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Chemical Oscillations.

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Volume 17 * Number 1
January 1978
Pages 1-72
International Edition in English
Chemical Oscillations
By Ulrich F. Franck[*]
Under constant external conditions, chemical reactions may also proceed in a rhythmic
manner. The kinetic mechanisms responsible for such oscillations prove to be unexpectedly
complicated. It can nevertheless be demonstrated that chemical periodicity is caused by certain
kinds of coupling between simultaneous reactions or transport processes. A general survey
of the chemistry and phenomenology of the principal chemical oscillations is followed by
a discussion of the situations leading to periodic reactions on the basis of the multivariable
kinetics of feedback systems. Autocatalysis and autoinhibition play an important role, as also
do kinetic instability and spatial propagation.
1. Introduction
Left to themselves under constant external conditions, normal chemical reaction systems equilibrate provided they are
not completely inhibited. Thermodynamically open systems
approach a likewise thermodynamically defined steady state,
as shown schematically in a plot of concentration uersus time
(Fig. la).
examples of chemical systems deviating considerably from
this pattern in their temporal behavior (cf. also Fig. 1b and
Ic). One of the most interesting deviations is the ability of
some chemical and physicochemical systems to undergo spontaneous oscillatory reaction (Fig. 1 d and 1e).
Although this phenomenon of rhythmic chemical, and especially electrochemical, reactions has long been known, thorough experimental and theoretical studies have only been
performed in the past 20 years. Several competent reviews
of this field have recently been published (see e.g. [‘-“I). In
the present article, recent discoveries and viewpoints will be
Fig. 1. Temporal behavior of thermodynamically open systems. X =concentration of starting compound. a) Normal non-periodic attainment of steady
state; b)attainment ofsteady stateat rhythmically changing rate; c) attainment
of steady state via attenuated oscillation; d) flip-flop oscillation; e) sawtooth
oscillation.
However, in their experimental work chemists, chemical
engineers, and biologists are also acquainted with numerous
[‘I
Prof. Dr. U. F. Franck
Institut fur Physikalische Chemie der Technischen Hochschule
Postfach der TH, D-5100Aachen (Germany)
Angew. Chem. Int. E d . Engl. 17,1-15 (1978)
t-
0
t-
o
t-
A
t-
o
J
Fig. 2. Non-periodic behavior patterns of systems with kinetic coupling.
a) Excitable bistability; b) overshoot behavior; c) excitable pulse formation.
R is an external stimulus acting transiently on the system (see also Fig.
21). The broken lines indicate subthreshold behavior.
1
evaluated to show that, in spite of the unexpected complexity
of their mechanisms, the kinetics of oscillatory chemical reactions can probably be reduced to a limited number of kinetic
principles, and is largely analogous to the kinetics of known
mechanical and electrical systems.
Closely connected with chemical periodicity are other
“anomalous” non-periodic modes of behavior, some of which
are illustrated in Figure 2. They generally occur on slight modification of systems capable of oscillation. Systems no longer
able to oscillate generally still possess bistability, i.e. they can
exist in two stable reaction states whose interconversion can
be triggered by external perturbations (“stimuli”). The study
of non-periodic behavior is of great value for the assessment
of oscillatory kinetics. It also underlines the close relationship
with the excitation processes of nerve and muscle cells, which
are themselves known to be capable of oscillatory behavior.
The phenomena mentioned have their origin in kinetic processes which are not observed in “normal” systems. A detailed
account presented in Section 3.3 shows that mainly kinetic
coupling exists between the effective reactions of the oscillatory
system and gives rise to various forms of chemical feedback.
As in mechanical and electrical systems, the generation of
oscillations in chemical systems is also a consequence of feedback processes. Studies should therefore concentrate on the
question how chemical feedback can come about, and what
form it must take for oscillations to occur.
Chemical oscillations in homogeneous systems are currently
under investigation in numerous laboratories. And yet, despite
the overwhelming body of detailed factual material and great
effort expended, not a single chemical oscillation has so far
be completely elucidated beyond all doubt. The reasons are
listed below:
Periodic chemical reactions occur only in extensively
coupled multivariable systems. Such systems are extremely
non-linear with respect to driving forces and the driven fluxes
and reactions.
Oscillations affect changes of states far removed from equilibrium, to which the classical theories of the thermodynamics
of equilibrium states and non-equilibrium states close to
equilibrium are no longer directly applicable.
Oscillatory chemical systems contain unstable states which
cause spontaneous rhythmically occurring spatial propagation
of changes of state. Hence homogeneous systems in a thermodynamic sense are no longer present.
Section 3 is devoted to these kinetic features and difficulties.
Before proceeding further, however, the principal chemical
and physicochemical oscillations and their specific phenomenology will first be considered.
sionally recorded are by no means due to particularly simple
linear processes.
In an open system with a regular flux of reactants the
oscillations can be maintained indefinitely. In closed systems,
where they were generally first observed, they are of limited
duration sufficiently distant from equilibration.
2.1. Chemical Oscillations in Homogeneous Systems
2.1.1. Purely Chemical Oscillations in Homogeneous Solutions
This class of chemical oscillatory reactions presently attracts
the greatest interest among both experimentalists and theorists.
Oscillations were previously thought to be impossible in homogeneous solutions. The unfounded assumption that periodic
behavior always requires the presence of a heterogeneous
structure was apparently based on the view that control of
the periodic change in the reaction can only be the result
of the action of fluctuating interfacial states as known from
heterogeneous catalysis.
t [minl-
Fig. 3. Bray oscillation: decomposition of hydrogen peroxide in the presence
of iodate and iodine. Upper curve, iodide concentration; lower curve, iodine
concentration (according to [ll]).
When Bray observed[’S6] in 1920 that the decomposition
of hydrogen peroxide in the presence of iodic acid and iodine
follows a periodic course (Fig. 3), much effort was devoted
to the search for the interfacial structures responsible for
the oscillations-they were suspected to lie in microscopic
oxygen bubbles[’, and also in dust particle^^^! Meanwhile,
after the discovery of further “homogeneous” periodically
reacting chemical systems, it is known that structural heterogeneity is not necessary for the occurrence of oscillations.
The Bray reaction proceeds according to the simple equation
Thus iodic acid and iodine are true catalysts. The usual
formulation as two partial equations
2. Phenomenology of Chemical and Physicochemical
Oscillations
The external appearance of the oscillations observed in
a large variety of systems is surprisingly uniform. The oscillograms either consist of periodic transitions of various degrees
of sharpness between two quasi-steady states, or they have
a sawtooth shape (Figs. 1d and 1e). Chemical oscillations
therefore resemble non-linear flip-flop oscillations more closely
than harmonic vibrations. The sinusoidal oscillograms occa2
with HzOZreducing the iodic acid and oxidizing the iodine
describes the actual reactions occurring only incompletely
or even incorrectly. Experimental results show that oxygen
evolution is not stoichiometric with the optically observable
conversion of iodine, and that the oxygen appears as product
of the iodine reaction (2) and not of the iodic acid reaction
(1) during the oscillation”!
Angew. Chem. Int. Ed. Engl. 1 7 , l - l S ( 1 9 7 8 )
Degn['ol interprets these experimental observations in terms
of an intermediate higher order branching reaction of a transient radical species.This would explain the non-stoichiometric
relationship between the liberation of iodine and O2 and
also the suspected autocatalytic nature of a partial reaction.
Another mechanism has been proposed by Matsuzaki, Nakajima, and Liebhafsky["], which comprises four essential reaction steps:
The third step is autocatalytic.
This reaction sequence corresponds to a mathematical
model of systems capable of oscillation which has been put
~ * has been
forward by Prigogine, Nicolis, and L e f e ~ e r [ 'l3I-it
designated as "Brusselator". The model (cf. Section 3.2) consists
of the following four sub-steps:
1) A
-+x
2) B+X
-+ D+X'
3) 2 x + x -+ 3 x
4) x
-+E
A+B
+
D+E
In the case of the Bray reaction, A and E would correspond
to HI03, X to H I 0 2 , X' to HIO, B to H202, and D to
02,the overall balance being:
B+D
because A = E.
The autocatalytic step (3) of the Brusselator and of the
reaction model of the Bray reaction is trimolecular. This is
the most problematic step of these models because trimolecular
reactions are extremely improbable, especially in a reaction
mixture like that of the Bray reaction for which 16 possible
reactions of 11 reactants have to be considered["] and in
which two bimolecular reactions (2 and 4) are in competition.
In 1951, Lynn['4.1 5 J found another homogeneous system
exhibiting concentration oscillations: the thermal decomposition of sodium dithionite.
In buffered (pH 24) aqueous solution, sodium dithionite
decomposes above 50°C according to the reaction:
2NaZS2O4+HZ0-+ 2NaHSO3+Na2S2O3
Experimental findings indicate an autocatalytic reaction
step suspected by Mason and Depoy['61to lie in the following
two-step radical reaction:
According to Wayman and Lem"'], HSO; takes up an
electron to form the reaction product HSO;.
Interest in chemical oscillations was awoken primarily by
the discovery of a highly impressive reproducible periodic
reaction by Belousov['8](1958). On attempting to oxidize citric
acid by bromate in the presence of Ce4+, he noticed that
the yellow color of the Ce4+ ion in the homogeneous reaction
mixture vanished and reappeared in a constant rhythm.
Credit goes mainly to Z h a b ~ t i n s k i i ["1' ~ ~for recognizing
the importance of this reaction for the understanding of periodic chemical and biological processes. In particular, he showed that the reactants could be modified and their oscillatory
ability thereby considerably improved. Thus the citric acid
can be replaced to advantage by other brominatable organic
compounds, especially by malonic acid. Zhabotinskii also
recognized that the Ce4+ ion serves as electron transfer agent
and can be replaced by redox systems having a correspondingly
high positive redox potential. The reactants of the "BelousovZhabotinskii reaction", currently the most thoroughly studied
process of its kind, are a solution of malonic acid, potassium
bromate, and cerium(1v) sulfate in 3~ sulfuric acid. After
an induction period of about IOmin, extremely regular oscillation sets in at a frequency of about two periods per minute
(at room temperature). They persist for about 15 to 30min
in a closed system, and can be maintained indefinitely in
an open system with a constant reactant flux.
Table 1 lists the components of the Belousov-Zhabotinskii
reaction. Potential organic brominatable components are compounds bearing a hydrogen readily substituted by bromine
in the proximity of a carbonyl group. The redox potential
of the electron transfer agent must exceed 0.9V[211,and the
Table 1. Modifications of the Belousov-Zhabotinskii reaction. Overall process for the most thoroughly investigated
variant:
2 B r 0 5 + 3 C H 2 ( C 0 0 H ) 2+ 2 H t
+ 3 C 0 2+ 4 H 2 0
+ 2 BrCH(COOH)2
Reducing agent
Oxidizing agent
Electron transfer agent
Acidic medium
Malonic acid
Bromomalonic acid
Citric acid
Malic acid
Gallic acid
Acetone
Acetonedicarboxylic acid
2,4-Pentanedione
2,5-Hexanedione
KBr03
K103IHzOz
Ce3 +Ice4
Mn2+/Mn3+
FerroinIFerriin [a]
Ru(bpy):+/Ru(bpy):+ [a]
Ru(phenj: +/Ru(phen): [a]
HxSOa
HClOa
+
+
[a] Ferroin = Fe(phen):+; ferriin= Fe(phen):+; bpy =2,2'-bipyridine; phen = 1,10-phenanthroline.
Angew. Chem. I n t . Ed. Engl. 17.1-15 (1978)
3
reaction medium must contain a strong acid. Moreover, bromination can be replaced by iodination with iodatehydrogen
peroxide["! The Belousov-Zhabotinskii reaction has not yet
been completely elucidated[" - 2 3 - "1.
At present some ten reactions of twelve reactants are under
discussion. Six of the ten reaction steps are regarded as essential partial reactions, two of them being considered as constituting an autocatalytic process. There remain the following five
reactions viewed as responsible for the oscillations in a narrower sense:
"9
1) BrO;+Br-+2H+
2) HBrOz+Br-+H+
3) BrO;+HBrOz+2Ce3++3H+
4) 2HBrOZ
5) nCe4++ BrCH(COOH)Z
-+
--t
-+
--t
-+
HBrOZ+HOBr
2HOBr
2HBr02+2Ce4++H20
BrO; + HOBr + H i
nCe3++Br-+
oxidation products
and another in which Ce3 is oxidized. The bromide concentration is relatively high in the reducing phase, and extremely
low in the oxidizing phase. Formation of bromomalonic acid
is more pronounced in the reducing phase than in the oxidizing
phase. The same applies to the evolution of heat. Bromine
appears briefly during the transition from the oxidizing to
the reducing phase.
Noyes e l
311 formulate the reaction sequence as follows: In the Belousov-Zhabotinskii reaction, bromination of
malonic acid can, in principle, proceed via two mutually exclusive pathways.
a) Pathway involving reduction of Ce4+ to Ce3+ in the
presence of bromide ion:
+
a1.[303
BrO;+2Br-+3H+ -+ 3HOBr
+ HzO
HOBr CHz(COOH)2 --t BKCH(COOH)~
+
Bromomalonic acid subsequently reduces the Ce4 present
to Ce3+,thereby decomposing to COz and bromide:
+
Reaction (3) is a stoichiometric autocatalytic reaction with
respect to HBr02. It is assumed to proceed in two steps
via the BrO; radical.
BrO; + H B r 0 2 + H +
2Br0;+2Ce3+ + 2 H +
__-
BrO;
+ HBr02 +2Ce3+ + 3 H +
-+
2BrO;+H20
2HBrO2+2Ce4+
-+
2HBrOZ+2Ce4+ + HzO
-+
Assuming the protonations to be very fast, a11 the steps
of the reactions given are of 2nd order.
BrCH(COOH)Z+4Ce4++2HZ0-+ Br-+4Ce3++HCOOH
+2C02 + 5 Hf
HCOOH + HOBr -+ Br - + COz + H + + HzO
The overall process in the reducing phase is:
BrO; +2CH2(COOH)z + 4Ce4+ -+
BrCH(COOH)2+4Ce3++ 3 C 0 z + 3 H + + H Z 0
b) Pathway involving oxidation of Ce3+ to Ce4+ in the
absence of bromide ion:
The oxidizing species is now bromate, which oxidizes Ce3+
and is itself reduced to HOBr. The latter then brominates
malonic acid :
BrO;
+4Ce3 +CH2(COOH)z+ 5 H
+
+
-+
BrCH(C0OH)z +4Ce4++ 3 H 2 0
Together, pathways a) and b) comprise the overall oscillation
reaction :
2BrOi +3CHz(COOH)z+2Hf
-+
2BrCH(COOH)2+3COz+4HZO
t [minl-
Fig. 4. Belousov-Zhabotinskiioscillation: bromination of malonic acid with
bromate in the presence of Ce4+.a) Br- concentration [24]; b) Ce3 concentration [79]; c) bromine concentration (spectroscopic measurement by W
Geiseler 179,811; d) concentration of bromomalonic acid (BMA) in a closed
system (according to [80]); e) temperature oscillation in a heat-flow calorimeter (measurements by W Geiseler) [79, 811. The synchronization lines
refer to the instability points of bromide concentration. Concentrations are
given in mol 1 '.
+
The presence of Br- thus determines which of the two
bromination pathways is feasible. Each individual reaction
pathway is assumed to proceed via intermediates not included
in the above partial equations. Bromous acid (HBr02) is
accordingly assigned a crucial role.
In the reducing phase (pathway a) the following intermediate
steps are assumed to occur in the first equation:
BrO; + Br-+2Hf --+ HBrOZ+HOBr
HBrOZ+ Br- + H + + 2HOBr
BrOB +2Br-+3H+
The oscillation process is presented in synoptic manner
with regard to the most important analytically accessible variables(Br-,Ce3 +/Ce4+,Brz, bromomalonic acid, and temperature) in Figure 4.
The oscillations consist in a kind of flip-flop alternation
between two reaction states: one in which Ce4+ is reduced
4
-+
3HOBr
In the oxidizing phase (pathway b), the reaction proceeds
via the above mentioned and formulated autocatalytic step
involving HBrO (reaction 3). This species disproportionates
to bromate and HOBr, which subsequently brominates
malonic acid:
Angew. Chem. Int.
Ed. Engl. 17, 1-15 (1978)
2BrOS +2 HBrOZ+4Ce3 + 6 H +
--+
2 HBr02
-+
+
~~~
4HBr02 +4Ce4+
+2H20
BrO;+HOBr+H’
~~~~
BrO;+4Ce3++5H’
--+
HOBr + CHz(COOH)2
-P
BrO; +4Ce3+ +CHZ(COOH)2+ 5 H +
-+
HOBr+4Ce4++
2 HzO
BrCH(COOH)Zt
H2O
BrCH(COOH)2+
4Ce4’ + 3 H 2 0
This is the equation of pathway b).
Numerous experimental results suggest that the formulations presented here provide an essentially correct description
of the individual reaction steps. However, various questions
remain open, particularly in connection with the actual origin
of the oscillations. A system of four simultaneous differential
equations derived from the above formulations, called the
“ O r e g o n a t ~ r ” by
[ ~ ~analogy
]
with the “Brusselator” and which
also has oscillatory solutions, takes the following form:
A+Y +X+P
X + Y + 2P
A + X --+ 2X+Z
2X +P+A
PY
Z--+--a+P
R ~ y / e i g h [441may
~ ~ , also belong to this class of periodic chemical reactions.
The kinetics of thermochemical oscillations is still poorly
understood. Once again, however, autocatalytic processes and
chemical feedback are known to play an important part. In
gas phase reactions, autocatalysis frequently involves branched
chain reactions which can occur both in the formation and
in the decomposition of intermediates. Gas reactions generally
span wide ranges of temperature. Under these conditions,
the temperature dependence of the rate constants leads to
marked non-linearity and non-stoichiometric coupling
between all the participating reactions.
Two oscillating thermochemical reactions have recently
been studied in particular detail, viz. the oscillations of cold
flames of propane at about 330°C (Fig. 5)f3’1and the thermochemical oscillations of carbon monoxide in oxygen at
600°C[41*
421. The kinetics of these reactions is apparently
very complicated. The most important reactions involve shortlived radicals which are very difficult or impossible to detect
analytically. In the oscillatory combustion of CO[421,18 reactions of 12 reactants are under consideration-eight simultaneous kinetic differential equations are obtained. An incomprehensible variety of material and thermal coupling modes
results.
P2
a+P
2.2. Oscillations in Heterogeneous Systems
A=BrO;; X=HBr02; Y=Br-; Z=2Ce4”; P=bromomalonic acid. A detailed discussion of the resulting differential
equations will be found in the specialist literature[I8-331.
2.1.2. Thermochemical Oscillations in Gas Reactions
Periodic reaction courses are also observed in a number
of gas reactions. A typical example is shown in Figure 5.
As with the periodic reactions occurring in solution, interfacial
structures play no part.
This Section is concerned with systems whose oscillatory
behavior is strictly linked with interfacial structures. Such
is the case when one of the oscillatory variables is electrical
in nature.
Oscillations in electrochemical systems concern changes
of state involving local currents between regions in different
electrochemical states. Such local currents can occur in closed
circuits only at interfaces present at electrodes and ion membranes.
The treatment of heterogeneous catalytic reactions requires
a knowledge of the structure of the interface because reactiondependent interfacial states are crucial to the occurrence of
periodicity.
2.2.1. Electrochemical Oscillations
5
10
t [rninl-
Fig. 5 . Temperature oscillation in the thermochemical oxidation of propane
in oxygen (CaHs:5011101%. P: 7SkNm-*) (according to [37]).
On oxidation of hydrocarbons in mixtures with oxygen,
“cold flames” occur at relatively low temperatures
( z400°C)[34-381.This kind of combustion frequently proceeds
with rhythmic emission of pale blue light and periodic evolutipn of heat.
Oscillations are also observed on combustion of hydrogen
~ u l f i d e l ~ ~ lofcarbon
and
m o n o ~ i d e [ in
~ ~oxygen.
- ~ ~ 1The
~ oscillationsofphosphorusvapor in contact with oxygen described by
Angew. Chem. I n t . Ed. Engl. I 7 , 1 - 1 5 ( 1 9 7 8 )
Periodic electrode processes are the longest known of all
chemical oscillations. As long ago as 1828, such reactions
were reported by Fe~hner[~’].
Around the turn of the century,
O s r ~ u I dengaged
[ ~ ~ ~in~a ~thorough
~
study of the oscillations
of potential and corrosion of chromium in hydrochloric acid
and of iron in nitric acid. He was the first to recognize a
surprisingly complete kinetic analogy between electrochemical
and electrophysiological oscillations[48]which is not limited
merely to the actual oscillations but extends over the entire
phenomenology of nervous stimulation. This was demonstrated by very impressive analogy experiments performed
in the 1920’s and 1930’s by Li//ie[49*501.
Systematic physicochemical research into chemical oscillations did not begin until about 1940 with Bonhoeffer’s fundamental studied’ - 531 on the “Ostwald-Lillie nerve model”
and “Periodic chemical reactions”.
We are now familiar with a large number of electrochemical
oscillations. Most of them are observed at electrodes on which
5
anodic or cathodic surface layers can form. Such electrodes
generally possess non-monotonic current-voltage curves,
which arise because surface layer formation or removal sets
in at a particular electrode voltage.
The current usually changes abruptly and the current-voltage curve shows the N-shaped course characteristic of surface
layer electrodes[54.551. With suitable external circuitry, i. e.
with a suitable position of the load line, the possibility of
bistability arises which is one of the prerequisites for the
occurrence of oscillations.
Fe in &SO,
0
1
E [VI
0 10
@
Zn in NaOH
Au in HCI
-
0
0
@
trs1-
1
E [Vl
-
5
t[sl-
sponds to positive feedback and the coupled regeneration process to negative feedback.
There is also another kind of electrode oscillation[561which
differs in its temporal course and frequency from passivationactivation oscillation, and occurs only under potentiostatic
conditions.
These oscillations are usually associated with electropolishing effects. Their frequency is relatively high, lying between
80 and 4000Hz (Fig. 7). They give rise to a flickering effect
which can readily be observed under a microscope. At certain
-2
-1
0
Co in CrQHCI
1
E [VI-
0
@
10
0
t[sl-
1
@
t[sl-
Fig. 6. Examples of electrochemical oscillations. Upper row, non-monotonic current-voltage curves; lower row,
galvanostatic (I=const.) or potentiostatic (E=const.) oscillations. a) Galvanostatic voltage oscillation at iron in
3.16 N HzS04+0.04 N HCI [56,59]; b) potentiostatic current oscillation at gold in 4 N HCI at Eh= + 1.75 V [82,54];
c) potentiostatic current oscillation at zinc in 4 N NaOH at Eh= - 1.1 V [82, 831; d) spontaneous (Z=O) voltage
oscillation at cobalt in 0.4 M CrOB+ 1 N HCI [82, 84, 851.
Some typical examples of non-monotonic current-voltage
curves and oscillograms recorded for surface layer electrodes
are presented in Figure 6.
Periodicity at these electrodes arises because the voltageindependent passivation-activation process is coupled with
a further process which, after a brief lapse of time, reverses
the change of state resulting from passivation or activation.
This mechanism will be examined more thoroughly in Section
3.3. It will also be shown that passivation-activation corre-
t
1
8
a&==---
t-
Fig. 7. Fast potentiostatic “flicker” oscillations [56, 831 a) at zinc in 1 N
H 2 S 0 4 at E h = - 0 . 2 9 V ; b) at iron in 1 N H2S04 at Eh= +0.3V; c) at
silver in 0.7 N KCN at Eh= + 0.53 V.
6
electrode potentials the flickering shows some degree of synchronization and interference colors appear, e.g. on zinc in
sulfuric acid. The mechanism of these oscillations is not yet
completely understood. It seems reasonable to invoke dynamic
dissipative structures which synchronize at certain potentials
and then become recognizable in the external current.
2.2.2. Oscillations in Membrane Systems
Electrochemical oscillations can occur at ion-conducting
membranes in the same way as at electron-conducting electrodes. The electrophysiological oscillations of nerve, muscle,
and sensory organ cells of living organisms are known to
take place at such membranes. Teorell’s observation[57,5 8 1
that even very simple porous ion exchange membranes exhibit
oscillatory behavior under constant external conditions
attracted considerable interest in this context.
The “Teorell membrane oscillator” has since been studied
in great detail with respect to its ion-transport kinetics. It
consists ofa highly porous membrane of ion exchange material
separating two compartments, which contain sodium chloride
solutions of different concentrations, and carrying a constant
electrical current (Fig. 8 a). The essential oscillatory variables
are the membrane potential E, determined by the membrane
resistance, and the hydrostatic pressure P which is set up
as a consequence of the hydrodynamic and electroosmotic
volume flow V between the compartments on the two sides
of the membrane.
Angew. Ckem. I n t . Ed. Engl. 1 7 , 1-15 (1978)
El
@
0
E'
E2
0
0
10
20
30
(0
t [minl-
Fig. 8. The Teorell membrane oscillator. a) Experimental set-up; b) current-voltage cur! L' 1 9 1 :
c) oscillations of voltage and pressure [58,59].
The voltage and pressure oscillations occurring in the Teore11 oscillator are shown in Figure 8c. The membrane may
consist of unglazed porcelain, porous sintered glass, porous
plastic, or pressed powders of quartz, metal oxides, or organic
ion-exchanger material.
Oscillations occur when the sign of the fixed ion charge,
the direction of the electric current, and the concentration
gradient of the salt solutions can be suitably ~ o m b i n e d [ ~ ~ . ~ ' l .
Under such conditions the membrane has a non-monotonic
current-voltage curve (Fig. 8b) as a result of reversal of the
direction of flux of the salt solution at a particular membrane
potential and the membrane resistance jumps to the value
corresponding to the modified influx of salt solution.
In Figure 8 b the load line I,=const. intersects the nonmonotonic membrane characteristic at three positions : hence
three steady states occur, of which the middle one is unstable
and the two outside ones are stable. The volume flux, which
is driven antagonistically by the pressure (P) and potential
(E), shifts the critical flux reversal potential E x in the same
way as the hydrostatic pressure changes with the volume
flux. The system becomes unstable at the limits shown in
Figure 8 b, the volume flux reverses, and the potential jumps
onto the other branch of the membrane characteristic. In
this way the state of the system alternates periodically between
the two points of instability (2; and Z ; ) . The oscillations
thus represent spontaneous alternating initiation of changes
of state of the membrane which are triggered by the hydrostatic
pressure, which itself depends upon the particular membrane
state. The Teorell membrane oscillator convincingly shows
how instability and kinetic coupling lead to oscillation. The
existence of the non-monotonic characteristic is a consequence
of positive feedback whereas the hydrostatic pressure effects
a negative feedback.
From a chemical and structural standpoint, two other artificial excitable membranes resemble the oscillatory nerve membrane even more closely. These are the bimolecular lipid membranes containing antibiotics, which were developed by Miiller
and Rudin[611(Fig. 9a), and the polyelectrolyte membranes
~ ~ *9 b).
~~~
produced and studied by S h a s h o ~ a [(Fig.
Both kinds of membranes are capable of oscillation and
exhibit electrophysiological excitation phenomena. Like the
Teorell membrane, they possess non-monotonic current-voltage curves. They are therefore bistable and have unstable
states. In the case of the lipid membranes this behavior is
caused by a pore-formation mechanism of the incorporated
antibiotic (e.g. alamethicin). S h a ~ h o u a [ assumes
~~]
that a
mechanism of cation-anion contact analogous to the p-n juncAnyew. Chem. I n t . Ed. Engl. 17. 1-15 (1978)
>
Y
10
@
0 10 20 30 40 50
0
E
2
1
t Irninl-
E ImVl--+
10
CI
C
@
0
1
2 3
EIVI-
L
0
0.1
0.2
t Is]--+
Fig. 9. Oscillations at biochemical membranes. a) Miiller-Rudin oscillator
[61]; left, current-voltage curve; right, galvanostatic voltage oscillation at
a phosphatidylcholine double-layer membrane containing lo-' g ml-' of
alamethicin and 10- g ml- of salmin. I = 3.2 pA cm- (according to experiments by G. Boheim [86]). b) Shashoua oscillator [62]; left, current-voltage
curve ofa polyglutamate-Ca2+ ion exchange membrane; right, voltage oscillation at a methacrylate-acrylic acid-Ca2 membrane.
'
+
tion in semiconductors operates in the polyelectrolyte membrane.
2.2.3. Chemical Oscillations in Systems with Heterogeneous
Catalysts
Oscillations in heterogeneous systems include the periodic
reactions observed in contact catalytic reactors. That unstable
reaction states occur in exothermic reactions of gases at heterogeneous catalysts follows from theoretical studies by Wagner1641,van HeerdenL6'], and Frank-Kamenetzkii1661.
Fig. 10. Oscillation in the oxidation of CO by air at a platinum-A1203
contact catalyst. a) Reaction-transport characteristic; r is the relative rate
constant in ppm C 0 2 in exhaust gas; the broken curve indicates hysteresis on
reverse order of measurement; b) oscillation at a single grain of catalyst; r refers
to the CO, concentration in ppm (according to [67]).
The most exhaustive experimental studies have been performed on the oscillatory behavior of the oxidation of carbon
7
monoxide and of hydrogen by atmospheric oxygen over platinum-alumina catalysts[67! Figure 10 shows an example of
such an oscillation in a heterogeneous reaction taken from
the work of Wicke et ~ 1 . r ~ As
~ ' .in electrochemical oscillatory
systems, non-monotonic reaction characteristics are found
once again. There exist an unstable reaction state and two
stable steady states.
Reactors of this kind are thus also bistable in the non-oscillatory state[67p
6 8 ] , and the oscillations are periodic alternations
between two quasi-stable reaction states. Two possible feedback modes exist in heterogeneously catalyzed reactions: 1)
by the non-linear temperature dependence of catalyzed exothermic reactions and 2) by kinetic action of the reaction products on the effectiveness of the catalyst, for example by changing the surface coverage. The first case is an example of positive
feedback and the second case one of negative feedback. Both
kinds of feedback can lead to oscillations.
Enzymatic reactions are known to be catalytic processes
in which the enzyme is regenerated in a cyclic manner. It
would appear that these reaction cycles do not always occur
at a steady rate but rather in rhythmic waves. The reason
is clearly seen to lie in the frequent substrate- or product-dependence of the enzyme effectiveness. Chemical feedback is then
possible and the condition for an oscillatory reaction course
is thus fulfilled.
In 1957, Duysens and Arnes~[''~observed that glycolysis,
i.e. the anaerobic degradation of glucose in living cells to
pyruvate, follows a periodic course. The oscillatory glycolysis
system can be isolated from cellular material by extraction
and studied in ~ i t r o [ ~ ~In* homogeneous
~*I.
phase this oscillation has the advantage that it can be relatively easily monitored
by observing the fluorescence of the NADH participating
in the reaction (Fig. 12).
2.3. Oscillations in Biological Systems
Rhythmic processes are characteristic of living organisms.
Elementary oscillatory phenomena were recognized and studied very early in electrophysiology, and hence much of the
terminology of oscillation and excitation kinetics comes from
neurophysiology.
2.3.1. Electrophysiological Oscillations
Oscillations of cells of nerve and muscle tissue and of sensory
organs are periodic excitation processes at ionic membranes
(Fig. 11) which possess a specific, potential-dependent selectivity for sodium and potassium ions. As a result, the nerve
membrane exhibits a non-monotonic current-voltage curve
like that of non-living ion membranes capable of oscillation
and therefore possesses bistability and instability. The nature
of the second or further kinetic variables has not yet been
t Imsl
-
t [msl-
Fig. 11. Oscillations at biological membranes. a) At an octopus nerve fiber
1871; b) at a single heart muscle fiber [SS].
unequivocally established for biological membranes capable
of oscillation. Divalent ions such as the calcium ion might
play an essential role.
2.3.2. Enzymatic Oscillations
Purely chemical oscillations have been known to occur
in living organisms since 1955, when Wilson and Calvin[691
first observed such processes in the dark reaction of photosynthesis. Meanwhile, oscillatory reaction courses have been
detected in numerous areas of biochemistry, in cell respiration,
in carbohydrate metabolism, in enzyme synthesis, mitosis,
morphogenesis, regeneration of damaged cells, and other
essential life processes. The common feature of all these oscillations is their enzymatic nature.
8
0
"
t [rninl-
Fig. 12. Oscillation in glycolysis. The reduced form of nicotinamide dinucleotide (NADH) participating in glycolysis exhibits readily measurable fluorescence permitting fluonmetric monitoring of the oscillation (according to
[891).
Although the mechanism of oscillation of glycolysis has
not yet been fully elucidated, a number of important chemical
and kinetic aspects of this interesting process are known.
Phosphofructokinase, an allosteric enzyme, seems to play
a critical role. This enzymf is inhibited by one of its substrates,
ATP, and activated by its products, ADP and fructose 1,6diphosphate. The chemical feedback necessary for oscillation
is thus ensured[73!
Yamazaki, Yokota,and Nakajima"'* 751 found the peroxidase
oxidation of NADH by atmospheric oxygen to be the first
periodic enzyme reaction which can be maintained indefinitely
in a thermodynamically open system.
2.4. Non-oscillatory Phenomena of Systems Capable of Oscillation
The special kinetic situation existing in oscillatory systems,
due primarily to non-linear relationships, to kinetic coupling,
and to instability, leads to typical temporal behavior when
oscillations no longer occur under modified conditions. Nonoscillatory phenomena of this kind can provide valuable clues
as to the nature of individual processes involved in the origin
of oscillations.
The surprising degree of similarity in the behavior of systems
capable of oscillation is apparent from the survey of their
phenomenology presented in Figure 13176*
"1;
this survey
could be readily extended. It also includes a very simple
electronic semiconductor system (neurodyn model) which was
developed on the basis of common kinetic features. It consists
of a tunnel diode (TD) with a non-monotonic current-voltage
curve and a NTC thermistor with the properties of coupled
Angew. Chem. I n t . Ed. Engl. 17, 1-15 ( 1 9 7 8 )
System
Set-up
Non
-monotonic
Bistebility
Pulse
formation
Oscilletion
Propagation
Zhabotinskii
Ostwald-Lillie
Model COO,S13
Fig. 13. Phenomenological analogy between chemical and physicochemical oscillatory systems. MA, malonic acid; P, product; Red. Pr.,
reduced products; .!?=dE/dt; Br- =dBr-/dt. The right- and left-hand curves in the column headed “Bistability” refer, respectively,
to excitation from the lower to the upper and from the upper to the lower stable steady state; the broken lines indicate the position
of the steady states
regeneration. The entire phenomenology of excitable and oscillatory systems can be simulated with this m 0 d e 1 ~ ~ ~ ~ ~ * 1 .
Systems no longer in oscillation usually still possess bistability (see Fig. 2a). Interconversion of the two stable reaction
states, which can be triggered by external perturbation, obeys
the so-called “all-or-none law” and therefore has threshold
character. Typically, such transitions between states propagate
spontaneously throughout the system once induced at a point
by local transport processes (“local fluxes”) set up at the
boundaries between regions in different reaction states resulting from the bistability of the system (cf. Fig. 15).
In some cases,the triggered system may return to the original
state in a spontaneous regeneration process coupled with
the triggering reaction. We then have the phenomenon of triggered reaction pulses (for example, action potentials of nerves),
which likewise propagate spontaneously.
The ubiquitous phenomena of overshoot (Fig. 2 b), refractarity, and accommodation, which are all consequences of coupled
regeneration, will not be considered in this article. The individual terms mean threshold alteration under superthreshold
conditions (refractarity) or subthreshold conditions (accommodation).
for periodic behavior to occur. The most important conditions
will now be briefly summarized:
a) Oscillatory processes concern temporal changes of state
which take place remote from thermodynamic equilibrium.
They occur as temporally unlimited phenomena only in thermodynamically open systems, i. e. systems with a material
flux.
b) Several kinetic variables participate in the periodic process. It follows from the phase shifts recognizable in the oscillograms that the variables are not directly interdependent but
merely coupled with one another kinetically, i. e. they can
be set independently but nevertheless mutually affect each
other’s rates of change. Hence no direct functions exist of
the kind y = f(x,...) but only a set of simultaneous differential
equations of the kind:
dx
_
dt - f*(X,Y> ...I
.
.
.
.
.
.
.
.
.
3. Theory of Chemical Oscillations
The phenomenology of the oscillatory systems shows that
certain thermodynamic and kinetic conditions must be fulfilled
Angew. Chem. I n t . Ed. Engl. 17.1-15 ( 1 9 7 8 )
c) Oscillatory systems are non-linear in their kinetic relationships, and they contain autocatalytic or autoinhibitory
processes.
9
3.1. Multivariable Kinetics of Chemical Oscillations
Periodic chemical reactions are thus seen to be understandable only on the basis of multivariable kinetics, describable
only in terms of sets of simultaneous differential equations.
Since two variables already suffice for a description of oscillations we shall limit the following brief treatment of multivariable kinetics to cases involving two variables:
The system under consideration will be described by two
simultaneous differential equations in the kinetic variables
x and y:
dx
- = f,(x,y);
dt
t
Y
Elimination of the common term t gives:
i.e. an expression describing the direction in which the state
of the system changes at each point x,y in an x / y coordinate
system. A time-free “state portrait” (phase plane) of the
system described by the kinetic differential equations is
obtained. The reaction vectors thus given for each point form
continuous trajectories which can be shown to intersect only
at discrete points. The intersections of the trajectories shown
are the steady states of the system. They may be stable or
unstable, depending upon whether the temporal changes along
the trajectories are directed into or out of the singular point.
Figure 14 shows the portraits of the surroundings of various
kinds of singular points occurring in kinetic systems. They
may be “nodes”, “foci”, or “saddle points”. The lines which
Fig. 14. Trajectories and isoclines surrounding singular points (=steady
states) in the two-dimensional portrait of state. a) Stable nodes; b) stable
foci; c) saddle point.
join states having the same direction of their trajectories in
the state portrait are called isoclines.
The portraits of non-linear coupled systems of particular interest in the present article are shown in Figure 15.
The cases shown are those of bistability, pulse formation,
and oscillation, which are readily interconvertible by variation
of the coefficients in the two kinetic differential equations.
Systems exhibiting bistability (Fig. 15a) possess three singular points, of which the middle one is a saddle point and
the other two nodes or foci.
10
XFig. 15. State portraits of coupled systems. a) Bistability; b) excitable pulse
formation (monostability);c) oscillation (astahility). S,= separatrix. The portraits correspond to the behavior of the neurodyn oscillator shown in the
bottom row of Fig. 13 (experiments by E . Lamme/[93]).
In the case of pulse formation (“monostability”) (Fig. 15b)
there exists only one stable node or focus and a “separatrix”
(&), which determines whether the system returns directly
(subthreshold stimulus) or via an extended route to the stable
singular state (superthreshold stimulus).
In the oscillatory case (Fig. 15c) there arises a boundary
cycle into which all the trajectories approach, whether they
come from without or within. The system moves indefinitely
on this closed loop, and its projections onto the x axis and
the y axis show the oscillations of these two variables. They
are seen have a phase lag of about one quarter of the entire
cycle. In simultaneously recorded oscillograms one variable
is seen to undergo maximum change when the other exhibits
minimum change at its point of inversion. The kinetic situation
in chemical oscillations is thus completely analogous to those
known in mechanical and electrical oscillatory systems.
It is seen in Figure 15c that oscillatory systems in the
interior of the boundary cycle contain a singular state which
is an unstable focus.
In order to assess the behavior of systems including two
kinetic variables, great importance attaches to the analysis
of the isoclines (Fig. 14). Like the trajectories they intersect
only at singular points; unlike the trajectories, however, which
are generally no longer calculable in elementary form by
integration, the isoclines are usually obtainable as relatively
simple functions of x and y from the relation
Angew. Chem. I n t .
Ed. Engl. 17.1-15 ( 1 9 7 8 )
where a=dy/dx=const. refers to the trajectory slope characterizing the relevant isocline.
By means of the isoclines, the trajectories or paths of state
can be approximately shown in the portrait of state and
the properties of the singular states determined; in particular,
it can be deduced from the position and shape of the isoclines
whether a given system of simultaneous differential equations
possesses oscillatory solutions or not.
3.2. The Problem of Stoichiometric Chemical Oscillations
The facts that chemical oscillations have hitherto always
been discovered merely by chance and that no oscillatory
system has yet been “synthesized” from known reactions,
together with the failure to completely elucidate even one
of the known oscillations, underline the difficulties besetting
the formulation of periodic reactions with stoichiometric reaction equations.
The question then arises as to the possibility of realistic
stoichiometric reactions being able to meet the known requirements for oscillation.
Only systems of two kinetic variables will be considered,
and the following limitations apply:
a) The realistic range of existence lies in the first quadrant
of the x/y coordinate system since x and y are concentrations
and cannot assume negative values.
b) The isoclines must intersect at least once in the first
quadrant in order that at least one steady state exists there.
c) The chemical reaction must be of second order or lower
since reactions of higher order are unlikely or unrealistic.
Bearing these limitations in mind, there remain surprisingly
few kinds of reactions which can be considered in setting
up stoichiometric reaction mechanisms. Only five normal noncatalytic and two autocatalytic reactions are left in the discussion :
a) Uncoupled reactions (containing only one variable):
A+X+P
X+X+P
A X + 2 X (autocatalytic)
+
b) Coupled reactions (containing both variables):
A+X+Y
X+X+Y
X+Y +P
X + Y + 2 X (autocatalytic with respect to X)
These seven kinds of equations yield seven types of kinetic
terms which can occur in the kinetic equations; it should
be remembered that X and Y are interchangeable.
-d _y -
dx
-=
dt
-kx
dt
dY
-=
dt
dx
dx
=
dt
+ k x (autocatalytic)
Angew. Chem. I n t . E d . Engl. 17,1-15 ( 1 9 7 8 )
+kx2
dx
+ k x j , (autocatalytic)
dt
+
3.3. The Feedback Concept of Chemical Oscillations
The occurrence of unattenuated oscillations in electrical
circuits and control loops as a consequence of feedback effects
is a well known fact. Application of this concept to chemical
reactions is unfamiliar since circuits and loops are not so
immediately obvious. However, on closer study of chemical
and physicochemical oscillation a very close analogy is seen
to exist.
Feedback occurs when a process acts kinetically upon itself.
It therefore consists basically in a closed chain of action.
Such a cycle can arise in two ways:
a) If the output of a transmission system acts upon the
input of the same system then we have a situation in which
an “effect” is influencing its own “cause”. Depending upon
whether self-enhancement or self-inhibition occurs the phenomenons is described as positive or negative feedback, respective-
Non-systemic
feedback
-kxy
-=
e) Indefinite oscillations are tior possible with the linear
and quadratic terms given above.
If the choice of reactions is limited to stoichiometric ones,
oscillations only become possible if terms of higher order
( >2) are present in the kinetic equations.
In thiscontext, thereader isreminded that,ofthe mathematical models of stoichiometric oscillations mentioned in Section
a trimolecular reaction
2.1.1, the Brusselator[12,’31includes
.
and the O r e g o n a t ~ r [ ~more
’I
than two kinetic variables and
a non-stoichiometric coefficient (P/(a P)).
A further realistic possibility for the generation of oscillations is provided by the presence of non-stoichiometric coupling processes, such as occur in chain branching and enzymatic
reactions.
+kx
-=
dt
Thus only linear and quadratic terms (x2,y2,xy)appear. Under
these conditions the isoclines are always second order curves,
i. e. conic sections; for the above terms they are hyperbolae,
parabolae, and straight lines.
Taking into account the permissible kinetic terms, isocline
analysis provides valuable information about the behavior
of stoichiometric systems.
a) If the kinetic equations are linear and do not include
an autocatalytic process, only one singular state occurs which
is always a stable node in the first quadrant.
b) If the xy term is also present, the singular state can
also be a focus, i. e. attenuated oscillations become possible.
c) If one of the linear terms is autocatalytic (dx/dr= +k.w),
then two singular states can occur in the first quadrant, of
which one is a stable node and the other a saddle point.
i. e. the system is excitable.
d) If quadratic terms are also present, three singular states
can form in the first quadrant; the middle one is a saddle
point and the others stable nodes. Bistability thus becomes
1 4=& 1
Systemic
feedback
Fig. 16. “Non-systemic” and “systemic” feedback in chemical aystems.
11
ly. The chemical analogy is stoichiometric autocatalysis (Fig.
16, top) in which the reaction product is also a reactant. This
form of feedback has no effect upon the properties of the transmission system or upon the rate constant of the reaction. It
is therefore designated “non-systemic” feedback.
b) Many feedback mechanisms act not upon the input but
instead upon the transmission system itself. The kinetic analog
is a reaction in which the reaction product influences the
rate constant (Fig. 16, bottom). Owing to its action upon
the properties of the system, this kind of feedback will be
termed “systemic”feedback. It is the commonest kind of kinetic
coupling in heterogeneous systems, e. g. at electrodes, contact
catalysts, and membranes. It can arise in homogeneous systems, e. g. as a result of the markedly non-linear temperature
dependence of the rate constants in thermochemical reactions,
and in enzymatic systems, e.g. owing to the dependence of
Fig. 17. Possible coupling modes in simply coupled chemical systems. a)
Intrinsic coupling (autocatalysis @, autoinhibition
b) cross-coupling
(cross-catalysis, cross-inhibition).
0);
the catalytic activity of allosteric enzymes upon their products
or substrates.
Systemicfeedback is usually non-stoichiometric (non-linear)
and thus much more favorable to oscillations than stoichiometric non-systemic feedback which is linear, being restricted
to bimolecular reactions.
In open systems, autocatalytic and autoinhibitory reactions
can produce positive or negative feedback, depending upon
whether the self-couplingaffects the formation or consumption
reaction of the coupled component. Four cases can occur
(Fig. 17a), of which the first and last lead to positive feedback
and the two middle ones to negative feedback.
Non-systemic or systemic coupling can also occur in kinetic
interactions between simultaneous reactions. Feedback then
occurs by “cross-catalysis”, a concept introduced into the
kinetics of chemical oscillations as long ago as 1948 by Bonh~efeer[~~].
Once again there are four possible coupling modes which
can lead to positive or negative feedback depending upon
the direction of action of the individual couplings (Fig. 17b).
With regard to the occurrence of oscillation, interest is
focused mainly on the kinetic behavior of chemical systems
with positive and negative feedback. This will be considered
with the aid of Fig. 18:
Backward coupling
Reverse activation
0
Reverse inhibition
I
I
Forward coupling
Forward activation
Forward inhibition
a
Fig. 18. Occurrence of typical kinetic characteristics by positive and negative feedback. Reverse activation (1) and forward
inhibition (4) correspond to positive feedback effects, and reverse inhibition (2) and forward activation to negative feedback effects. X=dX/dt.
12
Angew. Chem. Int. Ed. Engl. 17, 1-15 f 1978)
always observed[78! The oscillations arise from the antagonistic interaction of a relatively fast acting labilizing positive
feedback and a slower acting stabilizing (regenerative)negative
feedback. This situation can be represented in a simple feedback action scheme (Fig. 20). Hence the occurrence of oscillations depends not only upon the presence of both kinds of
feedback but also upon the correct ratio of the action times.
Assuming for the sake of simplicity that the formation
and consumption reaction are of first order, a plot of dX/dt
uersus X for these reactions gives two straight lines which
intersect at the steady state. Their slopes correspond to the
relevant rate constants. Self-coupling alters the slope of one
of the two lines according to the value of X : the formation
reaction in the case of backward coupling, and the consumption
reaction in the case of forward coupling. It can be seen in
Figure 18 that non-monotonic characteristics result from positive feedback and monotonic characteristics from negative
feedback. Thus, given sufficiently strong positive feedback,
bistability and instability arise. Positive feedback therefore
labilizes the system and is the actual cause of the propagation
phenomenon occurring on adequate (“superthreshold”) stimulation.
In contrast, negative feedback has a stabilizing effect and
always generates only one stable steady state. The bottom
row of Figure 18 shows the “dynamic”diagrams which describe
the overall behavior of the feedback systems with respect
to the formation and consumption of component X and which
illustrate the stability situation by means of the inserted arrows.
Characteristic temporal behavior patterns result for positive
and negative feedback, and can be used for experimental
determination of the kind of feedback operating in a particular
system (Fig. 19)[781.In the case of positive feedback the system
behaves in the manner of a capacitor. If the corresponding
characteristic is non-monotonic then the threshold character
of the triggering process is apparent.
In contrast, systems with negative feedback behave like
an inductance and exhibit characteristic “overshoot” phenomena (see Fig. 2b).
a
feedback
Positive
Formation
Fig. 20. Principle of antagonistic feedback in oscillatory systems.
As a supplement to Figure 13, the phenomenology of feedback systems and their causes are surveyed in Figure 21.
Each feedback loop can contain several variables in consecutive reactions. All these variables participate in the oscillatory
process. In most cases it is possible to decide experimentally
to which class of feedback a recorded oscillating variable
belongs[78! As shown by the examples in Figure 22, variables
of the positive feedback loop are generally recognizable by
pronounced discontinuities or resting points in their oscillograms which are due to the unstable states through which
Positive feedback
Monotonic
Negative feedback
Non- monntonir
t
t
I
I
f-7-
‘2
on off
fi
‘1
11
o
K-
X-
t-
t-,
X--
t-L
Fig. 19. Temporal behavior of electrochemical feedback systems [78] on sudden change of current. Systems with positive feedback have “pseudo-capacitive’’ properties, and systems with negative feedback “pseudo-inductive” properties.
On studying chemical oscillations with regard to their temporal behavior, characteristics of both kinds of feedback are
Angew.
Chem. I n t . Ed. Engl. 17, I-15 (1978)
the systems pass. Variables of the negative feedback loop
never exhibit such features owing to the absence of instabilities;
13
. I
Excitability
Bistability
instability
Propagation
Consumptp
formation
I 1
x-\
Positive feedback
llabilization I
X-
Stirn,,i,,c
.. x-
I
Neqative feedback
IStabilization I
\
1-
-
t
I
\
Overshoot
Refractarity
Accommodation
Stimulus
Oscillations
Pulse formation
Dissipative structures
tAntagonistic teedback
Fig. 21. Phenomenology of coupled systems and assignment to the various kinds of feedback
their oscillograms therefore usually have a relatively simple
sawtooth shape.
B.Z.
Iron i n HNO,
Brav Reaction
Reaction
Chemical and electrochemical oscillations occasionally
occur in chemical engineering, e. g. in heterogeneously cataTeorell Membrane
Variable of
positive
feed back
t-
negative
Variable of
feedback
,--3+wl/
t-
t-
HioTw
i
I
t-
L
w
t-
t-
t-
t-
Fig. 22. Simultaneous oscillograms of variables of oscillatory systems 1781. Variables belonging to positive feedback loops
generally exhibit flip-flop type oscillograrns with characteristic instability features (+), Oscillograms of variables of negative
feedback usually have a simple sawtooth shape without instability features (cf. Figs. 1 d and 1 e). B.-Z. reaction=BelousovZhabortinskii reaction.
4. Conclusion
At the close of an article about chemical oscillations it
is logical to ask what significance such reactions possess in
the various disciplines of science, and especially in chemistry.
Are they merely a fortuitous kinetic curiosity whose occurrence
is of more academic than practical interest, or are they of
fundamental importance in certain scientific, biological, or
technical processes.
In the theory of the thermodynamics of non-linear non-equilibrium processes oscillatory reaction courses are particularly
striking and interesting phenomena which can only be treated
by this theory, still very much in its infancy. It is mainly
here that theoretical p h y s i ~ s is[ ~concerned
~~
with chemical
oscillations.
14
lyzed oscillations (Section 2.2.3). However, they have little
engineering significance as far as improvement of overall processes is concerned. Instead, they represent an undesirable
complication.
Electropolishing processes exist which involve oscillatory
activity of the electrode materials to be polished, e. g. in the
electropolishing of steel in oxalic acid/hydrogen per~xide”~!
However, these are only exceptions in which oscillation kinetics is of engineering utility.
The significance of chemical oscillations in biology presents
a completely different picture. The rhythmic nature of processes in bionics, i. e. “the chemical engineering of living
organisms”, has considerable advantages over non-periodic
processes. Thus the oscillatory course of enzymatic reactions
on a boundary cycle around an unstable steady state leads
Angew. Chem. I n t . Ed. Engl. 17, 1-15(1978)
to stabilization of the overall process, resulting in enhanced
dynamic adaptability to changing external conditions and
the possibility of synchronization with simultaneous processes.
A rhythmic reaction course is of particular advantage if it
leads via reaction steps which are mutually exclusive for kinetic
reasons.
The orientation of living organisms according to time, i. e.
the explicit recognition of temporal variables detectable even
in single cell organisms such as Euglena, A c e t ~ b u l a r i a [ ~ ~ ~ ,
can be accomplished only by rhythmic processes. In other
words, “the biological clock” most probably has to be viewed
as an oscillatory chemical mechanism. In the field of enzymatic
processes such mechanisms are readily feasible owing to the
manifold possibilities of systemic coupling in the metabolism
of the living cell.
The ability of the excitable membranes of the nervous system
to undergo oscillations permit, as already mentioned, propagation and processing of information.
In the evolutionary process chemical periodicity is also
assumed[981to be operative in spatial differentiation-the
formation of defined structures. The activity of such structureforming oscillatory mechanisms can be directly observed under
the microscope, e.g. in cell division and in the regeneration
of damaged cells.
The elucidation of the diverse manifestations and effects
of chemical periodicity, which is clearly essential in living
organisms, will only become possible once the kinetics of
chemical oscillations can be understood in their elementary
form. And this is probably the true reason for the avid interest
recently shown in this area.
Received: April 22, 1977 [A 195 IE]
German version: Angew. Chem. 90, 1 (1978)
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