamic measure-
ments (and ab initio calculations) can assist in the study of depropagation. These
direct approaches not only help to identify the physically realistic model for a
given system, but can also provide independent measurements of the correspond-
ing reactivity ratios and equilibrium constants (thereby reducing the number of
adjustable parameters in the model).

Measurement of Reactivity Ratios

Reactivity ratios are typically measured by ﬁtting the terminal model composition
equation to experimental composition data. Depending on the chemical structure
of the comonomers, a variety of methods can be used to measure copolymer compo-
sition, including

H NMR, uv–visible spectroscopy, elemental microanalysis, and

Fourier transform infrared (FTIR) spectroscopy. For information on these proce-
dures, the reader is referred to Reference 64. In order to ﬁt the model to the data,
it is necessary to measure the copolymer composition for multiple initial values
of the comonomer feed ratios. As always, choosing an appropriate experimental
design is important for minimizing the statistical uncertainties in the parameter
estimates. It is worth noting that the common practice of choosing comonomer
feed ratios at regular intervals between f

= 0 and 1 is not the optimum design

for measuring terminal model reactivity ratios. Instead, Tidwell and Mortimer
(65) recommended multiple determinations at the following two feed ratios:

+ r

and

+ r

(18)

412

COPOLYMERIZATION

Vol. 9

Of course, the implementation of such a design presupposes knowledge of the
reactivity ratios we are trying to measure. However, one can normally obtain a
useful design using approximate reactivity ratios, as obtained from preliminary
experiments or literature values for related systems.

Having obtained the composition data it is then necessary to ﬁt it with the

terminal model, using the two reactivity ratios as adjustable parameters. For
many years, the standard method for ﬁtting the data was ﬁrst to linearize the
composition equation, as described by Fineman and Ross (66), thereby allowing
linear regression or indeed a simple graphical procedure. However, it is now gen-
erally accepted that this method transforms error structure in such a way that
linear regression is no longer valid (67), and instead nonlinear regression us-
ing the original composition equation should be adopted. Nowadays linearization
of the equation is in fact unnecessary, as modern mathematical software pack-
ages routinely perform nonlinear regression. In particular, a free program called
Contour (68,69) has been provided speciﬁcally for the estimation of reactivity ra-
tios. In implementing the nonlinear regression procedure, it is important to weight
the residuals according to the expected error structure in the data (70).

Finally, it should be noted that the above discussion refers to low-conversion

copolymerizations, for which composition drift is not signiﬁcant. For conversion-
dependent data, the appropriate integrated composition equation should instead
be ﬁtted to the data using an error-in-variables procedure (71) to deal with the
statistical uncertainties in the multiple independent variables.

Reactivity in Free-Radical Copolymerization

It is clear from the above discussion that reactivity in free-radical copolymeriza-
tion depends on a rich array of factors. There is strong evidence that the reactiv-
ity of the propagating radical is governed by the nature of both its terminal and
penultimate units. Depending on the system, these substituent effects in turn
arise from both steric and electronic factors, with the latter itself being a com-
plex result of sigma- and/or pi-effects on the stability of the propagating radical
and the monomer, as well as the possible stabilization of the transition structure
through partial charge transfer. A consequence of this latter effect is that the re-
activity of the propagation reaction can be dependent on the dielectric constant
of the reaction medium. The solvent (which may be the monomer) can also affect
the propagation reaction by forming a complex with the propagating radical or
comonomer, thereby altering its reactivity. In some cases this may result in an
altogether different propagation mechanism. In addition, monomer partitioning
can at times affect the apparent reactivity of the propagation step, as can depropa-
gation. This complex nature of propagation in free-radical copolymerization offers
huge potential for controlling the copolymer composition and microstructure (by,
eg, manipulating the feed composition, temperature or solvent type), and even
perhaps the stereochemistry (through the manipulation of radical or monomer
complexes). An understanding of substituent effects on the reactivity of the prop-
agation step is important for exploiting this potential for control.

The earliest attempts to describe substituent effects on reactivity were em-

pirical schemes, based on the terminal model. The most widely used of these are

Vol. 9

COPOLYMERIZATION

413

the Q–e (41) and Patterns (40) schemes, and as an example we will brieﬂy out-
line the former. In the Q–e scheme, the rate coefﬁcient for attack of radical i on
monomer j is given by the following equation:

= P

− e

(19)

Hence the terminal model reactivity ratios are given by

− e

)

(20)

In order to apply this equation, each monomer i is assigned empirically derived Q

and e

values, as obtained by ﬁtting equation 20 to sets of experimental reactivity

ratios. In this way, provided the Q and e values of the individual monomers are
known, reactivity ratios can be predicted for previously unmeasured combina-
tions of monomers. An extensive listing of Q and e values has been provided by
Greenley (72).

There are a number of shortcomings in the scheme. As noted above, the ter-

minal model does not provide a physically realistic description of copolymerization
kinetics and, as such, predictions (in this case of other reactivity ratios) based on
terminal model reactivity ratios may be subject to considerable error. Even in the
context of the terminal model, the assignment of independent Q and e values for
a “monomer,” regardless of whether it is acting as the monomer or radical and
regardless of what it is reacting with, is a major oversimpliﬁcation. In addition,
the experimental database of terminal model reactivity ratios used to obtain the
Q and e values was largely compiled nearly 50 years ago, using somewhat lim-
ited analytical methods. As a result of these problems, the predictions of the Q–e
scheme are at best qualitative, and are generally most successful for combinations
of monomers that are not subject to strong polar or steric effects.

More modern descriptions of reactivity in copolymerization have arisen from

the quantum chemical study of the addition of small radicals to alkenes. Such
small-radical models are applicable to propagation because the propagation re-
action is chemically controlled, and not generally inﬂuenced by substituents be-
yond the penultimate unit. On the basis of a large collection of both theoretical
and experimental studies of substituent effects in radical addition reactions, Fis-
cher and Radom (57) have recently presented a theoretical account of reactivity
framed in terms of the curve-crossing model (13). Under this model, the barrier
for an addition reaction is dependent on the exothermicity of the reaction, the
singlet–triplet gap of the double-bonded substrate and the possibility for low-lying
charge-transfer conﬁgurations in the transition structure, in turn estimated by
a consideration of the ionization energies and electron afﬁnities of the reactants.
These quantities are easily accessible from both experiment and high-level ab ini-
tio molecular orbital calculations, and the model thus offers a powerful approach
for predicting and understanding reactivity in free-radical copolymerization. For
more details of the curve-crossing model, and more fundamental aspects of reac-
tivity in radical addition reactions, the reader is referred to References 13 and 57
(also see C

OMPUTATIONAL

UANTUM

HEMISTRY FOR

REE

-R

ADICAL

OLYMERIZATION

414

COPOLYMERIZATION

Vol. 9

Termination in Free-Radical Copolymerizations

Detailed knowledge of free-radical copolymerization termination rate coefﬁcients
is of technical as well as fundamental importance. The radical–radical termina-
tion reaction in free-radical polymerization is the most complex reaction in the
polymerization process (73,74). Its termination rate coefﬁcient, k

, is inﬂuenced

by a multitude of different factors, which are not easily separated. Only within
the last 15 years have new methods become available that allow for an accurate
measurement of this rate coefﬁcient. The scatter of the termination rate coefﬁ-
cients given in the Polymer Handbook (75) reported for the same monomer at
the same reaction temperature is a direct manifestation of the inﬂuence of these
various parameters on k

. While the advanced techniques available to study ter-

mination processes in homopolymerizations and the current theoretical models
describing the termination process are discussed in the chapter “Free Radical
Polymerization,” the present section focuses on the use of kinetic data from free-
radical binary and ternary copolymerizations to clarify the termination process
as well as on giving practical guidelines for the estimation of termination rate
coefﬁcients in copolymerization.

Theoretical Background.

Modeling of the (average) termination rate co-

efﬁcient, k

,copo

, at low monomer conversions as a function of the initial monomer

feed in the reaction mixture has been thoroughly reviewed by Fukuda and co-
workers (76). The present chapter considers the most important models presented
in Reference 76.

Model A, which has been proposed by Fukuda and co-workers, assumes

diffusion-controlled termination to be related to the composition of the generated
copolymer according to equation 21

− 1

t,copo

= k

− 1

t 11

+ k

− 1

t 22

(21)

where F

is the mole fraction of comomoner i segments within the copolymer.

Similar in its underlying idea is one of the earliest models for the description

of the termination rate coefﬁcient introduced by Atherton and North (77):

t,copo

= k

t 11

+ k

t 22

(22)

In Model B, two types of free radicals, one carrying a monomeric unit 1, the other
with a monomeric unit 2 at the chain end, are considered to sufﬁciently account
for copolymerization k

according to

t,copo

= k

t 11

+ 2k

t 12

+ k

t 22

(23)

where the k

tii

’s are the homotermination rate coefﬁcients and the P

’s are the rel-

ative populations of the two types of the terminal free radicals (P

= 1 − P

which is referred to as Walling factor (9), describes the size of the cross-termination
rate coefﬁcient (k

t12

= k

t21

) relative to the geometric mean of the homopolymer-

ization rate coefﬁcients:

t 12

t 11

t 22

)

(24)

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COPOLYMERIZATION

415

It should be noted that a so-called geometric mean rule, which is formally identical
to equation 24 has been frequently used in free-radical gas kinetics to estimate
cross-recombination rate coefﬁcients k

from the associated homo-recombination

rate coefﬁcients k

and k

(78). Two other models given in Fukuda’s review (76),

C and D, in addition to the terminal unit also consider the penultimate unit at the
free-radical chain end as has ﬁrst been tried by Russo and Munari (79,80). These
models are associated with a greater number of individual termination rate coef-
ﬁcients. Given in equation 27 is the expression for k

,copo

of a binary copolymeriza-

tion, for which 10 termination rate coefﬁcients, k

tij

,kl

, need to be considered. k

tij

,kl

refers to the reaction of (two) radicals terminating in the monomer units ij and
kl, respectively. The P

’s and P

’s indicate relative populations of the four types

of “penultimate” free radicals (with chains ending in units 11, 12, 21, and 22, re-
spectively). These populations are available from the propagation rate coefﬁcients
and reactivity ratios as given below in equations 25 and 26 (15):

= 1 − P

/ ¯

( ¯

/ ¯

)

+ ( ¯r

/ ¯

)

(25)

and

= P

− P

+ f

(26)

The population P

is calculated analogously to P

t,copo

= 1

t ij

,kl

(27)

To reduce the relatively large number of termination rate coefﬁcients, Model C
replaces rate coefﬁcients for reactions between “unlike” free radicals, k

tij

,kl

(with

ij being different from kl), by the corresponding coefﬁcients for termination of the
“like” radicals, k

tij

,ij

and k

tkl

,kl

, via the arithmetic mean approximation in equation

28 (76):

t ij

,kl

= 0.5(k

t ij

,ij

+ k

t kl

,kl

)

(28)

Using equation 28 to eliminate all “unlike” termination rate coefﬁcients turns
equation 27 into equation 29, which contains only the four “like” (penultimate)
termination coefﬁcients:

t,copo

= k

t 11

,11

+ k

t 21

,21

+ k

t 22

,22

+ k

t 12

,12

(29)

Model D differs from Model C in that a geometric mean approximation (eq. 30)
is applied to reduce the number of penultimate termination rate coefﬁcients from
10 to 4:

t ij

,kl

= (k

t ij

,ij

t kl

,kl

)

(30)

416

COPOLYMERIZATION

Vol. 9

Equation 30 is the expression used by Fukuda and co-workers (76). Russo and
Munari in their pioneering study had an additional factor of 2 on the r.h.s. of
equation 30, with this formulation being identical to what is used in gas kinetics
to estimate cross-termination (78).

Substituting equation 30 into equation 27 yields the expression (eq. 31) for

,copo

according to Model D:

t,copo

= k

t 11

,11

+ k

t 21

,21

+ k

t 22

,22

+ k

t 12

,12

(31)

Fukuda and co-workers (76) presented Model D in a slightly different way (eq.
32), with the two penultimate rate coefﬁcients, k

t12

,12

and k

t21

,21

, being replaced

by two parameters

and

t,copo

= k

t11

,11

+ δ

)

+ k

t21

,21

+ δ

)

(32)

is deﬁned as

≡

t ji

, ji

t ii

,ii

(33)

Models A through D have been extensively tested toward their ability to describe
the termination rate coefﬁcients in various copolymerization systems.

Binary Copolymerization.

A large body of work has been carried out

to test the applicability of the Models A to D to copolymerizations of acrylate,
methacrylates, and styrenic monomers (76,77,79–82). It has proven advantageous
to study the copolymerization termination rate coefﬁcient in related monomer
families (eg within the acrylate or methacrylates families). For example, Buback
and Kowollik investigated dodecyl acrylate (DA) and methyl acrylate (MA) copoly-
merizations via the single-pulse pulsed-laser polymerization (SP-PLP) technique.
MA and DA have homotermination rate coefﬁcients that differ by orders of mag-
nitude [ie, log (k

)(L/mol/s) (1000 bar, 40

◦

C, conv.

< 0.1, DA) = 6.38 and log

)(L/mol/s) (1000 bar, 40

◦

C, conv.

< 0.1, MA) = 8.13]. The ability of the above-

mentioned models to describe the experimentally obtained copolymerization ter-
mination rate coefﬁcients can be tested by plotting log k

,copo

versus the comonomer

mixture composition, f

. Such a plot—together with the corresponding model

descriptions—is given in Figure 2.

Models A and B, which both consider only two types of terminating free

radicals, seriously fail to represent the experimental k

copo

data. The line referring

to Model B has been calculated under the assumption that equation 24 holds
with

= 1. Even major changes in do not result in a signiﬁcantly improved

quality of representing the experimental data by Model B. Also Model C, which
assumes diffusion control with four kinds of penultimate free-radical species, fails
to provide an adequate ﬁt of the measured k

,copo

data. Model D, on the other hand,

allows for an excellent representation of the experimental data. It should be noted
that the straight line through the DA–MA data in Figure 2 is not obtained just from
a linear least-squares procedure, but is derived from a ﬁt to Model D. The almost
linear change of log k

,copo

with monomer feed composition f should be particularly

Vol. 9

COPOLYMERIZATION

417

Fig. 2.

Copolymerization termination rate coefﬁcients at low conversions in MA/DA

copolymerizations at 40

◦

C and 1000 bar plotted against the mole fraction of MA. The lines

A through D correspond to the copolymerization k

models given in the text.

noted. This result bears important mechanistic information. Moreover, this linear
correlation may be useful for predicting or estimating copolymerization k

even in

situations where more than two monomers from the same family are polymerized
(see below).

The close-to-linear correlation that is seen in acrylate–acrylate copolymeriza-

tion systems is also found in intrafamily methacrylates copolymerizations. Studies
on such systems have been carried out in the past by Ito and O’Driscoll (83) by
rotating sector experiments at 30

◦

C and ambient pressure. The data given in their

pioneering study differ in the format in which they are presented: In the case of the
methyl methacrylates (MMA)–butyl methacrylates (BMA) system k

and k

val-

ues for the copolymerizations are given, whereas in the case of the MMA–dodecyl
methacrylates (DMA) system the termination rate coefﬁcients are given relative
to the homotermination rate coefﬁcient of MMA (ie, k

,copo

,homo

). The k

values

given for the homopolymerization of MMA and BMA are not in agreement with
the nowadays accepted values. (84) However, the difference of both values is close
to the difference which can be deduced from the more recent numbers. The same
observation holds true for the propagation rate coefﬁcients presented by these
authors. This observation indicates that their measurements are reliable in a rel-
ative way, but their absolute numbers are shifted. The data presented by Ito and
O’Driscoll are also best represented by Model D (see Fig. 3), which was found to
ﬁt the experimental data excellently, with the “like” termination rate coefﬁcient
being in between the homopolymerization rate coefﬁcients and very similar in
value.

The data for intrafamily copolymerization of acrylate- and methacrylates-

type monomers suggest that k

,copo

might be generally accessible from homoter-

mination data by employing equation 34, which has been termed the general

418

COPOLYMERIZATION

Vol. 9

Fig. 3.

Termination rate coefﬁcients from Ito and O’Driscoll (83) for dodecyl methacrylate

(DMA)–methyl methacrylate (MMA) (at 40

◦

C and 1000 bar, rescaled) and butyl methacry-

late (BMA)–methyl methacrylate (MMA) (at 30

◦

C and ambient pressure) copolymeriza-

tions. The data in Ref. 83 for the MMA–DMA system are given as relative rate coefﬁcients
k

,copo

,homo

. The k

,copo

values were calculated using k

,homo

from Ref. 84. The lines are best

ﬁt representations of Model D.

geometric mean rule (GGM) (79–81):

log k

t,copo

= f

log k

t,11

+ f

log k

t,22

(34)

This, however, must not be generally true for two main reasons: (1) The linear log
k

,copo

versus f correlation seen for the above systems will certainly be restricted to

situations where the same mechanism controls termination behavior. Under con-
ditions where k

,homo

varies with monomer conversion (which indicates a change

in the mode of termination rate control), the linear log k

,copo

versus f correlation

is no longer valid. At total monomer conversion between 20 and 35%, the log k

,copo

values for f

= 1.00 and 0.80 are almost the same (81), and linear plots as de-

picted in Figures 2 and 3 can no longer be drawn at this stage of the reaction. (2)
The theoretical concepts on which equation 34 is based (85) focus on the idea of a
purely diffusional controlled reaction. Cullinan and co-workers derived that the
composition dependence of the binary mutual diffusion coefﬁcient D

tends to be

linear between the inﬁnitely dilute extremes (85). These authors demonstrated
(and found agreement with the existing experimental data) that a modiﬁcation of
absolute rate theory based on thermodynamic aspects yields an expression which
predicts the concentration dependence of the binary mutual diffusion coefﬁcient
in terms of equation 35:

= D

(35)

which can be easily rearranged into equation 36

log D

= f

log D

+ f

log D

(36)

Vol. 9

COPOLYMERIZATION

419

The formal similarity with equation 34 is quite obvious. Understanding termina-
tion rate coefﬁcients at low conversions in purely translational diffusion terms
may lead to equation 34. However, this concept is not in agreement with the
ideas pioneered by Russo and Munari (79,80). These authors assume chemical
(most likely steric) effects to be important in the termination reaction. They sug-
gested the penultimate unit effect model (eq. 27). If the termination process was
governed by the same translational diffusion mechanism in the initial plateau
region for all monomers considered in this work, no curved dependencies of log
k

versus f should be obtained, since equation 34 (and eq. 36) always predicts

straight lines. If, however, the termination reaction is to some extent controlled
by chemical effects (Model D), curved dependencies may occur, depending on the
individual radical populations P

(which directly go into the modeling procedure).

Thus, copolymer systems with a pronounced curvature in the log k

,copo

versus f

plots provide the best argument against any oversimpliﬁcation of k

,copo

behavior.

Indeed, Fukuda and co-workers reported such dependencies for different copoly-
merizations, eg styrene and ethyl acrylate at 40

◦

C and ambient pressure (76).

More recently, Buback and Feldermann (82) reported curved dependencies in the
interfamily copolymerizations of MMA–DA and DMA–MA.

Ternary Copolymerization.

As technical polymerization processes with

more than two monomers are frequently met, it is an interesting question whether
a correlation corresponding to equation 34 will hold for terpolymerizations. The
resulting expression for terpolymerization k

would read

log k

= f

log k

t,1

+ f

log k

t,2

+ f

log k

t,3

(37)

where the monomer mole fractions f

add up to f

+ f

= 1 and where

is the homotermination rate coefﬁcient of monomer 3. Buback and Kowol-

lik test equation (eq. 37) for the intrafamily terpolymerizations of MA–BA–DA
and MMA–BMA–DMA (86). Remarkably, these two terpolymerization systems
adhered strictly to equation 37. It is worthwhile to plot binary low conversion
copolymerization data for the systems MA–BA, BA–DA, and MA–DA, together
with low conversion data on the MA–BA–DA terpolymerization system. Inspec-
tion of Figure 4 shows that all ternary copolymerization k

values are located on

a surface area that is indicated by the triangle (full lines) within the ﬁgure. The
termination rate surface is restricted to a certain set of initial mole fractions, as
they have to add up to 1; eg it is not possible to conduct a terpolymerization at the
mole fractions of 0.8 for MA and 0.5 for DA. This (quite obvious) restriction is the
reason for the triangular form of the termination rate surface.

The success of equation 37 in describing intrafamily terpolymerization termi-

nation rate coefﬁcients provides considerable temptation to propose an expression
(eq. 38) that may be useful (or even must be used because of missing alternatives)
for modeling k

of monomer mixtures that contain i monomeric species:

(38)

with the sum over f

adding up to 1.

420

COPOLYMERIZATION

Vol. 9

Fig. 4.

Low conversion termination rate coefﬁcients, k

,terpo

, for a MA–BA–DA terpoly-

merization at 40

◦

C and 1000 bar plotted vs the mole fraction of DA and MA in the initial

composition of the reaction mixture (86). The triangle indicates the surface on which the
termination rate coefﬁcients are located.

Equation 38 represents the generalized formulation of equations 34 and 37

in nonlogarithmic terms.

refers to the termination rate coefﬁcient of a mul-

timonomer copolymerization in the initial plateau region (at low and moderate
conversions, with this range depending on the particular type and concentration
of the monomers). k

is the homotermination rate coefﬁcient of monomer i in

this inital plateau region and f

is the mole fraction of monomer i in the reaction

mixture.

Controlled/Living Radical Copolymerization

Controlled/living radical polymerization (CRP) has been successfully applied to
many different radically polymerizable monomers. A recent book has summarized
this very rapidly developing ﬁeld, at least till 2001 (87). Because of limited space
available, only the most important issues will be discussed here.

Currently three systems seem to be the most efﬁcient processes for con-

ducting a CRP: nitroxide mediated polymerization (NMP) (eq. 1 in Fig. 5), atom
transfer radical polymerization (ATRP) (eq. 2 in Fig. 5) and degenerative trans-
fer systems (eq. 3 in Fig. 5) such as reversible addition–fragmentation transfer
(RAFT) or iodine degenerative transfer (IDT) processes (88–90). The key feature
of all CRPs is the dynamic equilibration between active radicals and various types
of dormant species.

Vol. 9

COPOLYMERIZATION

421

Fig. 5.

Three most common mechanisms of exchange between active and dormant species

in CRP.

The activation process (k

) for the dormant species (P

X) to form propa-

gating radicals (P

•

) may be spontaneous, ie, thermal (alkoxyamines in NMP)

(91) or catalytic (activation of alkyl (pseudo)halides by redox active transition-
metal complexes (92–94), Y, in ATRP). It may be also bimolecular [dithioesters
in RAFT (95) and alkyl iodides in IDT (96)]. The degenerative transfer process
may proceed through the addition–fragmentation chemistry or through hyperva-
lent radical intermediates of very short or sometimes longer lifetime. In the latter
case some retardation may occur. In all of these systems only radicals propagate
(k

). The conditions for the reaction are selected so that the reversible activation

or reversible transfer reaction provides a small steady concentration of growing
radicals, at concentrations similar to or smaller than in conventional radical poly-
merization (RP). Although termination (k

) does occur in all CRPs, the proportion

of dead chains is relatively small (typically between 1 and 10%) in comparison
with all living chains. The latter constitutes tiny amounts of growing species (ppm)
and a dominating proportion of dormant chains which are potentially active. In
NMP and ATRP, the formation of active radicals is accompanied by the genera-
tion of a persistent radical species, such as a nitroxide or a higher oxidation state
transition-metal complex, which can deactivate growing radicals reversibly. Both
processes follow persistent radical effect kinetics (97,98).

In all of these systems, propagation involves radical species, and therefore

chemoselectivities (reactivity ratios and transfer coefﬁcients), regioselectivities
(head-to-head units), and stereoselectivities (tacticity) determined for CRPs are
similar to those in conventional RP.

Developments in CRP were also applied to copolymerizations (99–102). Since

CRP is a living system, for the ﬁrst time segmented copolymers (block and
graft copolymers) were prepared in high yield via radical mechanisms. Simul-
taneous copolymerization results in preparation of statistical copolymers, with
comonomers displaying reactivity ratios similar to those in RP. However, since
nearly all chains start growing at the same time and continue till the end of the
reaction, drift in comonomer feed is reﬂected in composition variation along the
backbone of all chains and results in the preparation of tapered or gradient copoly-
mers. This is in contrast to RP, in which copolymers formed at the early stage of

422

COPOLYMERIZATION

Vol. 9

the copolymerization are rich in one comonomer and at the later stages are rich
in another comonomer.

In this section, some peculiarities of copolymerization conducted in CRP sys-

tems will initially be discussed and then examples of various types of copolymers
will be provided. They will include statistical, gradient, and alternating copoly-
mers, as well as block and graft copolymers. Some special systems will be also
presented and they will include preparation of stereoblock copolymers, and vari-
ous hybrid materials formed by either mechanistic transformation or by growing
well-deﬁned polymers from ﬂat, colloidal, or irregular surfaces.

Reactivity Ratios.

In CRP, monomer is consumed only in the reaction with

growing radicals. Therefore, in the ﬁrst approximation, composition of copolymers
in CRP and RP should be the same and in many cases the simplest four-reaction
copolymerization scheme should be applied, with all restrictions discussed earlier
in this article (103). The relative rates of comonomer consumption and conse-
quently apparent reactivity ratios in CRP should be the same as in RP. However,
some small discrepancies are expected, and indeed have been observed, in the
values of reactivity ratios from ATRP and conventional radical polymerizations
(104–107). There are a number of reasons for these discrepancies, sometimes as
obvious as conducting the copolymerization at different reaction temperatures
and in the presence of solvents. Additionally, differences in the rate of chain
growth can also alter the observed reactivity ratios. In a controlled polymeriza-
tion, polymer is slowly formed throughout the reaction and there is a continuous
increase in the molecular weight of the polymer, whereas in a conventional sys-
tem high molecular weight polymer is formed at the beginning of the reaction
and the composition of the polymer can be established at low conversion. This
difference in the rate of chain growth may also amplify the bootstrap effect (23).
The presence of a transition-metal catalyst in ATRP may further complicate the
picture by introducing the possibility of complexation of some monomers with the
Lewis acidic transition-metal catalysts (108–110). The ﬁnal, and perhaps most
important, cause of discrepancies in reactivity ratios observed in ATRP, are the
differences in methodologies of measuring reactivity ratios in controlled and con-
ventional radical polymerizations (106). In a conventional process the monomer
reactivity ratios are measured at low conversion with different monomer feeds.
However, in the ATRP, measurements at low conversions could be affected by the
structure of the initiator, which may preferentially react with one comonomer. For
this reason it is necessary to measure the cumulative composition of the copoly-
mers at a conversion higher than in a conventional process. However, as the cu-
mulative composition of a copolymer will always be equal to the monomer feed at
100% monomer conversion, the closer the reaction gets to 100% conversion, the
less precise the information that can be obtained on reactivity ratios.

In CRP, the classic copolymerization scheme (Fig. 6) should be supplemented

by the reversible activation steps for both initiation and propagation steps:

The values of the individual equilibrium constants should have an effect on

the total radical concentration, ie, the overall polymerization rate. However, the
reversible activation processes should not affect the relative rates of comonomer
consumption, once the steady state between radicals is established. Nevertheless,
during the initial stage of the reaction, cross-propagation reactions may not be
fully equilibrated and therefore concentration of both growing species in CRP may

Vol. 9

COPOLYMERIZATION

423

Fig. 6.

Simpliﬁed copolymerization scheme for CRP.

be different than in conventional RP. Thus, some discrepancies between CRP and
RP can be observed (107,111). In addition, diffusion phenomena may play some
role, especially in copolymerization of macromonomers (112–115). Nevertheless,
in a large majority of CRP systems examined the reported reactivity ratios are
very similar to those in conventional RP.

The penultimate effect on the kinetics of the activation/deactivation pro-

cesses may be quite signiﬁcant. For example, model studies of activation of various
dimers in ATRP copolymerization of methyl acrylate (MA) and methyl methacry-
late (MMA) showed the following relative values of k

act

for dimeric species: H–

MA–MA–Br, H–MMA–MA–Br, H–MA–MMA–Br, and H–MMA–MMA–Br to be 1,
4.6, 19, and 96, respectively. The back-strain effect resulting from the presence
of a MMA penultimate unit, and formation of a thermodynamically more stable
radical from a MMA–Br terminal unit, increased the values of k

act

∼5 and

∼20 times, respectively, in comparison to MA penultimate/terminal units. The
combined effects resulted in a

∼100-fold increase of k

act

for H–MMA–MMA–Br

relative to H–MA–MA–Br (116).

Statistical and Gradient Copolymer.

As shown in Figure 7, composition

along the chain varies in different ways for block, gradient, and random copoly-
mers. The composition does not vary in a block copolymer until after the crossover
point between blocks where there is an instantaneous composition change, a ran-
dom copolymer shows no continuous change in composition, and gradient copoly-
mers are copolymers in which the instantaneous composition varies continuously
along the chain contour (117–119).

In order to achieve this continuous change in instantaneous composition, all

chains must be initiated simultaneously, and must survive until the end of the

424

COPOLYMERIZATION

Vol. 9

Fig. 7.

Schematic representation of block, gradient, and statistical copolymers.

polymerization. Therefore, a living (ionic) or controlled/living radical polymeriza-
tion technique must be employed, as the signiﬁcant presence of chain-breaking re-
actions would lead to heterogeneity in composition as well as in molecular weight.

Because of facile cross-propagation, statistical (or nearly random) copoly-

merization is very easily achieved in free-radical systems, in contrast to ionic
reactions. The reactivity of many comonomers are relatively similar. For exam-
ple, in RP, methacrylates have similar reactivity to styrene and are

∼3 times

more reactive than acrylates. However, in anionic polymerization acrylates are
100 times more reactive than methacrylates and the latter much more reactive
than styrene. In cationic polymerization the opposite reactivity order is observed.
The copolymerization of monomers with similar reactivity should result in statis-
tical copolymers with no compositional variation during the polymerization. This
has been observed for copolymerization of the same type of monomers such as var-
ious styrenes, various methacrylates, and various acrylates. This is the case for
both CRP and conventional RP. However, in batch copolymerizations of different
classes of comonomers there is a continuous change of residual monomer compo-
sition in the reaction because one comonomer reacts faster than the other one.

The slow rate of initiation relative to propagation in conventional RP com-

posed of two or more classes of monomers leads to chains which have differing
compositions depending on when they were grown. This difference in reactivity
results in compositional variation among the chains and, in the extreme cases,
may result in a mixture of two homopolymers. In CRP, all the chains are initiated
early in the reaction and, under proper conditions, retain activity over the entire
course of the reaction. Therefore, changes in the instantaneous composition aris-
ing from variations in the relative concentrations of monomers are reﬂected along
all chains. In the extreme case of very different reactivity ratios, this may lead to
formation of block copolymers. At the end of the reaction, the cumulative compo-
sitions of the conventional and controlled reactions should be the same but in the
conventional case, a variety of compositions will be observed between the chains
while in CRP all chains will have a similar structure, a gradient of composition
along the chain (118,119). Such gradient (or tapered) copolymers have proper-
ties different to either block or random copolymers, making them candidates for

Vol. 9

COPOLYMERIZATION

425

applications such as blend compatibilizers, vibration-dampening materials, and
pressure-sensitive adhesives (117,119,120).

The shape of the gradient depends on the reactivity ratios, on the composition

of the monomer feed and on whether a mono- or multi-functional initiator was
employed. Forced gradient copolymers can also be prepared by CRP processes. In
the latter system, one monomer is fed into a reactor already containing another
monomer in a semibatch mode. In such a case gradient copolymers can also be
obtained from comonomers with the same reactivity ratios.

Because of the relative infancy of CRP, only a limited number of gradient

copolymers have been reported (102,119). Spontaneous gradient copolymers were
formed in ATRP copolymerization of styrene with n-butyl acrylate (nBA) (121,122),
and with MMA and nBA (123). Both systems had reactivity ratios similar to those
in conventional RP. Preparation of a forced gradient copolymerization was stud-
ied for styrene and MMA (99). The molecular weight of the copolymer was con-
trolled, polydispersity was low, M

< 1.25, and the instantaneous composition

of styrene relative to MMA in the backbone decreased as MMA was added. In
another study examining the forced gradient copolymerization of methyl acrylate
(MA) and styrene, changes in the feed rate of MA (0.085 vs. 0.050 mL/min) resulted
in differences in the instantaneous composition of MA in the copolymer. At a higher
feed rate, there was more MA incorporated into the chains at a given fractional
chain length (conversion) (119). The DSC and dynamic mechanical analysis indi-
cated that gradient copolymers have signiﬁcantly different properties than block
copolymers. The DSC traces showed that the forced gradient copolymer behavior
depends on the thermal history. The lower modulus (G

) of the forced gradient

copolymer compared to the statistical gradient copolymer demonstrates that ma-
terials with different properties were produced. The synthesis of forced gradient
copolymers of styrene and acrylonitrile as well as spontaneous gradients of styrene
and n-butyl acrylate were also reported (119,121,122).

Alternating Copolymers.

Alternating copolymers can be formed when

reactivity ratios for both comonomers are very low, approaching zero (109). An-
other option is to use large excess of one comonomer with r

∼ 0. Both of these

systems have been used in CRP. The ﬁrst successful attempts were based on
copolymerization of methyl acrylate and acrylonitrile with a large excess of
isobutene using ATRP. Well-deﬁned but low molecular weight alternating copoly-
mers were prepared. Subsequently, well deﬁned alternating structures of styrene
with either maleic anhydride or maleimides were prepared by NMP, RAFT, and
ATRP (124–126).

A more challenging synthetic task was to force monomers such as styrene

and MMA to form alternating copolymers. This was successfully accomplished
by complexing MMA, BMA, or MA with Lewis acids such as aluminumalkylchlo-
rides (108). Complexation signiﬁcantly increased the rate of cross-propagation.
By using RAFT as the CRP process, it was possible to prepare not only alternat-
ing copolymers but also block copolymers containing segments of both alternating
and homopolymer structures (126–128).

Block Copolymers.

CRP, as any controlled/living polymerization, can be

used for the preparation of well-deﬁned segmented copolymers, ie, block and graft
copolymers (qv) with various architectures (101). Block copolymers (qv) can be
formed by simple chain extension through the addition of the second monomer

426

COPOLYMERIZATION

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at the end of the polymerization of the ﬁrst one. They have been synthesized
from the isolated macroinitiators and have been also made by the transforma-
tion of the chain ends of macroinitiators prepared by another technique, eg ionic,
coordination, ring-opening polymerization, step growth, etc. (129–135). By using
multifunctional initiators, star-block copolymers have been made (136) (see also
B

LOCK

OPOLYMERS

, T

ERNARY

RIBLOCKS

Many block copolymers have been prepared by ATRP, NMP, and RAFT (99).

Since NMP can not yet be successfully used for the synthesis of polymethacry-
lates only block copolymers with styrenes, acrylates, and acrylamide segments
have been made. RAFT and ATRP can incorporate a broader range of monomers
into block copolymers. Efﬁcient blocking requires cross-propagation to be fast rela-
tive to the rate of polymerization of the second block. Since the rate of propagation
in NMP and ATRP depends on the product of the equilibrium constant and true
radical propagation constants, certain rules apply for successful block copolymer-
ization. The same is true for RAFT, which requires fast fragmentation of the ﬁrst
segment to achieve efﬁcient crosspropagation (137,138). In general, more stabi-
lized radicals, such as methacrylates, propagate via RP slower than those forming
less stabilized radicals (acrylates). However, equilibrium constants and rates of
fragmentation are much higher for stabilized radicals and therefore they domi-
nate the overall rate control. This means that blocking from polymethacrylates to
polyacrylates is efﬁcient for both ATRP and RAFT, but the opposite is not true.
Similar observations were reported for NMP and ATRP during preparation of
block copolymers that required blocking from polyacrylonitrile to poly(butyl acry-
late) (139) .

ATRP offers one synthetic setup to overcome this deﬁciency, via the halogen

exchange technique, which allows an approach to combat the crossover problem.
This technique utilizes a mixed-halogen system, ie, a bromo-containing initiator
and a Cu(I)Cl catalyst, to afford better control over initiation from the macroinitia-
tor and subsequent polymerization of the second methacrylates block, which is par-
ticularly useful for preparing acrylate–methacrylate block copolymers (140,141).
Halogen exchange provides a method for adjusting the equilibrium when moving
from a less reactive macroinitiator (A), with an apparently smaller equilibrium
constant, to successfully initiate polymerization of a monomer that has a larger
equilibrium constant (B in Fig. 8). Since ATRP equilibrium constants under com-
parable conditions for methacrylates are

∼1000 times higher than for acrylates

(142–144), cross-propagation from polyacrylates to polymethacrylates is inefﬁ-
cient in spite of

∼100 times higher propagation rate constants for the latter (145).

However, since alkyl bromides are

∼100 times more active than structurally sim-

ilar alkyl chlorides (146), the cross-propagation becomes successful.

Fig. 8.

Halogen exchange technique.

Vol. 9

COPOLYMERIZATION

427

Fig. 9.

Preparation of triblock copolymers with poly(butyl acrylate) central and

poly(methyl methacrylate) outer segments by ATRP.

When difunctional pnBA was used to prepare ABA block copolymers with

MMA using a Ni system, without halogen exchange, it was reported that the
mechanical properties of the resulting copolymers were poor compared to those
of the copolymers prepared by ionic methods, presumably because of the broad
polydispersity of the outer pMMA blocks (see Fig. 9) (147). Complete incorporation
of the macroinitiator did not occur until

>30% monomer conversion. The block

copolymers did, however, microphase separate into ordered structures (148).

In comparison, block copolymers prepared using a copper catalyst employing

the halogen exchange technique had predictable molecular weights and narrower
molecular weight distributions (141). There was no evidence of slow initiation
and the outer pMMA blocks were more uniform. When Moineau and co-workers
used this system, the mechanical properties were greatly improved relative to the
copolymers prepared with the Ni catalyst (149). It has been also reported that the
proper choice of solvent also improves block copolymerization (150).

Another approach to form similar materials is to add MMA to the growing

pBA chains before complete consumption of the BA to form gradient A segments.
The properties of these triblocks were different than those observed for “clean”
ABA blocks, with lowered glass-transition temperatures (T

) for the mostly hard

outer blocks as well as increased tensile strength and elongation, as shown in
Figure 10 (120).

The stress–strain curves recorded for the two triblock copolymer samples

during cold drawing of ﬁlms with a constant rate of 1 mm/min are shown in
Figure 11. The gradient copolymer has much higher tensile elongation (1000% vs.
300%) but much lower tensile strength (4.5 MPa vs. 7.3 MPa) than the copolymer
with pure homopolymeric blocks.

The ABA-isolated sample was a clean triblock, with a pnBA central block of

= 65,200, two blocks of MMA with an M

= 13,150, and an overall M

1.34. The ABA-sequential copolymer contained a clean central block of nBA of M

= 67,500, and two gradient end block copolymers containing 13 mol% nBA and
87 mol% MMA, with an M

= 10,600, and an overall M

= 1.24 (120). The

insert shows the small-angle X-ray scattering intensities for these samples, which
correspond to cylindrical morphology of pMMA hard segments. Since both types

428

COPOLYMERIZATION

Vol. 9

Fig. 10.

Temperature dependencies of the real (G

) and imaginary (G

) component of the

shear modulus measured at the deformation frequency of 10 rad/s. for the pure and tapered
triblock copolymers pMMA-b-pBA-b-pMMA and p(MMA-grad-BA)-b-pBA-b- p(MMA-grad-
BA) of approximately the same overall composition, MW and polydispersity. For comparison
DSC traces are shown, which help to localize the glass-transition temperatures (T

) of the

microphases.

of block copolymers had similar compositions and only varied in the sequence
distribution of the monomers, this type of structural control provides yet another
route to tailor make polymers with speciﬁc properties by CRP methods.

Graft Copolymers.

Graft copolymers can be prepared by three dif-

ferent techniques: grafting from the backbone, grafting through by using
macromonomers, and grafting onto the backbone.

CRP has been applied to the synthesis of each segment in the either en-

tire graft system or exclusively to the preparation of speciﬁc segments such as
backbone, macromonomer, macroinitiator, or the grafting process. For example,
NH

-terminated dendrons were grafted onto backbones made by CRP contain-

ing N-oxysuccinimide 4-vinylbenzoate (151). Alternatively macromonomers were
made by either ATRP or other process (coordination, or conventional RP) and
they were subsequently used for grafting through (152,153). In another approach,
macromonomers were made by group transfer polymerization, or other controlled
polymerization processes, but ATRP was used for the grafting through copolymer-
ization (112,114). Perhaps the simplest is the grafting-from process, which can
involve CRP for the synthesis of the backbone, for the grafting-from reaction, or
both (Fig. 12).

Vol. 9

COPOLYMERIZATION

429

Fig. 11.

The stress–strain curves recorded for the two triblock copolymer samples during

cold drawing of ﬁlms with a constant rate of 1 mm/min. ABA-isolated sample was a pure
triblock but ABA-sequential contained a pure central block of n-butyl acrylate of M

67,500, and two end gradient blocks containing 13 mol% n-butyl acrylate and 87 mol%
methyl methacrylate, with an M

= 10600, and an overall polydispersity of M

= 1.24.

The insert shows the small angle X-ray scattering intensities for these samples.

Grafting-From Technique.

Graft copolymers were among the ﬁrst materi-

als prepared by CRP. After establishing that a unimolecular TEMPO-based initia-
tor successfully controlled the polymerization of styrene (154), the grafting-from
approach was used to demonstrate that graft copolymers were possible using CRP
techniques (155). The multifunctional macroinitiator was prepared by free-radical
copolymerization of styrene with a vinyl benzyl TEMPO derivative using AIBN
at 60

◦

C. The macroinitiator was heated to 130

◦

C in the presence of more styrene,

which led to the preparation of a graft copolymer. The molecular weight of the
polymer increased from M

= 12,000 (M

= 1.86) to M

= 86,000 (M

2.01). After cleavage of the grafted side chains, analysis of the graft units gave
M

= 23,000 with M

= 1.26, in good agreement with the expected value of

21,000 (155).

Expanding on this idea, an alkene-functionalized TEMPO derivative was

copolymerized with propylene or 4-methylpentene using a cationic metallocene
compound to prepare an

α-oleﬁn-based macroinitiator for graft copolymerization

= 28,000 (M

= 1.80) or M

= 6080 (M

= 1.65)] (Fig. 13) (156).

Heating the propylene-based copolymer to 123

◦

C in the presence of 200 equiv

St increased its molecular weight to M

= 210,000, with M

= 2.0. The ﬁnal

molecular weights of the graft copolymers were dependent on the concentration
of St added to the reaction.

The grafting-from technique can also provide a route to graft copolymers

via ATRP. A poly[(vinyl chloride)-co-(vinyl chloroacetate)] copolymer, containing

430

COPOLYMERIZATION

Vol. 9

Fig. 12.

Three approaches to prepare graft copolymers.

Fig. 13.

Graft copolymerization of styrene with oleﬁns.

Vol. 9

COPOLYMERIZATION

431

Fig. 14.

Grafting from poly(vinyl chloride) by ATRP.

about 1% of the chloroacetate groups, was used to initiate the ATRP of St, MA,
nBA, and MMA, resulting in graft copolymers with well-deﬁned graft units (Fig.
14) (157).

H NMR analysis showed that the molecular weight of the macroinitiator

increased substantially after the ATRP graft copolymerization (M

= 47,400 to

= 112,700–363,000). The mole percent of the grafted monomer ranged from

50 to 80% in the copolymer, but GPC analysis was not useful for determining the
molecular weight of the copolymers (157). A detailed analysis of the ATRP of nBA
using

H NMR and FT-IR spectroscopy showed that with increasing reaction time,

the mole percent of nBA in the copolymer increased and the T

decreased. This

work demonstrated that commercially available macroinitiators can be used to
prepare graft copolymers via ATRP. In a similar way grafting was performed from
the defects in pVC (158).

Brominated polyethylene or ethylene styrene copolymers were also used as

initiators in the presence of a CuBr–PMDETA catalyst for the subsequent ATRP
grafting from copolymerization of acrylic monomers. In similar studies, syndio-
tactic pSt-g-pMMA, syndiotactic pSt-g-pMA, and syndiotactic pSt-g-atactic pSt
were synthesized by ATRP using brominated syndiotactic pSt as the initiator and
a CuBr–PMDETA catalyst. Both the graft density and the MW of the graft seg-
ments were controlled by changing the bromine content of syndiotactic pSt and
the amount of monomer used in the grafting-from reaction. The ATRP mechanism
for grafting was supported by NMR analysis of the end groups. The thermal prop-
erties of the graft copolymers depended on both the graft density and the graft
length (159).

Another

commercially

available

polymer,

poly(isobutylene-co-p-

methylstyrene-co-p-bromomethylstyrene), was used as a macroinitiator for
the ATRP of St to produce a graft copolymer (160). After 23 h at 100

◦

C, with

1 equivalent of catalyst relative to the number of moles of initiator, monomer
conversion was 81%; however, increasing the concentration of catalyst ﬁvefold
resulted in the same efﬁciency after only 11 h. No evidence of unreacted macroini-
tiator was found in the GPC analysis, indicating that nearly all of the chains
contained some grafted sites. Results from mechanical analysis showed that
when the wt% of St was high (28%), the graft copolymer exhibited no special
properties; however, with only 6 wt% pSt, the polymer was elastomeric and could
be reversibly stretched to 500% of its initial dimension (160).

It was demonstrated that the grafting of St and MMA using a similar back-

bone is better controlled by applying the halogen exchange method (161,162).
Mechanical and viscoelastic properties depend strongly on the morphologies of

432

COPOLYMERIZATION

Vol. 9

Fig. 15.

ATRP grafting from poly(ethylene-co-glycidyl methacrylate).

the samples, which are deﬁned by the proportion of the hard and soft segments
as well as by the uniformity of the hard pMMA grafts (161).

A commercially available poly(ethylene-co-glycidyl methacrylate) [p(E-co-

GMA)] polymer was transformed into an ATRP macroinitiator (163) by reacting
the p(E-co-GMA) with either 2-bromoisobutyric acid or chloroacetic acid to pre-
pare a backbone with multiple initiator sites. The resulting polymer was used
as a macroinitator for ATRP of St and MMA (Fig. 15). The consumption of both
monomers increased with time, as did the weight percent of the side chains in the
copolymer. GPC analysis of the cleaved pSt side chains showed a linear increase
of molecular weight with monomer conversion, and M

< 1.4 (163).

ATRP has been also used in the grafting-through process using hyper-

branched polyethylene macromonomers with methacrylate functionality prepared
by the living Pd-mediated process (164).

Inorganic macroinitiators can also be used for graft copolymeriza-

tions. Pendant vinyl functional pDMS was subjected to hydrosilation with
2-(4-chloromethylphenyl)ethyldimethylsilane to prepare a multifunctional ATRP
macroinitiator (133,165). The ATRP of St was carried out using a pDMS macroini-
tiator with an M

= 6600 and an M

= 1.76 to yield a graft copolymer with an

= 14,800 and an M

= 2.10. The increased polydispersity was attributed

to the variation in the number of initiating sites on the pDMS backbone.

H NMR

analysis showed that less than 5% of the total number of benzyl chloride moieties
were left unreacted and that the weight ratio of pSt/pDMS was 1.18 (133).

The ﬁeld of densely grafted copolymers has received considerable attention

in recent years. The materials (also called bottle-brush copolymers) contain a
grafted chain at each repeat unit of the polymer backbone. As a result, the macro-
molecules adopt a more elongated conformation. Examples of brush copolymers
have been provided within the context of deﬁning the level of control available
through ATRP (152,166,167). Synthesis of the macroinitiator involved the ATRP
of 2-trimethylsilyloxyethyl methacrylate followed by esteriﬁcation of the protected

Vol. 9

COPOLYMERIZATION

433

Fig. 16.

Synthesis of densely grafted brushes by ATRP.

alcohol with 2-bromopropionyl bromide provided a macroinitiator of well-deﬁned
structure (M

= 55,500, M

= 1.3) leading to a brush synthesized entirely by a

controlled process (Fig. 16). The macroinitiator was used for either the ATRP of St
or nBA, leading to the desired densely grafted structures. The grafting reactions
were found to be very sensitive to reaction conditions; the presence of additional
deactivator, high concentrations of monomer, and reduced temperatures were all
necessary to produce the desired materials without occurrence of any coupling
reactions.

Since the aspect ratio and size of the macromolecules were so large, indi-

vidual chains were capable of being observed by atomic force microscopy (AFM)
(153). The brushes with pSt side chains form elongated structures on a mica sur-
face with an average length of 100 nm, a width of 10 nm, and a height of 3 nm.
pnBA absorbs well onto the mica surface and forms spectacular single-molecule
brushes in which the backbone and side chains can be visualized using tapping
mode AFM.

Similarly, core-shell cylindrical brushes were prepared by conducting a block

copolymerization. A well-deﬁned backbone with a degree of polymerization of
∼500 was initially prepared by ATRP, which was followed by a transesteriﬁcation
and the subsequent grafting of pnBA chains using ATRP. A ﬁnal chain exten-
sion with St produced the block copolymers. The ﬁnal structure consisted of a
macromolecule with a soft pnBA core and a hard pSt shell (167). High resolu-
tion AFM micrographs of the block copolymer [pBPEM-g-(pnBA-b-pSt)] brushes
show a necklace morphology. The synthesis of well-deﬁned brush block copolymers
demonstrates the synthetic power of ATRP.

Other Systems.

CRP has been recently applied to grafting from many

surfaces, including colloidal particles, ﬂat wafer surfaces, and delaminated clay
nanolayers, in addition to irregular surfaces such as cellulose (ﬁlter paper), silica
(glass frits), and many other solid substrates (168). All of them can be consid-
ered as copolymers of biopolymers, inorganics, and well-deﬁned organic polymers.
Some of the advantages of using CRP for the synthesis include large range of
polymerizable monomers, undemanding reaction conditions, and simple function-
alization of surfaces. For example, hydroxy groups on the surface of cellulose, or

434

COPOLYMERIZATION

Vol. 9

essentially any oxidized polymer surface, can be esteriﬁed with

α-bromoesters to

provide ATRP initiating sites. In the same way silanol groups on glass or oxidized
silicon surface can be condensed with chloro- and alkoxysilanes containing simi-
lar bromoesters (169). Then any polymer can be grown with a high precision from
these functionalized surfaces. In this approach to material synthesis not only den-
sity of grafting and chain length can be controlled but also composition, yielding
responsive block, gradients, or statistical copolymers.

A special case of control over the structure of copolymers may include the ﬁrst

stereoblock copolymers made by CRP. By applying either RAFT or ATRP to poly-
merization of acrylamides in the presence of rare-earth triﬂates such as Y(OTf)

and Yb(OTf)

, it was possible to enhance isotacticity of N,N-dmiethylacrylamide

(DMA) from

∼50% meso to ∼90% meso dyads (170). At the same time control

of molecular weights and polydispersity was preserved. Similar results were ob-
tained for RAFT of N-ispropylacrylamide (171). The ATRP and RAFT of DMA was
applied to the ﬁrst one-pot stereoblock synthesis by radical mechanism. RAFT or
ATRP of DMA were started without Lewis acid to produce the ﬁrst atactic block.
Subsequently, the complexing agent was added at the desired conversion to con-
tinue the chain growth with the preferential isotactic placement (170).

Ionic Copolymerization

The copolymerization proceeding via an ionic mechanism is markedly different
from the free radical copolymerization (172). The reaction of ions with a vinylic
monomer molecule is generally governed by the polarity of the double bond and is
therefore much more selective than the radical addition reaction, in which radical
stability is the controlling factor. The reactivity ratios are signiﬁcantly different
from that measured in free-radical copolymerization (see Table 1) and cover a
much wider range. (Tabulated values of reactivity ratios in anionic and cationic
copolymerization can for instance be found in References 173 and 174.) The num-
ber of monomer pairs that undergo ionic copolymerizations is hence limited. In
most cases one of the reactivity ratios is larger than 1; the resulting copolymer-
ization favors the incorporation of one monomer into the polymeric chain leading
to extensive blocks. Because of the inverse polarity of anions and cations the
reactivity ratios exhibit an inverse behavior, that is, if r

> r

in an anionic

copolymerization, r

< r

will hold in a cationic copolymerization (see Table 1).

There is a tendency that the product r

·r

is close to one when two resonance

Table 1. Monomer Reactivity Ratios of Selected Comonomer Pairs for Different
Polymerization Mechanisms

Monomer

Free radical

Cationic

Anionic

Styrene

Vinyl acetate

0.01

8.3

0.015

0.01

0.1

Styrene

Chloropren

0.005

6.3

15.6

0.24

–

Styrene

Methyl methacrylate

0.52

0.46

10.5

0.1

0.12

6.4

Acrylonitrile

Methyl methacrylate

0.18

1.35

–

7.9

0.25

Vol. 9

COPOLYMERIZATION

435

stabilized monomers are involved. In such an ideal copolymerization the rela-
tive reactivities of the two propagating ionic species toward the two different
monomers are approximately the same. The product is, however, often found to
be much larger with one comonomer not stabilized, leading to block-copolymer
behavior. This characteristic is for instance observed in the copolymerization of
dimethyltrimethylene carbonate and pivalolactone in toluene initiated with K-
naphtalene, which proceeds via an anionic ring-opening polymerization (175).
The

C NMR analysis of the generated block copolymer only reveals signals cor-

responding to homodiads (176), demonstrating a well-deﬁned crossover reaction.
Less pronounced block copolymerization behavior in living ionic polymerization,
however, will lead to a gradient composition proﬁle along the chain (177). In the
organolithium-initiated anionic copolymerization of isoprene (or butadiene) and
styrene in hydrocarbon solvents, for instance, isoprene is polymerized almost ex-
clusively in the ﬁrst phase of the process, followed by polymerization of styrene,
producing a terminal polystyrene block. The products formed by such a process
are so-called tapered block copolymers, which are characterized by a fuzzy border
between the two blocks. If the fuzzy border region reaches the length of the whole
macromolecule, the polymeric material is called a gradient copolymer (see above).
To obtain block copolymers with a distinct border, one of the monomers is ﬁrst poly-
merized in the presence of an initiator to give an active homopolymer, which is
then put into the medium of another monomer, the polymerization of which results
in the formation of the second block (178). This sequential block copolymerization
has been utilized in living ionic polymerizations to synthesize a wide variety of
linear, and star-branched block copolymers [eg, carbocationic copolymerization of
isobutylene (yielding a rubbery block) with styrene (179),

α-methylstyrene (180),

or indene (181) (generating a plastic segment)]. Block copolymers containing crys-
tallizable blocks have improved properties since the interplay of crystallization
and microphase segregation of crystalline–amorphous block copolymers greatly
inﬂuences the ﬁnal equilibrium ordered states and results in a diverse morpho-
logical complexity. Because of the lack of suitable vinyl monomers that give rise to a
crystallizable segment by cationic polymerization, living cationic sequential block
copolymerization is unsuitable for the synthesis of these types of block copoly-
mers. An active-site transformation technique can be used to synthesize block
copolymers consisting of two monomers that are polymerizable only by two differ-
ent mechanisms (182,183). In this method, the propagating active center is trans-
formed into a different kind of active center, and a second monomer is subsequently
polymerized by a mechanism different from the ﬁrst one. The key process in this
method is the diligent control of end functionality, capable of initiating the sec-
ond monomer. Following this idea, poly(isobutylene-block-pivalolactone) diblock
and poly(pivalolactone-block-isobutylene-block-pivalolactone) triblock copolymers
were synthesized using ﬁrst cationic polymerization followed by anionic ring-
opening polymerization (182) and polyisobutylene-block-poly(tert-butyl methacry-
late) block copolymers were produced employing cationic polymerization followed
by anionic polymerization after quenching with thiophene and subsequent met-
allization, yielding an anionic macroinitiator (184).

The preparation of triblock copolymers can be conducted either via subse-

quent addition of two different monomers or synthesizing a bifunctional growing
chain. The success of any block copolymerization is always dependent on the ability

436

COPOLYMERIZATION

Vol. 9

Table 2. Reactivity Ratios in the Anionic Copolymerization of Styrene
and Isoprene in Different Solvents at Room Temperature, Using
C

Li as the Initiator (186,187)

Styrene

Isoprene

Solvent

Toluene

0.25

9.5

Triethylamine

0.8

1.0

Tetrahydrofuran

9.0

0.1

of the active center to attack the next monomer. This general problem can, how-
ever, be circumvented via end group alterations of the generated polymer blocks in
order to generate block copolymers of monomers that usually do not homopolymer-
ize [eg producing amphiphilic poly(tert-butyl acrylate)-block-polystyrene-block-
poly(4-vinylpyridine) (185)]. Ionic copolymerizations were throughout a long pe-
riod of time the major technique to generate block copolymers. However, they
seem to be superceded nowadays by the new living free-radical polymerizations,
because of their versatility and robustness (see above).

Another characteristic feature of ionic copolymerization is the sensitivity of

the reactivity ratios toward changes in the initiator, solvent, and temperature. The
reaction conditions have in general a dramatic effect on the r values, as illustrated
in Table 2 for the anionic copolymerization of styrene and isoprene in different
solvents (186,187).

The usage of different counterions also signiﬁcantly changes the copolymer-

ization behavior: employing an organosodium initiator in the copolymerization of
styrene and butadiene in toluene instead of an organolithium initiator leads to a
change of the reactivity ratios from r

= 0.004 and r

= 12.925 to r

= 0.42 and

= 0.30 (188), whereas usage of an organopotassium initiator yields the values

= 3.3 and r

= 0.12 (189). The reactivity ratios may also be inﬂuenced by the

presence of polar substances like ethers or amines in the reaction system. An in-
disputable advantage of ionic copolymerization over free-radical copolymerization
is hence the possibility to control efﬁciently the reactivity ratios by varying the
polymerization conditions.

These ﬁndings imply that the reactivity ratios are in general not only gov-

erned by the relative reactivities of the free ionic species, but additionally by
the equilibrium between free ions and ion pairs. The conventionally determined
reactivity ratios will therefore describe average reactivities. Considering an equi-
librium that precedes the actual propagation step (see eqs. 39 and 40)

(39)

(40)

the rate-determining steps are the propagation reactions of the intermediate
products with the rate coefﬁcient k

. The conventionally determined monomer

Vol. 9

COPOLYMERIZATION

437

reactivity ratios are hence not measures for the reactivities of the macro-ions
but rather for the concentration of the intermediates and the related rate coef-
ﬁcients. The different reactivity ratios in different solvent systems cannot solely
be explained by macroscopic polar or electrostatic effects (190). The concept of
preferential solvation of the growing ion by either the solvent or the comonomer
is helpful (191); however, the whole process is still far from being understood
in depth and additional effects (eg, complexation) have to be taken into account
(192). The effect of solvents on monomer reactivity ratios cannot be considered
independent of the counterion used. Whereas some comonomer systems show al-
tered reactivity ratios for different initiator systems, others exhibit no signiﬁcant
effect.

Temperature has a greater inﬂuence on the r

and r

values of ionic copoly-

merizations than in free-radical copolymerization because of the greater spread of
activation energies for the propagation reactions involving ions. There is no gen-
eral trend observable and the monomer reactivity ratios may increase or decrease
with temperature: in the isobutylene–styrene copolymerization r

increases by a

factor of 1.5 and r

increases by a factor of 3 when going from

−94

◦

C to

−30

◦

(193), whereas in acenaphtylene-

α-methyl styrene r

increases from 0.075 to 1.49

and r

decreases from 8.3 to 0.1 when going from

−70

◦

C to 20

◦

C (194).

In contrast to the free-radical copolymerization, the lack of termination in

living ionic copolymerization enables the direct determination of the rate coef-
ﬁcients of the cross-propagation step. The reaction rate of monomer B with the
living chain end a

−

is directly accessible via the concentration decrease of a

−

which may be traced via a suitable spectroscopic method. Alternatively, the con-
centration of B can be determined as a function of time with the concentration of
a

−

being held constant. The latter experimental technique requires extrapolation

of time toward zero, because the cross-propagation step is immediately followed
by homopropagation of B.

Alternatively, the propagation step in ionic polymerization can be described

using model reactions, arriving at an estimate of reactivity ratios. Mayr and co-
workers, for instance, have developed a reactivity scale for the electrophilic at-
tack of carbocations on nucleophiles (195–197). Employing (i) photometric and/or
conductometric measurements of the consumption of persistent carbocations, (ii)
photometric detection of laser ﬂash photolysis generated carbocations, and (iii)
competitive experiments yielding relative reactivities of different electrophiles
toward nucleophiles, they obtained consistent results and established a general
scale which assigns to all electrophiles and nucleophiles an electrophilicity param-
eter E (describing the carbocation) and a nucleophilicity parameter N (describing
the monomer), respectively. It could be shown via correlation analysis that the
reaction rate coefﬁcient k follows the linear free-energy relationship

log k(20

◦

= s(N + E)

(41)

with s being a slope parameter (198). For most

π-nucleophiles the s parameter

ranges between 0.8 and 1.1 and can therefore be generally neglected. It has been
shown that copolymerization in the cationic polymerization can only be expected
between monomers for which N

≈ −E holds (196).

438

COPOLYMERIZATION

Vol. 9

The copolymerization behavior of heterocyclic monomers following an ionic

mechanism (ie, ring-opening polymerization) is even more complex than that of
vinylic monomers (199). In cationic ring-opening polymerization, for instance,
there are often effects present which complicate the kinetic investigations: ex-
change reactions, macrocyclization, reversibility of polymerization (eg the ho-
mopolymerization step of the monomer with the smaller strain is often reversible)
and multiplicity of active species. The concept of describing the whole process via
the simple four-parameter equation for the calculation of the reactivity ratios is
hence not applicable. The reactivity of the monomer in cationic ring-opening poly-
merization is increasing with its basicity, but it is not inﬂuenced by the ring strain.
On the other hand, the active species (eg the onium ions) decrease their reactivity
with increasing basicity and an enhanced ring strain leads to an increased reac-
tivity. Further details of the ionic ring-opening polymerization may be found in
References 200–202.

The different homo- and cross-propagation steps within one ionic copoly-

merization system generally exhibit signiﬁcantly different rate coefﬁcients. The
resulting polymers therefore consist mainly of one comonomer species, constitut-
ing almost homopolymers. The industrial applications of ionic copolymerization
are therefore limited. One of the few examples is the anionic copolymerization of
butadiene and styrene in solution for the production of the so-called SBR (styrene–
butadiene rubber) (203).The copolymer consists approximately of equal amounts
of styrene, cis-1,4-butadiene, trans-1,4-butadiene, and 1,2-butadiene units. Stere-
oregularity, polymer composition, molecular weight, and branching can be de-
termined via the process control which governs the concentration and nature of
the active ionic species. Another industrially signiﬁcant process is the cationic
copolymerization of isobutylene with

∼2% isoprene using AlCl

/CH

Cl at

−90

◦

in boiling ethene (203–206). The incorporated double bonds may subsequently
be vulcanized to give an elastomer with good impermeability to water and gas,
weather stability, ozone resistance, and heat stability (207).

Cross-Linking

Monomers exhibiting two vinylic double bonds are often used as comonomers in or-
der to achieve a cross-linked structure of the ﬁnal polymeric chains (208,209). The
cross-linking reaction occurs during the reaction or subsequently, depending on
the reactivity of the second double bond. The bifunctional comonomer is generally
employed in low concentrations, because a high concentration of dienes leads to a
signiﬁcant amount of unreacted double bonds in the ﬁnal polymeric product, which
lower the resistance toward outside physical and chemical inﬂuences. However,
the extent of cross-linking is dependent on the amount of bifunctional monomer
BB (with B indicating the double bonds of the diene comonomer). The critical re-
action extent at which cross-linking occurs during a copolymerization with dienes
can easily be derived mathematically for the case of equally reactive double bonds
both in A and BB. The copolymerization of styrene and divinylbenzene (210), vinyl
acetate with divinyl adipate (211), and methyl methacrylate with ethylene glycol
dimethacrylate (212) are examples of such a behavior. At the double bond conver-
sion c there are c[A] reacted monomer units A, c[B] reacted double bonds of the

Vol. 9

COPOLYMERIZATION

439

monomer B and c

[BB] reacted monomer units BB (with [B]

=2[BB]). The num-

ber of crosslinks is simply the number of BB monomer molecules in which both
B double bonds are reacted, that is c

[BB]. The number of polymer chains is the

total number of A and B double bonds reacted divided by the average degree of
polymerization DP. The critical double bond conversion c

crit

at which cross-linking

occurs is reached when the number of crosslinks per chain is 1/2 and is therefore
described by

crit

[A]

+ [B]

[B]

·DP

(42)

for long chains. DP is the mass-average degree of polymerization (for details see
Ref. 213) that would be observed in the absence of any bifunctional comonomer.
Equation 42—denoted as Flory–Stockmayer equation (213)—has been found use-
ful for the prediction of the onset of the cross-linking, also denoted as the gel-point,
for equally reactive double bonds, especially when low concentrations of dienes are
employed (mole fraction of BB below 0.05). With increasing diene concentration,
equation 42 predicts smaller critical conversions than found experimentally (214).
This effect is called “delayed gelation.” The primary factor for this observation is
the signiﬁcance of the thermodynamic excluded volume effect on the intermolecu-
lar cross-linking reaction between the growing radical and the preformed polymer,
especially at higher molecular weights. Beyond the theoretical critical conversion,
a secondary factor is related to the intramolecular cyclization (215) which becomes
progressively important with conversion. The latter leads to the restriction of seg-
mental motion of the preformed polymer and causes steric hindrance.

The critical conversion in the copolymerization process in which the double

bond in comonomer A and the ﬁrst double bond in comonomer BC has the same
reactivity, but the second double bond in BC is less reactive, can be described by
equation 43:

crit

= 1 − exp

− 1

·x

·DP·r

(43)

with x

being the mole fraction of BC and r

being the reactivity ratio between

the B and C double bond. The copolymerization of methyl methacrylate and al-
lyl methacrylate would be an example for such a behavior (216). The resulting
copolymer will consist of copolymerized A and B double bonds. The relatively
unreactive C double bonds start to react, ie, crosslink, in the later stage of the
copolymerization.

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HRISTOPHER

ARNER

-K

OWOLLIK

HOMAS

P. D

AVIS

The University of New South Wales
M

ICHELLE

L. C

OOTE

Australian National University
K

RZYSZTOF

ATYJASZEWSKI

Carnegie Mellon University
P

HILIPP

ANA

Georg-August-Universit ¨at G¨ottingen

COTTON.

See Volume 5.

CRAZING.

See Y

IELD AND

RAZING

CREEP.

See V

ISCOELASTICITY

CRITICAL PHASE POLYMERIZATIONS.

See Volume 2.

CRYSTALLINE POLYMERS.

See S

EMICRYSTALLINE POLYMERS

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