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edited by JOHN W. MOORE
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Free-Radical Polymerization Using the Rotating-Sector Method A Computer-Based Study Stephen J. Moss University of Aston in Birmingham, Gosta Green, Birmingham B4 7ET, England Vinyl polymerization by a free-radical mechanism is of enormous commercial importance and great academic interest and has become one of the best understood of all reaction types during the last three d e ~ a d e s . This ~ . ~ topic therefore rightly occupies a significant place in most fust-degree courses in chemistrv. Since th&e is little doubt that detailed kinetic studies have provided the foundations for our present understanding of the mechanism, they must play a correspondingly large role in teachine and " eenerallv do so. Elementarv kinetic treatments of polymerization are properly restricted to steady-state conditions which leads to few difficulties for students while allowing the main features of the mechanism to be developed. However, there are limitations to this a~proach.In particular. kinctic experiments in the steady stat; b n ~ yyield composite values of the rate constants, while knowledge of the indiuidual rate constants is needed for real insights &to monomer reactivity. Unfortunately, perhaps, rate constants for the . . separate . propagation and termination reactions may only he determined by examining the kinetic behavior outside the steady state which presents more serious conceptual and mathematical problems. Although such a treatment is much more difficult for students to grasp, the problems may be reduced by placing the took in the context of a com~uter-basedstud^.^ This oaner . . discusses the prinriples of the approach and presents results from a ororram written in BASIC and used on a 48K Aoole .. I1 microcomputer.
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Kinetic Behavior The simplest mechanism for a free-radical polymerization involves only initiation, propagation, and termination. Initiation: Propagation: Termination:
-
I 2R' R'+M-M.' Ri M; + M M,+,' R P = kp[MIIMn.l Mzn (polymer) Rt= kt[M:I2 2M:
--
The polymerization rate is equal to the rate of propagation
--
Writing [M:] is given by
- R, = k,[MI[M:] (1) dt = n, the rate of change of radical concentration dn = R;- Rt= Ri - ktn2 -
dt In the steady state, dnldt = 0
(2)
Before examining the methods used to determine individual rate constants, i t will be convenient to define some useful properties of a chain reaction. (1) The kinetic chin length, F, is the average number of propagation
steps in each chain:
Y = RdR; I.e., (2) The auerage lifetime of the c h i n carriers 6)is the average time
between successive propagation steps. Since all the radicals present at any instant undergo a pcopagation reaction in the time interval t (from the definition oft),
(3) The aueroge lifetime of o chain between initiation and termi-
nation, 7,is clearly related to the two quantities defined ahove by r =it l.e., i = l/(kJli)1/2
(6)
Equations (5) and (6) show that the determinations of individual values fork, and kt are equivalent to determinations of the chain lifetime and the lifetime of the chain carriers. Two approaches are available. (a)
pre- and after-effectstudies;
(b) intermittent illumination-the rotating sector method.
Both methods require the ability to start the reaction a t an exactly known instant of the time so that photochemical initiation is essential. The principles of the pre- and after-effect are illustrated in Figure 1. During the light period, let Ri = 0 and solve eqn. (2) to obtain: n = (8/kJ1A t a n h ( t 6 ) (7) The amount of polymer produced a t any time, t , is then given
Sot
1
. In[cush ( t a ) ] (8) ktt The steady state is reached when t& >> 1, and cosh(t &) becomes equal to exp(-t&). Then, [PI, =
dt
=Y
' Ham, G. E. (Editor),"Vinyl
Polymerization," E. Arnold. London.
1967.
The rate of initiation, Ri, is usually measured by inhibition studies. Therefore, steady-state measurements suffice to determine values of k,lkt1/2.
6amford.C. H., Barb. W. G., Jenkins, A. D., and Onyon, P. E., "Kinetics of Vinyl Polymerization by Radical Mechanisms," Bulterworlhs. London, 1958. Barrow, G. M., J. CHEM. EDUC.,57, 697 (1980). Volume 59
Number 12 December 1982
1021
Figure 1.h e - and aner-effect in the polymerization of vinyl acetate. k, = 1000 dms mol-' s-' 0 = t X lo-O mol dm-$ s-' [MI = 10.8 mol dm-3 kt = 5.6 X 10' dm3 mol-' s-': R, = 4.57 X 10F mol dmw3s-' n. = 4.23 X 10-* mol dm-? 7 = 4.23 s: tt=b=20s.
Figure 3. Rmting-seetar plot f w slow flashing. tr = td = 100 s: 8 cycles ( O M details as for Fig. 1).
Solving the differential equation where no is the concentration of radicals when the light is switched off (i.e. a t t = 0), and is not necessarily equal to the steady-state concentration n,. The amount of extra polymer produced a t any time t after the-light is switched off is again obtained by integrating (nlt)dt
[PI, = ( l l k d ).l n ( l + noktt)
Figure 2. Rotating-sector method (schematic).
(12)
The simplest method of extracting the rate constants uses eqn. (11)in inverted form
This equation represents the linear part of the [PI versus time graph. Putting [PI, = 0, the line extrapolates back to
.
to = ln2/(0kt)'" = ? In2
(10)
When the light is switched off, the chain carriers are gradually removed hy termination reactions, and polymer continues to form but a t a continuously decreasing rate. Since 8 = 0 in the dark,
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Journal of Chemical Education
-t t - = - kd. t = llrate nt no
+
(13)
A plot of llrate_agaiust t for the dark period is linear with slope equal to k t t . In addition when no = ns, the time for the radicals to decay to half the initial value is easily shown to be equal to ? from eqn. (11). This result may be confirmed on Figure 1.
Figure 4. Rotatingsector plat. t = 1).
t. = 1 s:25 cycles (other details as for Fig.
Studies of the pre- and after-effect are experimentally difficult since very small amounts of polymerization must be measured in short time intervals. Nevertheless, such studies have been carried out on a few systems and have confirmed the validity of the approach. I t is much easier to measure average rates under intermittent illumination with a rotating sector than to measure rates during the pre- and after-effects.Figure 2 shows schematically the experimental arrangement. The average rate, R, under intermittent illumination is given by R =-'I -
tl
Pd
+ td +
=
A [J: +
nrdt
t(tl
td)
+
LCtd n&]
Figure 5. Romtingsector plot fw repM flashing. t, = details as for Fig. 1).
td = 0.2 s:60 cycles (olher
The full equation which results from the application of eqn. (14) is lengthy and rom~licated.The essential features of the
method m a y be made-clearer by considering two limiting CaSeS. Slow Flashing
If tr and t d are both long compared with r , then the steady-state radical concentration (n,) is reached very early in the light period, and the radical concentration falls to zero very early in the dark period (see Fig. 3). To apply eqn. (14), we write nl = n, (constant) and n d = 0.Then,
(14)
where Pl is the amount of polymer produced during a light period of duration ti, and P d is the amount of polymer produced during a dark period of duration td.
where R, is the steady-state rate in full illumination, and f is the sector factor, defined in eqn. (15).
Volume 59
Number 12 December 1982
1023
Rapid Flashing
A quasi-stationary state exists after the first few turns of the sector (see Figs. 4 and 5). Initially, the radical concentration rises more in a light period than it falls in the dark period. However as nl rises, the rate of decrease of n in the dark hecomes larger, and eventually the fall in radical concentration during the dark period exactly balances the rise during the light period. Note that the radical concentration can never reach the steady-state level. Consider now a limitine case where t~ and t~ - are both verv short so that negligible changes in radical roncentration uccur during the light and dark periods (see Fig. S),i.e.,
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nr = nd = R (= constant)
Radicals are therefore being removed a t a constant rate (= ktn,2). However, the system forms radicals only during the light period, i.e. it will behave as if the continuous rate of formation of radicals were Btll(tl td) = 0-f. Then, in the quasi-stationary state ken,= = 8 . f
+
n, = ns
.
Figm 6. R m k of an enended sbmy of mean rate as a funaim of flashing rate. Points are obtained using lhe computer program described here. Table 1. Integrated Rate Equations
(16)
f1J2
Light Pwicd
Dark Pericd
The overall rate is then given by -
RE& =
--n,t = R, .
(17)
f'i2
Comparing eqns. (15) and (171,
Ew/R&~ = f - l J 2
where C =
(18) Therefore, the average polymerization rate should increase with increase in the flashing rate. From the way in which eqn. (18) has been derived, it is clear g a t an essential requirement for observing a dependence of R on the flashing frequency is that radicals should disappear by a second-orcer reaction. If radicals are removed in a firstorder reaction, Rf& = RslOw;i.e., the whole basis of the method disappears.
Vinyl acetate
The General Case
Styrene
The average rate changes over from Rffftto & , a t sector speeds which depend on 7 for the reaction being studied. The exnerimental ~rocedureis to measure the averaee rate hv normal means over as wide a range of sector speeds as possible; tr must he varied over several orders of maenitude. ' The complete equation is obtained by applying eqn. (14) for a general flashing frequency, using the expressions for nt deriv@ in the pre- and after-effect cases. The final equation gives RIR, as a complex function of tl, f, and 7. Figure 6 shows the overall behavior. The rate constants may be extracted by comparing the results ohtained with standard curves calculated for the particular value off used and a range of values of 5. Alternatively, computer-fitting of the data may he used. Necessary conditions for obtaining satisfactory measurements of the individual rate constants hy the rotating-sector method are summarized below
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(1) The reaction scheme must be simple and well established. (2) Photoehemid initiation is required with zero (or very small) dark
reaction. (3) Radicals must be removed mainly by a bimolecular reaction. The Computer Program The program uses the integrated forms of the rate equations to calculate radical and polymer concentrations during alternate light and dark periods. The relevant eqns. are (I), (a), (I]), and (12) with modifications to incorporate the approoriate inteeration limits for each Deriod (Tahle 1). . Fieures 1. 3,4, and 5 were generated in this way; each run required about 50 sec of computing time.
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Input Required
(1) Initiation rate. Experimental values typically would be around mol dm+ s-'.
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Journal of Chemical Education
n, + no ---anda='h.InC n. - no
Table 2.
Absolute Rate Constants at 25%
1000
55
Methyl mahacrylate Methacrylonitrile me mlls of kpand h, 818 hn3 mol-'
143 26
5.6 X lo7 5.9 X lo7 1.2 X 10' 2.1 x 10'
8'.
(2) Rate constants k, and kt. Table 2 lists values for some common monomers. (3) Monomer concentration. (4) Ratio of dark to light period. (5) Duration of light period, specified in seconds or a s a multiple of the chain lifetime. (6) Number of cycles of light and dark. Output
The input information is repeated and the following calculated quantities provided. (1) Steady-state concentration of radicals (2) Steady-state rate (3) Chain lifetime ( 4 ) Scale limits for the axes The program may be used in several ways. The somewhat forbidding mathematics is made much more straightforward by using the program in tutorials or in lectures. However, an even more compelling approach is by individual students in a computer-hased study of the topic. Such a study may examine (1) basic ideas through the pre- and after-effect(Fig. 1); (2) limiting values of the mean rate at high and low flashing rates (Figs. 3 and 5, and eqn. (18)); (3) a mare extended study of mean rate against flashing rate (Fig. 6-the points were obtained with the program).
A listing of the program, written in BASIC for a 48K Apple I1 microcomputer, is available with samples of output from the author.
