Kinetics of Radical Polymerization

8/7/2019 Kinetics of Radical Polymerization

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KINETICS OF RADICAL POLYMERIZATION

Foundations

Raison detre. Worldwide synthetic polymer production amounts to of or-der 200 million tons per annum, approximately half of which is via radical means

(1). For those who produce polymer, the quantities of greatest importance are rateof polymerization, molecular weight andin the case of copolymerscompositionof the polymer product. Such information is furnished by kinetics. It follows from

these simple facts that it is vital beyond words to have good descriptions of the

kinetics of radical polymerization (RP). The purpose of this contribution is to

present such descriptions.

Fundamental Reactions. Any description of chemical kinetics must befounded on knowledge of the reactions that occur. The fundamental reactions of

radical polymerization are as follows: First, there must be a reaction that gen-

erates free radicals. Most commonly, this is achieved through thermally induced

decomposition of a deliberately added chemical initiator, for example, (di)benzoylperoxide (BPO):

C

O

O 2 C

O

OC

O

O

This process is said to produce primary radicals. Ideally, these add to

monomer, for example, tert-butoxy (eg, from the initiator di-tert-butyl peroxide)

with ethylene:

CH3 C

CH3

O OCH2 CH2+ CH2 CH2

CH3

CH3 C

CH3

CH3

A polymer molecule is formed by the continued occurrence of the addition

reaction, a process known as propagation:

CH2 CH2+ CH2 CH2CH2 CH2 CH2 CH2

There are three reactions that stop the growth of macroradicals. The mostintuitive is termination by combination, which occurs, for example, in styrene

1

Encyclopedia of Polymer Science and Technology. Copyright c 2010 John Wiley & Sons, Inc. All rights reserved.


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2 KINETICS OF RADICAL POLYMERIZATION

polymerization:

+C C

H

H

H

Ph

CC

H

H

H

Ph

C C

H

H

H

Ph

CC

H

H

H

Ph

Disproportionation is also a termination reaction in that it results in annihi-

lation of radical activity. It takes place, for example, in polymerization of methyl

methacrylate:

+C C

H

H

CH3

CO

CC

H

H

CO

CH3

OCH3 OCH3

+C C

H

H

CH2

CO

CC

H

H

CO

CH3

OCH3 OCH3

H

Finally, there are transfer reactions, as effected, for example, by dodecyl mer-

captan:

+C C

H

H

CH3

CO

OCH3

+C C

H

H

CH3

CO

OCH3

H S C12H25 H S C12H25

The small-molecule radical generated here is like a primary radical in that

it adds to monomer and thus initiates the formation of another polymer molecule.

In these ideal (and commonly realized) circumstances, small-molecule transfer is

different to termination in that it does not result in any loss of radical activity,

even though it is similar in that it limits growth. It is important to grasp that

the dead macromolecules produced by the above three reactions are the actual

polymer product of radical polymerizationas with all chain-reaction processes,

the radicals are but transient intermediates that have a very low concentration.

It is also important to be aware that all the above reactions occur simultaneously

in radical polymerization, as opposed to occurring in different time domains. As

will be seen, the kinetics of RP is shaped by the balances reached between these

different reactions.

Only a very basic description of RP chemistry has been presented here;

for more comprehensive details, the reader is referred elsewhere in this

Encyclopedia.

Reaction Scheme. Once apprised of a systems chemistry, the next stepis to translate this into a reaction scheme. Such is as follows for the above


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KINETICS OF RADICAL POLYMERIZATION 3

reactions:

Initiation: IRi 2R1 (1)

Propagation: Ri + Mkp Ri + 1 (2)

Combination: Ri + Rj(1 )kt

Di + j (3a)

Disproportionation: Ri + Rjkt Di + Dj (4a)

Transfer: Ri + XktrX R1 + Di (5)

Here I denotes initiator, M monomer, Ri (living) radical of degree of polymer-

ization i, Di dead polymer of degree of polymerization i, X transfer agent, and ka rate coefficient, the subscript to which signifies the reaction involved. Specific

points about this scheme will now be made in turn.

Initiation may occur in a variety of ways, not just by thermally induced

decomposition. For example, radicals may be generated photolytically (so-called

photoiniation) or they may be generated from monomer itself (so-called self-

initiation or autoinitiation) (2). For this reason, it is more appropriate to use the

rate of initiation, Ri, in kinetic equations rather than a rate coefficient. The sec-

ond point about initiation is that whatever the means and whatever the initiator,

there are always avenues by which primary radicals are squandered. For exam-

ple, in the case of BPO there may be decarboxylation of the benzoyloxy radical,

and the resulting phenyl radical may combine with a primary radical to produce

a species that will not regenerate radicals. For this reason it is standard practice

to speak of initiator efficiency, which is the fraction, f, of primary radicals that

actually add to monomer and thus successfully initiate polymerization. So in the

most common case of employing a thermally decomposing initiator one thus hasthat

Ri = 2f kdcI (6)

where kd is the rate coefficient for decomposition of initiator of concentration cI.

About combination and disproportionation, it needs to be noted that the

standard practice, because it is more convenient (see below), is to define the

sum of their rate coefficients as kt, the overall rate coefficient for termina-

tion. The individual values then follow via , the fraction of termination by

disproportionation.

Transfer may occur in many ways, not just involving a deliberately added

transfer agent. For example, it may also occur to monomer, solvent, and even

initiator (2,3). Usually one of these reactions will be dominant, and one may just

use ftr = ktrXcX for the overall frequency of transfer. For simplicity, this approachwill be followed here. Where it is the case that several transfer reactions occur to


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a significant extent, one should just use

ftr =

all XktrXcX (7)

in all the equations of this work. The other thing to note about transfer is that it is

assumed here to involve quantitative reinitiation of polymerization, ie, formation

of a R1 species. Where this is not the case, it is not difficult to adapt the equations

of this work appropriately (4). The same holds for the unusual situation where

primary radicals from initiator are much more slowly propagating than the rad-

ical from primary radical addition to monomer, and so one needs to distinguish

between these two species (4), as opposed to calling them both R1, as here (see

eq. 1). In fact, it is one of the aims of this article to provide a framework that may

easily be extended to account for the occurrence of extra reactions and/or species.

This notwithstanding, it is held that the scheme of fundamental reactions given

above is sufficient for understanding RP kinetics to a large extent.

Population Balance Equations. Recognizing that reactions 15 are el-ementary in nature, one may immediately write down from this scheme the fol-

lowing equations:

dcR1

dt= Ri + ktrXcXcR kpcMcR1 ktrXcXcR1 2ktcRcR1 (8a)

dcRi

dt= kpcMcRi 1 kpcMcRi ktrXcXcRi 2ktcRcRi ,i = 2, (9a)

dcDi

dt= 2ktcRi cR + ktrXcXcRi + (1 )kt

i 1j = 1

cRj cRi j ,i = 1, (10a)

Although these equations look forbidding, in fact they are just assemblages of

gain and loss terms from basic chemical kinetics, where t is time. As will be seen,

equations 810 are the foundations from which most important results follow.

The only new quantity in these equations is the overall radical concentration,

which under normal circumstances (eg, no long-lived primary radicals) is given

by

cR =

i = 1

cRi (11)

The well-known population balance equation for cR may be obtained either

through common sense or through summing equations 8a and 9a over all chain

lengths:

dcR

dt= Ri 2ktcRcR (12a)

This equation makes apparent that equations 810 have been written so thatthe IUPAC recommendation for writing rates of reaction (5) is adhered to, which


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in the present case entails that the overall rate of termination has a factor of 2

because it involves reaction of two radicals.

For the overall rate of propagation, one has that

Rp =

i = 1

kpcMcRi = kpcMcR (13)

It is standard practice to equate this to dcM/dt, the overall rate of con-

sumption of monomer, otherwise known as the rate of polymerization. Implicit

in this practice is the assumption that negligible monomer is consumed by re-

actions that start and stop polymer growth, for example, initiation and transfer.

This is known as the long-chain approximation (LCA), because it must be highly

accurate providing polymer chains are long.

Chain-Length-Dependent Termination. There are many factors thatcan complicate RP kinetics. With little doubt, the two most importantand byquite some distanceare chain-length-dependent termination (CLDT) (6) and

the variation of rate coefficients with conversion of monomer into polymer. The

latter will be discussed in due course. Although tremendously important, it is

only sparingly accompanied by new equations. On the other hand, the effect of

CLDT is captured by closed equations that are easier to use than suspected. For

this reason, CLDT is introduced now and the resulting equations will, as far as

possible, be presented in parallel with those from the so-called classical kinetics,

ie, kinetics in the absence of CLDT.

That termination must in general be chain length dependent in rate is so

simply and inescapably grasped that it is remarkable that this notion has onlybecome widely accepted in recent times (6). The argument runs as follows: Ter-

mination is diffusion controlled in rate (7); polymer diffusion depends on polymer

size, with big chains moving less rapidly than small chains (8); therefore, large

polymer molecules must undergo termination slower than do small radicals. For

this reason, equations 3a and 4a may be superseded by

Combination: Ri + Rj(1 )k

i,jt

Di + j (3b)

Disproportionation: Ri + Rjki,jt

Di + Dj (4b)

Because termination is a bimolecular reaction, the rate coefficient actually

depends on two chain lengths, viz those of both terminating chainshence the

notation kti,j.

The recognition of CLDT necessitates updating of the foundational equa-

tions of the preceding subsection (9):

dcR1

dt = Ri + ktrXcXcR kpcMcR1 ktrXcXcR1 2cR1

j = 1

k1,jt cRj (8b)


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dcRi

dt= kpcMcRi 1 kpcMcRi ktrXcXcRi 2cRi

j = 1

ki,jt cRj ,i = 2, (9b)

dcDi

dt = 2cRi

j = 1

k

i,j

t cRj + ktrXcXcRi + (1 )

i 1j = 1

k

j ,i j

t cRj cRi j ,i = 1, (10b)

dcR

dt=

i = 1

dcRi

dt= Ri 2

i = 1

j = 1

ki,jt cRi cRj = Ri 2ktcRcR (12b)

It is clear from equation 12b that the overall, or chain-length-averaged, ter-

mination rate coefficient introduced therein must be defined as follows (10):

kt =

i = 1

j = 1

ki,jt

cRi

cR

cRj

cR

(14)

The obvious issue raised by CLDT is that of what values to use for kti,j. Most

workers are happy to agree on a power-law dependence for homotermination rate

coefficients:

ki,it = k1,1t i

(15)

The physical basis for this is that polymer diffusion coefficients are typically

observed to have a power-law dependence on size (8). By now, there is a wealth of

empirical evidence backing equation 15 (6). More difficult to nail down has beenthe functional form of heterotermination rate coefficients, that is, kti,j where i =

j. A major reason for this is that it turns out not to matter: Simulations have

shown that in general one has identical trends in kt, independent of the form of

kti,j (11,12). Therefore the so-called geometric-mean model is often employed:

ki,jt = k

1,1t (ij )

/2 (16)

Although there is most likely no physical justification for using this model

(6), it has the pragmatic advantage of delivering closed expressions for kinetic

quantities, something that all other models do not, in which event one is left with

no option other than to work with equations 8b10b directly. Although this is

routinely feasible with modern computing capacity (13,14), obviously it is prefer-

able to have one-line expressions where possible. For this reason, results from

equation 16 will be presented in this work and recommended for use. Note that

for i = j equation 16 reduces to equation 15, and that the two equations together

represent a two-parameter description of CLDT: kt1,1, the rate coefficient for ter-

mination between monomeric radicals, stipulates the magnitude of kti,j values,

whereas , the exponent for variation of kti,i with i, quantifies the strength of the

CLDT, with = 0 returning chain-length-independent termination (CLIT). This

is the case throughout this work: all CLDT equations reduce, as they must, to the

corresponding CLIT equations in the limit of = 0. For this reason, such ana-logues are presented as equations b and a, respectively, with the same number.


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While the presented CLDT equations turn out to be very useful for describ-

ing much data, it should be mentioned at the outset that there is a caveat. It

is that, as will be explained in the section Variation of concentrations with time

(3.1.3.1), equation 15 turns out not to hold over the entire range of chain lengths.

Rather, is different in value for short and long chains (6,15). This has the con-sequence that where equation 15 is assumed in describing long-chain data, the

kt1,1 that is returned is not the true value, but rather is an underestimate that

pertains to the hypothetical situation of monomeric radicals behaving like long

chains of degree of polymerization unity (6,15).

Steady-State Polymerization

Steady-state polymerization (SSP) is when dcR/dt 0, that is, Ri 2ktcR2 (see

eq. 12b), meaning that

cR =

Ri

2kt

0.5(17)

In practice, this equation is accurate as long as there are no abrupt changes

in either the rate of initiation or the rate of termination.

Rate of Polymerization. Inserting equation 17 into equation 13 leads to

dcM

dt

= kpcMRi

2kt

0.5

(18a)

Using the more common index of fractional conversion of monomer into poly-

mer, x, this equation becomes

d ln(1 x)

dt= kp

Ri

2kt

0.5(18b)

It is stressed that these equations hold both for CLIT and CLDT.

Equation 18b recommends that data be plotted as ln(1 x) versus t. This

implicitly accounts for the effect of declining monomer consumption on the reac-

tion rate, whereas the latter is a source of nonlinearity in plotting cM or x versus

t (see eq. 18a). An example of this procedure is given in Figure 1 (16). Where a

plot of ln(1 x) versus t is linear, as in Figure 1, a straight-line fit delivers the

coupled parameter kp[Ri/(2kt)]0.5 as the slope. Where such a plot is nonlinear,

then kp[Ri/(2kt)]0.5 is varying with time, and its values are given by tangents to

a nonlinear fit of the data.

Chain-Length-Independent Termination. Equation 18 is the hallmark re-sult of classical kinetics and is so well known that its inclusion is, and always

has been, de rigueur in textbooks on polymer chemistry. For a thermally decom-

posing initiator (eq. 6), it results in the rate law Rp cM1.0cI

0.50.5 for the case of

CLIT, where is viscosity and kt 1/ (17). These orders are observed often andapproximately enough that this rate law can be paraded as a celebrated result in


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0.00

0.05

0.10

0.15

0 400 800 1200 1600

1.0% AIBN

0.3% AIBN

0.1% AIBN

ln(1-x)

t, s

Fig. 1. Results from three bulk polymerizations of methyl methacrylate at 70C, where

x is fractional conversion, t is time, and the wt % of 2,2 -azoisobutyronitrile (AIBN) is asindicated (16). Points: experimental results; lines: best fits to each set of results.

polymer science. At the same time, there are sufficient exceptions that one has to

conclude that this rate law cannot be the full truth.

Chain-Length-Dependent Termination. The steady-state solution of equa-tion 9b is

cRi =kpcMcRi 1

kp

cM

+ ktrX

cX

+ 2

j = 1

ki,j

tc

Rj

,i = 2, (19)

The difficulty here is the sum of termination frequencies. Only in the case of

the geometric-mean model, equation 16, is there a way around this dependenceofcRi on other cRi , thereby giving an alternative to iterative solution (13) of these

equations. For the case of negligible transfer and long chains, the result is (15,18)

cRi =Ri

kpcMexp

(2Rik1,1t )

0.5

kpcM(2 )/2i(2 )/2

,i = 1, (20)

Introducing this into equation 11 and then approximating the sum by an

integral, one obtains

cR =Ri

kpcM

2

2

(2Rik

1,1t )

0.5

kpcM

2

2

2/(2 ) 22

(21)

where denotes the gamma function. This may now be inserted into

equation 17, giving (15)

kt = k1,1t

22

2(2Rik

1,1

t )0.5

kpcM

2

2

2/(2 )(22)


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106

105

104

104 103 102 101 100

60 C, slope = 0.44140 C, slope = 0.445

dln(1x)/dt,

s

1

cAIBME, mol L1

Fig. 2. Data for the rate, expressed as d ln(1 x)/dt, of low-conversion, bulk polymer-ization of methyl methyacrylate as a function of concentration of 2,2-azoisobutyromethylester, cAIBME, at two different temperatures, as indicated. Points: experimental values (20);lines: linear best fits, with slopes as displayed.

From this, one can obtain an expression for rate via equation 18. The key point is

that (19)

kt (Ri)a(cM)

2a(k1,1

t )1 + a

, where a =

2 (23a)

This results in (see eq. 18a)

Rp (Ri)0.5 b(cM)

1 + 2b(k1,1t ) 0.5 b

, where b =a

2=

2(2 )(23b)

This is the rate law for SSP in the caseit is stressedof negligible chain trans-

fer. The parameter b quantifies the deviation from the classical reactant orders

(2b in the case of cM). Ifkt1,1 1/, then the order of Rp with respect to viscosity

is 0.5 + b.

As an example of using these expressions, some data (20) for dln(1 x)/dtversus cI is presented in Figure 2 as a loglog plot. The slope is less than the

classical (CLIT) value of 0.5, as predicted by equation 23b. This exemplifies that

CLDT is a reality and that, with careful experimentation, its effects can clearly

be seen. Taking b 0.50.44 from Figure 2, one obtains that = 2a/(1 + a) 0.2.

This is exactly as observed in other experiments and is in complete accord with

prediction for long chains (6). Note that only the slopes from Figure 2 have been

used to obtain this value of. From the absolute position of the points (or the in-

tercept of a linear fit), one may obtain, if all else is known, the value of kt1,1. This

exercise, which is illustrated elsewhere (9), requires only the equations above but

is algebraically more complicated. As with , it has been found to yield values of

kt1,1 that are in close accord with those from other experiments involving long

chains. The reader is reminded that such kt1,1 values are apparent rather than

true values of this quantity (see the sections Chain-Length-Dependent Termina-tion [1.5] and Variation of Concentrations with Time [3.1.3.1]).


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Number-Average Degree of Polymerization. Number-average degreeof polymerization, DPn, is the most commonly quoted index of polymer size. Since

polymer size is routinely correlated with material properties, one may argue that

DPn is the most important piece of information about a polymer. For RP the ap-

proach of Mayo (21) has stood the test of time as a description of DPn. It stemsfrom

DPn =rate of incorporation of monomer into polymer

rate of formation of dead polymer chains(24a)

SSP is implicit in equation 24. From the earlier laid foundations it follows

that

DPn =

kpcMcR

ktrXcXcR + 2ktcRcR + (1 )ktcRcR (24b)

Note that the LCA has been made here and that there is no factor of 2

with the combination rate, because it produces only 1 dead chain. This and the

following equations hold also for CLDT, as indicated. It is usual to invert this

equation and substitute equation 17 for cR. One obtains (3,22)

1

DPn=

ktrXcX

kpcM+

(1 + )(0.5Rikt)0.5

kpcM(25a)

This equation is often called the Mayo equation, and it has many guises,

because of the potential for writing the terms arising from both transfer (see

eq. 7) and termination in different ways. The Mayo equation may be used to pre-

dict DPn values. Alternatively, where DPn is measured, equation 25 may be used

to analyze data to obtain the values of rate parameters, as now explained.

For systems dominated by transfer, equation 25a simplifies to

Transfer limit:1

DPn=

ktrXcX

kpcM(26)

This suggests plotting 1/DPn versus cX/cM and obtaining ktrX/kp = CtrX as the

slope, where values ofCtr are known as transfer constants: the ratio of rate coeffi-

cients for transfer and propagation. Such plots are often termed Mayo plots, in

recognition of their first proposal and use (21). An example is given in Figure 3

(22).

Chain-Length-Independent Termination. In the event of CLIT (chain-length-independent termination), there is no variation of kt with cI, cM, and

cX. This means that even where the termination contribution to DPn is signifi-

cant, there will still be a linear variation of 1/DPn versus cX/cM, except that now

plots like those of Figure 3 will have a nonzero intercept, as for example in Mayosoriginal data (21). The constancy of kt means that variation of 1/DPn with cI

0.5


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0.000

0.004

0.008

0.012

0.000 0.004 0.008 0.012

60 C, slope = 1.4540 C, slope = 0.70

1/DPn

cDDM/cMMA

Fig. 3. Reciprocal number-average degree of polymerization, 1/DPn, from low-conversion,bulk polymerization of methyl methacrylate, MMA, as a function of concentration of do-

decyl mercaptan, cDDM, at two different temperatures, as indicated. Points: experimentalvalues (22); lines: linear best fits, with slopes as displayed.

will also be linear. For example, equation 25a may be written as

1

DPn=

ktrXcX

kpcM+

(1 + )ktRp

(kpcM)2

(25b)

This suggests plotting 1/DPn versus Rp (3). Perhaps confusingly, such plots

are also usually referred to as Mayo plots, and they are the most commonly used

method for determination of CtrM = ktrM/kp, the constant for chain transfer tomonomer. One measures DPn for varying rates of initiation in the absence of any

transfer agent, plots the data as 1/DPn versus either Rp or cI0.5, fits a straight

line, and, because cX = cM for the case of X being monomer, one may take CtrMas the intercept (23). This procedure is illustrated via sample calculations in

Figure 4. It is also evident that one can determine (1 + )kt from the slope of

such a linear fit (23), although for no good reason this practice seems largely to

have fallen into abeyance.

Chain-Length-Dependent Termination. For systems in which negligibletransfer occurs, equation 25 becomes

Termination limit: 1DPn

= (1 + )Ri

2kpcMcR(27a)

This equation is general. Introducing equation 21 for cR, one obtains (6,24)

DPn =

2

2

(2Rik

1,1t )

0.5

kpcM

2

2

2/(2 ) 22

2

1 +

(27b)

This gives DPn for CLDT (chain-length-dependent termination) in the ab-

sence of transfer.

First of all, it is instructive to evaluate DPn as a function of cI and plotthe results in Mayo form. This has been done in Figure 5. It is immediately


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0 10 0

1 104

2 104

3 104

4 10

4

5 104

6 104

0.0000 0.0002 0.0004 0.0006

CtrM = 1.0 104 , = 1

CtrM = 1.0 104 , = 0

CtrM = 0.5 104 , = 0

1/DPn

Rp, mol L-1 s-1

Fig. 4. Reciprocal number-average degree of polymerization, 1/DPn, as a function of rate

of polymerization, Rp, calculated using equation 25b with kt = 1 107 L mol 1 s 1, kpcM =5000 s 1 and varying Ri up to 2 10

7 mol L 1 s 1. Values of the constant for chaintransfer to monomer, CtrM = ktrM/kp, and the fraction of termination by disproportionation,, are as indicated for each set of values. The calculations illustrate how the interceptvaries with CtrM and the slope with (1 + )kt, thus enabling determination of these values.

0 100

1 104

2 104

3 104

4 104

5 104

6 104

7 104

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

= 0.5

= 0.2

= 0

1/DPn

cI0.5, mol0.5 L-0.5

Fig. 5. Reciprocal number-average degree of polymerization, 1/DPn, as a function ofsquare root of initiator concentration, cI

0.5, calculated using equation 27b with kpcM =

5000 s 1, = 1, fkd = 1 10 6 s 1 and varying cI up to 0.1 mol L

1. Points: calculationswith = 0 (and kt

1,1 = 1 107 L mol 1 s 1), = 0.2 (kt1,1 = 6 107 L mol 1 s 1) and =

0.5 (kt1,1 = 8 108 L mol 1 s 1) as indicated; curves: quadratic fits. Values ofkt

1,1 for each were chosen so that the fits have approximately the same slope as cI approaches zero.

evident that the effect of CLDT is to impart upward curvature. This is inter-

esting, because the observation of such a shape has traditionally been ascribed to

the occurrence of chain transfer to initiator (3). This is because cX = cI results in

a term in (cI0.5)2 in equation 25. However, it is clear that CLDT also results in a

Mayo plot of quadratic form. In some cases at least, this may be a more plausibleexplanation for such experimental results than is chain transfer to initiator.


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Also of interest is that equations 22 and 27b may be combined to yield (15,

18,24)

kt = k

1,1

t G(DPn)

, where G =

2

2 2 2

2

2

1 +

(28)

Because G is close to 1 in value, equation 28 gives that kt kt1,1(DPn)

,

which is identical in form to equation 15. In other words, the variation of ktwith DPn almost exactly mirrors the underlying variation ofkt

i,i with i. Not only

is this fascinating in itself, but it also gives rise to a commonly used and easily

exercised method for deriving CLDT rate coefficients: measure kt and DPn for

varying conditions (eg, different cI), do a loglog plot of the results, and take

as the slope and kt1,1 as the intercept (6). An example of this is presented later

in the section Determination of Termination Rate Coefficients by Multiple-Pulse

PLP. Molecular Weight Distribution. One of the major paradigm shifts in go-ing from micromolecular (or conventional) chemistry to macromolecular chem-

istry is that purity in size is no longer possible: even where all reactions are

perfectly under control, it is unavoidable to have a distribution of molecular

sizes. It is therefore of importance to have accurate descriptions of the molecular

weight distribution (MWD)or, equivalently, chain length distribution (CLD)

resulting from RP. This is especially so given that this distribution is now rou-

tinely measured by size exclusion chromatography (SEC) wherever polymers are

synthesized.

Chain-Length-Independent Termination. Equation 10a defines the in-

stantaneous number-chain length distribution, that is, the number of chains ofdegree of polymerization i being produced at any instant. While this equation

may be used as it stands, it is both convenient and instructive to simplify it.To do this, it clearly is necessary to possess an expression for cRi . This may be

obtained from equation 9a, a process in which it is convenient to introduce the

probability of propagation (of a radical), S:

dcRi

dt= 0cRi =

kpcMcRi 1

kpcM + ktrXcX + 2ktcRScRi 1 = S

i 1cR1 (29a)

Using equation 8a with Ri = 2ktcR2

, this becomes

cRi = Si 1(1 S)cR (29b)

This expression for the radical CLD can be useful in its own right. Here it is

introduced into equation 10a, giving

dcDi

dt= (ktrXcX + 2ktcR)(1 S)cRS

i 1 + (1 )kt(i 1)(1 S)2c2RS

i 2 (30)

This may be simplified to the two-variable expression

ni = Fn(1 S)Si 1 + (1 Fn)(i 1)(1 S)

2Si 2 (31a)


14/75


where ni is the number fraction of dead chains of length iie, the distribution of

equation 31a is normalized (meaning the sum from i = 1 to equals 1)and

Fn =ktrXcX + 2ktcR

ktrXcX + 2ktcR + (1 )ktcR(31b)

is the number fraction of chains formed by transfer and disproportionation. Equa-

tion 31a may also be derived via probability arguments. The advantages of the

derivation here (19) are that it shows how this equation is rooted in kinetics and

it accounts in a clear way for the balance between the disproportionation/transfer

and combination terms.

While equation 31a may nowadays be easily implemented via a spreadsheet

program, the latter was not always the case. Furthermore, it is not mathemati-

cally enlightening. It is therefore common practice to use the long-chain limit of

this equation. This is obtained by defining

=kpcM

ktrXcX + 2ktcR(32)

This is the average number of propagation events that a radical undergoesbefore it forms a dead chain, a quantity that is equal to the so-called kinetic chain

length if there is negligible transfer (25). Really is just an alternative parameterto S, as is evident from the following:

S =1

1 + 1 1 1 for large (33)

Thus in the limit (equivalent to S 1) one has that S = exp( 1),

which results in equation 31a becoming

ni = Fn1

exp

i

+ (1 Fn)i

1

1

exp(

i

) (34a)

Again, this equation is normalized (where now this operation involves inte-

gration from i = 0 to ). To most people, equation 34a is the more familiar form of

the so-called FlorySchulz distribution for the number-CLD from (radical) SSP.

Evaluations of both equation 31a (exact FlorySchulz) and equation 34a

(FlorySchulz with LCA) are presented in Figure 6. It is evident that even for as small as 24, the LCA holds excellently. For long polymer, this certainly justifies

using equation 34a as a description of MWD, even if, technically, equation 31a is

the exact expressionit is necessary only where the average chain length is very

small, for example, 10.

The other interesting aspect of Figure 6 is that it shows log ni versus i to

be linear. This is the reason behind a recommendation to plot MWD data as ln

ni versus i and then employ the slope of such a plot as a way of using the entire

MWD to analyze for underlying rate coefficients (26). However on closer scrutiny,

this is not such a leap forward (22). For one thing, equation 34a makes clearthat ln ni versus i is only linear when there is limited combination (ie, Fn is


15/75


106

105

104

103

102

101

100

0 20 40 60 80 100

S= 0.96, exactS= 0.96, LCAS= 0.90, exactS= 0.90, LCA

ni

i

Fig. 6. Number fraction, ni, of chains of length i for probability of propagation S = 0.96( = 24) and S = 0.90 ( = 9) calculated using equation 31a (exact) and equation 34a (LCA)with Fn = 1.

close to 1), a limitation that the Mayo approach does not have. Even where ln

ni versus i is linear, the slope is equal in value to 1/ (see eq. 34a), which is

just 1/DPn (compare eqs. 25 and 32). In other words, the so-called log CLD

approach is essentially equivalent to the Mayo approach. Finally, SEC data often

becomes very noisy when converted into number-CLD form, as implicit in what

now follows.

SEC does not deliver MWDs as ni. Rather, the so-called SEC MWD is w(log

i), where w is (relative) weight of polymer. Using equation 34a, the weight ofpolymer of degree of polymerization i is

ini

DPn= wi = Fw

1

i

exp

i

+ 0.5

1 Fw

1

i

2exp

i

(35a)

where

Fw =Fn

2 Fn=

ktrXcX + 2ktcR

ktrXcX + 2ktcR + 2(1 )ktcR(35b)

is the weight fraction of chains formed by transfer and disproportionation (cf. eq.31b). Note that Fw = in the event of negligible transfer. Also, equation 35a is

normalized, that is, it gives weight fraction. This is because DPn is the normal-

ization factor for the distribution ini. Equation 35a is converted into normalized

w(log10 i) as follows (26):

iwi =w(log10i)

ln10= Fw

i

2exp

i

+ 0.5(1 Fw)

i

3exp

i

(36)

Note that the form of any CLD, not just equation 34a, should be converted

as outlined above. For example, if one prefers to use the probabilistic form of theFlorySchulz distribution, equation 31a, then it is clear how one does so.


16/75


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

101 102 103 104

= 245, Fw = 1= 245, Fw = 0.25= 500, Fw = 0.25Experiment

w(log10i)

i

Fig. 7. Chain-length distributions presented as w(log10 i), where w is weight fraction

and i is chain length. Points: results from a low-conversion suspension polymerization ofstyrene at 80C (27). Curves: FlorySchulz distributions calculated using equation 36 withindicated parameter valuesnote how determines position and Fw shape (see text).

All of the equations presented here find application. However equation 36 is

the most useful, because it describes data as delivered by SEC. This is exempli-

fied in Figure 7. First of all Figure 7 illustrates the nature of the two parameters

in the FlorySchulz distribution: Fw (or equivalently Fn) determines the shape(width and height) of w(log i), while (or equivalently S) then stipulates the po-

sition. Thus it is facile to fit equation 36 to SEC data: one varies Fw until the

shape of the MWD is optimally described, and then one varies until the posi-tion is best overlayed. This makes clear that only a unique pair of -Fw values

will fit an MWD. In the case of the experimental data (27) of Figure 7, this pro-

cess yields an essentially flawless fit. It is surprising that use of equation 36 to

model MWD data is not more widespread, and that instead empirical functions

(eg, Gaussians) are often used, even though they are without theoretical basis

and are no easier to employ. It is important to grasp that the many rate coeffi-

cients and concentrations that play a role in determining MWD all reduce to just

two fitting parameters. What one can extract from these parameters depends on

what else is known. For example, if all else is known, then from one can obtain

the value ofkt. But ifktrX is not known, then this is not possible (see eq. 32).

Chain-Length-Dependent Termination. Now equation 10b must be usedfor the (instantaneous) number-CLD. Substituting in values of cRi as given by

equation 20, one can obtain the (normalized) result (24)

ni = FnCi 0.5exp( C

ip) + (1 Fn)A2pi1 exp( Aip) (34b)

Here C = (2Rikt1,1)0.5/(kpcM) = 1/ (where is now for CLDT, cf. equation 32

for CLIT), p = 1 (/2), C = C/p, A = 4C /(4 ) and Fn is as already defined.

The LCA is made in deriving this equation, and negligible transfer is assumed.

While mathematical acrobatics are required to derive the combination term of

equation 34b (24), it is quite easy to see the origin of the disproportionation term,because the termination frequency in equation 10b has form given by (kt

i,i)0.5


17/75


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

101 102 103 104

Experiment

= 0.19

= 0.50

w(log10i)

i

Fig. 8. Chain-length distributions presented as w(log10 i), where w is weight fraction

and i is chain length. Points: results from a low-conversion suspension polymerization ofstyrene at 80C (27). Curves: equation 34a with Fw = 0 (ie, 100% combination) and DPn =425 for = 0.19 ( = 135) and = 0.5 ( = 65).

(15), that is, i 0.5, and this must simply be multiplied by cRi , which has form

exp(Cip).

Equation 34b has three parameters: Fn as before, C equivalent to , and the

new parameter . It is converted into wi and then on into w(log i) exactly as out-

lined above. Some evaluations are presented in Figure 8, which illustrates two

things: (1) that the CLDT parameter acts to broaden MWDs (a point that will

be developed in the following section); and (2) that equation 34b provides a beau-tiful description of experimental data. Of course equation 34a also does this. The

difference is that the parameter values from fitting equation 34b are more mech-

anistically plausible. In the present example, Fw = 0.25 from fitting equation 34a

(see Fig. 7) is unreasonable for styrene polymerization, in which termination by

combination occurs almost exclusively, that is, Fw 0 (2). However, using this

value with equation 34b returns = 0.19 (28) (see Fig. 8), which is spot on for

CLDT of long-chain styrene (6).

Polydispersity Index (or Dispersity). From a CLD, all average molecu-lar weights can be calculated. For example, DPn is the first moment of normalized

ni. In the case of equation 34a, the result is

DPn = Fn + (1Fn)2 = v(2Fn) = v

2

2Fw

1 + Fw

(24c)

As indicated by the numbering, this is just a compact, two-parameter form of the

Mayo equation, which indeed it must be identical to. In the case of equation 31a,

the FlorySchulz distribution without the LCA, the result is

DPn = Fn 11 S

+ (1 Fn) 21 S

(37)


18/75


Since (1 S) 1 = 1 + (see eq. 33), it is evident that equation 37 is just

equation 24c with degree of polymerization 1 added for each chain-starting event

(recall that the LCA is just that such events can be ignored). Furthermore, both

equations make clear the well-known result that combination gives chains twice

as long as disproportionation and transfer.Weight-average degree of polymerization, DPw, is the first moment of nor-

malized wi, or the second moment of ni divided by its first moment (29). DPwis necessary to obtain expressions for the more commonly quoted polydispersity

index, PDI = DPw/DPn, which of course is the index of MWD broadness that is

quoted by most polymer scientists. IUPAC has recently recommended that this

quantity should be known as dispersity (30); however, for historical reasons this

work will remain with PDI. Results for the distributions recommended here are

FlorySchulz exact: PDI =1

2(1 + Fw)(2 + SFw) (38)

FlorySchulz with LCA:PDI =1

2(1 + Fw)(3 Fw) (39a)

CLDT(24):PDI =1 + Fw

2

( 6

2 )

[( 4 2

)]2+ 1 Fw

(39b)

Note that the Fw of equation 39b is , ie, the value in the case of no transfer

(see eq. 35b). Equation 39a is recovered from equation 38 in the limit of long

chains (S 1) and from equation 39b in the limit of CLIT ( = 0). Where the

LCA is valid (ie, eq. 39 holds), MWD broadness does not depend on , as discussed

earlier (see Fig. 7). Even in the case of equation 38, the effect of on PDI is trulyinsignificant except for very small , ie, S not close to 1.

Equation 39b is recommended for calculation of PDI and thence DPw. Eval-

uations of this equation are presented in Figure 9 (28) for the following pedagogic

purposes: (1) The classical limits of PDI = 1.5 and 2 for combination and dis-proportionation, respectively, are evident; (2) It is shown that PDI is a nonlinear

function ofFw (see eq. 39a), eg, 50% disproportionation takes PDImost of the way

to the value for 100% disproportionation; and (3) PDI increases with , ie, CLDT

broadens MWDs, as already seen in Figure 8.

While PDI is routinely quoted, it is actually a poor quantity to model. This

may be seen from Figures 8 and 9. The fit to the experimental data of Figure 8

has PDI = 1.72 (using equation 39a), whereas the fit of Figure 9 has PDI = 1.61

(using eq. 39b). And yet both fits could hardly be any better! If 1.72 is taken as

the PDI of the experimental data, then equation 39b says that = 0.32 is needed

to describe the data (assuming 100% combination). However, clearly this is far

too high to reproduce the entire MWD accurately.

In a similar vein, and for much the same reason, it is difficult to extract

accurate DPn from SEC data: where ni is highest, w(log i) is very small. Thus

DPn is also a poor quantity for kinetic analysis. An alternative approach is to use

values ofDPw/PDIin place ofDPn (22). This is because DPw can be more precisely

obtained from w(log i). But this approach requires advance knowledge of PDI. For

example, PDI= 2 in the case of transfer-dominated systems. However, in generalthere is not such certainty.


19/75


1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.2 0.4 0.6 0.8

= 1

= 0.5

= 0.2

= 0

PDI

Fig. 9. Polydispersity index, PDI, as a function of CLDT exponent for = 1 (100%disporportionation), 0.5, 0.2, and 0 (100% combination), as calculated using Equation 39b.

Considering all these factors, one must conclude that the most rigorous ap-

proach to kinetic analysis is to model (SEC) MWDs in entirety. This is recom-

mended.

Conversion. All the equations presented up to this point hold strictlyonly for polymer produced at an instant. However, the values of kinetically in-

fluential quantities change from instant to instant as a polymerization proceeds

and monomer is converted into polymer. At the very least, cM necessarily changes.

If this is all, then it is trivial to integrate equation 18b and obtain:

x = 1 exp( K

t), where K

= Kc0.5I and K = kp

f kd

kt

0.5(40)

It is also unavoidable that cI changes with time, decreasing according to the

standard first-order rate law cI = cI,0 exp(kdt). Where there is a significant induc-

tion time, tind, before polymerization commences, then cI,0 exp(kdtind) should be

employed as the value of cI,0. Using this expression for cI in equation 18b results

in

x = 1 exp

2K(cI,0)0.5

kd

1 exp

kdt

2

(41)

For kdt 1, this reduces to equation 40. Some evaluations of equation 41 are

presented in Figure 10. These make clear the result from equation 41 that there is

a limiting conversion of 1 exp[2K(cI,0)0.5/kd]. This is the phenomenon of dead-

end polymerization: where, because either cI is too low (meaning there simply are

not enough initiator molecules for all monomer to be converted into polymer) or kdis too high (meaning that the formed chains are too shortsee above expressions

for DPnfor all monomer to be incorporated), a system falls appreciably short

of 100% conversion. These effects are illustrated in Figure 10. At the same time,Figure 10 also makes clear that for standard parameter values, eg, those of the


20/75


0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80

kd= 1106s-1, cI,0 = 1x10

-3 M

kd= 1105s-1, cI,0 = 1x10

-3 M

kd= 1106s-1, cI,0 = 1x10

-4 M

x

t, days

Fig. 10. Fractional conversion of monomer into polymer, x, as a function of time, t, as

evaluated using equation 41 with kp(f/kt)0.5 = 1 10 1 L0.5 mol 0.5 s 0.5 and kd and cI,0as displayed.

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800

0.5 wt.% AIBME

0.2 wt.% AIBME

x

t, min

Glass effectGel

effect

Fig. 11. Fractional conversion, x, as a function of time, t, for bulk polymerization ofmethyl methacrylate initiated by 2,2-azoisobutyromethyl ester (AIBME) at 50C. Points:experimental values (20,31); curves: equation 40 with K = kp(fkd/kt)

0.5 = 1.48 10 4 L0.5

mol 0.5 s 1. This value is consistent with the known individual rate coefficients for thissystem, viz fkd 1 10

6 s 1 (20,31), kp 500 L mol 1 s 1, and kt 1.25 10

7 L

mol 1 s 1 (16). These values correspond to the first set of results (ie, the unbroken line)in Figure 10.

data in Figure 11, declining initiator concentration does not prevent all monomer

from being consumed. One generally only needs to watch out for this phenomenon

at high temperatures, where kd will be large.

Figure 11 shows some bulk polymerization data for methyl methacrylate

(MMA) (20,31) to which equation 40 has been fitted at low conversion. It is evi-

dent that both data sets are very well described up to about 13% conversion using

the one value of K. However, this equation is no longer valid beyond this point.

Broadly speaking, there are two effects that take place, as indicated in the figure.These are both important, and will now be discussed.


21/75


First, the effect that comes second, the glass effect. This sees the polymer-

ization all but cease at high conversion. A first impulse is that this could be the

phenomenon of dead-end polymerization. However for several reasons, this can-

not be the explanation: (1) The parameter values for the systems of Figure 11

predict a limiting conversion of 100%; (2) Figure 10 shows that a limiting conver-sion of approximately 85% will take days, not hours, to be reached; and (3) If the

limiting conversion were due to initiator consumption, then it would vary with

starting initiator concentration. However, there is no such variation (see Fig. 11),

and in fact the conversion of the glass effect always corresponds very closely to

that at which a system becomes glassy, hence the bestowed name.

Since small-molecule diffusion coefficients become very small as a polymer

solvent system becomes glassy, for many years it was believed that the glass effect

is due to the propagation reaction becoming diffusion controlled. However, when

eventually kp values were directly measured for such systems, it was found that

while kp does indeed start to decrease at this point, it does not do so by the many

orders of magnitude that are necessary to bring polymerization to a halt (32).

Thus a more plausible explanation for the glass effect is that initiator efficiency

plummets toward zero: the glassy polymer matrix keeps a pair of geminate pri-

mary radicals in the vicinity of each other for so long that they can hardly avoid

undergoing recombination (33).

The gel effect is an autoacceleration in rate that occurs at intermediate con-

version and is also known as the Trommsdorff effect or TrommsdorffNorrish ef-

fect in recognition of early workers to report on it (34,35). In fact, these are more

appropriate names than the gel effect: although there is acceleration in rate, as at

the gel point of step-growth polymerization, there is not gel formation. Right from

the beginning (36), it was recognized that this effect must be due to changing kt:as polymer is formed and a system becomes more viscous, the diffusion-controlled

nature of termination means that kt decreases. It turns out that the gel effect

is at its most pronounced in bulk MMA systems (as in Fig. 11); however, most

systems have at least some decrease of kt as polymerization proceeds.The above discussion makes clear that kt can be expected to change with

conversion potentially right from the beginning of a polymerization, while for

systems with a glass transition point, kp will start changing when this point is

reached. It is now recognized that f must depend on system viscosity, and thus it

too changes throughout the course of a polymerization (37), with the change being

cataclysmic at the onset of glassy conditions (33). Of course, equations 40 and 41

assume that all ofkt, f , and kp are constant in value. Thus these equations are

of limited utility.

If there were simple mathematical expressions available for the variations

of these rate coefficients with conversion, then integrated rate laws to describe

these situations might be possible. However certainly there is no agreement on

such expressions, and even if there were agreement, it is unlikely that the ex-

pressions would be simple, because quite clearly the described effects are of some

complexity. Therefore, the effects of conversion should be taken into account by

computer-based modeling that carries out numerical integration of equations 8

10, the foundational equations describing the kinetics of RP. There are countless

reports of such undertakings in the literature, without a consensus recipe havingemerged.


22/75


The method of moments is one approach that is often used for modeling the

course of a RP. It involves defining

k =

i = 1

i

k

cRi

(42a)

k =

i = 1

ikcDi (42b)

Thus k is the kth moment of the radical CLD, whereas k is the kth moment

of the dead polymer CLD. For example, since i0 = 1 for all i, one easily sees that

0 cR (eq. 11), and hence (eq. 12a)

d0

dt

= Ri 2kt00 (43a)

Using the above definitions together with equations 8a10a, one may show

that (38,39)

d0

dt=

i = 1

dcDi

dt= 2kt00 + ktrXcX0 + (1 )kt00 (43b)

d1

dt=

i = 1

idcRi

dt= Ri + ktrXcX0 + kpcM0 ktrXcX1 2kt01 (44a)

d1

dt=

i = 1

idc

Di

dt= 2kt01 + ktrXcX1 + (1 )kt201 (44b)

Differential equations for higher order moments may also be derived (and

are used), but the above sample is sufficient to give the flavor. Really, it is onlywith the double sums from combination that the derivations are at all tricky. In

equation 44b the terms for disproportionation and combination have been kept

separate rather than combined merely to emphasize their origin.

The point of equations 4344 is that they are an alternative to equations

8a10a in that, by definition, (d1/dt)/(d0/dt) is instantaneous DPn and 1/0 is

cumulative DPn. Similarly, (d2/dt)/(d1/dt) is instantaneous DPw and 2/1 is

cumulative DPw. Thus from numerical integration of a vastly reduced number

of equationsie, the necessary moment equations versus the full complement

of radical and dead polymer population-balance equationsone may still obtain

what might be regarded as the essential information for modeling RP, ie, rate

(eq. 13) and conversion, DPn, DPw, and any other average sizes that are desired.

This considerably speeds up the computing time that is required to model changes

with conversion. In the usual instance, the moments of the radical CLD are not

of interest per se (apart from 0); however, it is clear that determination of krequires knowledge of all k up to that k.

It is germane to ask whether the method of moments still has a place. (1)

From a pragmatic point of view, computers are now so fast that equations 810 may be numerically integrated on reasonable timescales. (2) In the previous


23/75


section, it was exemplified that it is better to model the entire MWD than just

averages of it. This of course requires using equations 810. (3) The method of

moments is only amenable to chain-length-independent termination. Since CLDT

is a reality (6), this also argues for using equations 810. Indeed, the very fact of

CLDT means that kt must change with conversion as cM changes (see above).(4) Because equations 810 follow directly from the reaction scheme, the risk of

error is reduced vis-a-vis the method of moments, for which appreciable mathe-

matics is required to derive the differential equations.

Regardless of whether one uses the method of moments or the raw popu-

lation balance equations, one must still specify how rate coefficients vary with

conversion. The importance of knowing and understanding these variations can-

not be overstressed (7), and attention should not be diverted from this pivotal

issue. Even the most mathematically and computationally sophisticated model

is of little use without accurate rate coefficients as input parameters. The next

section is mostly about the determination of such information.

Finally, if changes with conversion are so widespread and so important, it

is worth asking whether the effort put into understanding instantaneous rate

and MWD is justified. For one thing, the equations so derived are always the

platform for describing variations with conversion, as should be clear. Second,

the instantaneous equations will always apply at least for polymerization over

small intervals of conversion, eg, 10%, and sometimes they will apply over much

larger intervals, if variations are only relatively slight. Third, the instantaneous

equations still give understanding of the kinetics.

Non-Steady-State Polymerization

Equations from SSP always involve the propagation and termination rate coeffi-

cients coupled as kp2/kt. The historical driving force for non-steady-state (or in-

stationary) polymerization (NSSP) techniques is that, as will be seen in this sec-tion, they always return kp/kt (40). The idea is that by measuring both kp

2/ktand kp/kt for a system, combining these values enables kp and kt to be obtained

individually. It turns out this is a philosophically dubious practice, because CLDT

means that kt varies with Ri, and thus it is to be expected that kt will be dif-

ferent in SSP and NSSP experiments on seemingly identical systems (7,41). This

no doubt is part of the explanation for the large historical scatter in kp and ktdata.

This situation has been dramatically transformed by the advent of the NSSP

technique of pulsed-laser polymerization (PLP), which, as will shortly be seen,

enables facile and accurate determination of kp, independent of any knowledge

of all other rate parameters (4244). One might wonder whether this renders

other uses of NSSP as redundant, because kt may be easily determined from

SSP if kp is known. The answer is that attention has been shifted to exploiting

the advantages of NSSP techniques, in particular PLP, for studying termination,

which are that (1) they enable far more precise measurement of the tremendously

important variation of kt with conversion, and (2) they are uniquely positioned

for investigating CLDT.This section aims to substantiate the points made in the above outline.


24/75


Pulsed-Laser Polymerization.Fundamental Concepts. The idyll for PLP is sufficiently closely realized

that it can be considered a reality. It is that a laser pulse instantaneously cre-

ates a crop of uniformly sized small radicals from photoinitiator, and thereafter

there is no new radical generation (until the next pulse). In the absence of chaintransfer, the solution to equations 8a and 9a for this situation is (45,46)

cRi = cRexp( kpcMt)(kpcMt)

i 1

(i 1)!(45)

where

cR =cR,0

2ktcR,0t + 1(46a)

Here cR,0 is the concentration of primary radicals of size i = 1 created by the

laser radiation interacting with photoinitiator at t = 0. These equations havebeen written this way so as to emphasize the two parts: the radical CLD is a

Poisson distribution of variance kpcMt and DPn = kpcMt + 1 (eq. 45), whereas the

overall radical concentration decreases with time due to termination (eq. 46a).

Equation 45 has been evaluated using parameter values that are represen-

tative for PLP of (bulk) MMA. The results are presented in Figure 12 as i2cRi

versus log10 i. The reason for this mode of presentation is that this is how SEC

detects polymer (see above). Figure 12 may be used to illustrate two concepts

that are fundamental to understanding the deployment of PLP for determination

of propagation and termination rate coefficients:

(1) As long as there is negligible transfer, then to all intents and purposes the

radical CLD created by a laser pulse is a monodisperse population of size

kpcMt + 1 kpcMt, where t is the time elapsed since the pulse. Of course,

the CLD is not strictly monodisperse, because even a Poisson distribution

has a degree of polydispersity. However this decreases with time, as is illus-

trated in Figure 12 (remembering that the width of a SEC MWD reflects

PDI, see above), and for all but extremely small DPn this width can be

considered negligible. Indeed, simulations have confirmed that from a ki-

netic viewpoint, the radical CLD from a laser pulse behaves as if perfectlymonodisperse (46).

(2) Even though cR decreases strongly with time following a laser pulse, there

is actually a strong increase in SEC signal intensity of the evolving radical

CLD. This is due to two factors: (i) SEC (standardly) detects weight rather

than number of polymer chains; and (ii) separation is according to log i,

meaning that there is concentration of sizes together as i increases. Thus

in the example of Figure 12 there is actually about a 10-fold increase in

maximum SEC intensity in going from DPn = 21 to 501, even though cRdecreases by more than an order of magnitude during this time period (see

eq. 46a).


25/75


0.0 100

5.0 10-5

1.0 10-4

1.5 10-4

2.0 10-4

2.5 10-4

101 102 103

t= 0.004 s

t= 0.02 s

t= 0.1 s

i2cRi,molL-

1

i

Fig. 12. Evaluations of equation 45 with kpcM = 5000 s 1, kt = 1 108 L mol 1 s 1,and cR,0 = 1 10

6 mol L 1 at three different t, as indicated, to illustrate the evolutionof a radical chain-length distribution during pulsed-laser polymerization. Results are pre-sented as i2cRi versus log10 i, where cRi is the concentration of radicals of length i. Thenumber average chain length of each distribution is kpcMt + 1, ie, 21, 101, and 501 fromleft to right.

Determination of Propagation Rate Coefficients by Multiple-Pulse PLP.Consider carrying out a PLP in which laser pulses periodically strike at intervals

of td. This situation will be termed multiple-pulse PLP (MP PLP). When a new

pulse arrives, the radical CLD will consist of essentially monodisperse popula-tions of sizes kpcMtd, 2kpcMtd, 3kpcMtd, and so on from previous pulses: This is

one of the messages of Figure 12. Owing to the new pulse, these populations will

be suddenly subjected to a vastly increased frequency of termination, because of

the creation of a whole host of new radicals. Thus there will be a marked increasein the production of dead chains around sizes kpcMtd, 2kpcMtd, 3kpcMtd, and so

on. Therefore if the dead-chain CLD is determined at the end of the experiment,

it will contain identifiable features at these chain lengths. By reading off these

values, it is a simple matter to determine kp, because td (the inverse of the laser

pulsing frequency) and cM are easily controlled and known. This was the revolu-

tionary idea published by Olaj and co-workers in 1987 (44).

A theoretical validation of this idea is presented in Figure 13, which shows

a set of results from numerical solution of equations 8a10a (47) for the situation

of MP PLP. Several features warrant mention: (i) Even though the number of re-

maining radicals is relatively small when a new pulse arrives, their SEC signal is

more than large enough to generate observable SEC features from the enhanced

rate of termination that is consequent upon a pulse. This is as foreshadowed by

Figure 12. (ii) The chain length kpcMtd and its integer multiples lie a little to the

low molecular weight side of the CLD peaks. The reason for this was realized

right from the beginning (44): The radical CLD consists of Poisson distributions

and termination is not instantaneous. Without either one of these conditions,

kpcMtd corresponds to the peak positions. How then to determine kp? Workershave remained with Olaj and co-workers initial suggestion of using the position


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0

1

2

3

4

5

6

7

101 102 103

w(log10i)

i

Fig. 13. Chain-length distribution, presented as w(log10 i), where w is weight fraction

and i is chain length, from simulation of a multiple-pulse PLP with kpcM = 2000 s 1, kt =1 109 L mol 1 s 1, cR,0 = 5 10

8 mol L 1, td = 0.1 s, = 1, and ktrX = 0 (47). The(dotted) vertical lines show the positions of the chain lengths (left to right) kpcMtd, 2kpcMtd,and 3kpcMtd.

of the point of inflection on the low molecular weight side of each CLD peak (44).

It is visually evident from Figure 13 that these features must give kpcMtd, and

hence kp, accurately. However, one should always remember that this is not an

exact result, and so it introduces a small but unavoidable degree of systematic

error into values ofkp obtained by this technique.

So much for the theory; what about reality? A typical set of results is pre-sented in Figure 14 (48). It is clear that in essence the theoretical expectations

of Figure 13 are observed. Of course there are differences. The most important

of these is that SEC column broadening acts to disperse the sharp features of

the theoretical CLD. Nevertheless, these features are still clearly visible in theactual CLD, and it has been confirmed that this broadening does not undermine

kp determination (49). This study also illustrates how the sharpness and relative

intensities of the PLP peaks are a function of the parameters kt, cR,0, and td, but

that this also does not affect kp determination (49), something that is also implicit

in comparison of Figures 13 and 14.

In Figure 14, the derivative trace of the CLD is also presented. The first

maximum in this trace, which is the maximum corresponding to kpcMtd, is the one

that is most strongly evident. The higher order maxima (or points of inflection in

the CLD) are usually of lower intensity in the first place (eg, Fig. 13), and they are

rendered even more feeble by SEC broadening. Thus standard practice is to use

only this first maximum in the derivative trace for numerical determination of kpand to use the second and any further maxima that are evident as a consistency

check: Their position should be close to the appropriate multiple of the position

of the first peak for the obtained value of kp to be considered robust (43). It is

illustrated that the data of Figure 14 satisfies this criterion.

It took only a short time before the multiple-pulse PLP method was recom-

mended by IUPAC as the method of choice for kp determination (43). Reviewstestify to how widely and successfully it has been deployed (42,50). Major reasons


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0.0

0.5

1.0

1.5

2.0

-10

-5

0

5

10

102

103

104

w(log10i)

dw(log10i)/dlog10i

i

Fig. 14. Results from a multiple-pulse PLP of bulk methyl methacrylate at 40C withtd = 0.1 s (48). Full line: w(log10 i), where w is weight fraction and i is chain length;broken line: dw(log10 i)/dlog10 i; short, dotted vertical lines: positions of chain lengths (leftto right) 460, 2 460, and 3 460, making it clear that the positions of the maxima inthe derivative trace are (close to) equal to multiples ofkpcMtd.

for this are the intuitive nature of the method and the fact that it is independent

of knowledge of essentially all other rate parameters.

Determination of Termination Rate Coefficients by Single-Pulse PLP.Variation of Concentrations with Time. Buback bestrides this domain like

a colossus. His idea, first published in 1986 (51), was simply to monitor polymer-ization kinetics on a subsecond timescale following a single laser pulse. For this

reason, the technique is known as single-pulse PLP (SP PLP). Because Ri = 0during this period, equation 12 takes the very simple form dcR/dt = 2ktcR

2. At

the time this method was developed, CLDT was not definitively established, so it

was not unreasonable to assume, as was done, that kt is independent of time.

Thus equation 46a is obtained. Inserting this into equation 13 then leads to

cM

cM,0 = (2ktcR,0t + 1) kp/(2kt)

(47)

Here cM,0 is the monomer concentration when the laser pulse strikes at t = 0.

An example of using equation 47 is presented in Figure 15 (52). The first

thing to note about this figure is the timescale of the cM measurements: There are

hundreds per second, made by online NIR spectroscopy (53). The second thing to

note is that equation 47 fits the data well, which means the values of the fitted

parameters kp/kt and ktcR,0 can be regarded as genuine. This illustrates the point

made in opening this section that NSSP experiments yield kp/kt. By introducing

kp from MP PLP experiments on identical systems, one thus obtains kt. The third

thing to note about Figure 15 is the relatively small change in cM, and hence x,that takes place during a measurement of kt. Thus a very detailed map of the


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Fig. 15. Relative monomer concentration, cM/cM,0, versus time for single-pulse PLP in-volving copolymerization of an equimolar mixture of methyl acrylate and dodecyl acrylateat 40C, 1000 bar and 5% conversion (52). The signal is from coaddition of the results ofthree experiments. The difference between the measured data and the (best) fit of equa-tion 47 is shown in the residuals (res) plot. Reproduced from Ref. (52), copyright 1999, withkind permission from American Chemical Society.

variation of kt with conversion may very easily be built up, simply by carryingout a succession of SP PLP experiments on the one polymerizing system. This is

exemplified in Figure 16 (42,52). With other techniques, it simply is not possible

to probe the conversion dependence of kt in such a precise and finely controlled

way. For this reason, this technique has by now been used to study many, many

systems in this fashion (42). Bear in mind that the scatter of the kt values in

Figure 16 is very small by the historical standards of RP kinetics.

A passing comment on the lack of variation of kt with x in the results of

Figure 16 is warranted. This stands in contrast to the results of Figure 11 where,

as discussed, the acceleration in rate around x 0.2 is due to strong decrease

of kt. This illustrates the diversity of termination behavior that there is in RP

systems and hints at the challenges that lie in trying to specify and understand

the variation of kt with conversion (7), something that is of prime importance in

modeling RP kinetics (see the section Conversion).

The second great boon of the single-pulse PLP technique is its ability to

probe CLDT. This is because, as already explained, the radical CLD is both

monodisperse (to good approximation) and evolving in a known way with time

(see Fig. 12). Thus the termination rate at any instant reflects kti,i

for i at thatinstant, and as time goes on the gamut of i are covered. So by analyzing SP PLP


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106

107

0 0.2 0.4 0.6 0.8

kt,Lmol1s

1

x

Fig. 16. Single-pulse PLP measurements of termination rate coefficient, kt, as a functionof conversion, x, for bulk polymerization of dodecyl acrylate at 40 C and 1000 bar (42,52).

data in a detailed way from instant to instant, an exceptional opportunity for

measuring kti,i across a large range ofi is offered.

While this elegant notion has been keenly appreciated almost since the ad-

vent of SP PLP (54), for a long time its implementation met with difficulty. The

problem is that cM(t) data are relatively insensitive to the variation of kti,i with i.

Thus even the high precision cM(t) obtained using NIR spectroscopy may be well

fitted with quite different CLDT models (55). Notwithstanding this, there were

a number of notable successes in obtaining kti,i from so-called SP PLP NIR data

(55,56). However, it was always evident that things would be much better if the

radical concentration could be measured directly in conjunction with SP PLP, be-

cause then one would effectively only be using the first derivative of the data (ie,

dcR/dt) to probe CLDT, as opposed to having to use the second derivative in the

case ofcM(t). In other words, cR(t) is a direct route, cM(t) an indirect one. In 2004,

this aspiration was achieved when EPR spectroscopy was coupled with SP PLP

(57).

Substituting i = kpcMt + 1 into the power-law model for kti,i (eq. 15), one

obtains an expression for kt(t) that when inserted into dcR/dt = 2ktcR2 yields a

relatively easily integrable expression. The final result is (46)

cR,0

cR 1 =

2cR,0k1,1t

kpcM(1 )[(kpcMt + 1)

1 1] (46b)

This correctly reduces to equation 46a in the limit = 0. Of more interest is

that the long-time limit of equation 46b is

cR,0

cR 1 =

2cR,0k1,1t (kpcM)

(1 )t1 (48)

By using i = kpcMt, equation 48 may also be derived, as it was by manyworkers, as holding at all times. The problem with this is that it cannot hold in


30/75


Fig. 17. Log plot of (cR,0/cR 1) versus time for SP PLP EPR data of bulk n-dodecylmethacrylate at 0C and 12.9% conversion (58). Points: experiment; line: integrated ratelaw (46) with kt

1,1 = 1.5 107 L mol 1 s 1, S = 0.65, L = 0.17, and ic = 50. Reproducedfrom Ref. (58), copyright 2006, with kind permission from American Chemical Society.

the limit t = 0, because using i = 0 in equation 15 gives an infinite termination

rate coefficient (46).

In one way or another, equation 48 has played a part in all efforts directedat obtaining kti,i via SP PLP. In the specific case of the so-called SP PLP EPR

method, ie, SP PLP monitored by EPR spectroscopy, equation 48 suggests plotting

data as log[(cR,0/cR) 1] versus log t, because then a straight line with slope 1 should result. An example of using this approach is presented in Figure 17 (58).

While it is clear that there are linear regions in the plot, there are two surprises:

(1) At very early times, there is a downward curvature in the plot. It turns out this

is perfectly described by equation 46b (46,58), ie, this curvature does not at all

represent varying with i, but simply is a consequence ofkti,i being finite rather

than infinite at t = 0. (2) Of more significance is that the data tilt upward at long

times, suggesting a transition from a high value of (ie, low value of slope =

1 ) at small-chain lengths to a low value at long-chain lengths. In fact suchcomposite model (15) behavior was exactly what had been theorized just before

the advent of the SP PLP EPR method:

ki,it = k1,1t i

S ,i ic

ki,it = k1,1t (ic)

S + L i L ,i > ic (49)

The key feature of this model is that at chain length ic there is a

crossover from small-chain termination behavior, with power-law exponent S, tolong-chain behavior characterized by L (15). It turns out that there is an


31/75


equivalent of equation 46b for the composite model (46), and Figure 17 shows

an example of fitting this equation to data. As is evident, the data are flawlessly

described across a timescale of three orders of magnitude, a process that yields

the four parameter values kt1,1, S, L, and ic (58).

Incidentally, comparison of equations 15 and 49 makes clear that methodsthat consider only long-chain data must yield an apparent value of kt

1,1 that is

equal to kt1,1(ic)S + L, where in this expression kt

1,1 is the true value (6,15).

Since S > L, it is evident that the apparent value (from long-chain data) un-

derestimates the true value, often by close to an order of magnitude (6,15). A

strength of the SP PLP EPR method is that, because it probes also short chain

lengths, it delivers the true value ofkt1,1 and thus avoids this issue.

By now the SP PLP EPR procedure is finding widespread use (59), and to-

gether with other methods it is enabling a picture of CLDT of hitherto unimagin-

able detail and clarity to emerge. It is as follows (6): (1) Methacrylates have L 0.16, in accord with theory (60), ic 50100 and S 0.50.65, where the higher

value ofS holds for n-alkyl pendant groups and the lower value for spherical

ones. The data of Figure 17 evidence some of this. (2) Acrylates seem to have L 0.20.25, ic 2030, and S 0.81; however, there is some uncertainty in

these results because of possibly perturbing effects from midchain radicals (see

Section 4). SP PLP EPR, which has the added advantage of being able to distin-

guish different radical types, is being used to address this issue (61). (3) As would

be expected from small-molecule termination studies (62), the variation of kt1,1

with viscosity is as anticipated, ie, kt1,1 1 (59,63), and the (true) value ofkt

1,1

is consistent with the simple Smoluchowski description (6,59,62):

k1,1t = 142(D1 + D1) (50)

Here D1 is monomer diffusion coefficient, is the radicalradical separation at

which the reaction of termination occurs, and the fraction 0.25 accounts for spinmultiplicity. In all important features, all the above is as was posited when the

composite model was proposed (15).

It should be evident from all this why SP PLP EPR has so quickly become

the acme of methods for studying termination in mechanistic detail (59). How-

ever, this does leave one point to address: If the EPR version of SP PLP so beau-

tifully uncovers the chain-length dependence of termination, why did the NIR

version with CLIT (ie, eq. 47) function so well for so many years? There are sev-

eral reasons: (i) SP PLP NIR is best suited to fast propagating monomers, because

these give larger changes in cM. High kp means that i quickly becomes long, which

means that is small throughout most of the time period of a measurement. As

already mentioned, with cM data it is difficult to distinguish between 0.15

and 0. (ii) Fast propagating monomers usually also have high ktrX. If trans-

fer occurs on the timescale of an SP PLP measurement, then the monodispersity

assumption breaks down. Instead there is a constant generation of small free

radicals, which leads to a relatively time-invariant kt (46).

Molecular Weight Distribution. After some time, it was realized that not

just the rate of polymerization but also the chain-length distribution from SP PLPshould bear the footprint of CLDT in a decipherable way (64). The reasoning is as


32/75


0

0.1

0.2

0.3

0.4

0.5

101 102 103

= 0.25 fit

= 0.15 fit

= 0.05 fit

Experiment

w(log10i)

i

Fig. 18. Modeling of the PLP data of Figure 14 with equation 51. Points: experimental

values for i < 320, offset by 0 (lower set), 0.05 (middle), and 0.1 (upper); lines: best fits ofequation 51 with = 0.05 (lower), 0.15 (middle; values offset by 0.05), and 0.25 (upper;values offset by 0.1), = 0.63 (65) being used in each case. Values are offset so that thethree fits can be visually distinguished.

follows: Because of the monodisperse nature of the radical CLD in SP PLP, dead

chains of length i can only arise from disproportionation of radicals of length i and

combination of radicals of length i/2, ie, they may only be formed at either 1 or 2

unique instants. Thus one may relatively easily formulate the rate of formation

of chains of length i: The contribution from disproportionation involves kti,i and

cR at t = i/(kpcM), whereas that from combination involves kti/2,i/2

and cR at t =i/(2kpcM). For these values one may use equations 15 and 48b, respectively. The

end result from substituting these into equation 10b is (46,64)

cDi = cR,0Ci

1 +

C

1 i1

2+ (1 )

cR,0C

4

i

2

1 +

C

1

i

2

1 2(51)

Here C = (2cR,0kt1,1)/(kpcM) is the PLP analogue of the quantity encountered ear-

lier with SSP (see eq. 34b).

Figure 18 presents results from using equation 51 to model a portion of the

MP PLP data of Figure 14 with = 0.63 (65) and various . Of course, it is not

strictly valid to do this, because this experiment was not an SP PLP. However,

the chains from prior to the first peak must almost all arise from termination

involving only radicals with i < kpcMtd. Thus to good approximation this portion

of the CLD should be as if from a true SP PLP. Indeed, it is evident from Figure

18 that equation 51 furnishes an excellent description of this data. Instead, the

problem is that it provides too easy a fit: As is shown, one may choose almost

any sensible value of and still end up with a thoroughly satisfactory fit, not to

mention that fitting is insensitive to the value of (64).

What may be done about this? Obviously independent determination of the

value ofC would eliminate this problem, because for a particular C there will

only be one pair of and C values that fits a data set. However, in practice, itis difficult to do any better than to come up with a reasonable range of C values


33/75


0

1 105

2 105

3 105

4 105

0 1000 2000 3000 4000 5000

= 0, C= 2.4 103

= 0.2, C= 8.0 103

= 0.5, C= 3.2 102

i2cDi,molL-1

i

Fig. 19. Illustrating the chain length distributions of dead polymer that result fromsingle-pulse PLP: evaluations of equation 51 with = 1 and (as displayed) three different, with C chosen in each case so that output values are in the same vicinity. Results arepresented as i2cDi (which is proportional to SEC signal) versus i, where cDi is the concen-tration of dead chains of length i produced by one laser pulse. The C values are physicallyreasonable, eg, cR,0 = 1 10

7 mol L 1, kt1,1 = 2 108 L mol 1 s 1, and kpcM = 5000 s

1

give C = 8 10 3 (see text).

in advance, because both cR,0 and kt1,1 are tricky to determine. So what must

be done is to try to realize true SP PLP conditions, ie, to carry out an MP PLP

experiment in which td is very long. The desirability of this may be seen from the

large-i limit of equation 51 (66):

i2cDi = i cR,0

C(1 )2( + [1 ]2 ) (52)

This equation shows that when SEC signal is plotted versus i, its limiting form

(for SP PLP) is simply that ofi (66), as is intimated by the sample evaluations of

equation 51 that are presented in Figure 19. This enables easy determination of

, independent of C, somewhat analogous to the way in which the t1 variation

acts to nail down with SP PLP EPR.

Unfortunately, there are still obstacles to the implementation of this elegant

theory. These stem from the necessarily long timescale. This makes it difficult

to avoid the occurrence of chain transfer, which undermines the theory behind

equation 51. The unbounded increase of i2cDi with isee Figure 19is also an

issue. This is a paradox if ever there was one, but it was foreshadowed in Figure

12, and is explained by equation 52. It can lead to large weights of polymer at and

beyond the exclusion limit of a SEC setup, which is dangerous for the instrument.

It is often remarked that an MWD must contain a full record of the underly-

ing kinetics of a system, and thus that it must be a route to complete understand-

ing of polymerization kinetics. At the very least, the logic here relies on different

kinetics giving measurably different SEC signals. It has been seen in this sec-

tion that this is a false assumption: The data of Figure 18 look like they have adistinctive shape, but in fact this shape may be acceptably reproduced with


34/75


0

1 107

2 107

3 107

0 0.2 0.4 0.6 0.8 1

cR,molL1

t, s

Fig. 20. Radical concentration, cR, as a function of time, t, for multiple-pulse PLP, as

evaluated using equation 46a with = 1 10

7

mol L

1

and: upper (unbroken) line:kt = 1 107 L mol 1 s 1, td = 0.1 s; middle (dashed) line: kt = 1 10

8 L mol 1 s 1,td = 0.1 s; lower (dotted) line: kt = 1 10

8 L mol 1 s 1, td = 0.3333 s. The horizontallines of matching pattern show the associated values of cR,max from equation 53, therebyillustrating the fast attainment of pseudostationary conditions in MP PLP.

a range of values that is meaninglessly large. Upon reflection, this theme

has been implicit throughout this work. For example, in SSP the MWD from

chain-length-independent disproportionation is absolutely identical in shape tothat from chain transfer (see the section Chain-Length-Independent Termina-

tion [2.3.1]). Figures 7 and 8 provide another example. One must conclude from

this that it is inadvisable to rely on MWD modeling alone for elucidation of RP

kinetics (40). It would seem to be better practice to employ methods based on c(t)measurement, with MWDs being introduced as supplementary information. And

this is all aside from only c(t) methods being suitable for the determination of

how termination rate coefficients vary with conversion (40), eg, Figure 16.

Determination of Termination Rate Coefficients by Multiple-Pulse PLP.Multiple pulse PLP experiments are carried out for hundreds or even thousands

of pulses. Of these, it takes only a handful at most for the radical concentration

to reach what is termed a pseudostationary state: the loss of radicals by termi-

nation between pulses is exactly counterbalanced by the creation of new radicals

from a laser pulse, resulting in a periodic variation of cR. This is illustrated in

Figure 20, which shows repeated evaluation of equation 46a for three sets of pa-

rameter values. By repeated evaluation is simply meant that after every time

period of td has elapsed, an amount this being the concentration of radicals

created by a laser pulseis added to cR to give the new value of cR,0 for use in

equation 46a over the next time period of td. Clearly this mimics MP PLP. What

Figure 20 proves is that for all intents and purposes an MP PLP may be regarded

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Kinetics of Radical Polymerization