Chapter 4
Theory of Polymerization Reactions
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Classification of Polymerization
Reactions
• Condensation (Step)
• Addition (Chain)
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4.1 Free Radical Polymerization
• Initiation
• Propagation
• Termination
■
■
combination
disproportionation (double-bond formation)
• Chain (active center) transfer
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Kinetics of Free Radical Polymerization
• Concentration of free radicals does not
change with time (quasi-steady state
assumption)
• Rate limiting reaction is propagation reaction
Rp = −
dM
*
= k p MM
dt
An expression for M* is needed. The quasi-steady
state approximation is used.
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MULTICHAIN CONDENSATION
POLYMERIZATION
Consider the condensation polymerization of
A-B with a small amount of a multifunctional
monomer, R-Af. As the polymerization
reaction approaches completion linear species
will disappear in favor of multichain polymer
molecules.
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MULTICHAIN CONDENSATION
POLYMERIZATION
The polymer formed upon complete reaction
will consist of f chains attached to a central
unit R.
R [ -A (B-A) y ] f
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MULTICHAIN CONDENSATION
POLYMERIZATION
For example: ε-aminocaproic acid
[ H2N - (CH2)7 - COOH ]
plus a tetrabasic acid
[ R-(COOH)4 ]
would yield:
R-[CO - { HN - (CH2)7 - CO } y OH ]4
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MULTICHAIN CONDENSATION
POLYMERIZATION
The total number of units in the entire
molecule, x , depends upon the sum of the yvalues of the individual chains.
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MULTICHAIN CONDENSATION
POLYMERIZATION
A size , x, much larger or smaller than average
is only likely to occur if several (or all) of the
chains in the molecule are abnormally large
or small. Such cooperation among
statistically uncoordinated compounds will be
comparatively rare.
Hence the distribution will be narrower than
ordinary linear polycondensation.
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MULTICHAIN CONDENSATION
POLYMERIZATION
Let p = probability that an A group has
reacted. The probability that a particular chain
contains y units is py (1-p)
The probability that the f chains have lengths
y1, y2, y3 ...... yf respectively is:
py1 py2 py3 . . . . pyf (1-p)f
now:
y1+ y2+ y3 . . . . yf = x-1
(1)
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MULTICHAIN CONDENSATION
POLYMERIZATION
If the central unit, R, is counted as one of the
total x, the probability may be expressed as:
p(x-1) (1-p)f
Hence, the probability that the molecule
contains exactly x units distributed over f
chains in any manner whatsoever equals the
product of this expression and the total
number of combinations of y which fulfills
equation (1).
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MULTICHAIN CONDENSATION
POLYMERIZATION
This is simply the total number of combinations of (x1) + (f-1) taken (f-1) at a time: or
N! / M! (N-M)! ], so
N x,f = [ ( x + f - 2 )! / ( f - 1 )! ( x - 1 )! ] p(x-1) (1-p)f
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MULTICHAIN CONDENSATION
POLYMERIZATION
Similarly, the weight distribution can be shown to
be:
W x,f = [ x (x + f - 2)! / (f - 1)! (x - 1)! ]
p(x-1) [(1-p)f+1 / (fp+1-p)]
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MULTICHAIN CONDENSATION
POLYMERIZATION
By a number of approximations it can be shown
(Schaefgen and Flory):
<xw> / <xn> = 1 + 1 / f
This indicates that as f increases the
distribution sharpens.
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4.3.4 Polyfunctionality and Gelation
The presence of polyfunctional units nearly
always presents the possibility of forming
chemical structures of macroscopic
dimensions or infinite networks.
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Polyfunctionality and Gelation (con’t)
Examples include phenolic resins,
polyurethanes, gel formation in diene
polymers, vulcanization of elastomers etc.
In this section we will be concerned with
defining the critical conditions for the
formation of infinite networks and molecular
weight distributions for non-linear polymers
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Polyfunctionality and Gelation (con’t)
Polyfunctional Condensation Polymerization
Consider a reaction involving:
where reaction can only occur between A and
B. This would lead to structures like:
A
A
A
A
B
B
A
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Polyfunctionality and Gelation (con’t)
B
BA
AB
A
B
B
A
A
B
B
B A
AB
B A
A
AB
B A
A
A
B
B
A
A
B
B
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Polyfunctionality and Gelation (con’t)
Major Assumptions:
1. Principle of equal reactivity.
2. Intramolecular condensation not permitted.
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Polyfunctionality and Gelation (con’t)
Purpose:
To define the conditions under which an
infinitely large chemical structure or an
infinite network will occur.
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Polyfunctionality and Gelation (con’t)
Definition: α (a in your textbook) is defined as
the probability that a given functional group of
a branch unit leads, via a chain of
bifunctional units, to another branch unit, ie:
A
A
A +
A +
B
B
A
A
A
A
[B
BA
i = 0 to
A
∞
]
i
B
BA
A
A
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Polyfunctionality and Gelation (con’t)
Let: PA be the probability that an A group has
reacted.
PB be the probability that a B group has
reacted.
ρ be the ratio of A's (reacted and unreacted)
belonging to branch units to the total number
of A's.
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Polyfunctionality and Gelation (con’t)
The probability that an A group is connected to
a sequence of units as depicted above is
thus:
PA[PB(1- ρ)PA]i PB ρ
a is thus given by
for i=0 to
α = Σ [PAPB(1- ρ)]i PA PBρ
Evaluation of the summation yields the
important result:
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Polyfunctionality and Gelation (con’t)
α = PA PB ρ /[1-PA PB (1- ρ)]
If we substitute r=A/B, or PB = rPA it follows
that:
α = r PA2 ρ / [1-r PA2(1- ρ )]
α = PB2 ρ / [r -PB2(1- ρ)].
or:
The parameters ρ and r are fixed by the feed to
the reaction. The concentration of unreacted
end groups A or B, may be determined
analytically along the reaction coordinate.
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Polyfunctionality and Gelation (con’t)
Special Cases:
1. No A-A groups present.
ρ=1
α = r PA2 = PB2/ r
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Polyfunctionality and Gelation (con’t)
2. The concentrations of A and B are equal.
ρ = 1, PA= PB = P
α = P2 ρ / [1 - P2(1- ρ)]
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Polyfunctionality and Gelation (con’t)
It is important to note that the above treatment
is not completely rigorous. For example, the
polycondensation of R-Af and R'-Bg cannot
be handled.
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Polyfunctionality and Gelation (con’t)
In addition, differing reactivities such as the
secondary OH in glycerol which is less
reactive than the primary OH, are not taken
into account.
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Critical Value of α
• At some critical value of α, incipient formation
of an infinite network will occur depending
upon the functionality of the branching unit.
• Consider the following: For a trifunctional
branching unit, each chain which reacts with
this unit is succeeded by two more chains.
Similarly, if both of these react with the
branching unit, four chains are produced and
so forth. Schematically:
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Critical Value of α (con’t)
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Critical Value of α (con’t)
If α < 0.5, there is a less than even chance that
each chain will lead to a branch unit and thus
to two more chains. In fact, there is a greater
than even chance that it will end at an
unreacted functional group.
Eventually, termination of chains must outweigh
continuation of the network through
branching. When α < 0.5, all molecular
structures must be limited; ie. finite in size.
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Critical Value of α (con’t)
If α > 0.5, each chain has a better than even
chance of reproducing two more chains. An
infinite network is now possible.
Hence, α = 0.5 represents the critical condition
for the incipient formation of an infinitely large
network in a trifunctionally branched system.
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Critical Value of α (con’t)
General Statement:
If f = the functionality of the branching unit,
gelation will occur when a ( f-1) exceeds
unity. In another form:
α c = 1 / ( f-1)
Comment: It is important to recognize that an α
> 0.5 does not mean that all the material is
combined in an infinite structure. Both gel
(infinite network ) and sol (molecules of finite
size) exist in varying amounts
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Problem 4.8 (Grulke, 1994)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Monomer Composition
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4.4 Ring Opening Polymerization
• See specific examples in Grulke, Table 4.11
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4.5 Stereospecific and Coordination
Complex Polymerization
• Coordination catalysis essential for some
commodity polymers (e.g., polypropylene)
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4.6 Ionic Polymerization
• Can be used to produce monodisperse
polymers
• Control of reaction kinetics extremely
important
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