7466
J. Phys. Chem. 1992,96,1466-1412
Convective Instabilities in Traveling Fronts of Addition Polymerization John A. Pojman,* Richard Craven, Akhtar Khan, and William West Department of Chemistry and Biochemistry, University of Southern Mississippi, Hattiesburg, Mississippi 39406-5043 (Received: April 17, 1992) Traveling fronts of polymerization have been observed in unstirred solutions of methacrylic acid and benzoyl peroxide. Three types of convective instabilities have been observed. (1) The heat released by the exothermic reaction decreases the density of the reacting solution but changes in the composition tend to make the density increase. The net change in density is negative. Simple convection results, which causes a downward propagating front to remain perpendicular to the gravitational vector even as the tube is tilted. (2) However under some conditions of concentrationand temperature, long slender “fingers” of polymer are observed to sink from the solid polymer front. The appearance of structures analogous to “salt fingers” in ocean layers of different temperaturesand salinity are analyzed in terms of the theory of double-diffusive convection. Similarities with convection in directional solidification are considered. (3) Pulsating fronts have been observed which result in a striated material. The energy of activation of the fronts was determined and used to show that a convective instability instead of a pure thermal one is the cause of the pulsations.
Introduction An autocatalytic reaction in an unstirred vessel can support a constant velocity wavefront resulting from the coupling of diffusion to the chemical reaction. A flame front is a common example in which heat is the autocatalytic species that diffuses into unreacted regions, stimulating a reaction that produces more heat. Numerous reactions in solution have been described in which a front of chemical reactivity propagates through the medium from the site of an initial concentration perturbation.’-’’ Although there has been extensive work on traveling fronts, little has been done with polymer reactions. However, traveling wave fronts in populations of short self-replicating RNA variants have been created in thin capillary tubes.” Traveling fronts have been studied in synthetic polymerization reactions under high pressure by workers in the former USSRI2-” and recently under ambient conditions. Pojman and EpteinI9postulated that any exothermic traveling reaction front could cause conditions for double-diffusive(multicomponent) convection if the sum of the partial molar volumes of the products was less than the sum of the partial molar volumes of the reactants; i.e the product solution is more dense than the reactant solution. Or, it may be thought of as having a thermal expansion (ApT < 0) and an isothermal contraction (Apt > 0). This mechanism has been invoked to explain the orientation d e pendence of front velocities in the iron-nitric acid system. The convection is called “double-diffusive” or “multicomponent” convection2s22 and occurs in a wide variety of physical systems, including ocean layer mixing, crystal and zone cent r i f ~ g a t i o n .The ~ ~ effect on traveling fronts in aqueous systems has recently been studied.6q19.25-26 Free-radical polymerizations are very exothermic and form products that are more dense than the monomer. Thus, double-diffusiveconvection should occur in traveling fronts in a polymerization reaction. Solid-state or gasless combustion reactions (thermites) bear strong similarities with traveling polymerization reactions. Examples include the synthesis of intermetallic compounds (e&, nickel and aluminum)27and the synthesis of borides (niobium and boron).28 The reactants are placed in a cylinder as powders, and one end is heated to an ignition temperature, initiating a traveling front. Merzhanov et al. r e p ~ r t e d that ~ ~ sin~the ~ reaction of Nb + B, the velocity exhibited periodic pulsations and that the product possessed a layered structure with the number of layers equal to the number of pulsations. A great deal of analytical and numerical analysis has been done to reproduce these observation^.^'-^^ Bayliss and M a t k o w ~ k ypredicted ~~ that there should be a period-doubling route to chaotic propagation. They performed a bifurcation analysis in which they studied the changes in qualitative behavior of the system as a function of the bifurcation parameter that was related to the energy of activation and the difference in temperature between the reacted and unreacted
regions. However, experimental studies of thermite reactions are Micult to perform. htremely high temperatures (2000 K)must be dealt with, and rapid observations must be ma& because fronts propagate with velocities on the order of 1 ~ m / s . ~ ~ Traveling fronts of addition polymerizations are very similar to condensed phase combustion because both involve a thermal propagation mechanism. Therefore, polymer fronts provide an opportunity to test the predictions of nonlinear propagation phenomena. Traveling fronts are ideal for thisstudy because both the activation energy and temperature difference between reacted and unreacted materials can be adjusted. The effective activation energy can be adjusted by adding a promoter such as N,N-dimethylaniline.I8 The solution temperature can be controlled by immersing the reaction tube in a constant temperature water bath. We report here OUT observations of “fwcrs” of poly” formed at the interface of descendingfronts and of pulsating fronts, both of which are the result of convective instabilities. Experimeatrl Section A solution of methacrylic acid and benzoyl peroxide was p l a d into a test tube. Fronts were initiated by the addition of a p proximately a half mililiter of Nfl-dimethylaniline (DMA) to the top of the solution. constant velocity fronts appeared as slowly moving (-1 cm/min) regions of solid polymer. All experiments were performed in a hood behind a shatterproof shield because some tubes have exploded. Methyl ethyl ketone peroxide (MEKP), also known as 2-butanone peroxide (Aldrich), was used as a 50 wt % solution in dimethyl phthalate. Practical grade cobalt(I1) naphthenate (Fluka) was used as a solution in mineral spirits and was approximately 8% cobalt. All other chemicals were of reagent grade (Aldrich) and used without further purification. It was found that the presence of the inhibitor in the methacrylic acid did not affect the velocity.I8 All reactions were performed in 2.2 X 25 cm culture tubes unless otherwise indicated. Gel permeation chromatography was performed with a Waters GPC, eluting THF through four Ultrastyrogel columns. Samples were prepared by converting the polymer to poly(methy1methacrylate) using diazomethanein order to make the samples sohble in tetrahydrofuran. The columns were calibrated with polystyrene standards. Video images were digitized on a Macintosh IIcx with a RasterOps 364 video board, and the gray scale was measured on a contrast-enhanced image using ImageAnalyst (Automatix) software. Temperature measurements were made with four microthermocouples (Omega) and a Strawberry Tree A/D board on a Macintosh IIcx. Results Immediately under the descending front, vigorous fluid motion can be observed. Some of this is caused by bubble formation
Q022-3654/92/2096-1466S03.oo/o 0 1992 American Chemical Society
Traveling Fronts of Polymerization
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7467 the center, the temperature is almost 50 OC higher than at the wall. Fingering did not normally occur during the entire propagation. It would often appear and then stop during the course of propagation. Fingering would never occur all the way until the front reached the bottom of the tube. During some reactions, as the front reached the bottom of the tube, the front would pulsate, altemating between accelerating and slowing down. During the acceleration phase, vigorous fluid motion could be observed under the front. The polymer formed is striated with a length scale of 0.1 cm, as seen in Figure 7.
Figure 1. Photograph of a traveling front of methacrylic acid polymerization in 2.2 cm diameter tube with [benzoyl peroxide] = 1 .O g/100 mL. The tube has been tilted 45O to the vertical. Note that the front is perpendicular to the gravitational vector.
during the polymerization as the temperature reaches 195 O C , I 8 exceeding the boiling point of methacrylic acid (163 OC). However, more interesting is the rolling motion of the low viscosity polymer. As monomer reacts, it rapidly heats up and increases its volume. Because of the solid mass of polymer above it, it can only swell and push down. At lower initiator concentrations, less fluid motion exists. The effect of the tube orientation with respect to the gravitational field was investigated. If the tube in which a front is propagating downward is tilted 4 5 O to the vertical, the reacting solution ascends along the solid front. This material quickly forms a solid front perpendicular to the gravitational vector, as shown in Figure 1. If the tube is returned to its original orientation, the front levels off. The velocity of the front was not affected. If the tube is inverted so that the front would be propagating upward, vigorous antisymmetricconvection of the unreacted solution ensues. The increased fluid motion removes sufficient heat to stop the reaction. With higher concentrationsof benzoyl peroxide (>O.S g/100 mL), descending "fingersn of polymer are observed which break off to form drops of solid polymer raining down the tube. Figure 2 shows photographs of the evolution of fingering. Notice that the fingers occur only in a central core. Figure 3 shows the axisymmetric hole left in the polymer. Figure 4 shows a diagram of the temperatures and concentrations of benzoyl peroxide which result in fingering. There is a minimum and a maximum concentration necessary for fingering. The appearance of fingering is very sensitive to the temperature of the solution. We have taken samples of the fingers after they have settled on the bottom of the tube and analyzed their molecular weight distribution via gel permeation chromatography. The fingers have a substantially high molecular weight, with material extending above 106 g/mol, as compared to los g/mol for the bulk polymer (Figure 5). Figure 6 shows the temperature profile of the front at four different distances from the center of a 2.2 cm (i.d.) tube. Near
Msctrssiaa The density of poly(methacry1ic acid) is greater than that of methacrylic acid itself.% Consequently, the polymer formed under the descending front could sink were it not for thermal effects. The large amount of heat released by the reaction increases the temperatureand decreases the density of the polymerizing solution immediately below the front. The thermal expansion more than exceeds the isothermal contraction. It is this large thermal expansion that causes the polymerizing material to rise when the tube is tilted. Such convection is completely expected from simple buoyancy. The stability analysis for the density gradient induced by the descending front is the same as for hot, salty water overlying cold freshwater. In both situations, two components are present in the solution that have opposite effects on the density of the solution. The salt plays the same role as the polymer. Consider hot, salty water above cold freshwater as depicted in Figure 8a. The system may appear to be stable, if the density decreases with height. Yet, it may not be. Imagine that at the interface between the layers, a small parcel of the upper solution were to deviate from its position by descending into the cold, fresh region. Because the temperature and concentration are higher than in the surrounding region, heat and salt diffuse out. The heat leaves at a greater rate, because of its larger diffusivity. Now the parcel is cool and dense; because of its higher salt concentration, it sinks. The surface area of the parcel increases, accelerating the loss of heat and the corresponding increase in density. Similarly, if a parcel of cold freshwater protrudes into the hot, salty layer, heat diffuses in faster than the salt. This leaves the parcel less dense than the surrounding layer, and the parcel will rise. .These results are known as "salt fingers", which appear as long slender regions of alternately descending and ascending fluid. The fingering in the polymer system is different from fingering in other systems because fingers can only form in one direction (Figure 8b). The solid polymer prevents fingers of unreacted monomer from rising. Therefore, the fluid that is displaced by the descending fingers must be replaced by flow of the bulk solution. We have observed antisymmetric fluid motion (down the center and up one side) which causes the fingers to curve toward the sides of the tube. This system is analogous to convection induced in directional solidification. Nguyen et al." studied upward solidification of lead-thallium alloys. The density gradients were exactly reversed from our case but analogous; the solid/liquid interface was ascending with an upward decreasing thermal gradient and an upward increasing solutal density gradient. Because of the opacity of the liquid, they were not able to observe the fluid flow directly. However, by analyzing the striations left in the solid, they concluded that convection had occurred. Depending on the size of the crucible, the "pattern has a central axisymmetric core and an outer annulus"?' Because of the very high thermal conductivity of the liquid metal, the mechanism driving this convection is closer to the Binard-Marangoni process than to a double-diffusive one. Nonetheless, further study of the convection in traveling polymer waves should provide insight into systems such as the lead-thallium one that are difficult to study directly. McCay and McCay3*studied upward solidification of ammonium chloridewater solutions in which the solute decreased the density of the solution. They observed ascending fingers, which are analogous to the descending ones in our polymer system.
7468 The Journal of Physical Chemistry, Vol. 94, No. 18, 1992
Pojman et al.
Figure 2. Evolution of descending 'fingers" of polymer in a solution of methacrylic acid and benzoyl peroxide. The experiment was performed in a 2.2 cm (i.d.) tube but only the central 1 cm wide region is shown. Ambient temperature was 26 OC. [Benzoyl peroxide] = 1.0 g/100 mL.
Figure 9 shows the density gradients created by this process. They are the same as those created by a descending polymerization front. In both systems, a statically stable (but dynamically unstable) net density gradient is formed. Higher concentration of initiator decreases the fingering; the vigorous fluid motion at the front does not allow the formation of fingers. This may also reflect the fact that at high initiator concentrations, the thermal gradients may be larger. Higher
temperatures mean that more time is required for the heat to diffuse out of a parcel of polymer/monomer solution than at lower temperatures. If the heat cannot diffuse out sufficientlyto increaSe the density to a value greater than the surrounding solution before the front "catches up", then no fingering will be ~bserved.'~ The large molecular weight difference between the fingers and the material left in the front is caused by the temperature difference between the front and the fingers. High temperature does
The Journal of Physical Chemistry, Vol. 96, No. 18, I992 7469
Traveling Fronts of Polymerization
I
I
I
Figure 3. (a, top) Cross section of a polymer sample from a front that exhibited (left) and did not exhibit (right) fingering. Notice the axisymmetric hole created by polymer fingering. (b, bottom) Longitudinal section of same sample in part a. The diameter of each sample is 2.2 cm. Same conditions as Figure 2. 10
3 n m a M m ~ n a m A
A
m ~a ~
a Aa
L
annulus
I
c.
;;,::::... :,
C
. A .
20
25
.A
. . . . A . A
0.0
A A A
30
35
,
40
Temperature ("C)
Figme 4. Concentrationsof benzoyl peroxide and temperatures at which fingering occurs. The temperature of the monomercinitiator solution was controlled by equilibrating the solution in the 2.2 cm (i.d.) tube in a water bath (fO.l "C). The front was allowed to propagate within 2 cm of the water level but not under it, because the increased heat loss extinguishes the front. 0 indicates fingering was observed during some part of the propagation; A indicates no fingering was observed.
not favor high molecular weight material nor complete conversion. At elevated temperatures, the initiator can decompose before all the monomer can react (initiator "b~mout")?~Even if the initiator is not all consumed, the average molecular weight decreases at
20 25 30 Elution Volume (mL) Figure 5. Relative molecular weight of the fingers and the polymer in the solid front as determined by gel permeation chromatography (GPC). Annulus indicates material adjacent to the fingering zone; edge material was taken from the region adjacent to the tube wall. Molecular weights indicated are referenced to polystyrene standards.
IS
higher temperatures because the same amount of monomer is distributed over more chains that are formed as the initiator decomposes more rapidly than at low temperature. High temperature also affects the molecular weight by competing with the "gel e f f e ~ t " . ~As , ~the ~ degree of polymerization becomes high, the viscosity increases. T h e diffusion limited termination reactions
Pojman et al.
7470 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 200
u^
150
0 W
I
HOT, SALTY WATER
0.25 cm 0.70 cm A ‘ 1 . 0 c m
___-
4
100
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Monomer Solution I
COLD, FRESH WATER
Solid Polymer I
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-
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POLYMER SOLUTION
0
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Position (cm) Figure 7. Gray scale versus position along the direction of propagation of solid polymer formed by a pulsating front. [Benzoyl peroxide] = 1.0 g/ 100 mL. The regularly spaced peaks in gray correspond to striations left in the polymer. The video image was digitized on a Macintosh IIcx, and the gray scale measured on a contrast-enhanced image using ImageAnalyst software.
are slowed, thus increasing the overll rate of reaction and the length of the molecules. High temperature decreases the viscosity, increasing the rate of termination. The polymer fingers sink due to their heat loss. Thus, they have the same initiator concentration as the other material along the front, but their temperature is considerably lower which keeps the viscosity high and the termination rate low as well as keeping the rate of initiation low. In the polymer system, the axisymmetric distribution of fingers is not due to fluid motion but is related to the radial thermal gradient. Initially we thought that fingering would not occur in 1.5 cm diameter tubes because a critical tube size was required for the fluid motion induced by the descending fingers. However, if methyl ethyl ketone peroxide (MEKP) is used instead of benzoyl peroxide, fingering can occur along the entire width of the front, in both 1.5 and 2.2 cm (i.d.) tubes (Figure 10). The same is true for tert-butyl peroxide. The appearance of fingering is enhanced by the addition of 1 drop of cobalt naphthenate solution to 20 mL of MEKP/monomer solution. Cobalt naphthenate is a standard promoter for the decomposition of Notice that the hole left in the polymer in Figure 10 tapers toward the bottom of the tube because of the pressure buildup that compresses the gaseous products and reduces the volume. It might appear that the promoter enhances fingering by increasing the rate of polymerization initiation in regions farther from the solid polymer zone. This would give the fingers more time to form before the front “catches up”.19 Yet, promoting the decomposition of BPO with DMA did not cause fingering in narrow tubes nor across the entire front in wide tubes. We propose that with benzoyl peroxide, the polymerization occurs too rapidly for fingers to form except in the central region
I
Figure 8. (a, top) Mechanism for double-diffusive convection. Apc and L\pr have opposite signs, and the net density gradient appears to be stable. However, if a small parcel of the warm, salty solution enters the lower section, because the heat diffuses faster than the salt, the parcel is left with a greater density than the surrounding medium. The buoyant force pulls it down. If a small parcel of cold freshwater enters the warm, salty solution, heat will diffuse in faster than the salt. The parcel will be less dense than its surroundings and rise. (b, bottom) The mechanism for fingering beneath a descending polymerization front. Immediately below the solid polymer is a region of polymer/monomer solution. The polymer increases the density, but the large temperature increase caused by the polymerization decreases the density. However, if a small parcel of the hot, polymer solution enters the lower section, because the heat diffuses faster than the polymer molecules, the parcel is left with a greater density than the surrounding medium. The solid polymer prevents the formation of ascending fingers, and so antisymmetric fluid flow (indicated by the curved arrow) is induced to replace the descending material.
Psolute Pthermal Pnet Figure 9. Thermal (pthcml), solute (pwlutc),and net (pnet)density gradients induced by the traveling front and by directional solidification of ammonium chloride. The traveling front is descending, but the directional solidification is proceeding upward. In both cases, the net density gradient is statically stable, but dynamically unstable.
where the high temperature prevents the gel effect by keeping the viscosity low. It is in this central region from which the polymer has not solidified that fingers can descend. In narrow tubes (1.5 cm i.d.) fingers never form because the temperature of the front is too low. However, it is not clear why fingering occurs in narrow tubes and across the entire front in systems with MEKP and tBPO nor why a promoter such cobalt naphthenate encourages fingering. Further, no fingering has been observed in fronts with cumyl peroxide as the initiator over a range of 1-40 g/100 mL at 24.7 “C. We believe that the appearance of fingering is a function of the temperature (both of the unreacted solution and the front), density, and viscosity of the polymer solution at the front as well as the rate of front propagation. The initiator can affect all of these except the initial temperature of the solution.
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Traveling Fronts of Polymerization +*
The Journal of Physical Chemistry, Vol. 96, No. 18,1992 7471
n
-
E
15
.
W
E
0
.C. .d c1
v)
Fingering 1.1 cdmin
1.5 cm-
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E =6Okl/mol
Effect of Fingering am h t Velocity Fingering has been observed to increase the rate of front propagation in the iron-nitric acid frontsz6by increasing the transport of the autocatalytic species. However, we have observed fingering slow down the rate of the polymer fronts (Figure 11). The fingers remove heat from the front, lowering the temperature and the rate of thermal diffusion. Thus, the appearanceof fingers makes it difficult to study the dependence of the front velocity on solution temperature. polsrting h
t The front pulsations are observed after fingering as the front nears the bottom of the tube. Pulsations were not observed in 1.5 cm tubes. The front accelerates and simultaneously vigorous turbulent fluid motion occurs directly under the front. A layer of bubbles is left in the solid polymer, most likely consisting of methacrylic acid vapor and COP The front slows down while the
g100 mL
E-= 33 kl/mol
-
From Figure 5 it can be seen that the polymer formed in the annulus surrounding the fingering zone has a great deal of low molecular weight material. Unreacted methacrylic acid can be seen (it is yellow) in the annulus indicating that the initiator has 'burned out". The other factors described above cause a lower molecular weight. Therefore, the region in which fingers formed must have had an even lower molecular weight and lower viscosity that allowed fingers to form. The fingering process usually ceases before the front reaches the bottom of the tube. Chechilo and Enikolopyan observed material breaking away from a descending front of methyl methacrylate polymerization, which they could eliminate by applying pressure of greater than 5000 atm.13 Therefore, we suspected that a pressure buildup caused by the COz released by the benzoyl peroxide decomposition could be the cause. We measured the pressure and found an increase of 2.5 atm. It is not clear if this would be sufficient to differentially compress the monomer more than the polymer and reduce the density difference between a growing finger and the surrounding solution.
I6
14
Figure 11. Position of a front with a benzoyl peroxide concentration of 4.0 g/1o0 mL at 24.0 OC is shown as a function of time in a 2.2 cm (i.d.) tube, with and without fingering. Note the lower velocity when fingering OCCUlTed. 1.51
F i i 10. Fingering across the entire front of methacrylic acid polymerization with methyl ethyl ketone peroxide (8.0 mL/100 mL) as initiator with 1 drop cobalt naphthenate solution/20 mL in a 1.5 cm (i.d.) tube. The inset shows a cross section of the front and the hole formed by the fingering. Notice that the hole tapers from a diameter of 1 cm at the start of the fingering zone down to 0.4 cm at the end of this 4 cm long section.
12
10
Time (min)
_ I -
. d
0.0
0.00325
0.0033
0.00335
I
0.0034
1JT w-9 Figan 12. Natural log of the front velocity versus 1/T is shown for benzoyl peroxide concentrations of 0.5 g/ lo0 mL benzoyl peroxide and 8.0 g/lo0 mL. Three line are fit to plausible regions of linear behavior, and the respective energies of activation indicated (E, = -slope/R). Experimental conditions are the same as in Figure 4.
fluid motion subsides. The cycle is repeated with a frequency of about 10 pulsations/min. Matkowsky and Shiva~hinsky~~ calculated that adiabatic thermal front propagation is stable when
w h m E, is the activation energy of the front, Tois the temperature of the unreacted material, and Tmxis the temperature of the reacted zone. We measured the E, of the front. It was not possible to insert the entire tube into a water bath (unless it was >50 "C) because the large thermal conductivity of water removed enough heat to quench the front. Instead, the tube was inserted in a bath such that the front remained 2 cm above the water line. Figure 12 shows the logarithm of the velocity plotted versus l/Tfor the two different concentrations in which no fingering was observed. The other concentrations did not show any consistent trend, because of the extensive fingering. The consistency of the data is less than would be desired, but it was not possible to perform experiments at higher temperatures than 40 "C with 8.0 g/100 mL BPO because the entire solution would polymerize homogeneously. Three lines are fit to the data, providing an estimate of the upper limit of the E, of 60 kJ/mol. The stability parameter for To= 298 K, T,, = 463 K, and E, = 60 kJ/mol, a = 2.7, is well below the stability limit. The pulsations observed occur as the front nears the bottom of the tube. However, if the front is unstable, it should be unstable
1412 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 during the entire time of propagation. Further, because of the radial temperature gradient, we would expect the regions of the front would pulsate with different frequencies in accord with the temperature gradient. Neither of these phenomena were observed. Therefore, we propose the following mechanism for the pulsations: As the front nears the bottom of the tube, some properties of the fluid change, perhaps because of fingers that have fallen and redissolved. If the thermal conductivity of the solution decreases, then heat from the front cannot escape rapidly enough before convection sets in. The vigorous fluid motion removes heat, cooling the front and slowing its propagation. The cooler front does not induce as much fluid motion, so it heats up, accelerating and repeating the cycle.
Conclusion Traveling fronts of methacrylic acid polymerization provide a test of the theory proposed for convectiveeffects on some traveling fronts. Because there is a large thermal expansion caused by the exothermicity of the reaction as well as an isothermal contraction during polymerization, fronts can exhibit a double-diffusive instability resulting in polymer "fingers" analogous to salt fingers in ocean layer mixing and directional solidification. The fingering is very sensitive to the initiator concentration, the solution temperature, and the type of initiator used. Fronts with benzoyl peroxide have fingering only in the center of a 2.2 cm diameter tube and none at all in 1.5 cm tubes. However, fingering occurs in narrow tubes if methyl ethyl ketone peroxide is used instead of benzoyl peroxide. Contrary to previous work on convection and traveling fronts, the fingering slows down the fronts instead of increasing the propagation velocity. Fronts exhibit pulsations that are the result of a convectivefluid motion and not a purely thermal instability. The energy of activation of the fronts was determined and shown to be less than the critical value for a pulsating instability in a thermal front. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. We acknowledge support from the National Science Foundation's Mississippi EPSCoR Program. We also thank Bernard Matkowsky for helpful discussions. Registry No. Methacrylic acid, 79-41-4.
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Pojman et al. (6) Bazaa, G.; Epstein, I. R. J . Phys. Chem. 1985,89, 3050-3053. (7) Hanna, A.; Saul, A.; Showalter, K. J. Am. Chem. Soc. 1982, 104, 3838-3844. (8) Field, R. J.; Burger, M. Oscillations and Trawling Waws in Chemical Systems; Wiley: New York, 1985. (9) Harrison, J.; Showalter, K. J . Phys. Chem. 1986, 90,225-226. (10) Szirovicza, L.; Nagyp61, I.; Boga, E. J . Am. Chem. Soc. 1989, I I I, 2842-2845. (1 1) Bauer, G.J.; McCaskill, J. S.;Otten, H. Proc. Narl. Acad. Sci. USA. 1989,86, 7937-7941. (12) Chechilo, N. M.; Enkolopyan, N. S.Dokl. Phys. Chem. 1974,214, 174-176. (13) Chechilo, N. M.; Enikolopyan, N. S . Dokl. Phys. Chem. 1976,230, 840-843. (14) Chechilo, N. M.; Enikolopyan, N. S.Dokl. Phys. Chem. 1975,221, 392-394. (15) Enikolopyan, N. S.;Kozhushner, M. A,; Khanukaev, B. B. Dokl. Phys. Chem. 1974,217,676478. (16) Khanukaev, B. B.; Kozhushner, M. A.; Enikopyan, N. S. Dokl. Phys. Chem. 1974,214, 84-81. (17) Surkov, N. F.; Davtyan, S.P.; Rozcnhcrg, B. A.; Enikolopyan,N.S. Dokl. Phys. Chem. 1976,228,435-438. (18) Pojman, J. A. J. Am. Chem. Soc. 1991,113,62844286. (19) Pojman, J. A.; Epstein, I. R. J . Phys. Chea. 1990, 94,49664972, (20) Huppcrt, H. E.; Turner, J. S.J. Fluid Mech. 1981, 106, 299-329. (21) Turner, J. S.Annu. Rev. Fluid Mech. 1985, 17, 11-44. (22) Turner, J. S . Buoyancy Effecrs in Fluids; Cambridge University Prm: Cambridge, 1973. (23) Langlois, W.E. Annu. Rev. Fluid Mech. 1985, 17, 191-215. (24) N w n , P.; Schumaker, V.; Hahll, B.; Schwedes, J. Biopolymers 1969,7,241-249. (25) NagypB1, I.; Bazaa, G.;Epstein, I. R. J . Am. Chem. Soc. 1986,108, 3635-3640. (26) Pojman, J. A.; Epstein, I. R.; Nagy, I. J . Phys. Chem. 1991, 95, 13061311. (27) Anselm-Tamburini, U.; Munir, 2. A. J. Appl. Phys. 1989, 66, 5039-5045. . .. . (28) Puszynski, J.; Degreve, J.; Hlavacek, V. Znd. Eng. Chem. Res. 1987, 26, 1424-1434. (29) Merzhanov, A. G.;Filonenko, A. K.; Borvinskaya, I. P. Sou. Phys. Dokl. 1973. 208. 122-125. (30) Shkdiiky, K. G.;Khaikin, B. I.; Merzhanov, A. G. Combust., Explos. Shock Waves 1971, I , 15-22. (31) Kaper, H. G.;Leaf, G.K.; Margolis, S.B.; Matkowsky, B. J. Combust. Sci. Technol. 1987, 53, 289-314. (32) Matkowsky. B. J.; Sivashinsky, G.I. SZAM J . Appl. Marh. 1978,35, 465-478. (33) Frankel, M. L. Phys. Lerr. A 1989, 140, 405-410. (34) Bayliss, A.; Matkowsky, B. J. J. Comp. Phys. 1987, 71, 147-168. (35) Bayliss, A.; Matkowsky, B. J. SZAM J . Appl. Marh. 1990, 50, 437-459. (36) Poly(methy1 methacrylate), a related compound, is 22% more dense than its monomer. (37) Nguyen Thi, H.; Billia, B.; Jamgotchian, H. J . Fluid Mech. 1989, 204, 581-597. (38) McCay, M. H.; McCay, T. D. J. Thermophys. 1988, 2, 197-202. (39) Davytan, S.P.; Zhirkov, P. V.; Vol'fson, S.A. Russ. Chem. Rev. 1984, 53, 150-163. (40) Norrish, R. G.W.; Smith, R. R. Narure 1942, 150, 336-337. (41) Trommsdorff, E.; KBhle, H.; Lagally, P. Makromol. Chem. 1948, I , 169-198. (42) Updegraff, I. H. Unsaturated Polyester Resins. In Handbook of Composites; Lubin, G., Ed.; Van Nostrand: New York, 1982; pp 17-37.
