J. Phys. Chem. 1985, 89, 22-32
22
FEATURE ARTICLE
Sustained Oscillations and Other Exotic Patterns of Behavior in Isothermal Reactions
P. Gray* and S . K. Scott
School of Chemistry, University of Leeds, Leeds LS2 9JT, UK (Received: June 19, 1984)
- -
The aims of this Feature Article are to show how simple or complex isothermal autocatalytic reactions can give rise to oscillatory
behavior by considering virtually the simplest possible example (A + 2B 3B; B C) under the simplest and most readily
realizable of circumstances (cstr). Although this is only a two-dimensional scheme it vividly and clearly represents all the
key aspects of many real systems, including the birth, growth, and extinction of stable oscillations. Moreover, it is a “robust”
model and it does not collapse when reversibility and parallel alternative routes are added. In the recent past, mathematical
schemes have usually been either too elaborate to be understood or have failed to satisfy such basic requirements as the principle
of detailed balance. Yet other models have failed to conserve mass or have not represented stable oscillations. Such fatal
flaws are absent not only from this prototype but also from its comparison (A + B 2B; B C saturating). The springs
of chaotic behavior in deterministic chemical models are also touched on, but it should be noted that real cstr must fall short
of perfect homogeneity and that there may still be a gap between expectation and experimental proof of chaos.
- -
I. Introduction
Ever since the revival of interest in oscillatory behavior in
isothermal systems, elaborate mechanisms and complex models1**
have been the order of the day. Fundamental aspects have often
been obscured. By contrast, the nonisothermal field has been well
served ever since the mid-1950s by studies3s4 in depth of the
simplest possible example of thermal feedback. These efforts have
been rewarded: significant new propertiesS were still being found
into the 1980s. The aim of this feature article is to perform the
same service for isothermal systems with chemical feedback
(isothermal autocatalyses) in the continuously flowing, well-stirred,
“tank” reactor (cstr). There is no substitute for the hard and
careful work on individual systems that is necessary to establish
detailed mechanisms, but by showing the rich variety of behavior
that basic models with very simple features can generate, we may
curb excessive elaboration.
I . Advantages of Open Systems. In closed systems there is
no exchange of matter with the surroundings, and chemical reactions proceed to equilibrium. Closed systems are often very
convenient experimentally. They offer economy of materials and
ease of precise control. Because their final state is one of equilibrium, chemical thermodynamics is a powerful aid to interpretation. On the other hand, stationary states other than the
final one are not in reach, multistability is impossible, and oscillatory behavior cannot be sustained. In open systems, where
there is a continuous supply of fresh reactants and a matching
removal of products, truly steady states are easily realized and
oscillatory conditions can be maintained indefinitely.
2. Advantages of Good Stirring. Chemical reactions in open
systems abound in nature (streams, fires, living organisms, single
cells), and they are commonly associated with nonuniform concentration and temperature fields. Their mathematical treatment
is correspondingly arduous, abounding in partial differential
equations.
The simplest of open systems to treat theoretically is one in
which conditions are uniform, i.e. one with “perfect mixing”. It
is also one of those that can be approximated experimentally, in
(1) Gray, P. Ber. Bunsenges. Phys. Chem. 1980, 84, 309.
(2) Gray, B. F. Spec. Per. Rep.: React. Kine?. 1975, 1 , 309.
(3) van Heerden, C. Ind. Eng. Chem. 1953.45, 1242.
(4) Uppal, A.: Ray, W. H.; Poore, A. B. Chem. Eng. Sci. 1974, 29, 967;
1976, 31, 205.
(5) Balakotaiah, V.; Luss, D. Chem. Eng. Sci. 1983, 38, 1709; Chem. Eng.
Commun. 1981, 13, 1 1 1 .
0022-3654/85/2089-0022$01.50/0
the continuously flowing, well-stirred, “tank” reactor (cstr): good
stirring can come close to perfect mixing. The distribution of
residence times is simple and is fully characterized by its mean
value; partial differential equations are banished and replaced by
ordinary ones. Finally, in stationary states, we normally have to
deal only with algebraic relations.
Since the pioneering work6 of Denbigh in the 1940s, reinforced
in recent years by the Bordeaux school,’ solution chemists have
turned increasingly to the cstr to study oscillatory reactions.
Historically, these approaches led to the discovery of oscillatory
behavior before that of multistability whereas the easiest route
to the interpretation of these phenomena starts with the idea of
multiple stationary states. Both phenomena, however, require some
form of mechanistic feedback, and for both autocatalysis provides
a simple chemical mechanism.
3. Autocatalysis as Isothermal Feedback. The feature common
to nearly all isothermal oscillatory reactions is autocatalysis. (The
obverse property of autoinhibition is considered briefly in section
VII.) Autocatalysis is shared not only by the Belousov-Zhabotinskii reaction,”I0 a wide range of halide-based oscillators,” and
the arsenite plus iodate reaction’* but by numerous enzyme
systemsI3 and by chain-branching reactions in the gas phase.I4
All mechanistic schemes include autocatalytic steps (or autocatalytic combinations of elementary steps). However, even the
simplified forms15J6of these schemes (Brusselator, Oregonator)
are still complicated, and there is still a need to look at even simpler
prototypes in a logically ordered way. Accordingly, we consider
throughout steps with the overall stoichiometry A B. We may
also take this as written to represent a reaction satisfying the rate
law = ka. When the same change is catalyzed by a species Y
-
(6) Denbigh, K. G. Trans. Faraday SOC.1944, 40, 352; 1947, 43, 648;
1948, 44, 263, 479; J . Appl. Chem. 1951, 1, 227.
(7) Vidal, C.; Pacault, A. “on-Linear Phenomena in Chemical
Dynamics”; Springer-Verlag: New York, 1981; pp 10-14.
(8) Belousov, B. P. Sb. Ref. Radiat. Med. 1959, 1 , 145.
(9) Zhabotinskii, A. M. Dokl. Akad. Nauk. S S S R 1964,157, 392; Biof
izika 1964, 9, 306.
(10) Noyes, R. M.; Field, R. F.; Koros, E. J . Am. Chem. SOC.1972, 94,
1394.
(11) Epstein, I. R.; Orban, M. ‘Oscillations and Travelling Waves in
Chemical Systems”; Field, R. J., Burger, M., a s . ; Wiley: New York, 1984.
(12) (a) Lintz, H.-G.; Weber, W . Chem. Eng. Sci. 1980, 35, 203. (b)
Papsin, G . A,; Hanna, A.; Showalter, K. J. Phys. Chem. 1981, 85, 2575.
(13) Aarons, L. J.; Gray, B. F. Chem SOC.Rev. 1976, 5, 359.
(14) Semenov, N . N. “Chain Reaction”; Clarendon Press: Oxford, 1935.
( 1 5 ) Prigogine, I.; Lefever, R. J. Chem. Phys. 1968, 48, 1695.
(16) Field, R. J.; Noyes, R. M . J . Chem. Phys. 1974, 60, 1877.
0 1985 American Chemical Society
Feature Article
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985 23
-+
and has a rate satisfying the expression kay, we may represent
this as A Y B Y. Autocatalysis corresponds to catalysis
by the product B itself. Two exemplary
span the whole
range of behavior likely to be encountered:
+
rate = k,ab
quadratic autocatalysis
cubic autocatalysis
rate = k,ab2
(la)
(1b)
These may be represented by the “chemical” equations
A+B-2B
and
A
+ 2B
-
(IC)
3B
(14
stract, but the algebra is not much more demanding than that
involved in locating multiple stationary states.
Sustained oscillations are possible in these two-dimensional
models. Section V presents the qualitative, and some quantitative,
aspects but does not dwell on the additional algebraic effort
required.
Chaos requires a third dimension. The circumstances in which
there is a chaotic response to very small periodic perturbations
(e.g. in pumping speeds or bath temperatures) are considered in
section VI. Section VI1 shows how, when decay rates are not so
simple as first order, even quadratic autocatalysis is sufficient to
generate much of the above behavior. Finally, section VI11
considers some of the wider implications of these and related
modern studies of similar systems.
Normally these will arise from a subscheme of elementary steps.
For instance, the arsenite-iodate reaction produces iodide ions
with an experimentally determined rate law of the formI2v2*
11. Cubic Autocatalysis in Open Systems: Kinetic Model and
Mass Balance in a Well-Stirred Reactor
d[I-]/dt = ( k , + k2[I-]][I-][I03-][H+]2
Suppose there is a chemical reaction which can be represented
by a first-order step
This does not, however, require us to believe that it requires a triple
encounter as an elementary step. We return to questions of this
kind in section VIII.
It is important to note (i) that reversibility makes little difference
to behavior until the equilibrium constant ([B]/[A]), becomes
small and (ii) that less simple stoichiometry A + nB ( n + m)B
can readily be incorporated if required.20*21
In a closed vessel, the rate of an autocatalytic reaction22passes
through a maximum and product concentrations plotted against
time lie on S-shaped curves. Such behavior is typical of a very
wide range of systems indeed, ranging from many gas-solid reactions and solidsolid interconversions to the growth of biological
populations constrained by a limited food supply and the spread
of infectious diseases. Lotkaz3 provides a classical survey still
attractive today.
We shall concentrate mainly on the cubic rate law (lb), although in section VI1 we study simple or quadratic autocatalysis.
Steps like (1 b) lie at the heart of many complex models. We shall
see that it does not need a complicated pattern of auxiliary reactions to generate very exotic behavior indeed, despite much
folklore to the contrary.
4. Arrangement of Text. The present article is concerned with
studying the various patterns of behavior arising from the simplest
model schemes based on reactions ICand Id. Where appropriate,
we shall mention briefly experimental observations of particular
features. The recent feature article by EpsteinZ4offers an admirable and up-to-date survey of the chemistry of some real
systems: we set out a complementary account of the necessary
analysis.
Section I1 develops the concept of multistability and its relation
to discontinuous jumps from one stationary state to another and
the origin of hysteresis. These ideas are familiar to chemists in
other contexts and can be vividly illustrated by means of a flow
diagram. In section I1 we consider the simple situation where the
catalyst B is indefinitely stable, undergoing no subsequent reaction.
The composition of such systems requires only one variable for
its complete description at all times. In section I11 we uncouple
the concentrations of A and B by allowing the first-order decay
B-C
-
to remove the catalyst. Two variables are now necessary to
describe the evolution of the system. Section IV develops the
techniques of local stability analysis. These ideas are more ab(17) w a y , P.; Scott, S . K. Chem. Eng. Sci. 1983,38,29.
(18) Gray, P.; Scott, S. K. Chem. Eng. Sci. 1984,39, 1087.
(19) Scott, S. K. Chem. Eng. Sci. 1983,38, 1701.
(20) Lin, K. F. Chem. Eng. Sci. 1981,36,1447.
(21) Scott, S . K. J . Chem. Soc., Faraday Trans. 2, in press.
(22) Boudart, M. “Kinetics of Chemical Processes”; Prentice-Hall: Englewood Cliffs, NJ, 1968.
(23) Loth, A. J. ‘Elements of Mathematical Biology”; Dover Publications:
New York, 1956.
(24) Epstein, I. R. J. Phys. Chem. 1984,88, 187.
A
-
rate = k3a
B
(2)
(we choose the subscript 3 for this step to remain consistent with
work already p ~ b l i s h e d ’ ~ - ’ ~ Suppose
~ ~ ~ ) . also that this is accompanied by autocatalysis so that we can add either of the steps
I C or Id. Then, if these are the only steps in the mechanism, the
composition of the system at any time is fixed if we specify either
the concentration of A or the concentration of B, because of the
additional requirement
[A10 = [AI + [BI
(3)
(assuming the system contained pure A initially), where [AIo is
the concentration of A at t = 0. For this system, even if there
was an initial nonzero concentration of B, the reactant and catalyst
concentrations cannot vary independently. These are one-parameter systems. In many physical situations, however, the
catalyst may not be indefinitely stable but will undergo further
reaction, decomposition, or poisoning. The simplest decay of B
is via a first-order process
B
+
C
rate = k2b
(4)
This adds another dimension to the system, as there are now three
species concentrations and still only one relationship between them.
Most importantly, the concentrations of A and B may now vary
independently of each other-a requirement for oscillatory behavior.
Almost all of the special features of these autocatalytic systems
can be highlighted by studying the simplest case in the simplest
circumstances. These correspond to the situation where the autocatalytic reaction swamps the uncatalyzed step.
Accordingly, we shall study the system
A
-
+ 2B
B
3B
C
rate = klab2
rate = k2b
(5a)
(5b)
Such schemes have zero rate when the concentration of B is zero
(Le. at zero conversion). We may alter this by allowing a nonzero
initial concentration of B, bo # 0.
Before proceeding to the mass-conservation equations for this
mechanism, we note that the product klao2has units of s-l. It
is the inverse of a characteristic chemical time scale
which plays an important role in determining the behavior observed
under closed and open conditions. A second characteristic time,
for the lifetime of the catalyst, is given by the inverse of the
first-order rate constant k2
t2
= l/k2
(7)
(25) (a) Gray, P.; Scott, S . K. J. Phys. Chem. 1983,87,1835. (b) Gray,
P.; Scott, S . K. Ber. Bunsenges. Phys. Chem. 1983,87,379. (c) Gray, B. F.;
Gray, P.; Scott, S . K. J . Chem. Soc., Faraday Trans. I , in press.
Gray and Scott
The Journal of Physical Chemistry, Vol. 89, No. 1, 1985
24
Stable catalysts decay slowly and have long values of t2; unstable
catalysts are characterized by short lifetimes.
1. Mass Balance Equations in Open Systems. For the kinetic
mechanism ( 5 ) we may write down the two mass-balance equations
for the concentrations of A and B:
where k f = l/tresis the inverse of the mean residence time. Note
that we do not have to consider a third differential equation for
the concentration of the product C as this is uniquely determined
at all times by the concentrations of A and B in the reactor and
the inlet conditions
c = (a0 bo) - ( a b )
+
+
Equation 8 may be compactly recast in terms of a small number
of dimensionless groups
a = a/ao dimensionless reactant concentration
@ = b/ao
dimensionless catalyst concentration
y. R is a cubic curve which passes through a minimum at the
origin, corresponding to zero rate at zero conversion; it shows an
inflection point at y = ]I3,reaches a maximum at y = 2 / 3 , and
falls to zero again at y = 1, corresponding to complete conversion
of A to B.
The maximum value of R = 4/27 corresponds to a real maximum reaction rate
y = (ao - a ) / a o dimensionless extent of conversion
Po = bo/ao dimensionless inlet concentration of catalyst
T,,,
= t,,,/t,h
= t2/tch
i2
T
Figure 1. Cubic autocatalysis, with infinitely stable catalyst B: (a)
dependence of reaction curve R and flow line L on extent of conversion
y = 1 - a with no inflow of B; (b) variation of stationary state with
residence time corresponding to (a) showing extinction or washout; (c)
dependence of R and L on y for nonzero inflow of B; (d) corresponding
variation of ylr with ires
showing ’ignition”, extinction, and hysteresis.
dimensionless residence time
dimensionless catalyst lifetime
dimensionless time
= l/tch
This yields
da 1-a
_
- -a@ + d7
Tres
The two dimensionless measures of the reactant concentration a
and y are simply related by
a=l-y
- -
In the limit of perfect catalyst stability ( k 2 0; t 2
system is reduced to one of a single variable with
p
= 1
+ Po - a = y + 8,
(10)
m)
our
(11)
For the general case of k2 > 0, y and @ are no longer equivalent
measures (for Po = 0) nor even directly related.
Nevertheless, an investigation of the extreme situation of an
indefinitely stable catalyst, with simplification (1l), provides us
with valuable insights which will be of use in understanding the
more general case.
2. Stationary-State Extents of Conversion with Infinitely
Srable Catalyst: Flow Diagrams. For systems in which the
catalyst does not decay ( r 2 m) and for which there is no catalyst
B in the inflow, we may use eq 11 to obtain a single mass-balance
equation. In terms of the dimensionless extent of conversion y,
this has the form
-
(note under these conditions d y l d r = -da/dr = d@/dr.) We
denote the two terms on the right-hand side of (12) by R and L.
These represent the rate of chemical conversion of A to B and
the net rate of inflow of A, respectively.
Figure l a shows how R and L vary with the extent of conversion
This finite bound on the rate distinguishes our model from some
of the improperly formulated schemes26or “pool chemical” approaches in which the concentration of A is assumed constant and
for which the reaction rate can become infinitely large.
The flow line L is a straight line passing through the origin with
a gradient equal to l/Tre,. For slow flow rates (long residence
times), L is relatively flat; for fast flows, L is steep. Thus, the
effect of varying the flow rate or residence time in our system is
to vary the position of L relative to R (which is unaffected by
changes in r,,).
Stationary states of the reaction occur when the two rates R
and L are in exact balance and d y l d r becomes zero. Such
conditions are represented by the intersections of R and L on the
flow diagram (Figure la). Any flow rate for which L intersects
R three times corresponds to a system with three possible stationary states (multistability). We call these y ] , y2, and y3 in
order of increasing extent of conversion. The lowest intersection
is at the origin y 1 = 0. Here there is no reaction, and the outflow
and inflow consist of pure A. The other two stationary states lie
in the ranges
0
< y2 < y2 and ‘/z < y3 < 1
As the flow rate is increased, the flow line L gets steeper. This
causes y2and -3
, to move closer together. For the special residence
time
T,,,
=4
1
i.e. -tre, = 1 / k l a o 2
4
R and L are tangential and y2 and y3 merge at the value ]I2. For
any shorter value of r, yzand y3disappear and the system jumps
to the only remaining intersection yIcorresponding to no reaction.
This jump is known as extinction or washout.
The dependence of the stationary-state extent of conversion on
residence time swept out in this way is shown in Figure lb.
( 2 6 ) Karmann,
K.-P.: Hinze, J. J . Chem. Phys. 1980, 72, 5467.
The Journal of Physical Chemistry, Vol. 89, No. I, I985
Feature Article
3. Effect of Adding Catalyst to Inflow: Infinitely Stable
Catalyst. If the concentration of B in the inflow is not zero, two
new features arise. The mass-balance equation is now
The changes brought about by nonzero Bo occur only in the reaction rate term R . The rate of chemical conversion of A to B
is now no longer zero for y = 0 and hence R does not pass through
the origin but is "lifted up" to a finite rate (Figure IC). This also
means that R and L, which are unaffected by the value of Po, no
longer intersect at the origin.
There may now therefore be two flow rates at which R and L
become tangential, given by
Between these values there are three intersections of the two curves
and, hence, three stationary states. As the residence time is
decreased toward the upper tangency, y2and y3 merge and extinction occurs, as before. At the lower tangency, corresponding
to longer residence times, y1and y2 merge and the system jumps
to y3. This is known as ignition. The variation of the residence
time causes the stationary-state extent of conversion to sweep out
the S-shaped curve shown in Figure Id.
4 . Stability of Stationary States: Relaxation Times and
Slowing Down. The stability of a given stationary state may be
tested by seeing if a small perturbation Ay = y - y,, grows
(unstable stationary state) or decays to zero (stable stationary
state). The time dependence of A y is given by
AT = A ~ o / ( l +
In general, the time taken for the perturbation to decrease from
Ayo/ento Ayo/e"+' is given by
7*,+,
111. Systems with Catalyst Decay: Patterns of Stationary
States (Isolas and Mushrooms)
Superficially, the change in the system brought about by allowing the catalyst to have a finite, rather than infinite, lifetime
might be expected to be rather modest. In fact, profound alt e r a t i o n ~ are
' ~ made to the patterns of stationary states and their
stability.
We now have three chemical species in our scheme
-
+ 2B
<< IXAYol
(15)
the displacement varies exponentially according to
Ay(7) = Ayo e x p ( - ~ / ~ * )
(16)
where 7. = -1/X is the characteristic relaxation time, the time
taken for AT to decay or grow by l / e of its initial value. If X
is negative, 7* is positive and the perturbation decays as 7 becomes
larger and larger. For positive values of X, the perturbation
increases exponentially.
Using this, we find that the highest and lowest stationary states
(yl and y3) are stable while the middle solution y2 is unstable.
Furthermore, when Bo = 0 (no catalyst in inflow), we have for
the relaxation times
7 r = -7rcs for y1
rate = k,ab2
3B
C
rate = kzb
There is one relationship linking the three concentrations a, b,
and c of the most general form
a,+b,+c,
sum of
lr(AYo)*I
= 7*en
This shows how the relaxation time lengthens at each stage as
the perturbation decays. This longevity of transients is called
"slowing down" and was first computed by Heinrichs and
S ~ h n e i d e r .Slowing
~~
down in the arsenite-iodate reaction has
recently been observed experimentally by Showalter et aLZ8
B
If the initial perturbation Ayo is very small, so that
(17)
This shows that the perturbation no longer decays by an exponential law. The time dependence depends upon the size and the
sign of the initial perturbation: those for which Ayo is positive
decay; perturbations for which Ayo is negative diverge from the
stationary state.
We may still define a relaxation time 7* for stable systems (Ayo
> 0) as the time taken for Ay to decay to l / e of Ay,. This gives
7* = 2(e - l ) / A y ,
A
where X and p reflect the dependence of dy1d.r on the extent of
conversion
W 0 7 )
25
=
a+b+c
sum of
(18)
concentrations = concentrations
in inflow
in reactor
arising from the reaction stoichiometry. We thus have two independent concentrations in our system. For convenience we shall
only consider situations in which there is no final stable product
C added to the inflow, co = 0.
I . Stationary-State Solutions and Flow Diagrams: Isolas for
Systems with No Catalyst Inflow. A stationary state is achieved
when the rates of change da/d7 and dP/d7 become simultaneously
zero. We may add eq 9a and 9b to yield a relationship between
CY,,
and P,,, the dimensionless concentrations of A and B in the
stationary state. For a system with no autocatalyst in the inflow,
Po = 0, this yields
Comparing this expression with eq 1 1 , we see that P, is no longer
equal to the dimensionless extent of conversion of the reactant
but is reduced by the factor 1 7,,,/72. This difference reflects
7 0 = l / y 3 ( l - 2y3) for y3
the fact that some of the B formed from A has subsequently been
converted to C. The longer the residence time, the greater the
As we approach the extinction tangency, we have y2 y3
quotient 7,/72 and, hence, the higher the probability that a given
1/2. For these values, the relaxation times become infinitely large
molecule will be converted to the final product C.
as X tends to zero. In fact, infinite values cannot be a c h i e ~ e d , ~ ~ , ~ ' If we now substitute for @, from (19) into ( 9 ) ,we may obtain
although there is a lengthening of 7. (Le. perturbations decay more
for the stationary-state concentrations
slowly). As X approaches zero, inequality 15 cannot be satisfied
no matter how small Ayo is made. We must then consider both
terms on the right-hand side of (14). For X = 0, and noting that
when y2 = y3 = ll2,we replace (16) by
p =
7r
= l / y 2 ( l - 2 y 2 ) for y2
+
--
(27) Heinrichs, M.; Schneider, F. W . J. Phys. Chem. 1981.85.21 12; Ber.
Bumenges. Phys. Chem. 1980,84, 857.
(28) Ganapathisubramanian, N.; Showalter, K.J. Phys. Chem. 1984.87,
1098; J . Chem. Phys. 1984,80, 4177.
R
=
L
26
Gray and Scott
The Journal of Physical Chemistry, Vol. 89, No. 1, I985
of the isola correspond to the two residence times for which L is
tangential to R, i.e. for which
I
Figure 2. Cubic autocatalysis with catalyst of finite lifetime, no B in
inflow: (a) dependence of R and L on stationary-state extent of conversion ysa;(b) variation of ysswith irefor unstable catalyst ( T <
~ 16),
only the axis yy, = 0 represents intersections of R and L;(c) variation
of ymwith rre for i2= 16 showing birth of an isola at yI =
and 7,= 16; (d) variation of ysswith T~~ for stable catalyst (i2> 16) showing
isola with two points of extinction.
We may represent the reaction rate and flow terms R and L as
functions of ysson a flow diagram (Figure 2a). Again, R is a
cubic curve passing through the origin yss = 0 and falling back
to zero at the state of complete conversion yS = 1. The flow term
L is still a linear function of ysspassing through the origin, but
now the gradient is a more complicated function of the residence
time:
2
gradient of L = dL/dy
For the shortest residence times (highest flow rates) we may
neglect the term rr,/rZ as small compared with unity. The
gradient then is proportional to 1/rreS,and the flow line becomes
less steep as the residence time decreases.
At the longest residence times (lowest flow rates), 7,,/i2 is very
much larger than unity, so the gradient becomes proportional to
rIa. The flow line now becomes steeper as the residence time
increases. This differs from the stable catalyst case studied in
increases.
section 11, where the gradient always decreases as ires
In between these extremes of long and short residence times,
the flow line achieves a minimum slope. This occurs when the
mean residence time is equal to the characteristic lifetime (or
half-life) of the catalyst, Le. when 7, = 72. The minimum gradient
is given by
It is the relative positions of this flow line with minimum slope
and the line passing through the origin which is tangential to the
reaction rate curve R that determine the pattern drawn out by
the dependence of the stationary state as etc. on residence time.
Systems with relatively unstable catalysts are characterized by
low values of T ~ .For these the minimum gradient of L, 4/i2, is
relatively high so the flow line is always steep. If L always lies
above the tangent line (which has a gradient of I/J, R and L
cannot have any intersections other than at the origin. The stationary state yss= 0 is unique for all residence times. This
dependence of ysson T,,, is shown in Figure 2b, where there are
no nonzero solutions.
Under conditions where the catalyst is more stable, r2 becomes
higher. The flow line of minimum gradient may now lie below
the tangent line, and there will then be three stationary-state
intersections over some range of residence time. The dependence
of yss on ires for this situation is shown in Figure 2d. The two
nonzero extents of conversion y2 and y3 lie on a closed curve or
isola. There is both a lower bound and an upper bound on the
residence times over which these solutions exist. The two ends
There are two such residence times, given by the roots of this
quadratic, because L passes twice through tangency as T , , ~is
increased: once as the gradient is decreasing before reaching its
minimum and once as the gradient is increasing after its minimum.
The borderline between parts b and d of Figure 2 w u r s when
the loss line with minimum gradient exactly coincides with the
tangent line. The flow line L may then just touch the reaction
rate curve when rrep= i2.For this residence time, there is a second
This represents the “birth”
stationary-state solution, yss =
of an isola as a point in the yss-~res
plane and is shown in Figure
2c.
The two ends of an isola are points of extinction of the nonzero
extent of conversion. One of these accompanies a decrease in
residence time (as found for the S-shaped curve in the previous
section when the catalyst is infinitely stable), and the other accompanies an increase in T,,.
2. Stationary-State Patterns with Catalyst Znfow: Isolas and
Mushrooms. If some catalyst B is included in the inflow, with
a concentration Po = bo/ao,the relationship among CY,6, and y
at the stationary state becomes
Substituting this result into 9, we obtain instead of eq 20
R
+
(1 - yss)(yss Po)’ =
I(1 +
rres
):
2
yssp L (22)
We now no longer have yss = 0 as a solution.
Again the effect of a nonzero concentration of autocatalyst in
the inflow is to alter the reaction rate curve R, lifting it up in the
neighborhood of the origin (Figure 3a). Thus, the lowest intersection corresponds to a nonzero extent of conversion. There
are also now two lines passing through the origin which are
tangential to some part of R: one corresponds to extinction and
one to ignition.
The flow line L still passes through a minimum gradient when
rIes= i2.We must now consider the position of this line with
respect to the two tangents (Figure 3b-f).
If the catalyst is highly unstable, L is always steep and always
lies above the upper extinction tangent. The only intersection is
the one at low extents of conversion y,. This branch of solutions
(Figure 3b) reaches a maximum when r,,, = i2.
For the special case that the minimum slope of L corresponds
to the upper tangent, an isola is born as a point (Figure 3c). With
a slightly more stable autocatalyst, L may pass beneath the upper
tangent but not reach the lower one. This leads to an isola (Figure
3d) which sits above the lowest (nonzero) branch of stationary
states. The size of the kola increases as the catalyst lifetime
becomes longer, as does the height of the maximum along the y,
branch. A second special case arises when i2is such that L can
just touch the lower ignition tangent. The bottom of the kola then
just touches the lower branch of stationary states (Figure 3e).
If the catalyst stability is further increased or if more catalyst
is added to the inflow, the flow line may pass below the lower
tangent. The lower two stationary states y1 and y2 then merge
and disappear over some range of residence time. The resulting
pattern (Figure 3f) is called a mushroom. There are now two
extinction points and two points of ignition.
Experimental observations of isola and mushroom dependences
of the stationary-state extent of conversion on residence time have
recently been made for the arsenite-iodate system by Ganapathisubramanian and Showalter.28 In order to provide a suitable
removal step for the autocatalyst I-, these authors employed an
additional flow of solvent through their reactor. (This is sufficient
to enrich the pattern of stationary states but does not provide the
The Journal of Physical Chemistry, Vol. 89. No. 1 , 1985 27
Feature Article
equations (9) and two independent variables a and j3. This gives
a total of four partial differential coefficients which determine
the sign and character of the two exponents A+ and A- via the
quadratic equation
"
(3)
aa d r
I
i.e.
X2 - (Tr)X
+ det = 0
Depending on the signs and magnitudes of these partial derivatives, which are evaluated at the stationary state, A+ and Xmay be real or complex with positive or negative real parts. The
time dependence of the perturbations is then described by
Figure 3. Cubic autocatalysis with catalyst of finite lifetime, nonzero
inflow of B: (a) dependence of R and L on 7%;(b) unstable catalyst, no
tangency of R and L; (c) birth of isola, minimum slope of flow line
corresponds with upper (extinction) tangency; (d) isola, minimum slope
of L lies between tangencies; (e) birth of mushroom, L touches lower
tangent; (f) mushroom.
decoupling necessary to allow oscillations.) Good agreement
between experiment and predictions based on a similar analysis
to that described above was obtained by using known kinetic
parameters.
3. Effects of Including the Uncatalyzed Step A
B. Our
simple model so far ignores the uncatalyzed conversion of A to
B. We have in effect assumed that this always proceeds very much
more slowly than the catalyzed step. The main effects of including
the additional step
A
B
rate = k,a
-
-
are qualitatively very similar to having a nonzero inlet concentration of the a u t o c a t a l y ~ t .Again
~ ~ ~ the lowest branch of stationary-state solutions is lifted up above the y = 0 axis.
There is, however, one important change. It occurs at long
residence times. Here dependences of yy,on 7, shown in Figures
2 and 3 all have a unique solution, and this solution tends to zero
conversion as 71a becomes infinitely long. This suggests that, in
the absence of an uncatalyzed component, there would be no net
reaction even in a closed vessel. Hence, the system would not
approach its state of chemical equilibrium. This remains the case
in the cstr even if the concentration of autocatalyst in the inflow
is nonzero.
If, however, the uncatalyzed reaction is included, the lowest
stationary state does not fall back to zero. Instead, this branch
tends to complete conversion yss= 1 as 7,- tends to infinity, as
required by equilibrium considerations. Nonetheless, the isola
and mushroom patterns remain. They do not disappear, because
the behavior at all but the longest residence times is only very
slightly influenced by including the uncatalyzed step.
IV. Character of Stationary States for Systems with Catalyst
Decay: Bifurcation Diagram in Parameter Space and Birth of
Sustained Oscillations
In section 11.4 we analyzed the stability of the different stationary states to small perturbations. The determining quantity
turned out to be the sign of a single coefficient X which was
obtained by differentiation of the single governing equation with
respect to the single variable y.
For a system in which the catalyst decays we may examine the
stabilities in an analogous way.29 We now have two governing
Aa = p 1 exp(X'7)
+ p 2 exp(X-r)
AS = q1 exp(X'7)
+ q2 exp(X-7)
(24)
where p l , p2, q l , and q2 are constants.
1. Systems with No Catalyst in Inflow, Po = 0. When the
concentration of B in the inflow is zero, the lowest stationary state
corresponds to no conversion (a1= 1, = 0, y1 = 0). Both the
roots X+ and A- are real and negative. This means that the
exponential terms in (24) decay monotonically to zero as 7 tends
to infinity. A useful, alternative way of looking at this approach
to the stationary state is to plot the variations of a and j3 against
each other. In these a+ phase-plane diagrams the stationary
states are represented by points, known as singular points or
singularities, with coordinates a,,and p,,. There may be one or
three of these depending on the number of solutions of eq 20. The
variations of a and p in time now correspond to motions or trajectories across the plane. The monotonic decay in time to aI and
PI, governed by real, negative values of A+ and X-, appears as a
monotonic movement across the a-p phase plane to the point of
a = y1 = 1, j3 = PI = 0. This monotonic approach characterizes
the singularity as a stable node (Figure 4a).
The middle solution (a2,p2, or y2) is always such that Xf and
A- are both real and always have opposite signs. Although one
of the experimental terms decays as T increases, the other, with
a positive value for A, increases with time. The stationary state
is unstable, and the system cannot reside there. The motion across
the a-j3 phase plane corresponding to these values of X is shown
in Figure 4b. All but two trajectories diverge from the singularity
which is known as a saddle point. (The trajectories have the same
configuration as the contours of the saddle displayed by the
transition state on a potential energy surface in the theory of
chemical reactions.)
The sign and character of the roots A+ and A- for the highest
stationary-state extent of conversion (y3,a3,f13) vary considerably.
At the shortest residence times at which the isola exists, XI and
X2 are real and both negative, so we have a stable node on the
fast-flow side. As T, is increased, the roots may become complex
conjugates with negative real parts. When this occurs, Aa and
A0 decay via a series of damped oscillatory overshoots and undershoots. This sort of response characterizes a stable focus (sf),
as the variation of j3 with respect to a appears as a spiral (Figure
4c).
For longer residence times, the roots are still complex, but the
real parts may change from negative to positive. The perturbations
now grow in time, with a spiraling away from the stationary state
in the a-/3 phase plane (Figure 4d). This is known as an unstable
focus (uf).
The roots finally become real but remain positive before the
right-hand extinction of the isola. The perturbations now grow
monotonically (exponentially) in time. The stationary state is an
unstable node (un; Figure 4e).
(29) Andronov, A. A,; Vitt, A. A,; Khakin, S. E. "Theory of Oscillators";
Pergamon: Oxford, 1966.
28
Gray and Scott
The Journal of Physical Chemistry, Vol. 89, No. I , 1985
I
I
’
,
I
(r)
Figure 4. Different characters of singularities (stationary states) in a-,3
phase plane: (a) stable node, sn; (b) saddle point, sp; (c) stable focus,
st; (d) unstable focus, uf; (e) unstable node, un.
2. Systems with Catalyst in the Inflow, Po > 0. Including some
catalyst in the inflow not only leads to a wider range of possible
patterns for the dependence of the stationary states on the residence
time but to a wider range of characters displayed by the lowest
of the three solutions. For small nonzero inlet concentrations of
@, a1may change from a stable node to a stable focus for some
range of 7ra. As Po gets larger, the lowest solution may also
become unstable (uf or un). This instability is only found for
mushroom patterns and on the branch of y1at long residence
times. The y1 branch at short residence times (to the left of the
mushroom in Figure 30, the so-called flow branch, is always stable.
The middle stationary state yz is still always a saddle point.
The uppermost branch shows the same variety of character and
stability displayed when there is no catalyst in the inflow.
3. Dependence of Stationary-State Patterns and Character
on the Parameters r2 and Po. What pattern is displayed by the
dependence of the stationary-state concentrations on residence
times (isola, mushroom, etc.) depends upon the dimensionless
values of the catalyst lifetime 7 2 and the inlet concentration Po.
We may subdivide each of these patterns according to whether
their difference branches show changes from stable to unstable
solutions as rre8is varied. Thus, we find nine different possible
case^.^*^^^ These are presented in Figure Sa. Each of these
corresponds to different ranges of values of r2and P,,. These ranges
can be portrayed as areas on a 72+O plane as shown in Figure
5b. Thus, if Bo = 0.1 and 72 = 40, the system will display pattern
viii-a mushroom which has unstable portions along both the
uppermost branch y3and the lowest branch yl as well as the saddle
point y2.
0.1
b
0
(30) Escher, C. E.; Ross, J., private communication.
(31) Boissonade, J.; De Kepper, P. J . Phys. Chem. 1980, 84, 501.
1,15
0.1
Figure 5. (a) Nine possible patterns for the dependence of the stationary-state extent of conversion on residence time: solid lines represent
stable solutions (sn or sf), dashed lines give unstable states (un,uf, or sp),
and the symbol represents point of Hopf bifurcation; the middle branch
is always a saddle point. (b) Values of catalyst inflow concentration Po
and lifetime T~ at which the above patterns are found.
that developed for nonisothermal reactions in the chemical engineering literature.
1 . Birth of Limit Cycles. If the concentrations a and P of the
reactant A and autocatalyst B do oscillate regularly, then the
variation of (Y with respect to P in the phase plane appears as a
closed loop or limit cycle. (This is most easily seen by drawing
out the changes in (Y and /3 with time as the system moves round
the limit cycle in Figure 6a in the direction of the arrows. The
resulting concentration histories are as shown in Figure 6b.)
Such limit cycles originate at points where a stable focus
changes to an unstable focus. This is known as a point of Hopf
bifurcation.3293 The mathematical condition for this is that the
real parts of the roots X+ and X- (see section IV) pass through
zero. This provides an analytical criterion which can readily be
applied. When there is no autocatalyst in the inflow (Po = 0),
the residence time at which Hopf bifurcation occurs is given by
V. Sustained Oscillations in Autocatalytic Reactions: Limit
Cycles
Systems that have a stable focal character show a damped
oscillatory approach (or “ringing”) to their final stationary state.
There are a number of chemical reactions, however, which display
long-lived oscillatory sequences that become apparently indefinitely
sustained when carried out in open systems.
Some previous discussions of such oscillatory behavior have been
based on the ”cross-shaped” diagrams of De K e ~ p e r ,and
~ ] some
are presented in terms of the switching between two different
branches of the S-shaped curves. The present model allows a clear
and rigorous analysis to be employed without either oversimiplification or loss of physical significance. This approach mirrors
0.05
7,,
=
T*,,~
N
=
j/4723/2(1
+ (1 - 472-1/2)1/2)2
723/2(1- 2 7 ~ - ’ -/ ~272-l - ...)
There is no such change possible for systems in which the half-life
of the catalyst is too short, such that 72 < 16 (Le. 7 2 ‘ 1 2 < 4).
The initial growth in size of the limit cycle is proportional to
the square root of the difference between T~~ and 7*,,. The
complete analytical expression can also be derived,I8at the expense
of some algebraic effort, for our simple model. The “amplitude”
of the corresponding oscillations also grows in this way and can
(32) Hopf, E. Ber. Verh. Saechs. Akad. Wiss. Leipzig,Math.-Phys. KI.
1942, 94, 3.
(33) Hassard, B. D.; Kazarinoff, N. D.; Wan, Y.-H. “Theory and Applications of Hopf Bifurcation”; Cambridge University Press: Cambridge,
England, 198 1.
Feature Article
The Journal of Physical Chemistry, Vol. 89, No. 1 , 1985 29
Figure 6. (a) Limit cycle in a+ phase plane. (b) Sustained oscillations
corresponding to motion around the limit cycle.
be represented as a projection onto the response (cu,)-constraint
( T , ~ diagram
)
(Figure 7) as an envelope originating at T*,. This
representation is extremely compact and conveys a livelier feeling
for the growth of the oscillations than that given by Uppal et al.;4
they display only the upper limit of the extent of conversion or
the switching between branches of stationary states, which would
require a different diagram for each residence time.
2. Stability of Limit Cycles. The appearance of a limit cycle
in the a+ phase plane does not guarantee sustained oscillations
in the concentration histories. Just as stationary states may be
stable or unstable, so may limit cycles. An unstable cycle does
not lead to physically realizable oscillations. Instead, it acts as
a “watershed” in much the same way as the separatrices of a saddle
point influence the motion across the phase plane. Any point
starting within an unstable limit cycle tends to the stable stationary
state, which also lies inside. A system with initial conditions
outside the cycle must move to a different stationary state. Thus,
whenever unstable limit cycles arise, their size is important.
Stable limit cycles correspond to physically observable sustained
oscillations. Their sizes are representations of the oscillatory
amplitude, although they give no indication of the period. For
our system with no B in the inflow, Hopf bifurcations occur always
at some residence time so long as
72
2 16
(25)
For the limit cycles to be stable, however, we require that
r2 > ca. 28.5
The frequency wo of the oscillations at the birth of the stable limit
cycle is finite and nonzero. At the point of Hopf bifurcation we
have purely imaginary eigenvalues for the stationary state
A’ = f i w o
3. Computed Results. The general route to evaluating the size
and period of a stable limit cycle for given values of the residence
time etc. is to integrate the parent equations (9) starting at a
suitable initial point. A series of such “numerical experiments”
establishes the dependence of the oscillatory behavior on these
different parameters. In the vicinity of the critical residence time
T*,,,
the concentrations vary sinusoidally with relatively high
frequency and small amplitude. As the residence time is increased,
the pulses become more angular and are separated by longer
periods.
4. Phase Portraits. The numerical computations also allow
us to complete the phase portraits around the stationary states.
Different combinations of (i) the number of solutions and (ii) their
character and stability along with (iii) the occurrence of stable
or unstable limit cycles give rise to the different patterns. In all,
a total of nine rearrangements are displayed by the cubic autocatalysis with first-order decay of B. These are shown in Figure
8. Stable limit cycles surround either unstable stationary states
or unstable limit cycles; an unstable limit cycle must enclose a
stable stationary state.
Any system will show changes between the different portraits
as T ~T ,~or, Bo is varied. For instance, the appearance of sustained
oscillations about the highest extent of conversion accompanies
a change from pattern b to portrait e (Figure 8) as an Hopf
bifurcation occurs.
5 . Extinction of Oscillations. A stable limit cycle may dis-
Figure 7. Dependence of a, on T~~ for a system with = 40 and Po =
1/15, showing variation in character and stability of stationary states and
the amplitude of sustained oscillations about the unique unstable focus.
The symbol 0 represents point of Hopf bifurcation.
Figure 8. Different phase portraits in a-P plane corresponding to different combinations of stable and unstable stationary states and limit
cycles: (O), sn or sf; ( X ) , sp; (O), un or uf. Stable limit cycles are shown
by solid curves, and unstable ones by dashed curves.
appear in a number of different ways. If the residence time is
decreased back through the point of Hopf bifurcations (P,), the
limit cycle shrinks back to zero size. The corresponding oscillations
decrease smoothly to infinitesimally small amplitudes as the
unstable focus becomes stable.
There are three possible fates for the cycle as T~~ is increased.
First, the oscillations may attain a maximum amplitude and then
decrease in size again as the residence time approaches a second
point of Hopf bifurcation. The two other routes lead to the sudden
quenching or extinction of fully developed oscillatory behavior.
Either the stable limit cycle “collides” with an unstable cycle
growing from inside and surrounding the same stationary state
(see e.g. Figure 8) or it may grow sufficiently large to touch the
separatrix of a neighboring saddle point. In the latter case a
homoclinic orbit is created exactly at the point of touching.34a
Both of these mechanisms lead to a ”bursting” of the stable cycle
and a dramatic change in the behavior of the system. mas elk^^^^
has provided an interesting commentary on some of the bifurcations of isothermal systems.
VI. Aperiodic Behavior in Autocatalytic Systems: The
Springs of Chaos
The simple, two-variable scheme studied so far leads to regular
oscillatory behavior. The oscillations have constant amplitude
and frequency for given values of 72, T,,,, and Bo. The majority
of experimental observations of periodic behavior fall into this
class. However, under some circumstances, more complex patterns
can be found. Orban and E p ~ t e i nhave
~ ~ discovered transitions
(or bifurcations) from apparently regular, large oscillatory excursions to ones with one large peak followed by one small peak
and so on, in the chlorite-thiosulfate reaction. Subsequent
transitions to trains with yet smaller pulses interspersed between
(34) (a) Merkin, J. H.;
Needham, D. J. Proc. R. SOC.London, Ser. A, in
press. (b) Maselko, Chem. Phys. 1982, 67, 17.
(35) Orban, M.; Epstein, I. R.J. Phys. Chem. 1982, 86, 3907.
30 The Journal of Physical Chemistry, Vol, 89, No. I , 1985
Gray and Scott
the large excursions occur as the residence time is increased.
Furthermore, certain experimental conditions give rise to nonsteady
behavior which has no apparent periodicity. Such aperiodic or
chaotic states have also been observed for the Belousov-Zhabotinskii reaction by various authors.36
Chaos requires that a system has at least three independent
variables. A purely homogeneous, isothermal chemical scheme
would require three independent concentrations. A very readable
introductory study of a chaotic response for a nonisothermal system
with thermal feedback having two independent reactions coupled
by temperature variations has recently been presented by Jorgensen
and A r k 3 ’
However, is is still an open question whether chaos arising from
an homogeneous chemical mechanism has been observed experimentally or whether it comes from the imperfect control of external features, such as small variations in the ambient temperature
of the flow rate. Let us consider that the use of a peristaltic pump
might superimpose a small periodic perturbation on the flow rate,
so that the residence time is given by
The determinant vanishes at the turning points of the S-shaped
dependence of the stationary-state concentrations on T:,. Thus,
eq 30 can only be satisfied if Po < I / * .
In the limit of zero inlet concentration of the autocatalyst, eq
29-31 arc satisfied by
= rEs/(l
E COS
ires
2xw7)
where t is a small number and 7ES is the mean value of T,,,.
Substituting this into eq 9 has the effect of introducing the time
explicitly into the right-hand side of the governing equations
dB - cup
P +Po - P
_
--
+
dT
7:
4 P o - P ) cos 27rwt
(26)
K
=2
These equations are called nonautonomous. It is possible to show
that for certain values of the various parameters 7 2 , 7Es, Po, t, and
w this system will describe aperiodic behavior. We first examine
the autonomous equations obtained in the limit of vanishingly small
e, t
0. These are the same as eq 9; however, it will be convenient
here to study the case where 72 is not constant but varies with
the residence time in such a way that
-
-
is a constant. Multiplying through (26) by 7Es and introducing
0
the new time variable t ‘ = 1/72, = t/tres, we find as E
da
= -7:,ap*
(1 - a)
~
S
=
S
1
Bo - KPs,
we require
Tr(J) = ( 1
+ 2K)pS,2- 2(1 + Po)Pss+ ( I + .)/.E,
=0
(29)
where p,, is given by the solution of
(36) See e.g.: Vidal, C. In “Chaos and Order in Nature”; Haken, H., Ed.;
Springer-Verlag: New York, 1981. Swinney, H . F. Phys. Rea. Leu. 1982,
49, 245.
(37) Jorgensen, D. V.; Aris, R. Chem. Eng. Sci. 1983, 38, 45
(38) Keener, J. P. SIAM J . Appl. Math. 1982, 67. 25.
P,, = y4
7Es
and r2 are equal,
Again we find that a characteristic half-life of the autocatalyst
such that
16k2 = klaoZ
is of great significance. In section I11 we have shown that this
is the minimum value of 72for which isolas arise in the 7ss-7res
plane. It is also the threshold value of 7 2 for which Hopf bifurcation can occur (section V), and now we see that it is the
minimum value for chaotic behavior.39 This novel result is clearly
of great interest.
VII. Quadratic Autocatalysis (Simple Branched-Chain
Reactions) with Complex Termination
We have so far discussed exclusively the cubic autocatalysis
(1b). The simpler rate law (la) with a quadratic dependence on
concentration may be more widespread: it is certainly also of
interest and of importance. The step
A+B-2B
has already found a p p l i ~ a t i o n l ~inq ~elementary
~
studies of isothermal branched-chain reactions and also in many biochemical
or biological fields.
When this rate law is coupled with a first-order decay (4) of
the autocatalyst B for systems with no catalyst inflow (Po = 0),
there may be two stationary-state solution^.'^ One of these
corresponds to no conversion, ass= 1 and p,, = 0, and exists for
all residence times. The other solution corresponds to nonzero
extents of conversion and exists for residence times such that
7res
= klaOtres
> 72/(72
- 1)
(32)
where
+
The dependence of yss = 1 - asson 72, governed by (28) is an
S-shaped curve provided Po is less than I/*. Mushrooms and isolas
are not found when 7 Z s and r2 are linked by eq 27.
The conditions under which the perturbed, nonautonomous
equation (26) with t # 0 will have chaotic solutions are those for
which the trace and the determinant of the Jacobian matrix of
the autonomous system (28) become zero simultane~usly.~~
Thus,
noting that
= 16 and
From eq 27 we see that K = 2 implies that
hence
7 2 = 16
7:s
7:,
7Es
72
= k,ao/k2
is the appropriate form of the characteristic catalyst lifetime.
When both these stationary states are realistic, i.e. when ( 3 2 )
is satisfied, the upper nonzero state is stable (either a stable node
or a stable focus (see section IV)) and the lower state with zero
conversion is an unstable saddle point. For short residence times,
such that inequality 32 is reversed, the zero solution becomes stable
and the physically unrealistic nonzero state (which now corresponds to negative catalyst concentrations, p,, < 0) becomes a
saddle point.
The effect of including some catalyst in the inflow (Po > 0)
is to remove the bistability found above. A unique nonzero stationary-state extent of conversion exists for all residence times.
This solution is always stable, although it may change character
from node to focus as the residence time is varied.
Irrespective of the inlet concentration of the autocatalyst (Po
L 0), there is no Hopf bifurcation with this rate law and first-order
removal of B; hence, this system cannot display sustained oscillations or stable limit cycles. Similarly, there is no possibility of
a transition to chaotic behavior even if the flow rate is perturbed
periodically.
We shall now consider the effects of coupling a quadratic
autocatalytic step with a removal or deactivation of the catalyst
B whose rate increases less quickly than linearly with the catalyst
concentration
B
C
rate = k2b/(l + rb)
(33)
-
(39) Gray, P.; Knapp, D.; Scott, S. K., paper in preparation.
The Journal of Physical Chemistry. Vol. 89, No. 1, 1985 31
Feature Article
Such a rate law may arise for instance if the conversion of B to
C occurs on a surface where the number of free sites available
for adsorption may become significantly reduced (LangmuirHinshelwood kineticsa) or if the conversion is catalyzed by an
enzyme which might be immobilized within the reactor (Michaelis-Menten kinetics41).
This modification introduces sufficient nonlinearity to the
system to allow a stable focus to become ~ n s t a b l eand
~ ~ for
,~~
sustained oscillatory behavior to grow from such a point of Hopf
bifurcation.
In dimensionless form, the mass-balance equations for this
scheme are
da _
- -4 -(11
a)
d7
71,
+
+
where p = rag is the dimensionless measure of the saturation term
in (33).
For simplicity we shall concentrate on systems with no catalyst
in the inflow, Po = 0. Equation 34 then has three stationary-state
solutions. One of these is the state of no conversion, am = 1 and
P,, = 0. The other two solutions are given by the roots of a
quadratic equation, one of which corresponds to positive concentrations of autocatalyst and the other to negative values of &.
Only the positive root is physically acceptable. Again we find
that this solution does not exist for all residence times, only those
for which inequality 32 is satisfied. This requirement on T , is~
determined only by the catalyst lifetime 72 and is independent of
the size of p .
The dependence of ym = 1 - asson residence time is shown in
Figure 9. The stabilities of the stationary states along the two
branches are also indicated. The state of zero conversion is a stable
node (sn) at the shortest residence times but changes to a saddle
point as the second, nonzero branch emerges. The upper solution
is, at first, a stable node. As the residence time is increased, there
are changes in character between stable node and stable focus
(sf). Ultimately, there is a Hopf bifurcation at a critical residence
~ a stable limit cycle emerges to surround an unstable
time T * and
focus. Sustained oscillations are then observed in the concentrations of A and B computed by integrating eq 34.
Hopf bifurcation is only possible for systems in which the
dimensionless value of the saturation term is greater than the
characteristic lifetime of the catalyst, Le. for which
P
> 72
The bifurcation is always supercritical; Le., it gives rise to stable
limit cycles and sustained oscillations whose amplitudes grow
smoothly from zero as the residence time is increased beyond T * , ~
The dependence of the critical residence time T*= on the saturation
term p is shown in Figure 10 for various values of T ~ In
. general,
we find a peninsula within which oscillatory behavior is found.
These peninsulas move to longer residence times as the catalyst
becomes more stable, Le. as T~ lengthens. The point of Hopf
bifurcation is given a p p r o ~ i m a t e l yby~ ~
VIII. Discussion and Conclusions
The results presented in the previous sections demonstrate
clearly that complex kinetics and nonisothermal effects are not
necessary requirements for a system to display a vast range of
(40) Langmuir, I. J . Am. Chem. SOC.1918, 40, 1361.
(41)Michaelis, L.; Menten, M. Biochem. Z . 1913,49, 333.
(42) Merkin, J. H.; Needham, D. J.; Scott, S.K., submitted for publication
in Proc. R . SOC.London, Ser. A .
$*
‘J
Tres
Figure 9. Quadratic autocatalysis with surface termination of catalyst:
dependence of stationary-stateextent of conversion and its character on
residence time. Hopf bifurcation occurs at
leading to sustained
oscillations and stable limit cycles. The symbol 0 represents point of
Hopf bifurcation.
~
(35)
Although this expression is strictly valid only in the limit of large
T~ (and hence, from (35), of large p), it is good to better than 5%
accuracy over much of the range of p for r 2 as low as 4.
1
I
0
1
I
‘Ogl,&P
3
Figure 10. Dependence of residence time for Hopf bifurcation on saturation term p for various values of catalyst lifetime r2. The dashed line
is the approximate bifurcation locus in the text.
complex and interesting (perhaps even bewildering) behavior. The
analyses require a certain amount of algebraic manipulation, but
the use of flow diagrams in particular helps illustrate many of
the stages much more than any purely mathematical approach.
We conclude this article with a list of discussion points which will
draw out the major conclusions and indicate areas of future research.
(1) Kinetically elaborate models are not necessary to model
extremely varied chemical behavior. Even the simplest situation,
a single-step reaction (Id), may lead to two different patterns for
the dependence of the stationary-state extent of conversion on
residence time. One finds either unique solutions or multiplicity,
with associated ignition, extinction (“washout”), and hysteresis.
(2) When the autocatalyst B is not infinitely stable but decays
in a first-order manner (k2 # 0) there may be three patterns of
stationary states vs. T,,: unique solutions, isolas, and mushrooms.
Isolas have two points of extinction but no ignition; mushrooms
have two extinctions and two ignitions, with two regions of hysteresis.
(3) Considerations of the local, dynamic stability further
subdivide these patterns. There are two forms of unique solutions,
differing as to whether the stationary state may become unstable
or not, three types of isola, and four varieties of mushroom. These
different patterns divide the P0-q plane into nine regions.
(4) The points at which a stable focus becomes unstable are
the same as those at which limit cycles emerge in the phase plane.
Stable limit cycles correspond to conditions under which sustained
oscillations are observed in the species concentrations. Unstable
limit cycles may also arise. These play a similar role to the
separatrices of a saddle point acting as a frontier around a stable
stationary state. Initial conditions starting within the cycle tend
32 The Journal of Physical Chemistry, Vol. 89, No. 1, I985
toward that state, while those starting outside move to a different
singularity.
( 5 ) The onset and disappearance of oscillations may arise in
a number of ways as the conditions (e.g. the residence time) are
changed. The most straightforward of these has a smooth growth
of the limit cycle and, hence, of the oscillatory amplitude from
infinitesimally small values. If the limit cycle becomes too large,
it may touch the separatrix of a saddle point. The oscillations
then cease abruptly and the system moves to the lower (stable)
stationary state. If the residence time is then decreased again,
the system may jump back into large oscillations as the stable
solution merges with the saddle point.
An alternative method by which oscillations are extinguished
is the merging of a stable and unstable limit cycle which both
surround the same stable stationary state. Decreasing the residence
time reveals a range of conditions over which there is hysteresis
between a steady reaction and an oscillatory mode. The system
eventually jumps back into fully fledged oscillations.
(6) All the above is achieved with the simple, irreversible autocatalytic step of Z e l ’ d ~ v i c hand
~ ~ a decay reaction. These
schemes are not elementary steps but are adequate representations
of some real systems over certain ranges of concentrations. To
match the behavior over wider ranges, we may add additional
reactions, but the principal interesting features still survive. The
simplest embellishment2” of the mechanism is to invoke the lower
order, quadratic autocatalysis (IC) and the uncatalyzed process
( 2 ) simultaneously with the cubic step (Id). We may also recognize that the reverse reactions may have nonzero rates.
The conditions under which multiple stationary states may exist
over some range of the residence time for this augmented scheme
are readily established by considering the tangency of the R and
L curves on a flow diagram. If we at first ignore the reverse
reactions, R and L become tangential twice, corresponding to
extinction and ignition, provided
where k,, k,, and k, are the rate constants for the cubic, quadratic,
and uncatalyzed steps, respectively. If the quadratic step is neglected (kq 0), multiplicity requires
k,ao2 > 27ku
(37)
-
while if the uncatalyzed step can be ignored, the condition becomes
kcao > kq
(38)
The inclusion of these extra steps reduces the range of k, and a.
over which multiple stationary states can exist, but it does not
remove multiplicity completely. The reverse reactions also help
to restrict the conditions for more than one solution, but again
multiplicity can still be a feature of these autocatalytic systems.
We may note that the principle of detailed balance requires that
each of the steps IC, Id, and 2 have the same equilibrium constant
K,, because they have the same stoichiometry. If kq is very small,
then inequality 37 becomes for the reversible system
kcaOZ> 27(1 K;’)2ku
+
( 7 ) The rigorous analyses employed in the present paper may
lead to results which differ quantitatively and even qualitatively
from the “pool chemical” approaches. In the latter, the concentrations of certain species, in particular the reactants, are assumed
to vary so slowly compared with the time scale of the experiment
or oscillations that they may be treated as constants. In this way,
Karmann and Hinze have concluded incorrectly%that multiplicity
is not a real feature of the system considered in ( 6 ) . B. F. Gray
has shown4 that requirements on the magnitude of different rate
(43) Zel’dovich, Ya. B. Zh. Tekh. Fiz. 1941, 11,493. Zel’dovich, Ya. B.;
Zysin, Y. A. Zh. Tekh. Fiz. 1941, 11, 501.
(44) (a) Gray, B. F.; Aarons, L. J. Symp. Faraday SOC.1974, No. 9, 129;
J . Chem. Phys. 1976, 65, 2040. (b) Gray, B. F.; Morley Buchanan, T. J .
Chem. SOC.,Faraday Trans. I , in press.
Gray and Scott
constants for the pool chemical approximation to be valid for the
“Bru~selator”’~
are inconsistent with the conditions for the onset
of oscillatory behavior (Hopf bifurcation) but that the model of
Field and NoyesI6 is not inconsistent. The use of flow systems
and the machinery of sections 11-VI1 provide a much safer route
to the analysis of chemical systems.
(8) The kinetic schemes studied have displayed sustained oscillations only in open systems. In the limit of infinitely long
m, it is a “common-sense” expectation that
residence times, t ,
the model should go over smoothly to the required behavior of
a closed system, i.e. that there should be a unique stationary state
of stable nodal character which tends toward the chemical
equilibrium composition.
For the simplest models of sections 111-VI1 this is not, however,
the case. The cubic rate law has a unique stable node at this limit,
but this corresponds to a state of no conversion rather than the
complete conversion of equilibrium. With the quadratic catalysis
of section VII, the felony is further compounded: not only does
the system tend to a state different from chemical equilibrium
in this limit, it also retains its focal character. For the simple
first-order decay of B, or with values of the saturation term such
m: for larger
that p < 72rthe solution is a stable focus as T~
values of p, the system is an unstable focus surrounded by a stable
limit cycle.
The inclusion of the lower order and uncatalyzed steps, even
with the smallest nonzero rates, puts this right, however, and all
the systems approach equilibrium as t, becomes infinite.
(9) The complexity of the behavior exhibited by these schemes
arises from the balance between the reaction term R and the flow
term L. The cubic rate law leads to a highly nonlinear form for
R which is typical of many real systems. Although this complexity
in R may arise from a termolecular step, it might equally be the
result of a subsequence of simpler processes. It cannot be emphasized too strongly that our approach does not require belief
in a single elementary reaction.
The cubic form for R requires only a linear dependence of L
on the concentration to lead to the array of exotic patterns outlined
in the previous section. If L can become a more complex function
itself, then simpler dependences of R would suffice.
(10) An interested reader will find many similar analyzes applied to a simple exothermic reaction in the chemical engineering
l i t e r a t ~ r e . ~ ~Recent
, * ~ studies include the discoveries of both new
patterns in the dependence of stationary-state extents of conversion
on residence time5 and new phase-plane portraits.46 Isothermal
oscillations in biological systems have been demonstrated by
Chance and co-authors4’ and discussed by various a ~ t h o r s . ~ ~ - ~ ~
Further investigationsof the cubic autocatalytic rate law of sections
111-V are being made by Lignola and D’Anr~a’~
using the catastrophe theory approach of Golubitsky and Keyfitz” as adapted
by Balakotaiah and LUSS.~
-
-
Acknowledgment. We thank Professors B. F. Gray of Sydney,
I. Epstein of Brandeis, C. Escher of Aachen, and particularly K.
Showalter of West Virgina for stimulating criticism and comments.
We are also grateful to the SERC and the EEC,to British Gas
for research grant support, and to the ARGC for the award of
a Queen Elizabeth I1 Fellowship to S.K.S.
(45) Vaganov, D. A.; Samoilenko, N. G.;Abramov, V. G. Chem. Eng.Sci.
1978.33, 1282.
(46) (a) Williams, D. C.; Calo, J. M. AIChEJ. 1981,27, 514. (b) Kwong,
V. K.; Tsotsis, T. T. AIChEJ. 1983, 29, 343.
(47) Chance, B.; Eastbrook, R. W.; Ghosh, A. Proc. Narl. Acad. Sci.
U.S.A. 1964, 51, 1244.
(48) Higgins, J. Ind. Eng. Chem. 1967, 59, 19.
(49) Edelstein, B. B. J . Theor. Biol. 1970, 26, 227.
(50) Seelig, F. F. J. Theor. Biol. 1971, 30, 485.
(51) Sel’kov, E. E. Eur. J . Biochem. 1968, 4,19.
(52) Lignola, P.-G.; D’Anna, A. Proc. R. SOC.London, Ser. A , in press.
( 5 3 ) Golubitsky, M.; Keyfitz, B. L. SIAM J . Marh. Anal. 1980, 11, 316.