J. Phys. Chem. 1984,88, 6667-6670 at the surface. To achieve this the value of yu is iterated with a Newton-type iteration. After each integration step across a lattice layer, the potential must be examined and, occasionally, the Newton-step length must be reduced to prevent overflows. Calculation ofthe Polymer Profile. Using the Scheutjens-Fleer theory, we followed their line in the evaluation of the polymer
6667
concentration profile, except for the modification that we used In +01 as the iteration variable to find the zero of +! +O,-l, where +I is obtained from 40i using (5). As eq 29 of Roe’s paper’ shows the derivatives of the partition function with respect to &, a term -abi - p)has to be added to the left-hand side of this equation, which is solved, using a Newton-iteration method.
+
Single Molecule Gas-Phase Polymerization Kinetics of Vinyl Acetate. Nonsteady Measurements H.Reiss* and M. A. Chowdhury Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: July 23, 1984)
This paper continues our recent work on the kinetics of free-radical chain polymerization in the gas phase. As in the previous work, we avoid having the involatile polymers condense out of the vapor phase by employing a cloud chamber technique which allows us to study the kinetics under conditions such that there are so few polymers growing simultaneously that they cannot encounter one another. We detect the arrival of individual product molecules by having them nucleate macroscopic drops of monomer liquid. The arrival of product molecules is so discrete that signal averaging must be employed in the determination of the rate. As in the previous work, the polymer is poly(viny1 acetate). In the present study we employ a nonsteady technique in which the reaction is initiated by a pulse of photons. Among other things, this approach makes it possible to determine the degree of polymerization of the polymer capable of participating, along with monomer molecules, in the binary nucleation process and therefore makes it unnecessary to rely on a difficult nucleation theory. The monomer supersaturation has been deliberatelychosen to be high, so that high enough rates could be achieved to allow the signal averaging to be performed by hand. As a result, the polymers responsible for nucleation are small, containing only about six monomer units.
1. Introduction
In a recent paper’ a method was introduced, capable of measuring the kinetics of gas-phase chain polymerization reactions by detecting the arrival of single polymer molecules. This method involved the use of an upward thermal diffusion cloud chamber which maintained monomer vapor in a steady, supersaturated state. Ultraviolet photons, admitted to the chamber, produced free radicals and initiated chain polymerization. The polymer chains grew to a critical size at which they were capable of participating in condensation nuclei leading to the formation of drops of liquid monomer. The drops fell through a laser beam, scattered strong light signals, and were counted. The rate of drop formation was identical with the rate of production of polymers of the critical size and was therefore a measure of the kinetics of polymerization. Reference 1 contains a detailed verification of the validity of this method and its underlying mechanism and should be consulted for this purpose. The critical degree of polymerization (necessary for participation in a condensation nucleus) depends on the degree of supersaturation of the monomer vapor. By adjusting the supersaturation, it is possible to “tune” to the arrival of polymer molecules of a specified size. The series of events, outlined above, will only occur faithfully in the manner described, if conditions can be arranged such that each condensation nucleus contains only one polymer molecule. Thus, the number of chains growing simultaneously must be so small that there is little probability of two polymer molecules encountering one another. This, in fact, is the reason for hoping that the gas-phase process can be sustained. In other attempts to study such gas-phase kinetics it has not been possible to avoid polymer-polymer encounters and abundant condensation of involatile small polymer molecules. The enormous sensitivity of the nucleation phenomenon enables one to detect the arrival ( 1 ) Chowdhury, M. A.; Reiss, H.; Squire, D. R.; Stannett, V. Macromolecules 1984,17, 1436.
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of single molecules and, therefore, to work under conditions (if they can be established) such that growing polymers cannot encounter one another and escape the vapor phase. In ref 1 emphasis was placed on ( 1 ) demonstrating that the observed droplet formation was due to the polymer process just described and (2) showing that conditions could be established such that only the “single polymer” condensation process occurred. As a somewhat secondary goal, the measured data were used to estimate the various kinetic constants. As explained in ref 1, the study was a “bootstrap” one, in which confirmation of the postulated phenomenon depended on the compatibility of experimental observations with one another, as well as with theory. This “bootstrap” character was mandated by the fact that no other method of comparable sensitivity was available to provide independent confirmation of the underlying process. When referred to these goals, the studies of ref 1 may be considered successful. The monomer involved was vinyl acetate, and with this monomer, the results of ref 1 did indicate that conditions for single polymer nucleation had been established. Furthermore, the above-mentioned compatibilities of experiment with experiment and of experiment with theory were observed. The experiments of ref 1 were conducted under steady illumination (under a steady flux of UV photons). Among other things, the critical polymer size had to be deduced from an approximate theory of nucleation.2 The accuracy of that theory could be assessed for the case of homogeneous nucleation of the monomer i t ~ e l f but , ~ one cannot be assured of its quantitative validity in the case of binary nucleation involving both monomer and polymer. Nevertheless, its application indicated that, in ref 1, polymers having degrees of polymerization of the order of 30 were responsible for nucleation. As explained in ref 1 , many improvements can be effected in subsequent studies. Even the degrees (2) Reiss, H.; Chowdhury, M. A. J . Phys. Chem. 1983,87,4599 (3)Chowdhury, M.A. J . Chem. Phys. 1984,80,4569.
0 1984 American Chemical Society
6668 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 of supersaturation quoted in ref 1 could be in error by as much as 10%. In the present study, we make some of these improvements. However, there is still room for much more, and we discuss these matters below. The principal advance of the present study involves the use of pulsed, nonsteady reactions, which allow us, among other things, to determine the size of the polymer involved in the nucleus without having to rely on a necessarily imprecise nucleation theory. In addition, we make further progress toward the independent determination of the various rate constants. Attention is again confined to the formation of poly(viny1 acetate). The experiments which form the subject of this paper involve the same system as in ref 1 and equipment for UV illumination, drop detection, and counting. However, the cloud chamber (described in ref 3) has been improved so that precise temperature measurements are possible (and therefore accurate specifications of supersaturation), and also, more inert gaskets have been used. An important difference is the fact that a pulse of UV light, short in comparison with the time required for the production of polymer, rather than a steady beam is admitted to the chamber. The rate of appearance of critical size polymer is then monitored by counting the drops which appear after the pulse. The overall production of drops is even less in the pulsed system than in the steady-state experiments, and as a result, the “reaction noise” is even more pronounced. It is therefore necessary to signal average during each small interval of time following the pulse. We discuss this feature in the following sections. In the present section we develop the theory which describes the response to the pulse. As in the studies of ref 1, the gas-phase polymerization of vinyl acetate is described by the following reaction
Re
+ CH2=CHX
-
+ hv
---*
R1 =
(4)
in which (Y
For x is
= k,M
+ kd
(5)
> 1 the equation for the rate of change of R, with time dR,/dt = kpMR,-l
- aRx,
R,(t=O) = 0, x
x
>1
(6)
etc.
(1)
in which X represents the acetate group. We ignore both chain transfer and bimolecular chain termination, because of the relative unimportance of the former at the monomer concentrations typical of the experiment and because the latter is absent in view of the small number of polymer radicals which are permitted to grow simultaneously. However, it is necessary to consider the escape, by diffusion, of growing free radicals from the UV beam. In ref 1 we treated this process as a first-order one, characterized by the rate constant kd, and provided some rationale for this assumption. This rationale is really inapplicable for the nonsteady case (and again we shall have more to say about this later), but for the present, in lieu of a better assumption, we continue to assume that diffusive escape is a first-order process governed by a rate constant kd which is independent of polymer size. Our hope is that it will be a crude approximation, capable of delineating some of the main features of the overall process as they emerge from the experiment. In the future it will surely be necessary to develop a more correct (and far more difficultly amenable to mathematical analysis) description of the escape (and return) phenomena, if the method is to be finally reduced to a truly quantitative tool for the determination of rate constants. We denote by R, the concentration of growing free radicals of degree of polymerization x. The initial concentration of free radicals of size x = 1 is determined by the light pulse. Thus
R,(t=O) = kiMZT = 6
(2)
in which t denotes time, ki is the rate constant for initiation, M is the monomer concentration, Z is the light intensity, and T is the time of duration of the pulse (in our experiments, usually less than 1 s and considerably shorter than the time required to produce polymer). After the pulse, the equation governing the rate of change of R 1 with time is
+ kd)R1
(3)
in which k, is the rate constant for propagation and kd is the
>1
(7)
The set of equations comprised by eq 3 and 6 together with the initial conditions eq 2 and 7 can be solved, one by one, the solution for R,, being substituted into the differential for R , so that a solution for R , is obtained, etc. Alternatively, and more expeditiously, it is possible to formulate a differential equation for a generating function from which the various R, may be derived and immediately arrive at the general formula for the time dependence of R,. In either case, the following simple general result is easily obtained.
R,-1 =
PYx-2tX-2e-ar ~
(x
(8)
- 2)!
in which y is written for k,M. Now, if a single polymeric radical of size x causes nucleation, instantaneously upon its appearance, the rate of nucleation will be given by
J = k,MR,-,
Re
R-CH,-CHX,
dR,/dt = -(k,M
constant, mentioned earlier, for escape by diffusion. Integration of eq 3 subject to eq 2 leads to the result
The initial condition is obviously
11. Theory of the Nonsteady Process
CHz=CHX
Reiss and Chowdhury
(9)
no other process being available to destroy radicals of size x. Substituting eq 8 into eq 9 and taking account of the definition of y, we find J
Prx-’ t rZe-a1 =(x - 2)!
eK tx-2 e-ai
where K is independent of time. This is the result we shall use. Taking the logarithm of both sides of eq 10, we find
or In J = K
+ (x - 2) In t - cut
where K is given by K=lnI+
1
(1 3)
If an experiment can be conducted in which both the supersaturation and the temperature are maintained constant, while only I is varied, then the change in In J will be given by the change in In I; Le., a plot of In Jvs. In I will exhibit a slope of unity. Not only will the observation of this dependence lend credibility to the assumed mechanism, but, in particular, it will confirm the single polymer nature of the nucleus. A somewhat similar method of confirmation was employed in ref 1. 111. Experimental Section
The upward thermal diffusion cloud chamber employed in these experiments is described in ref 3. The light source was a 1000-W Xe short-arc lamp (Oriel). A collimated beam of UV light (full spectrum of the lamp with only the band between 260 and 280 nm being effective for initiation) from this source passed through an optical system consisting of a water filter (to remove infrared), an iris, and a quartz lens. To provide a reporducible pulse of UV light, the shutter of a Canon FTb camera body was employed.
Polymerization Kinetics of Vinyl Acetate
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6669
r\
A
0)
0)
b e
I
a
$
a
4
1
20 4 [l 0i
40
60
80
100
120
0 01 a J 0
-. I
I
20
40
I
I
I
60
80
100
120
-0
Time (s)
Figure 1. Signal averaged rate of nucleation vs. time. S = 3.40, T = 274.0 K, elevation of He-Ne laser beam = 0.5 (reduced height), and light intensity I = Io. The solid line is the least-squares-fittedcurve.
The pulse had a duration of 1s and passed into the vinyl acetate vapor in the cloud chamber at an elevation which included the level of maximum supersaturation. Nucleation and drop formation were monitored by means of signals scattered from a He-Ne laser beam placed below the UV beam and at right angles to it. The scattered light signals were counted by means of a photomultiplier, amplifier, discriminator, and a counting circuit described in ref 1. We take this count rate as proportional to the actual rate of nucleation and therefore to the actual rate of arrival of individual polymer molecules of the critical size (see section IV). Because of the inherent noise (mentioned earlier) at the very low counting rates, typical of these experiments, it was necessary to resort to signal averaging. For this purpose a large number of runs, each initiated by a UV pulse admitted under indentical conditions of supersaturation, temperature, light intensity, and pulse duration, were performed. Figure 1 shows a result averaged over 40 such runs. Each point in the figure is the average number (for these 40 runs) of counts, observed in a particular 1-s interval, following the pulse. Figure 2 is a similar plot but involves only 20 runs. Employing the information obtained from the studies of ref 1, we established conditions under which only the single polymer nucleation process would be active in a steady-state experiment. Because of the much smaller number of growing chains in the pulsed experiments, the single polymer process is expected to also prevail.
IV. Discussion The data of both Figures 1 and 2 were obtained at a temperature (UV beam level) of 274.0 K and a supersaturation S = 3.4. This value of supersaturation was considerably higher than the highest value (S = 2.6) of the studies of ref 1. According to the theory of ref 2 the degree of polymerization required for a vinyl acetate polymer participating in the formation of a binary nucleus should lie, for this temperature and supersaturation, somewhere between 6 and 7. This is much smaller than the value of x typical of the studies of ref 1 . In those studies the likely critical size was in the neighborhood of 30. The solid curve in Figure 1 is a least-squares fit of the data of that figure to eq 10. As the curve shows, that equation does accommodate the data rather well. A larger number of runs should hopefully reduce the variance so that the kinetic parameters can be determined more accurately. We will report data based on a larger number of runs in future experiments where we will be able to perform the signal averaging electronically, rather than tediously by hand, as in the present case. The least-squares fit to eq 10 yields a value of x = 6.8. This value lies between 6 and 7 as required by the theory of ref 2. However, in spite of this dramatic agreement, the limited accuracies of both the theory and the experiments are such that we
0 IY
Time (s)
Figure 2. Signal averaged rate of nucleation vs. time. S = 3.40, T = 274.0 K, elevation of He-Ne laser beam = 0.5 (reduced height), and light intensity I = 0.5011,. The solid line is the least-squares-fitted curve, and the dashed line is the theoreticai curve using least-squares-fittedparameters of Figure 1, corrected for the reduced light intensity.
must still consider the agreement to be only nominal. Nevertheless, it is encouraging. This example does, however, indicate how such pulsed experiments can be used to determine the critical degree of polymerization without relying on a difficult theory of nucleation. Of course, in the final analysis, a more accurate model than that embodied in eq 10 must be used for this purpose. In particular, the device of assuming the existence of kd must be abandoned. The data of Figure 2 are based on runs in which the supersaturation and temperature were the same as in the runs of Figure 1 (S = 3.4, T = 274.0 K), but where the light intensity was reduced to 0.501 of the value used in the runs of Figure 1. The scatter of points in Figure 2 is considerably greater than the scatter in Figure 1 since signal averaging is much less effective when 20 rather than 40 runs are involved. Nevertheless, a least-squares fit to eq 10 was performed which resulted in the solid curve of Figure 2. The value of x derived from this fit was 4.0 which compares favorably with the value of 6.8 obtained by fitting the data of Figure 1 to eq 10. Assuming the validity of the model upon which we have based our equations, the critical degree of polymerization, x, should be identical in Figures 1 and 2, since both temperature and supersaturation are the same in those figures. The least-squares fit in Figure 1 leads to a value of a = 0.1 12, while that in Figure 2 provides a value of a = 0.055. Again, the a’s should be identical, but given the nominal validity of the model, the agreement must be considered satisfactory. The least-squares values for K, unfortunately, can only be very inaccurately connected to I unless the “noise” in the signal is averaged out very completely. This is apparent from eq 13 in which the second term on the right, which does not contain I , is seen to be dominant. Noise which affects the value of x will therefore obscure the effect of a change in I . However, only Kin eq 10 should change in passing from Figure 1 to Figure 2; both x and a should remain the same. In fact, since the ratio of the light intensity in Figure 2 to that in Figure 1 is 0.501, K for Figure 2 should be -12.2 as compared with the value of -11.53 obtained from the least-squares fit of Figure 1. We are, of course, assuming that noise has ceased to play a role. Employing this value of -12.2 for K i n Figure 2 and using x = 6.8 and a = 0.1 12 (the same values found in the data of Figure l ) , we can plot the curve of J vs. t , for the conditions of Figure 2, predicted from the data of Figure 1. The dashed curve in Figure 2 is calculated from eq 10 in this manner. It exhibits reasonable agreement with the least-squares fit (solid curve) to the data of Figure 2 , itself. It is significant that the maxima of the two curves are well matched in both position and height; Le., the solid curve of Figure
J . Phys. Chem. 1984, 88, 6670-6675
6670
2 has about half the rate of the curve of Figure 1. In this connection it should be noted that the time t* at which the maximum occurs when x and a are maintained constant (temperature and supersaturation are constant) is easily shown, via eq 12, to be given by t* = (x - 2)/a (14)
The agreement of the positions of maxima for the two curves thus indicates that both a and x are the same in the two cases, and the deviation in the least-squares-fitted values may be experimental errors resulting from the less complete signal averaging based on only 20 runs. It should be possible to control the cloud chamber conditions so that single polymer nucleation occurs under different degrees of supersaturation (different values of M) but at the same temperature. Values of a obtained from such experiments could then be plotted vs. M to yield a straight line with slope, according to eq 5, equal to k,. The intercept of this plot at M = 0 should then yield kd. We have not yet accomplished this, but if we assume, to a first approximation, that kd may be set equal to zero, then we can derive, from a alone, a value for k,. A value so derived from the present data is k, 3 X cm3 molecule-’ s-’. Values of k, measured4 for the free-radical chain reaction in liquid vinyl cm3 molecule-’ s-]. Again acetate are of the order of 5.4 X there is nominal agreement.
-
(4) Matheson, M. S.; Auer, E. E.; Bevilacqua, E. B.; Hart, E. J. J . Am. Chem. SOC.1949, 71, 2610.
In closing, it must be reemphasized that the device of assuming the existence of kd can only be very approximate for these pulsed experiments. A growing free radical which diffuses out of the UV beam may continue to grow at different supersaturations and different temperatures as it wanders throughout the chamber and may then return to be counted within the optically defined volume of observation. An accurate assessment of the kinetics therefore requires knowledge of the time-position history of each counted polymer. Such information can only be made available through a detailed solution of the complicated boundary value problem involving both diffusion and reaction (and very complicated boundary conditions). In fact, since the critical degree of polymerization depends upon supersaturation, polymers may nucleate at different sizes at different elevations in the chamber. If these nucleation events occur at a position directly above the volume of observation, the drops which they produce will be counted. This is the reason why, in the measurements of the present paper, the count rate is taken to be proportional rather than equal to the rate of polymerization. We have commenced work on the numerical solution of the above-mentioned boundary value problem. The solution is absolutely necessary if the cloud chamber technique is to be evolved into a precise method for measuring the kinetics of gas-phase polymerization. Acknowledgment. This work was supported by NSF Grant No. ATM82-13871. Registry No. Vinyl acetate, 108-05-4.
Collisional Deactivation of K(5*PJ) by HPmIdentlficatlon of the Primary Quenching Channel King C. Lin, Alan M. Schilowitz, and John R. Wiesenfeld* Department of Chemistry, Cornell University, Baker Laboratory, Ithaca, New York 14853 (Received: April 4, 1984)
Pulsed photodissociation of KI at 193 nm was used as the source of K(~’PJ)in a series of experimental studies of collisional deactivation. By comparison of the yield of ground-state K(4’S,,J in the presence and absence of H2 and Dz, it was possible to demonstratethat chemical reaction plays no significant role in the deactivation process. Of the available quenching channels, that leading to the intermediate S2S1 and 32DJ states appears to dominate. The possible importance of near-resonant electronic-to-vibrational energy t r a d e r is discussed and an application to the refinement of alkali metal lasers presented.
The term “superalkali” has been coined to describe alkali metal atoms in their electronically excited states.] The low ionization potential characteristic of alkali metals permits extensive delocalization of the single ns electron in the ground state during the course of collisions. The resulting high reactivity2 (made manifest by large cross sections occasionally > I nm2) is also to be expected for the superalkalis as the unpaired electron lies even closer to the ionization potential. Whether such extraordinary reactivity is actually realized has primarily been a matter of conjecture for it is experimentally difficult to differentiate between chemical reaction of the superalkali, M*, and substrate, AB M* AB MA B and the competing quenching process
+
-.
+
(1) Bersohn, R. In “Molecular Energy Transfer”; Levine, R., Jortner, J., Eds.; Wiley: New York, 1976; p 154. Barker, J. R.; Weston, R. E. J. Chem. Phys. 1976, 65, 1427. Bersohn, R. In “Alkali Halide Vapors: Structure, Spectra, and Reaction Dynamics”; Davidovits, P., McFadden, D. L..Eds.; Academic Press: New York, 1979; p 345. (2) Davidovits, P.; In “Alkali Halide Vapors: Structure, Spectra, and Reaction Dynamics”; Davidovits, P., McFadden, D. L., Eds.; Academic Press: New York, 1979; p 331.
0022-3654/84/2088-6670$01.50/0
+
+
M* AB M AB Further difficulties in assessing the relative importance of specific deactivation channels arise when multiple reactive and/or quenching pathways become available for the deactivation of a superalkali atom. This is often the case for alkali metals in excited states above the lowest-lying n2PJ level. Early experiments concerned with characterization of superalkali atom kinetics centered upon the measurement of total deactivation cross section^.^,^ The use of variable-wavelength alkali halide photodissociation as a source of excited atoms permitted the production of superalkalis with a range of kinetic energies. Deactivating molecules could be grouped into four broad classes depending upon their electronic structure. Those molecules such as the alkanes with lowest unfilled orbitals of u* symmetry were generally inefficient in quenching M*. Those in which the lowest unoccupied orbital is a*,but which cannot react exothermicaIly to yield chemically distinct products, quench M* somewhat more efficiently. The most efficient deactivators provide +
(3) Earl, B. L.; Herm, R. R. Chem. Phys. Left. 1974, 22, 95. (4) Earl, E. L.; Herm, R. R. J . Chem. Phys. 1974, 60,4568.
0 1984 American Chemical Society
