Untangling Complex Systems, page 42
steady state. If we follow x as a function of time, we record a trace like that shown in Figure 8.2d.
The phenomenology depicted in Figure 8.2 is called excitability. A system is excitable if perturbing it in a small amount, it rapidly returns to the steady state, but if it is given a significant perturbation,
it makes a large amplitude excursion before returning to the initial state. An excitable system has
two essential characteristics: (I) a threshold of excitation and (II) a refractory period. The threshold
of excitation is the smallest value of perturbation that causes the system to make a large excursion
before recovering the resting state. The refractory period is the time needed to recover the steady
state after a perturbation, which is the time that must elapse before the system is ready for receiving
another stimulation.
8.3.2 oscillaTions
The second important situation is shown in Figure 8.3, and it occurs when the intersection of the two nullclines lies on the middle branch of the S-shaped curve, which is an unstable steady state as
inferred from the signs of the first derivatives, f and g. Even a tiny perturbation pushes the system away from the steady state: from P to and suddenly to because
0
P 1
P 2
f changes very quickly. Then,
it moves more slowly along the x-nullcline until it reaches point P . From there, the system jumps to 3
the other branch of the x-nullcline (point P ). This branch is in the negative region of
4
g( x, y), so the
system moves along the x-nullcline toward P as
, the system
5
y slowly decreases. After reaching P 5
makes a fast transition back to P , and the cycle repeats. It is clear that we have periodic temporal
2
oscillations, as shown in Figure 8.3b, called relaxation oscillations. If we exclude the initial transient stage soon after the perturbation, then the dynamics is the same no matter what the perturbation is.
8.3.3 in PracTice
From a practical point of view (Epstein and Pojman 1998), after finding a bistable system, con-
firmed by the experimental observation of a hysteresis loop, and after choosing Y as a candidate for
negative feedback, we must look for two effects to happen. First, the inflow of a small amount of
Y to our reactor input stream should shrink the range of bistability measured as a function of some
tunable parameters, for instance, the flow rate k (like in Figure 8.4). Second, although the range of 0
bistability should narrow, the states of the two branches should retain their identities. If this occurs,
204
Untangling Complex Systems
Oscillations
Y
Steady state II
Steady state I
Bistability
k 0
FIGURE 8.4 Cross-shaped phase diagram3 for a bistable system where species Y is added at different flow rates, k .0
at a certain point by increasing Y the bistability vanishes and oscillations start (see Figure 8.4). As shown in the cross-shaped phase diagram of Figure 8.4, often the region of oscillation widens as
Y increases, although this may depend on the system we are studying. If by adding an even small
amount of Y, the states of the two branches begin to approach one another in character and blend
into one, it means that the negative feedback is too fast with respect to the reactions that give rise to
the bistability. In such case, it is necessary to try different species for the negative feedback.
The first success of the methodology explained in this paragraph dates back to 1980s when De
Kepper joined the Epstein’s group at Brandeis University. After a careful investigation, the team
proved that the chlorite-iodide reaction [8.9] is autocatalytic in iodine and gives rise to a bistable
chemical system in a CSTR (Dateo et al. 1982):
ClO−
−
+
−
+
+
→
+
+
2
4I
4H
Cl
2I2 2H2O [8.9]
In the presence of excess chlorite, the iodine is oxidized further to iodate according to the following
reaction:
5ClO−
−
−
+
+
+
→
+
+
2
2I2 2H2O
5Cl
4IO3 4H [8.10]
The introduction of arsenite generates the necessary negative feedback required to observe oscil-
lations in a quite wide range of conditions. In fact, arsenite reacts with iodate and produces iodide
(Epstein and Pojman 1998):
IO−
−
+
→ +
3
H
3 3AsO3
I
H
3 3AsO4 [8.11]
It was also found that the chlorite-iodide system can oscillate by itself in a flow reactor, though in a
much narrower range of conditions than the full chlorite-iodate-arsenite system (Dateo et al. 1982).
In fact, reaction [8.10] and/or [8.12] may provide the necessary feedback to allow it to oscillate.
IO−
−
+
+
+
→
+
3
I
5
H
6
I
3 2
H
3 2O [8.12]
3 Boissonade and De Kepper (1980) proposed an abstract model based on two independent variables summarizing the relationships between bistability and limit cycle. They discussed both the linear and nonlinear stability analysis, obtaining a cross-shaped diagram like that shown in Figure 8.4. Epstein and Luo (1991) demonstrated that the same cross-shaped phase diagram may be obtained by replacing the two variable ordinary differential equations of Boissonade and De
Kepper, with a single differential delay equation in which the delayed negative feedback of the y variable is mimicked by replacing y( t) in equation [8.8] with x( t − τ).
The Emergence of Temporal Order in a Chemical Laboratory
205
8.4 PRIMARY “OSCILLATORS”
The oscillatory chemical reactions have rather complicated mechanisms because they usually
involve many species and many steps. Therefore, it usually requires a great experimental and com-
putational effort to determine the complete set of elementary steps that specifies how an oscillatory
chemical reaction occurs. Often, to elucidate the main properties of the oscillatory phenomenon, it
is satisfactory to focus on the most important processes, overlook the details and propose an abstract
model. For defining a taxonomy of chemical oscillators and partition all the synthetic oscillating
chemical reactions in families, it is useful to find out a “core” set of reactions that produce the essen-
tial oscillatory dynamics. Each minimal set of reactions necessary to observe chemical oscillatory
phenomena will be defined as a “primary oscillator.” If we focus our attention on homogeneous
isothermal reactions, we may sort them into five main families of “primary oscillators.” The first
four “primary oscillators” will show the antagonism between a positive feedback and a coupled
delayed negative feedback restoring the initial state. The fifth “primary oscillator” will rely, only,
on properly delayed negative feedback events.
8.4.1 oregonaTor model: The “Primary oscillaTor” of coProducT auToconTrol
A family of chemical oscillators is based on the overall reaction of bromination and oxidation of
an organic compound by acidic bromate in the presence or not of a catalyst. The catalyst is usually
a one-electron redox system of the type M( n+ )1+/M n+ or [
n+1 +
n+
m ](
)
ML
/[ML m] where M represents a
transition metal and L is a bidentate ligand. Its redox potential is between 1.0 and 1.5 V. When a
catalyst is involved in the reaction, the chemical oscillator is referred to as Belousov-Zhabotinsky
(BZ) reaction. In the mid-1970s, it was found that even in the absence of a catalyst the reaction
between acidic bromate and some organic molecules (like phenol’s and aniline’s derivatives) may
exhibit oscillations (Orbán et al. 1979). In both catalyzed and uncatalyzed oscillations, a relevant
role is played by the intermediate bromide ion, Br− (Ruoff et al. 1988). In fact, the redox state of
the system depends crucially on its concentration. It has been demonstrated that when the bromide
concentration is higher than a critical value, [Br−] c (
−
5 10 6)[BrO−
= ×
3 ] , the system is in its reduced
c
state and the main reaction occurring in solution is
BrO−
−
+
+
+
+
→
+
3
2Br
H
3
RH
3
RBr
3
H
3 2O [8.13]
wherein RH is the organic substrate. Reaction [8.13] derives from a bunch of four elementary steps:
Br− + BrO−3 + 2H+ → HOBr + HBrO2
Br− + HBrO2 + H+ → 2HOBr
[8.14]
3(Br− + HOBr + H+ → Br2 + H2O)
−
+
3(Br2 + RH → RBr + Br + H )
In [8.14], three important intermediates appear: hypobromous acid (HOBr), bromous acid (HBrO ),
2
and bromine (Br ). Br brominates the organic substrate. While the elementary steps [8.14] proceed,
2
2
the concentration of Br− is progressively consumed. As it becomes smaller than its critical value,
[Br−] c, the entire system shifts to its oxidized state and another set of elementary reactions takes place:
2(BrO−
+
•
3 + HBrO2 + H
→ 2BrO2 + H2O
4
•
n+
+
( n 1
(BrO
+ )+
)
2 + M
+ H → M
+ HBrO2
[8.15]
2H
HBrO
−
+
2 → HOBr + BrO3 + H
HOBr + RH → RBr + H2O
206
Untangling Complex Systems
In [8.15], we discern the involvement of an autocatalytic production of bromous acid if we sum the first
and the second steps. In the case of uncatalyzed brominations, the organic substrate replaces the reduced
metal ion. By summing the elementary steps in [8.15], we obtain that the overall transformation is
BrO−
n+
+
( n 1
+ )+
+
+
+
→
+
+
3
4M
RH
H
5
RBr 4M
2H2O [8.16]
wherein the catalyst is oxidized. When the concentration of Br− drops below its critical value, the
concentration of HBrO increases autocatalytically of about five orders of magnitudes. The pro-
2
cesses [8.14] and [8.15] take place under different conditions in the same system, and a solution
reacting by [8.14] will of necessity convert itself to one reacting by [8.15]. If we want to have oscil-
lations, there must be a process producing bromide and pushing the system from the set of reaction
[8.15] to [8.14]. The oxidized form of the catalyst M( n+ )1+, produced in [8.15], reacts with the organic
species. In the case of malonic acid as organic substrate, the reactions are (Noyes et al. 1972):
6 ( n 1
M + )+ CH
n+
+
+
(
) +
→
+
+
+
2 COOH
2H O
6M
HCOOH 2CO
6H [8.17]
2
2
2
4 ( n 1
M + )+ BrCH COOH
2H O
Br− 4M n+ HCOOH 2CO
5H+
+
(
) +
→
+
+
+
+
[8.18]
2
2
2
As the concentration of bromomalonic acid ([
(
BrCH COOH) )
2 ] increases, the reaction [8.18]
becomes always more relevant. The Br− produced in [8.18] is consumed by the high concentration
of HBrO according to the second reaction of [8.14]. However, when the rate of [8.18] becomes
2
sufficiently great, [HBrO ] is quickly depleted and it drops of many orders of magnitude. This
2
turns off process [8.15], it turns on process [8.14] and completes the cycle, which will start again.
Experimental evidence (Varga et al. 1985) suggest that additional bromide may come from inter-
mediate bromo-oxygen species. Moreover, when bromination of the organic substrate is slow or not
possible, Br piles up and it may produce bromide by hydrolysis
2
Br +
→
+
− + +
2
H2O
HOBr Br
H [8.19]
in both uncatalyzed and catalyzed systems.
The basic features of the mechanism proposed by Field, Körös, and Noyes (FKN) are included
in the abstract model known as Oregonator advanced by Field and Noyes who were working at the
University of Oregon (Field and Noyes 1974a). Such model consists of five steps that involve five
chemical species: X = HBrO
( )+
2, Y =
−
Br , Z
n
=
+
2
1
M
, A =
−
BrO3, P = HOBr, and O = all oxidizable
organic species:
A + Y → X + P
X + Y → 2 P
A + X → 2 X + Z [8.20]
2 X → A + P
Z + O → ffY
The concentrations of A, P, and O are assumed to be constant as that of H+, whereas those of X, Y
and Z are changeable. The first two steps of [8.20] represent the first two steps of [8.14]. The third
step of [8.20] represents the autocatalysis of HBrO . The fourth is the HBrO disproportionation.
2
2
If the autocatalytic process plays the action of positive feedback, the disproportionation is part of
the negative feedback. In fact, disproportionation becomes important when X accumulates, and it
works against a further increase of X. However, the disproportionation alone is not strong enough
to compete with and inhibit the “explosive” autocatalytic stage. The crucial process of delayed
negative feedback is the coproduct autocontrol provided by the fifth and the second process of the
Oregonator (Luo and Epstein 1990). As X increases autocatalytically, also Z grows (see the third
The Emergence of Temporal Order in a Chemical Laboratory
207
step of the Oregonator). Then, Z transforms to Y through the fifth step (where f is a stoichiometric coefficient), and Y rapidly consumes X (see the second step). When Y is high and X very low, the first step regenerates X. The transformation of Z to Y provides the essential time delay between the positive and the negative feedbacks to have limit cycles and hence observe oscillations.
TRY EXERCISES 8.2, 8.3, AND 8.4
There are other oscillating reactions whose mechanisms involve coproduct autocontrol as the domi-
nant form of negative feedback. For instance, the reaction involving trypsin, 4-[2-aminoethyl]ben-
zenesulfonyl fluoride, and aminopeptidase M (Semenov et al. 2015); the oxidation of benzaldehyde
by air catalyzed by CoBr (Colussi et al. 1990); the bromate-iodide reaction in acidic solution (Citri
2
and Epstein 1986), and the H O –KSCN–CuSO reaction that oscillates only above pH 9 (Orbán
2
2
4
1986). The latter, also known as Orbán reaction, was the first example of a homogeneous, liquid
phase, halogen-free system that oscillates even under batch conditions. If we mix hydrogen per-
oxide, potassium thiocyanate, and copper sulfate in alkaline solution, we can observe oscillations
in color (between yellow and colorless states), redox potential, and O evolution. If we also add
2
luminol, we can record chemiluminescent oscillations. The original mechanism proposed in 1989
(Luo et al.) was composed of 30 reactions and 26 independent variables. These reactions may be
partitioned in three groups (Orbán et al. 2000). The key steps of the first group regard the alkaline
decomposition of H O catalyzed by Cu(II):
2
2
H
2+
−
( )
2O2 + Cu
+ OH → HO2Cu I + H2O
[8.21]
HO
( )
−
+
n
{ −}
•
2Cu I + SCN
→ Cu SCN
+ HO2
n
The intermediate copper-peroxide complex (HO2Cu(I)) is yellow and is responsible for the color
