ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS phần 5 potx

MAIN REACTIONS USED IN STEP-GROWTH POLYMERIZATION 245
The reaction of carboxylic acids with anhydrides is also commonly used for
the cross-linking of diepoxide prepolymers. The reaction has to be carried out at
higher temperature than previously and, in the presence of tertiary amines, occurs
through ring-opening of the oxiranes by the carboxylate generated from anhydrides:
R-COO
O
R
′
R-COO-CH
2
-CH-R′
O
+
At high temperature other reactions occur, which make the structure of the result-
ing networks extremely complex. Indeed, secondary hydroxyls formed upon ring-
opening of epoxides can in turn react with the oxiranes of the precursor:
R
′
HO
RO
+
R
′
O
ROH
Such reactions increase the density of cross-linking of the network formed.
7.5.5. Substitution Reactions on Silicon Atoms
Only a minor portion of industrially produced polysiloxanes is obtained by chain
polymerization of cyclosiloxanes (octamethylcyclotetrasiloxane). Most are syn-
thesized by water-induced hydrolysis of dialkyldichlorosilanes followed by self-

condensation of the disilanol formed. The starting monomer is dimethyldichlorosi-
lane, which is prepared by copper-catalyzed reaction of methyl chloride on metal
silicon. The hydrolysis of the chlorinated derivative
Cl Si
CH
3
Cl
CH
3
2H
2
O 2HCl
HO Si
CH
3
O
H
CH
3
++
corresponds to a nucleophilic substitution.
In the presence of bases, the condensation occurs by nucleophilic substitution,
and the result of the self-condensation of silanol groups is poly(dimethylsiloxane):
Si
CH
3
OH
CH
3
OSi

CH
3
CH
3
H OH
n
(n − 1)H
2
O
nHO
+
Depending on whether the silanol function is carried by a mono-, di-, or trivalent
monomer, one may have termination, polymerization, or cross-linking. The reac-
tivity of silanols is closely related to the nature of the alkyl groups, the number
246 STEP-GROWTH POLYMERIZATIONS
of hydroxyls carried by the silicon atom, and the size of the polysiloxane carry-
ing them. Thus, (CH
3
)
2
Si(OH)
2
is the most reactive among dialkylsilanediols. The
mechanism of the acid-catalyzed condensation of silanols (by HCl originating from
the first step) can be represented by
~~~~
Si OH
H
+
~~~~

Si O
H
H
+
Then
~~~~
Si O Si
~~~~
H
2
O~~~~
Si OH
~~~~
SiO
H
H
+
+
+
As the oligodimethylsiloxanediols gain in size, the reactivity of their terminal
silanols decreases due to their tendency to establish intramolecular hydrogen
bonding.
Si O
H
Si O
H
Such interactions only exist after the condensing oligomer has reached a certain
size, with the cyclization being impossible when they are still too small.
In the presence of bases (Et
3

N), condensation proceeds by nucleophilic substi-
tution:
~~~~
Si O
~~~~
SiHO
+
~~~~
Si O Si
~~~~
HO+
~~~~
Si OH
B
BH
+
~~~~
Si O
+
7.5.6. Chain-Growth Polycondensation
Conventional step-growth polymerizations occurs in the initial phase through con-
densation/addition of monomers with each other and then proceeds via reactions
of all size oligomers with themselves and with monomers. In such a process the
precise control of the polycondensate molar mass is elusive—in particular, in the
initial and intermediate stages where only oligomers are formed. The polyconden-
sate molar mass indeed builds up only in the final stage and its dispersity index
increases up to 2.
MAIN REACTIONS USED IN STEP-GROWTH POLYMERIZATION 247
In an attempt to better control both molar masses and the dispersity in polycon-
densates, a new concept of polycondensation has been recently proposed that pro-

ceeds in a chain polymerization manner (Chapter 8). In a context where monomers
would have little option but to react first with an “initiating” site and then with
the polymer end-group and would be prevented from reacting each other, all the
requirements would be met to bring about so-called chain-growth polyconden-
sations. Under such conditions, the polycondensate would increase linearly with
conversion and be controlled by the [monomer]/[initiator] ratio and its mass dis-
persity index would be close to unity.
Yokosawa and co-workers have proposed two approaches to such chain-growth
polymerization of X–AA–Y-type monomers:
(a) Specific activation of propagating end-groups and concomitant deactivation
of those carried by the monomer through substituent effects;
(b) Phase-transfer polymerization with the monomer being stored in a separate
solid phase.
The polycondensation of phenyl-4-(alkylamino)-benzoate carried out in the pres-
ence of phenyl-4-nitrobenzoate acting as initiator and a base is a perfect illustration
of approach (a) theorized by Yokozawa.
in tetrahydrofuran,
at room temperature
base
O
O
O
2
N
+
O
O
RNH
Initiator
Monomer

(
)
O
N
O
2
N
R
O
O
n
base =
N
C
8
H
17
Et
3
Si
CsF 18-crown-6
___
M
n
≤ 22,000 M
w
/M
n
≤ 1.1
Reaction mechanism:

O
O
RNH
N
R
O
O
base
strong deactivation
strong electron-donating group
inactive
strong activation
O
O
O
2
N
electron-withdrawing group
reactive
O
2
N
N
O
RO
O
weak electron-
donating group
reactive
weak deactivation

monomer
O
N
O
2
N
R
O
O
(
248 STEP-GROWTH POLYMERIZATIONS
The base serves to abstract a proton from the monomer and generate an aminyl
anion, which in turn deactivates its phenyl moiety. This anion reacts preferentially
with the phenyl ester group of phenyl-4-nitrobenzoate and the amide group formed
has a weaker electron-donating character than the aminyl anion of the activated
monomer. The reaction of monomers with each other was thus efficiently prevented
so that well-defined aromatic polyamides could be obtained up to 22,000 g/mol
molar mass and with a dispersity index of 1.1.
The case of solid monomers that are progressively transferred to an organic
phase with the help of a phase transfer catalyst and thus placed in a situation to
react with the polymer end group is an illustration of approach (b).
This concept of chain growth polycondensation is new in synthetic polymer
chemistry but not in Nature. In the biosynthesis of many natural polymers, Nature
takes indeed full advantage of this concept: for instance, DNA is obtained via a
polycondensation of deoxyribonucleoside of 5

-triphosphate with the 3

-hydroxy
terminal group of polynucleoside with the help of DNA polymerase.

LITERATURE
G. Odian, Principles of Polymerization, 4th edition, Wiley-VCH, New York, 2004.
M. E. Rodgers and T. E. Long (Eds.), Synthetic Methods in Step-Growth Polymers, Wiley,
New York, 2003.
8
CHAIN POLYMERIZATIONS
8.1. GENERAL CHARACTERS
Chain polymerizations proceed differently from these occurring by step growth.In
the latter case, polymers grow by reaction (condensation or coupling) with either a
monomer molecule, an oligomer, another chain, or any species carrying an antago-
nist functional group. Each condensation/addition step results in the disappearance
of one reactive species (whatever its size) from the medium, so that the molar
mass of such a “condensation polymer” is due to increase in an inverse proportion
to (1 −p), where p is the extent of reaction. The reaction between these antag-
onist functional groups that can be carried indifferently by monomer molecules
or growing polymer chains brings about the formation of the constitutive units
of polycondensates through covalent bonding. Two reactive functional groups are
consumed after each condensation/addition step.
Unlike the case of polycondensations and polyadditions, in chain-growth poly-
merizations, very long macromolecules can be formed just after induction of the
reaction, and active centers are generally carried by the growing chains. The gen-
eral scheme describing chain growth is the same as for other chain processes: after
production of a primary active center (P
∗
) by an initiator (I) or a supply of energy
to the system, this species activates a monomer molecule (M) through transfer of
its active center on the monomer unit thus formed:
A −→ P
∗
P

∗
+M −→ PM
∗
Organic and Physical Chemistry of Polymers, by Yves Gnanou and Michel Fontanille
Copyright  2008 John Wiley & Sons, Inc.
249
250 CHAIN POLYMERIZATIONS
This first step called initiation and often consisting of two phases is followed
by a propagation (or growth) step, during which macromolecules grow by chain
addition of monomer molecules to the newly formed PM
∗
species. Upon reaction
with a “fresh” monomer molecule, the active center carried by the growing chain
is transferred to the last generated monomeric unit, and so on:
PM
∗
+M −→ PMM
∗
(written PM
∗
2
)
PM
∗
n
+M −→ PM
∗
(n+1)
In most systems, propagation is very fast and corresponds to an exothermic phe-
nomenon whose overall activation energy is generally positive; in some cases, the

reaction may run out of control and even become explosive. Termination reac-
tions, when existing, may self-inhibit polymerizations getting out of control by
deactivating growing chains. These terminations occur irrespective of the degree
of polymerization of the growing chains:
PM
∗
n
−→ PM
n
In addition to terminations, certain systems can undergo other chain-breaking reac-
tions, such as chain transfer represented as follows:
PM
∗
n
+T −→ PM
n
+T
∗
T
∗
+M −→ TM
∗
TM
∗
+nM −→ TM
∗
n+1
T is called transfer agent, but transfer can occur to monomer, polymer, initiator,
or any molecule present in the reaction medium.
This transfer phenomenon blocks active chains in their growth and generates new

active centers (T
∗
) that are able to initiate the formation of novel macromolecules.
Chain transfer prevents the obtainment of polymeric chains of high molar masses
but can be used to control molecular dimensions when targeting oligomers or
samples of low molar masses. In certain conventional chain polymerizations, the
three steps of initiation, propagation, and termination as well as transfer can occur
simultaneously, which means that each initiated chain propagates and undergoes
termination or perhaps transfer, independently of events occurring in its surround-
ing. In other words, the time required for the formation of a chain can be lesser
than one second in certain systems, whereas the corresponding half-polymerization
time can be equal to several hours.
Chain-growth polymerizations are distinguished from one another, depending
upon the types of active centers that initiate and propagate the polymerization
process. Thus, four families of chain polymerizations are generally considered:
•
Free radical polymerizations, whose propagating active centers involve free
radicals,
POLYMERIZABILITY 251
•
Anionic polymerizations, which require nucleophilic reactive species,
•
Cationic polymerizations (“symmetrical” of the preceding ones), whose prop-
agating species are electrophiles,
•
Coordination polymerizations, whose active centers are complexes formed
by coordination between monomer molecules and transition metal atoms.
These four important methods of polymerization exhibit their own peculiarities.
Certain monomers can be polymerized (until today) by only one of them; this is
the case, for example, of vinyl acetate or acrylic acid, which can be polymerized

only by free-radical means. On the contrary, styrene can be polymerized by any of
the aforementioned methods of polymerization.
8.2. POLYMERIZABILITY
Polymerizability is the faculty of an organic compound (monomer molecule) to
undergo polymerization. Two conditions must be fulfilled to this end:
•
Compliance with thermodynamic constraints
•
Existence of an adequate reaction
The polymerizability of a monomer can be evaluated by means of the rate constant
of polymerization which varies with the method of polymerization chosen.
8.2.1. Compliance with Thermodynamic Constraints
Like any other reaction of organic chemistry, chain polymerizations are equilibrium
reactions that can be schematically represented as follows:
PM
∗
n
+M
K
−−−→
←−−−
PM
∗
n+1
The equilibrium between growing polymer chains and the monomer is determined
by the thermodynamic conditions. By definition, at equilibrium
G = 0
Therefore, one has
G = G
0

+RT ln K = H
0
−TS
0
+RT ln K = 0
where G
0
, H
0
,andS
0
represent the standard variations of free energy,
enthalpy, and entropy, respectively, corresponding to the transition undergone by
monomer molecules in their standard state (pure liquid, gas, or unimolar solution)
becoming the monomer units of polymeric chains, in their novel standard state
(amorphous solid state or solution in unimolar concentration).
252 CHAIN POLYMERIZATIONS
From the above equilibrium, the equilibrium constant can be written as
K = [PM
∗
n+1
]/

[PM
∗
n
][M]

If the concentrations of species PM
∗

n
and PM
∗
n+1
are assumed practically identical,
which is reasonable (at a first approximation) at equilibrium for values of n higher
than a few monomeric units, one can write
K = 1/[M]
which gives
RT ln[M] = H
0
−TS
0
R ln[M] = (H
0
/T ) −S
0
and corresponds to
T
c
= H
0
/

S
0
+R ln[M]
equ

or

ln[M]
equ
= (H
0
/RT
c
) −(S
0
/R)
In these last two equations, the c index after T denotes ceiling conditions corre-
sponding to the monomer concentration at equilibrium [M
equ
]. Indeed, in most of
polymerizations, the variation of entropy is negative since the transition from the
monomer to the polymer state corresponds to a decrease in the degrees of freedom
of the system; thus, the entropy term is unfavorable to the polymerization process.
For the latter to occur, it should be compensated by a negative value of the polymer-
ization enthalpy, which implies that chain polymerization reactions are exothermic
processes. When the temperature is raised, the entropy term increases as well until
becoming equal, in absolute value, to the enthalpy term. The polymerization can
then no longer proceed.
The maximum temperature beyond which the monomer concentration cannot be
lower than a reference value, taken in general equal to the concentration of the
pure monomer, is called ceiling temperature. For example, in the case of styrene,
it corresponds to 8.6 mol·L
−1
. It should be emphasized that certain authors take a
monomer concentration of 1 mol·L
−1
as reference value, which entails a value of

T
c
higher than the one resulting from the preceding convention. The definition of
the ceiling temperature is thus fully arbitrary since there exists for any tempera-
ture considered a certain monomer concentration in equilibrium with the growing
chains.
In the case of liquid vinyl monomers and related ones, the value of the enthalpy
of polymerization is generally in the range −30 to −155 kJ·mol
−1
; it is definitely
lower (in absolute value) for heterocycles.
POLYMERIZABILITY 253
The two terms (enthalpy and entropy) affect the value of the ceiling temperature,
but for different reasons; in the case of vinyl and related monomers, H
0
—which
reflects the energy difference between the π bonds in the monomer molecule and
the σ bonds in the polymer chain—closely depends on the number and the nature
of the substituents carried by the double bond; these substituents determine the
rigidity of the polymer chain and, in turn, the value of the entropy term. However,
the relative variations of the entropy term with the nature of the polymer are less
significant than those characterizing the enthalpy term, and hence the latter is more
prominent.
The values of H
0
and S
0
found in handbooks (Polymer Handbook, Com-
prehensive Polymer Science, etc.), were actually taken from primary publications.
However, these values often correspond to states of matter which differ from one

monomer to another and, in addition, were determined by different means. It is
thus inappropriate to present these values in a same table since they cannot be
valuably compared. The readers willing to determine either the ceiling tempera-
ture or equilibrium concentration under given conditions for a particular monomer
are requested to refer to primary publications whose references can be found in
Polymer Handbook. As an example, the well-known case of α-methyl styrene is
discussed below from data drawn from the article in Journal of Polymer Science,
25, 488, 1957:
H
0
=−29.1kJ·mol
−1
,S
0
=−103.7J·mol
−1
·K
−1
[M]
bulk
= 7.57 mol.L
−1
Based on these values, a ceiling temperature of 334 K (i.e., 61
◦
C) was calculated
for pure α-methyl styrene in the total absence of polymer.
The above example illustrates the necessity to carry out certain polymeriza-
tions at relatively low temperatures for conversions to reach completion. Should a
particular monomer be characterized by a rather low ceiling temperature, the ther-
mal decomposition of the corresponding polymer would occur at low to moderate

temperature.
8.2.2. Reaction Processes Compatible with Chain Polymerizations
A chain polymerization implies that the active species formed upon addition or
insertion of the last monomer molecule is of the same nature as the original one.
Such chain growth also entails the formation of at least two covalent bonds between
other monomer units. In view of the previously mentioned thermodynamic con-
straints, a negative variation of the free enthalpy of polymerization is another
imperative to fulfill. These two conditions considerably restrict the variety of the
organic compounds that can be polymerized, and only two main categories of
monomers meet these criteria:
•
Monomers carrying unsaturated groups whose high negative value of H is
due to the transformation of π bonds into σ bonds under the effect of an
254 CHAIN POLYMERIZATIONS
addition reaction:
C
CC
C
O
C
CC
~~
~~
CC
~~
~~
~
~
CO
~~

•
Cyclic strained monomers, which can be opened under action of an active
center, by nucleophilic substitution, addition–elimination on carbonyls, and
so on; here, the negative enthalpy of polymerization results from the release
of the cycle strain:
X
X
~~
~~
Oxiranes → Polyethers
Lactams → Polyamides
Cyclosiloxanes → Polysiloxanes
Cycloalkenes → Polyalkenamers, etc.
Depending upon the electronic structure of the molecular group responsible for
the polymerization, monomer molecules can be susceptible to an attack by free
radicals, nucleophilic species, electrophilic species, or coordination complexes. In
all cases, the polymerizability (measured by the rate constant of propagation, k
p
)
is determined not only by the reactivity of the monomer (M) but also by that of
the active center PM
∗
n+1
resulting from its insertion,
PM
∗
n
+M
k
p

−−−→ PM
∗
n+1
The effects induced by the substituents of the polymerizable function on the
two reactivities play often in opposite directions. Generally, the reactivity of the
monomer outweighs that of the corresponding active center; in other words,
the higher the monomer reactivity and the lower that of the active center, the higher
the corresponding rate constant of propagation. The reasons will be discussed when
considering each type of polymerization.
It is indisputable that, at the present time, vinyl and related monomers are by
far the most used (in particular from the economic point of view); this is why
examples will be generally taken from this family of compounds.
8.3. STEREOCHEMISTRY OF CHAIN POLYMERIZATIONS
A vinyl monomer possesses a plane of symmetry and is thus achiral. Upon poly-
merization, sp
2
-hybridized carbon atoms are transformed into sp
3
ones, and this
STEREOCHEMISTRY OF CHAIN POLYMERIZATIONS 255
process generates an asymmetry that is particularly noticeable when the inserted
monomer molecule is located at the growing chain end:
~~~~CH
2
-CH
A
-CH
2
-HC*
A

However, this asymmetry is only observed for active centers ∼∼∼HAC* in their
final configuration—that is, when the carbon atom carrying the active center is sp
3
-
hybridized. It is therefore not the case of active centers such as carbon centered
free radicals or free ions, which are sp
2
-hybridized. For such systems, the final
configuration of tertiary carbon atoms is fixed only after insertion of a monomer
molecule—that is, next to the asymmetrical carbon atom of the penultimate unit.
Among the parameters that determine the final configuration of the last unit
inserted at chain end (or the penultimate one if the final configuration is not
attained), the stereochemistry of previously inserted monomeric unit is obviously
a significant one. Two repeating units are necessary to define any such stereo-
chemistry that requires conditional probabilities. Depending upon the number of
preceding monomer units exerting an influence on the configuration of the last
unit added, the mechanism of monomer addition indeed follows either zeroth-,
first-, or second-order Markovian statistics. If a simple probability, P
m
, is sufficient
to describe the various additions and the structures formed—either meso (m)or
racemic (r)—the process is said to follow zeroth-order Markov (or Bernouillian)
statistics. If the last linkage in the chain—either m or r —controls the addition and
the stereochemistry of the monomer to be added, the mechanism is called first-order
Markov process. Limiting our discussion to the case of zeroth-order Markovian (or,
more usually, “Bernouillian”) statistics, one can thus define P
m
as the probability
of formation of an m dyad—that is, the insertion of two successive units of the
same configuration ([R] or [S])—and define P

r
as the probability of formation of
an r dyad, with
P
r
= (1 −P
m
)
and
(P
r
+P
m
) = 1
From these equations, one can write the probability of existence of longer sequen-
ces, such as that of triads, and so on.
For isotactic triads (mm =i), one has P
i
=P
m
2
For syndiotactic triads (rr =s), one has P
s
=(1 −P
m
)
2
For heterotactic triads (mr =rm =h), one has P
h
=2 P

m
(1 −P
m
)
For rmrr pentads, one has P =P
m
(1 −P
m
)
3
,etc.
256 CHAIN POLYMERIZATIONS
(mm) = P
m
2
(mr + rm) = 2P
m
(1-P
m
)
(rr) = (1-P
m
)
2
P
m
0
0
1
1

P
tr
Figure 8.1. Probability (P
tr
) of formation of i, h,ands triads, according to Bernouillian statistics.
In Figure 8.1 are plotted the variations of the probabilities of existence of the
various types of triads against P
m
. To check whether the addition of such monomer
follows Bernouillian statistics, one generally resorts to NMR and compare the
relative intensity of meso and racemic dyads and iso-, syndio-, and heterotactic
triads with calculated values.
When the relative configuration of the last added unit is controlled by the con-
figuration of the last inserted dyad (and not that of the last monomeric unit), the
statistics is more complex, reflecting a peculiar mechanism of polymerization.
With ionic or coordination polymerizations, different families of active species
(free ions and ion pairs, for example) may be simultaneously involved, each one of
them propagating with its own statistics. Analysis of the probabilities of existence
of the various sequences is even more difficult to interpret in these cases.
The identification of the type of configurational statistics thus gives extremely
valuable information about the intimate mechanism of the propagation.
8.4. ‘‘LIVING’’ AND/OR ‘‘CONTROLLED’’ POLYMERIZATIONS
In certain conventional polymerizations, the three steps of initiation, propagation,
and termination occur simultaneously, in a ceaseless movement that ends with
the total consumption of the initiator and/or the monomer. In other words, new
chains appear at all times, grow, and eventually stop growing as a result of one of
the chain-breaking reactions (termination or transfer). The lifetime of a propagating
center can be very short compared to the total polymerization time. Such a situation
causes inevitably a great disparity in the degrees of polymerization of the various
chains constituting a sample.

‘‘LIVING’’ AND/OR ‘‘CONTROLLED’’ POLYMERIZATIONS 257
A completely different situation arises when the propagating active centers are
not subject to transfer or termination and the initiation step is short compared to
that of propagation. Although contemplated a long time ago, it was only in the
1950s that Szwarc succeeded in his search for termination/transfer-free polymer-
izations with his work on anionic polymerization. He called such systems “living,”
assimilating the initiation of polymerization to the “birth” of chains, the propa-
gation to their “growth,” and the termination/deactivation of growing species to
their “demise”; Szwarc carried even further the analogy with biological systems,
identifying temporarily inactive species to “dormant” ones.
If the efficiency of the initiating system is total and the time necessary to create
the chains is short compared to that of the propagation, all the chains “are born” and
“grow” simultaneously until all the monomer is consumed. Under such conditions
the polymer samples formed exhibits a little dispersity of their molar masses.
Because the number of chains is determined by the number of molecules of ini-
tiator, the degree of polymerization of chains can be easily expressed as a function
of the monomer conversion (p), the initial monomer concentration [M]
0
,andthe
(monovalent) initiator concentration [I]:
X
n
=
[M]
0
p
[I]
If the monomer conversion is total, this expression reduces to
X
n

=
[M]
0
[I]
which represents
M
n
=
mass of produced polymer
number of moles of initiator
Remark. When an initiator gives rise to a bivalent propagation (bivalent
initiator or its precursor), the predictable molar masses are equal to the double
of those calculated using the above relations.
As for conversion, its expression can be easily deduced assuming that all active
species [M
∗
n
] exist and propagate under an unique form and all the conditions
mentioned above are respected. The rate of monomer (M) consumption can then
be written as
−
d[M]
dt
= k
p
[M]
∞

n=1
[M

∗
n
]
where k
p
is the rate constant corresponding to addition of the monomer molecule
onto the growing species. In this equation, the total concentration in active species
258 CHAIN POLYMERIZATIONS
is written for simplicity [M*] without any index and is identified with [I]:
∞

n=1
[M
∗
n
] = [M
∗
] = [I]
which gives, after integration,
ln
[M]
0
[M]
= k
p
[M
∗
]t
Conversion can be easily deduced:
p = 1 −exp


−k
p
[M
∗
]t

since
p =
[M]
0
−[M]
[M]
0
thus
ln
[M]
0
[M]
=−ln(1 −p)
The persistence of active centers even after consumption of all the monomer allows
one to trigger further chain growth by incremental addition of monomer and/or to
synthesize complex macromolecular architectures that would be inaccessible by
conventional polymerizations.
For systems that are not (strictly speaking) “living” and may be subject to
chain-breaking reactions with possible interruption of chain growth, most of the
advantages of truly “living” polymerizations may, however, be preserved, provided
that transfer and termination are minimized. Indeed, if the latter reactions occur only
to a limited extent and the initiation step is short compared to that of propagation,
polymer chains of controlled size and relatively well-defined complex architectures

can nonetheless be obtained. Such polymerizations are called “controlled.”
Remark. “Living” polymerizations are not necessarily “controlled.” Poly-
merizing systems subject to a slow or incomplete initiation, as well as those
with a propagation step faster than the homogeneous mixing of the reagents or
faster than the rate of exchange between different active species, enter in this
category of uncontrolled and yet living polymerizations. A high dispersity in
the size of the resulting polymer chains is observed.
Obviously, polymerization systems that exhibit at the same time a “living” char-
acter and afford chains and architectures of controlled size and structure are in great
demand. Among the specific characteristics, one can mention a low dispersity of
chains. Such a narrowing of molar mass distributions with the degree of polymer-
ization can be calculated.
‘‘LIVING’’ AND/OR ‘‘CONTROLLED’’ POLYMERIZATIONS 259
Let [M
∗
1
], [M
∗
2
], [M
∗
3
], ,[M
∗
n
] be the concentrations of active species corre-
sponding to degrees of polymerization indicated in index, the rate of disappearance
of the species M
∗
1

can be written
−
d[M
∗
1
]
dt
= k
p
[M][M
∗
1
]
with the same for species [M
∗
2
]and[M
∗
n
]:
+
d[M
∗
2
]
dt
= k
p
[M]


[M
∗
1
] −[M
∗
2
]

+
d[M
∗
n
]
dt
= k
p
[M]

[M
∗
n−1
] −[M
∗
n
]

On the other hand, the average number (d
ν) of monomeric units consumed by an
active chain during an interval of time dt can be expressed as follows:
d

ν = k
p
[M]dt =−
d[M]
[M
∗
]
Identifying k
p
[M]dt with dν in the expression of the rate of disappearance of [M
∗
1
]
species, one obtains
d[M
∗
1
]
[M
∗
1
]
=−d
ν
which gives, upon integration,

M
∗
1
I

d[M
∗
1
]
[M
∗
1
]
=−

t
0
dν ⇒ ln
[M
∗
1
]
[I]
=−
ν
and thus
[M
∗
1
] = [I]e
−ν
Introducing [I]e
−ν
in the expression of the rate of disappearance of the species
[M

∗
2
]gives
d[M
∗
2
] =

[I]e
−ν
−[M
∗
2
]

dν
which is a differential equation of the following type:
dy = (ae
−x
−y)dx or
dy
dx
+y = ae
bx
whose solution is
y = axe
bx
260 CHAIN POLYMERIZATIONS
The variation of [M
∗

2
] as a function of ν can then be written
[M
∗
2
] = [I]νe
−ν
Repeating the same reasoning for the variation of [M
∗
3
], one obtains
d[M
∗
3
] =

[I]νe
−ν
−[M
∗
3
]

dν
an expression which is of the type
dy
dx
+y = P(x)ae
bx
whose solution is

y = e
ax

P(x)dx
Thus, the variation of [M
∗
3
] can be deduced as follows:
[M
∗
3
] = [I]

v
2
2

e
−ν
and in the general case of [M
∗
i
]
[M
∗
i
] = [I]
ν
(i−1)
e

−ν
(i −1)!
The molar fraction of the species of degree of polymerization i is thus written
f
i
=
[M
∗
i
]
[I]
=
ν
(i−1)
e
−ν
(i −1)!
As for the mass fraction of the species having a degree of polymerization i, it can
be easily deduced [if one identifies the mass of fragment I of the initiator (M
a
)
with that of a repetitive unit (M
0
)]:
W
i
=
M
0
(i +1)f

i
M
0

∞
i=1
(i +1)f
i
=
(i +1)f
i
ν +1
which can be also written
W
i
=
νe
−ν
(ν +1)
·
ν
(i−1)
(i −1)!
(i +1)
FREE RADICAL POLYMERIZATION 261
The expressions of M
n
and M
w
can thus be easily deduced:

M
n
= M
0
ν +M
a
= M
0
(ν +1)
M
w
= M
0
X
w
= M
0
∞

i=1
iw
i
= M
0
νe
−ν
ν +1
∞

i=1

i
2
ν
(i−2)
(i −1)!
The above sum can also be written

ν +3 +
1
ν

e
ν
, giving for M
w
M
w
= M
0
(ν
2
+3ν +1)
(1 +ν)
and thus
D
M
=
M
w
M

n
= 1 +
ν
(ν +1)
2
The samples obtained under such conditions exhibit a Poisson-type distribution of
their molar masses. This type of distributions is obtained when one distributes in
a random way m objects in n boxes, with m >>> n.
8.5. FREE RADICAL POLYMERIZATION
8.5.1. Reminders on Free Radical Reactions
Free radicals can be regarded as resulting from the homolytic rupture of covalent
bonds. They are generated by using either physical (thermal, radiative, etc.) exci-
tation or chemical (oxydo-reduction, free radical addition, etc.) means. If they are
not stabilized by particular substituents, their lifetime (about one second in normal
polymerization conditions) is extremely short due to a very high reactivity. Their
hybridization state is generally trigonal (sp
2
) except for those carrying substituents
of large size developing steric hindrance.
Free radicals can be involved in the following six reactions, all occurring in the
polymerization processes:
Combination R
•
+
•
R

−→ R–R

Disproportionation 2 R–CH

2
–CH
2
•
−→ R–CH
2
–CH
3
+R–CH
=
CH
2
Abstraction/transfer R
•
+R

X −→ RX +
•
R

Addition R
•
+H
2
C
=
CR
1
R
2

−→ R–CH
2
–
•
CR
1
R
2
Fragmentation RA
•
−→ R
•
+A
Rearrangement R

R

R
•
−→
•
R

R

R
262 CHAIN POLYMERIZATIONS
Free radicals can be stabilized by resonance and electron-withdrawing effects.
When their stabilization is sufficient—in particular, due to the existence of many
canonical forms—they can become persistent and be isolated, like the following

free radicals:
NO
NN
NO
2
NO
2
NO
2
Diphenylpicrylhydrazyl (DPPH)
Tetramethylpiperidyloxyl (TEMPO)
For TEMPO, the most suitable representation features a 3-electron N-O bond
which explains why this free radical cannot dimerize by its nitrogen or oxygen
atom. The free radicals of this family (known as “nitroxyl” radicals) are usually
employed to reversibly trap growing transient radicals and thus ensure a control of
the propagation step (see Section 8.5.8).
Free radicals have thus a marked tendency to participate in chain reactions, more
particularly in addition and abstraction reactions.
8.5.2. General Kinetic Scheme of Free Radical Polymerization
This kinetic scheme describes the initiation step by a molecule (initiator I) releasing
free radicals by homolytic rupture of a covalent bond (dissociation reaction).
Initiation:
Activation
I
k
d
−−−→ 2R
•
then
R

•
+M
k
i
−−−→ R–M
•
with
R
•
+R
•
k
c
−−−→ R–R as side reaction.
Because of their proximity when they appear in the reaction medium and the high
value of the rate constant of combination (k
c
), a non-negligible fraction of R
•
FREE RADICAL POLYMERIZATION 263
radicals generated by the initiator are lost in termination reactions and thus do not
initiate polymeric chains: the proportion really active is called efficiency factor or
efficiency (f ) of this initiator.
To establish the kinetic equations, it is considered—what was experimentally
established—that all reactions occurring in free radical polymerizations are first-
order with respect to each reactive species.
For the initiation step:
R
d
=−d[I]/dt = k

d
[I] =
1
2
d[R
•
]/dt
R
i
=+d[RM
•
]/dt = k
i
[R
•
][M]
Since the first “leg” of the initiation step (activation of initiator) is generally slow
due to a high value of the corresponding energy of activation (about 120 kJ·mol
−1
),
it determines the global kinetics of initiation; thus
R
i
=+d[RM
•
]/dt = 2fR
d
= 2fk
d
[I]

The coefficient 2 takes into account the fact that two R
•
radicals are simultaneously
formed by decomposition of one molecule of initiator (I).
Propagation:
RM
•
+M −→ RMM
•
RM
•
n
+M
k
p
−−−→ RM
•
n+1
In a first approximation—confirmed experimentally—one assume that the rate
constant of propagation (k
p
) is nearly independent of the degree of polymerization.
The rate of propagation (R
p
) is roughly equal to the total rate of polymerization
(R
pol
) since all monomer molecules except one per chain (that implied in initiation)
are consumed during this step:
R

p
= k
p


n
RM
•
n

[M]
Because of the low selectivity of free radical reactions, the chain growth can be
stopped at any moment by termination reactions; two polymeric radicals are neu-
tralized either by combination or by disproportionation in the process:
RM
n
•
RM
m
•
+
RM
n+m
k
c
k
disp
RM
n
+ RM

m
The two reactions can occur simultaneously and thus the rate constant of termina-
tion (k
t
) corresponds to a weight average of the individual rate constants (k
c
, k
dis
).
264 CHAIN POLYMERIZATIONS
The overall rate of termination (R
t
)isgivenby
R
t
= k
t


n
RM
•
n

2
Remarks
(a) In the above equation, k
t
is the rate constant of the two types of bimolec-
ular termination reactions.

(b) A factor of 2 is often found in the literature in the expression of the rate
of termination to take into account the fact that 2 polymeric radicals are
consumed by a same termination reaction. This reasoning is unjustified
and is equivalent to count twice the reactive species participating in the
termination process, which is appropriately described through the square
of the concentration in free radicals. It is recommended to be careful
when using k
t
values found in the literature.
Because of the respective values of rate constants of termination (k
t
∼10
7
to
10
8
L·mol
−1
·s
−1
) and propagation (k
p
∼10
2
to 10
4
L·mol
−1
·s
−1

at 60
◦
C) reactions,
it is recommended to work with particularly low instantaneous concentrations in
free radicals ([RM
•
n
] ∼10
−8
M), in order to favor propagation over termination
reactions. It is difficult to measure such low value of [RM
•
n
], except by using a
spectrometric technique as sensitive as electron spin resonance (ESR). Assuming
that the number of active chains remains constant—which is true only during
short intervals of time—, one can calculate the rate of polymerization even if
[RM
•
n
] is experimentally inaccessible and thus unknown. This assumption implies
that the rate of appearance of RM
•
n
is equal to their rate of disappearance, which
corresponds to steady-state conditions; one can accordingly write R
i
=R
t
,which

corresponds to
2fk
d
[I] = k
t


n
RM
•
n

2
This equation can be solved for [

n
RM
n
•
]:


n
RM
•
n

=

2f [I]k

d
/k
t

1/2
Introducing this expression into the equation of the rate of polymerization gives
R
p
= k
p

2f [I]
k
d
k
t

1/2
[M] ∼ R
pol
=−d[M]/dt
FREE RADICAL POLYMERIZATION 265
where R
pol
is the overall rate constant of polymerization.The above equation can
be rewritten as
−
d[M]
[M]
= k

p

2f [I]k
d
/k
1/2
t

dt
which corresponds to
ln
[M]
0
[M]
= k
p

2f [I]
k
d
k
t

1/2
t
assuming that [I] is constant and does not vary over the period t. The general
equation of polymerization can also be expressed under the form
R
pol
= constant [I]

1/2
[M]
or
R
pol
= constant [M]R
1/2
i
The last equation shows that the overall rate of polymerization is primarily deter-
mined by the rate of initiation. This is not surprising and can be accounted for by
examining the energies of activation of the various steps:
R
p
2
= 2(k
2
p
/k
t
)k
d
f [I][M]
2
from which one obtains
2E
ao
= 2E
ap
+E
ad

−E
at
or
E
ao
= E
ap
+E
ad
/2 −E
at
/2
where E
ao
, E
ap
, E
ad
,andE
at
represent the energies of activation of the overall
polymerization, the propagation, the dissociation (first part of initiation), and the
termination, respectively. The values of E
ap
and E
at
could be experimentally deter-
mined for a certain number of systems. For example, in the case of the free radical
polymerization of styrene one has:
(E

ap
−E
at
/2) ∼ 27 kJ·mol
−1
The energy of activation of dissociation for most of the initiators functioning by
homolytic rupture of a covalent bond ranges between 120 and 170 kJ·mol
−1
;since
the experimental determination of E
ao
gives a value in the range of ∼100–125 kJ·
mol
−1
, one can deduce that it is E
ad
which contributes the most to the value of
E
ao
. The generation of primary free radicals is thus the step that determines the
global kinetics of the whole process.
266 CHAIN POLYMERIZATIONS
Remark. Energies of activation of reactions that generate free radicals by
redox systems (∼50 kJ·mol
−1
) are much lower than those corresponding to
homolytic ruptures; polymerization kinetics are likely to be affected by such
a difference.
8.5.3. Initiation of Free Radical Polymerizations
8.5.3.1. Generation of Initial (‘‘Primary’’) Free Radicals. Most of the free

radical initiators (generators) used are unstable molecules that can homolytically
dissociate (I →2R
•
) under thermal effect, due to the presence of a weak covalent
bond.
The homolytic dissociation of a covalent bond is all the weaker since:
•
The electronegativity of the covalently bonded elements is high.
E
d
O – O
<E
d
N –N
<E
d
C – C
•
The stabilization (by electron-donor and/or resonance effects) of the radicals
resulting from the dissociation is high (see Table 8.1).
Table 8.1. Dissociation energy of C–H bonds and stabilization energy of the
corresponding hydrocarbon free radicals
Dissociation energy Stabilization Energy of Radical
Molecule R–H E
d
(C–H) (kJ·mol
−1
)R
•
(kJ·mol

−1
)
H–CH
3
426 0
H–CH
2
–CH
3
393 33
H–C(CH
3
)
3
376 50
H–CH
2
–CH
=
CH
2
324 102
H CH
2
322 104
It must be stressed that radicals generated by dissociation are all the more
reactive because their formation is difficult; thus, in Table 8.2, methyl radical is
the most reactive among the represented radicals.
Organic peroxides and hydroperoxides are very commonly used at the labora-
tory scale as well as at the industrial level. Their instability can be characterized by

their half-life time (t
1/2
)—that is, the time necessary to their half-decomposition at
a given temperature—or by the temperature at which they exhibit a given half-life
time (see Table 8.2); from these half-life times it is possible to easily find the cor-
responding value of k
d
using the kinetic equation of decomposition of the initiator
[I] = [I]
0
exp(−k
d
t)
corresponding to
ln[I]
0
/[I] = k
d
t
FREE RADICAL POLYMERIZATION 267
Table 8.2. Half-life times of organic peroxides
Temperature (
◦
C)
for a Half-Life of Half-Life Times (hours) for various temperatures
10 h 1h 1min 40
◦
50
◦
60

◦
70
◦
80
◦
90
◦
100
◦
110
◦
120
◦
130
◦
140
◦
150
◦
2,5-Di(tert-
butylperoxy)-2,5-
dimethylhexyne
128 149 191 — — — — — — — 90 27 8.1 1.7 0.9
Tert -butyl peroxide 126 147 186 — — — — — — — 75 22 6.0 1.4 0.6
Cumyl hydroperoxide 122 147 200 — — — — — — 115 42 13 4.8 2.0 0.8
Tert -butyl
hydroperoxide
121 140 179 — — — — — 165 42 12 3.2 1.0 0.3
2,5-Di(tert-
butylperoxy)-2,5-

dimethylhexane
118 137 172 — — — — — — 135 33 8.5 2.3 0.7 0.2
Dicumyl peroxide 117 134 170 — — — — — 100 25 6.6 1.7 0.5 0.1
Tert -butyl perbenzoate 109 125 163 — — — — — 135 30 7.8 2.2 0.6 0.2 —
2-2-Bis(tert-
butylperoxy)butane
103 125 168 — — — — 55 16 4.8 1.6 0.6 — —
Tert -butyl
diperphthalate
103 122 160 — — — — — 78 19 4.8 1.4 0.5 — —
Tert -butyl peracetate 101 120 160 — — — — 165 43 11 3.3 1.1 0.3 — —
2,5-Dibenzoylperoxy-
2,5-dimethylhexane
100 119 158 — — — — 135 37 10 2.8 0.9 0.3 — —
Tert -butyl permaleate 81 105 155 — — — 35 12 4.2 1.3 0.6 0.3 — — —
Tert -butyl
perisobutyrate
78 95 130 — — 120 28 6.7 1.8 0.5 0.1 — — — —
Bis(4-chlorobenzoyl)
peroxide
76 94 133 — — 67 18 5.5 1.5 0.5 0.2 — — — —
Tert -butyl per-2-
ethylhexanoate
76 92 124 — — 140 28 5.7 1.4 0.4 0.1 — — — —
Benzoyl peroxide 72 92 133 — — 45 13 3.7 1.2 0.4 0.1 — — — —
Succinyl peroxide 67 90 142 — — 19 7.0 2.5 1.0 0.4 — — — — —
Acetyl peroxide 68 86 122 — — 30 8.0 2.2 0.6 0.2 — — — — —
Propionyl peroxide 64 81 118 — 80 17 4.5 1.2 0.4 0.1 — — — — —
Lauroyl peroxide 62 80 116 — 70 15 3.7 1.0 0.3 — — — — — —
Decanoyl peroxide 62 79 115 — 67 13 3.5 1.0 0.3 — — — — — —

Octanoyl peroxide 62 78 114 — 63 13 3.3 0.9 0.3 — — — — — —
Bis-(3,5,5-
trimethylhexanoyl)
peroxide
60 77 112 — 47 9.9 2.6 0.7 0.2 — — — — — —
Tert -butyl perpivalate 55 73 110 — 19 5.0 1.5 0.4 — — — — — — —
Bis-(2,4-
dichlorobenzoyl)
peroxide
55 72 106 — 20 4.7 1.4 0.3 — — — — — — —
Tert -butyl
per-neodecanoate
48 66 99 40 8.0 1.7 0.4 — — — — — — — —
Isopropyl
peroxydicarbonate
47 62 95 30 6.0 1.2 0.3 — — — — — — — —
Cyclohexyl
peroxydicarbonate
45 60 93 26 4.2 0.9 0.2 — — — — — — — —
Acetylcyclohexane-
sulfonyl peroxide
32 46 67 5 0.4 0.05 — — — — — — — — —
268 CHAIN POLYMERIZATIONS
and for [I]
0
/[I] =2wehave
k
d
= 0.693/t
1/2

In general, free radicals initiators are used under conditions of half-life times of
about 10 hours. The decomposition of peroxides can be single-step or multistep;
for example, dicumyl peroxide (DICUP) decomposes as follows:
CH
3
CH
3
CH
3
CH
3
2
k
d
CO
COO
CH
3
CH
3
C
whereas the decomposition of benzoyl peroxide requires two successive steps with
a last fragmentation step:
C
O
OO
C
O
2CO
2

+
k
d
C
O
O
2 2
Thesameistruefortert-butyl peroxide:
(CH
3
)
3
C–O–O–C(CH
3
)
3
k
d
−−−→ 2(CH
3
)
3
C–O
•
−→ 2(H
3
C
•
+CH
3

–CO–CH
3
)
In the last two cases, the two reactions can occur successively only if the initially
generated radicals did not succeed in adding a monomer molecule to initiate the
polymerization.
When the reaction medium requires the initiator to be water soluble (polymeriza-
tions in emulsion, in aqueous solution, etc.), mineral peroxides such as potassium
persulfate are often utilized:
K
+
,
−
O
3
S–O–O–SO
3
−
, K
+
k
d
−−−→ 2SO
•
4
, K
+
Azo compounds also are very much used; and depending on whether they carry
hydrophilic groups or not, they can be water- or organosoluble; it is the case of
azobis(isobutyronitrile) (AIBN)

H
3
CC
CH
3
CN
N
CH
3
C
CH
3
CN
N
N
2
+ 2
H
3
CC
CH
3
CN
which is organosoluble, whereas its dicarboxylic homolog
C
CH
3
CN
N
(CH

2
)
2
HOOC
C
CH
3
CN
N
(CH
2
)
2
COOH
is water-soluble.
FREE RADICAL POLYMERIZATION 269
It is sometimes necessary to generate free radicals at low temperature, which
implies that reactions with low activation energy such as oxydo-reductions are
used. Depending upon the needs of the reaction medium, one can use either
hydrophilic, hydrophobic, or mixed systems of initiation; for example,
S
2
O
8
2−
+S
2
O
3
2−

−→ SO
4
2−
+SO
4
−
•
+S
2
O
3
−
•
Fe
2+
+H
2
O
2
−→ Fe
3+
+HO
−
+HO
•
R–OH+Ce
4+
−→ RO
•
+Ce

3+
+H
+
RO–OH +Fe
2+
−→ Fe
3+
+HO
−
+RO
•
or even
C
O
OO
C
O
NR
2
+
+
C
O
O
NR
2
+
C
O
O

,
+
C
O
O
,
NR
2
O
C
O
Photochemical initiation resulting from the activation of monomer molecules
by photons alone is difficult to achieve; generally, a molecule is added in the
reaction medium which will be used as intermediate between the photon and the
monomer molecule to activate. Free radicals can be generated by intramolecular
scission; an example is given below for benzoin ethers:
C
O
CH
OR
+
HC
OR
hn
C
O
Another possibility is the intra- or intermolecular abstraction of H
•
as with ben-
zophenone in association with an amine:

BP*
+
R
2
NCH
2
R′
exciplex
[BP NR
2
CH
2
R′]*
R
2
NCHR′
hn
C
O
BP*
C
OH
+
δ
−
δ
+