J. Phys. Chem. 1992,96, 1511-1514 toward micelles associated with a radical. In such casea,the effects of redistribution are probably not very important. However, at lower hydrophobe/SDS ratios where the hydrophobe is never fully consumed before full conversion of the acrylamide, these processes will play an increasingly important role. Considering all of the points discussed here, it is clear that any copolymer sample taken at high conversion levels will be polydisperse, both in composition and in molecular weight distribution. Indeed, those copolymers prepared at high ratios of hydrophobe/SDS will also contain a high proportion of acrylamide homopolymer since the hydrophobe is exhausted before the acrylamide monomer (Figure 9).
Conclusions The results reported here show that despite the presence of high levels of surfactant (35cmc) and low levels of a hydrophobic comonomer ( 0.247. The two saddle-node points occuring within the bulge of the crescent enclose an isola. The latter is born at the isola point at the lower minimum at Bo = 0.061 338, and it is transformed into a mushroom at the upper minimum at Po = 0,088925. The domain of the mushroom, 0.088925 < bo < 0.122, is characterized by four saddlenode points. The steady states lose their stability through a set of Hopf bifurcations emanating from the left horn of the crescent. The locus of Hopf points is shown by the dotted curve. These results agree with the analysis of Gray and Scott.12 2. Inbomogeneops Cpse: Quantitative Sbift of Bifurcation Set 5. The set of bifurcations which arises from the homogeneous limit is labeled &, to distinguish it from the emergent set E,. The dependence of xh on the mixing parameter x is shown in Figure 2b far the homogeneous and inhomogeneous cases,where x = 0.15. (16) Brodkey, R. S. Chem. Eng. Commun. 1981,8, 1-23. (17) The degree of inhomogeneity x summarim a great amount of hydrodynamic detail of the incompletely stirred CSTR. The problem of quantitatively predicting the value of this model parameter for any particular experiment has not been solved. (1 6) Seydel, R. From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis; Elsevier: New York, 1988. (19) Seydel, R.; BIFPACK, A Program Package for Calculating Bifurcations; State University of New York at Buffalo: Buffalo, NY, 1985; (second version).
Inhomogeneity of a
The Journal of Physical Chemistry, Vol. 96, No. 3, I992 1513
CSTR with Autocatalator
1.5 r
1
p
b
0.031
I
o."~'"""''''""'"""'' 0
0.05
0.1
0.2
0.15
0.25
0
0
0.06
e
0.12
0.18
0.24
e
1.5 2[
0.1
0.05
0.5
0
0.05
0.1
0.15
0.2
0.25
6
e, mushroom; f, oscillations; g, monostability. (b, bottom) Illustration of the shift of the bifurcation set Ehwith degree of inhomogeneity x = 0.0 and 0.1 5. The dynamics at points K,L,Mare described in the text.
I
0'" 0
" "
0.05
" " " " " "
0.1 X
0.1
0.01 8
Figure 2. (a, top) Bifurcation set for the homogeneous autocatalator. Full lines are loci of saddle-node points, and dotted curves, loci of Hopf points. The Bo points indicated by the letters mark the following dynamic domains: a, monostability; b, isola point; c, isola; d, birth of mushroom;
o.8
0
0.15
0.2
Figure 3. The response (product yield) at point L (Figure 3) as a function of the segregation x: Bo = 0.17; B = 0.028.
Increasing inhomogeneity causes Z,,to be displaced to a higher Bo and to shrink slightly. This figure illustrates how x may act as a bifurcation parameter: for instance, the system located at point Kin parameter space oscillates when homogeneous or slightly inhomogeneous but ceases to do so above a critical value of the segregation x. This phenomenon has been called "phase death".'Ob Conversely, oscillations are induced at point M. At point L,the system's response to increasing x is illustrated in F i i 3: a steady state becomes oscillatory and finally reverts to a steady state. Similar behavior was found in experiments on the chlorite/iodide rea~tion,'~.~ It is evident from Figure 2b that in a similar fashion an isola may vanish or be born or a mushroom may be produced from a monostable state or be converted into an isola. In addition to this bifurcation behavior, Figure 3 also illustrates that away from the bifurcation points the quantitative dynamical state attributes, i.e. steady-state concentrations and limit cycle amplitude
Figure 4. (a, top) At x = 0.06, a new bifurcation set Z,,consisting of saddlenode lines,has emerged near &. (b, bottom) The evolution of 2, with segregation x: a, x = 0.049; b, x = 0.052; c, x = 0.06; d, x = 0.07; e, x = 0.15. f, & is drawn for x = 0.049 and moves only little on the scale of the graph.
(and frequency; not shown here), are also functions of the segregation x . The quantitative shift of 2 h is thus seen to have both quantitative and qualitative consequences. These are summarized in the classification scheme of Figure 5 . 3. InhomogeneousCase x > x,,it: The Emergent Bifurcation Set E,. Beyond a critical value x,,it * 0.049 of the mixing parameter, a new bifurcation set, labeled 2,, emerges near the old one, as shown in Figure 4a. The evolution of this set as a function of x is illustrated in Figure 4b. At birth x = 0.049; 2, is limited by two closely spaced saddlenode curves, shown as a short oblique line, which confer an extremely narrow range of bistability on the single stable state in this region. With increasing x, this structure expands rapidly and deforms into an asymmetric crescent of saddle-node points. In addition, a new loop of Hopf points, nanower than that of the 9set, develop at high x. As x increases further, the emergent set Zeoverlaps and engulfs the homogeneous set Zh, and finally its lower bulge is truncated as the bovalues of the set become unphysically negative. The emergent set gives birth to emergent domains of isola, mushroom, hysteresis, and oscillatory dynamics. A domain of tristability (when 0 is used as the control parameter) exists whenever the two bifurcation sets &, and 2, overlap each other, as in Figure 4b where there are regions in Bo space where 4, 6,and 8 saddle-node points coexist.
Discussion The classification scheme, Figure 5 , summarizes our conclusions. The quantitative stirring effects (Class A) have been known for the longest time.' Steady-state concentrations (Class A.l) and oscillation attributes (Class A.2) are known to be sometimes sensitive functions of the stirring rate.211 The calculations (Figure 3) reproduce both of these aspects. The quantitative shifts of hysteresis limits in CSTR studies (Class A.3) are now well establi~hed.~ They have been successfully modeled using macro4and micromixingsv6calculations. All three subclasses A.1, A.2, and A.3 are essentially understood through the ~imulations.~ The
1514 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
A
A’
Quantltatlve effects:
A.1
stable oterdy stater
A.2
ore~~~at~onr
A.3
shin of bifurcation rot I,,
Bifurcations induced by shift of 2,: orclllrtionr, ph-• doah, IaOIO,
6
...
Emergent blfurcation set 2,:
6.1
Z, mmargrntret
6.2
blfurcrtlonr from IlmH cycle#In 4 (0.0. cham).
Figure 5. Classification scheme of stirring effects in CSTRs.
bifurcation behavior that constitutes class A’ and the quantitative shift of bifurcation sets constituting class A.3 are, it should be stressed, aspects of the same physical reality. The classification into A and A’ merely distinguishes quantitative and qualitative
Ali and Menzinger effects of the same cause. Systematic studies of class A’ effects have so far been restricted to few model calculations.6 The calculated bifurcation sets (Figures 2b and 4) for the coupled autocatalator model make the distinction between the qualitative stirring effects belonging to class A’ and those belonging to class B quite obvious. Experimental observations, however, are frequently presented as response diagrams, from which this distinction cannot be drawn without considerable further work. The diagnosis that could distinguish between cases A’ and B effects requires knowledge of the bifurcation set, either from experiment or from computations, neither or which is easily accessible, particularly when the reaction mechanism is not well established. There are reasons to believe, however, that for weak inhomogeneities (x
