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Electrical Dissipative Structures in Membrane-Coupled Compartment Systems.

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Annalen der Physik. 7. Folge, Band 41, Heft 4/6, 1984, S. 267-279
J. A. Barth, Leipzig
Electrical Dissipative Structures
in Membrane-Coupled Compartment Systems
Sektion Physik der Humboldt-Universitiit zu Berlin
Abstract. Reaction-diffusion systems with charged particles are studied. Conditions for the
arising of electrical dissipative structures in a compartment system consisting of two boxes separated
by a membrane are derived. The appearance of a polar dissipative structure is proved for a simlpe
capacitor model in combination with a simple second order chemical kinetics which leads to an analytically solvable problem. Electrical dissipative structures can in principle be considered as non
equilibrium electrical batteries. The theoretical efficiency of such batteries is estimated.
Elektrische Dissipative Strukturen in Membrangekoppelten Systemen
Inhaltsubersicht. Es werden Reaktions-Diffusiomsystememit geladenen Teilchen studiert.
Bedingungen fiir die Entatehung elektrischer dissipativer Strukturen in einem Kompartment-System, bestehend aus ewei durch eine Membran getrennten Zellen werden abgeleitet. Die Entatehung
einer polaren dissipativen Struktur wird fiir ein einfaches Kondensrttorenmodell in Kombination mit
einer einfachen chemischen Kinetik, das analytische Losbarkeit gestattet, nachgewiesen. Elektrische
dissipative Strukturen konnen im Prinzip als elektrische Batterien fern von Gleichgewicht betrachtet werden. Der theoretische Wirkungsgrad einer solchen Batterie wird berechnet.
1. Introduction
The appearance of dissipative structures in reaction diffusion systems with neutral
particles has been studied by many workers [l- '01. However many interesting chemical
reactions in solutions as e. g. the Belousov-Zhabotinskij-reactioninvolve charged particles. Therefore reactions-diffusion problems with charged particles have found much
interest in the last time [8-131. The electrical dissipative structures described in this
paper lead to spontaneously arising electric voltages beyond critical values of the parameters of the system. I n principle this seems to be a new method for the conversion of
chemical energy into electrical energy, which may be of interest as a possible source of
electric voltages and currents. Nonequilibrium electrical batteries based on this effect
are thinkable as well as chemical sensors. However this is still on the level of theoretical
speculations. It is the aim of this paper to study the effect for a simple analytically solvable model.
Let us consider a system of two compartments separated by a membrane. It is assumed that the chemical substances are homogeneously distributed within the boxes.
If the balance equation, describing the temporal development of the concentration of
the species i in the box j denoted here by cij, is derived we have to take into account
that chemical reactions take place within the compartments and the charged particles
Ann. Physik Leipzig 41 (1984) 4/5
are transported through the membrane by concentration and electric potential differences. This leads t o the phenomenological description :
where Di is the diffusion coefficient of the substance i and zi their valence. The complementary equation is obtained by exchanging the numbers of the boxes (1f-* 2).
Note, that equation (1.1)is valid only if (eqJi)
kBT since it is obtained by linearization. The rate of the chemical reaction Rii will be in general a nonlinear function of all
concentrations cii, i = 1 ... s. As a consequence of the charge conservation law the
chemical reaction rates have t o fulfil a n additional condition:
2 ~ i R i j= 0 ,
i= 1
j = 1, 2
If the thickness of the boxes is small compared with the thickness of the membrane,
the system can be treated as a plate capacitor, where within the boxes, which are the
plates of the capacitor, the electric potential is constant and within the membrane,
which is the dielectricum, the electric potential increases linearly (Fig. 1).The electric
potential difference is known from electrostatics
where C is the capacity of the capacitor, Q the charge of the plate and V the volume of
the box. F is Faradays number. The purpose of this paper is t o investigate the appearance of dissipative structures in such two compartment systems. For neutral particles
such analysis has been done especially by authors from Brussels school [14-161. Our
special interest is paid to the influence of the ionic interaction between the charged
particles on the appearance of dissipative structures in such simple model systems.
box 1
box 2
Fig. 1. The capacitor model
2. A Capacitor Model with Two Reacting Species
Let us consider for example a system with two reacting ionic species, now for simplicity denoted by Xi+, X i , where i = 1,2 is the number of the box. Thedifferential
equations describing the temporal development of the species read in dimensionless
- - - d+(X,+ - X ; t )
X,+)LIP, rl+(X,+,X,)
et al., Electrical Dissipative Structures
2 69
to is a characteristic reaction time, co denotes a characteristic concentration and 1 is the
length of the box. The other three equations are obtained if the number of the box and
the sign of the charge is changed (1+-+ 2 ;
++ -). The potential difference will be
A l-p = - e (Z+PC$ + 2-Pc,) v
k,T C
2-X;) =
Because of the global electroneutrality, that means that the total charge of the whole
system is zero, the four variables are not independent. By a suitable transformation we
find three independent variables in the case of z+ = + 1 and z- = -1
Q = - [ ( X , + - X , ) - (X$- X , ) ]
= -(z+X,'
Q is the charge of one plate of the capacitor, P has the meaning of a chemical gradient,
while X is the total number of cations which is equal to the total number of anions.
A transformed' set of equations is obtained
rl(X,+,X ; ) - r2(X,+,Xgj
a,P = -dQ - D P - = d X Q
where D is the aum of the diffusion coefficients d, and d- and d = d+ - d- is the
difference of the diffusion coefficients. Now we assume that the chemical reactions
have the simple form of a polynomial of third order with arbitrary coefficients
j+x: +f-X<+f++(Xi+)Z+f+-XifXi
+ f - - ( x i ) 2+ f++ + (Xi+l3 + f++ ( X t x i
q ( X i + , X i )= f o +
+f+--Xi+(XF)* + f - - - ( X i ) 3
Using the assumption (2.6) the set of equations (2.5) has the form
a,x = 2f0 + (f+ + f - ) X
+ -(I+++
+ +f++
+f++- +f+--
+f---)X3 +.f(x,P,Q)
Ann. Physik Leipzig 41 (1984)4/5
2 70
+ (f++-.f--) x + 1 (3f+++ +f++--f+-- - 3f_--) x*
b = f+ + f- + (f+++ f-+ + f--) x + q(f+++
+ f++- + f+-- + f---) x*
=f+ -f-
The function F ( P , Q ) is a nonlinear homogeneous function of third order in the variables
P and Q. Furtherj(X, P,Q ) unites all terms where products of the variables X , P and Q
and products of P and Q occur.
Now we are interested in the stationary states, which are'possible in such a reaction
diffusion system with charged particles. One steady state is easily obtained: Q ( O ) =
P(O)= 0, X(O) is calculated from the last equation of (2.7) in which fix,P, &) will be
zero. This stationary state (which is called homogeneous steady state) corresponds to
a homogeneous distribution of the ions in the compartments. The system is locally
electroneutral, that means the charge of each plate is zero, the electric potential difference vanishes.
x, = x+= x,
dg, = 0
Electrical dissipative structures may arise if the homogeneous steady state is unstable
with respect to perturbations. Then the system will change into another steady state,
where the electric potential difference does not vanish.
In order to investigate the stability of the homogeneous stationary state a linear
stability analysis is done, where it is assumed that the steady state values are exposed
to small disturbances of the form
Q = Q(O)
+ SQ
P = Po)+ SP
+ SX
SP ~ X P ( F ) SX
Because of Q(O) = Po)= 0 the equations are uncoupled. It follows that the equation
for the eigenvalues p splits into a product of a quadratic and a linear equation, so that
one eigenvalue p3 is directly determined. The total number of reacting particles is stable,
= f+
+ f- + (f+++ f+- + f--)
Therefore the stability analysis is reduced to the investigation of the sign of the solution
of the quadratic eigenvalue equations:
[b - (1
P1,2 =
The stability behaviour for constant X(O), shown in Fig. 2, is determined by three
functions in the space of the chemical parameters a and b : the vertical line Re@) = 0,
the straight line p = 0 and the parabola Im(p) = 0. The structure shown in Fig. 2 is
et al., Electriml Dissipative Structuree
topologically stable with respect to variations of the parameter d+, d- and 7. I f the
sign of the difference of the diffusion coefficients d changes, a has to be replaced by
If (2.10) and (2.8) are compared, it is seen that p3 and the chemical parameter b
are equal. Because of the required stability of the total number of cations resp. anions
(2.10), only the part where b < 0 is of interest. For b > 0 it is necessary to pump
additional energy from outside into the system in order to keep X constant. But this
case is not considered in the present paper since it is of less interest.
For the assumed simple chemical kinetics (2.6) it is easy to determine all possible solutions for the homogeneous steady state X(O). Thereby it has to be taken into account,
that X(O)has to be positive. The requirement that all concentrations have to be positive
leads to restrictions for arbitrary eligible coefficients in the chemical reaction rates.
Fig. 2. The stability of the uniform stationary state without external load
So it is necessary that fo > 0 and f+,f- > 0. For the reason of finity of the stationary
state it is required that the coefficient of the highest power in the chemical reaction
rate is negative. If all these restrictions are taken into account, it has been shown that
it is impossible to get three homogeneous steady states. That means that homogeneous
bistability as it is known from neutral particles [17] does not exist in a two component
system with charged particles. If only one uniform steady state exists the condition
that b < 0 is always fulfilled. It is seen that the condition (2.10) is no sharp restriction
for such simple chemical reaction kinetics.
Because of the fact that for our chemical kinetics always b < 0, there exists only a
small region in the space of the chemical parameters where the homogeneous steady
state will be unstable with respect to small disturbances supposing that X(O)is constant.
Instability will occur only if the chemical parameters of the system cross the straight
Ann. Physik Leipzig 41 (1984) 4/5
2 72
line p = 0 (see Fig. 2). If the uniform stationary state of the considered reactiondiffusion system is unstable with respect to small disturbances, a dissipative structure
will arise. The new appearing steady state is characterized by an inhomogeneous distribution of positive and negative charged particles in the two compartments. This leads
to a charging of the plates of the capacitor, a voltage arises. In the case of a quadratic
chemical kinetics (f+++ = f++- = f+-- = f--- = 0) it is possible to calculate these
inhomogeneous steady states analytically. The function F ( P , Q ) in the second equation
of (2.7) will be zero, therefore the first two equations of (2.7) may be treated as a linear
system of equations in the variables P and Q. This system will only have a nontrivial
solution if the determinant of the matrix of the coefficients will be zero. From this
condition it is possible to calculate the value of X ( l ) . Using the connection between Q
and P and the last equation of (2.7) P(l)and Q(I) are obtained (compare section 3).
From this solution it is seen, that in a certain region of the parameters it is possible to
get two solutions for X ( l ) and therefore four solutions for P(l)as well as Q(l). In other
regions of the space of the parameters only one solution for X ( l )and two solutions for
P ( l )or Q(l) respectively are obtained. In the last case the system is bistable because
of the fact that the two solutions are equal in magnitude but opposite in the sign of
P(l)and Q(l). That means there are two directions of the polarity of the arising voltage
possible, the system is similar to an electric flip-flop. In dependence on the initial conditions the system will change into one or the other polar state. In the region of the parameters where more than two inhomogeneous steady states exist, the system is multiple
stable. For illustration let us consider e.g. dissociation reactions with a velocity influenced by the cations.
X + f X - t A
Here B is an energy-rich molecule and A a low-energy decay product. The chemical
kinetics of this reaction is assumed to be (i = 1, 2)
T $ ( X $X
, i ) =A -X t X i
+ ( B - C X t X i ) (Xi+)'
That means:
f++= B, f+++-
Assuming strong pumping with B-molecules ( B 9 C X $ X c ) we may neglect one backward reaction and put simply C = 0.
The analysis of this example has been presented elsewhere [18]. Fig. 3 shows the
bifurcation diagram in the space of the chemical parameters A and B in form of a cusp.
The critical parameter Acrit and Bcritmay be calculated from the requirement of the
0.59 B
Fig. 3. The bifurcation diagram: a) without external load, b) with load given bya resistance8 = 0.01
(The other parameters are given in the text).
et el., Electrical Dissipative Structures
2 73
reality of all variables and the positivity of the total number X of cations or anions.
It has been shown, that the arising voltages are higher for increasing B. Only if the
diffusion coefficients are unequal in magnitude a dissipative structure may arise [18].
These electrical dissipative structures may be of some interest for the generation of
electric voltages and currents since in principle they are converters of chemical energy
into electrical energy. The,source of the electrical energy is the assumed difference between the chemical potentials of the B-molecules and the A-molecules.
3. Electrical Dissipative Structures under Load
If an electrical dissipative structure has been built up, electrical energy is stored in
the system in form of the potential difference between the two compartments. This
energy depot may be used by an external device e.g. a resistance. This resistance is
realized by two electrodes which are fixed to the compartments (see Fig. 5) and pick
5 0 1 AQ'rnV
40 -
0.9 B
Fig. 4. The potential difference between the capacitor plates in dependence on the bifurcation p r a meter B, ( a : A = 0.01;b : A = 1).
Fig. 5. The capacitor model with an external
up the charges for the transport through the external conductor. It is assumed that
external current of ions is proportional to the voltage and therefore also proportional
to the charge of the plat+ of the capacitor.
If these additional transport terms are inserted into the set of equations (2.1), then
after the transformation of the variables (2.4) a set of equations is obtained, which
describes a two compartment system where a current flows through an external resistance.
Introducing the notation
S = S+ 8- H = S+ - S-.
2 74
Ann. Physik Leipzig 41 (1984) 4/5
We get for the simple chemical reaction rates a polynomial of third order:
+ (f++ f-) x + -1ij-(f++ + - f-4 x2
+ 1 + + f+-- + f---) X 3 + f(X, P,Q )
a,x = 2fo
The homogeneous steady state is the same as for a capacitor without load, because of
the vanishing voltage there is no current through the conductor
Q ( 0 ) = p(0)= 0 - X(0)
For the purpose of investigation of the influence of the external resistance on the behaviour of the system the linear stability analysis of the homogeneous stationary state
is made in the same way like in the former case. Therefore only the solutions of the
quadratic eigenvalue equation determine the stability behaviour of the system.
/Fig. 6. The stability of the uniform stationary state with load given by a resistance S
et al., Electrical Dissipative Structures
2 75
Let us discuss the influence of the parameter S, describing the charge transport and the
parameter H on the perpendicular Re@) = 0, the straight line p = 0 and the parabola Im(p) = 0 which are important for the'stability behaviour (Fig. 6). The perpendicular Re@) = 0 moves in direction of larger values b, the straight line p = 0 is turned
around the point a, = qd
H ; b, = D . This point is simultaneously the vertex of the
parabola along which the intersection point of all three curves moves if S varies (pointed
line in Fig. 6). There exists a region which is never touched by the line p = 0 with increasing S. This is the domain of batteries which are stable with respect to increasing
load. But this domain lies in the two quadrants where b > 0. As seen in the former
chapter for simple chemical kinetics always the condition b < 0 is fulfilled. Therefore
for every point where an electrical dissipative structures is built up, a critical value of
the parameter S exists, where the dissipative structure will break down. This critical
point corresponds to a border value of the possible load. For a simple quadratic form
of the chemical rate function it is again possible to calculate the inhomogeneous steady
state in dependence on the charge flow through the conductor analytically in the same
We get :
X(1) =
D(f+ + f-) - d ( f + - f-)
+ d2 - D2 + S D - H d - S(f+ +f-)
+ + f--) Dl26
(f++ -f - - ) d + (D2-d2)l2C-(f+++f+-+f--)D-((f++f-)D/2C+(f+++f+-+f--)
+ + f--1 DIG
In order to demonstrate the influence of the external resistance let us study our example
which has been investigated in the former chapter (2.12). Fig. 3 b shows the bifurcation
diagram in the space of the chemical parameters. It is seen that the cusp will be larger
for increasing 8.That means that the region where the system is multiple stable will
increase. This is also seen in Fig. 7. With increasing 8 the critical value Bcrltwhere the
dissipative structure appears will be larger end for large S a transition of the system from
the bistable regime to the multiple stable regime is possible. If a fixed value of the
chemical parameter B is considered it is seen that the arising voltage will be smaller
with increasing flow of charges through the conductor and at a critical value of S the
voltage breaks down.
Physik Leipzig 41 (1984)416
2 76
These investigations show that the utilization of the electrical energy, provided by
the dissipative structure, by an external resistance is possible. The model system consisting of two compartments separated by a membrane may be considered as a nonequilibrium battery.
Fig. 7. The potential difference between the capacitor plates in dependence on the external resistance S : a) S = 0, b) S = 0.5, c) S = 1, H = 0, A = 1.
4. The Efficiency of Noneqdibrium Batteries
The appearance of an electrical dissipative structure is connected with the conversion
of chemical into electrical energy. This is an energy transformation in a large distance
to the thermodynamic equilibrium, whereas the present energy transformation processes
technically used are founded on the laws of the classical equilibrium thermodynamics.
The possibility of a technical utilization of dissipative structures for energy transformation is discussed now only by some authors [19--211. I n the following section the
physical foundations of possible methods for energy conversion and the efficiency of
nonequilibrium batteries based on electrical dissipative structures are discussed. The
essential role for isothermic energy transformations is played by the free energy
The balance equations for the change of the free energy reads:
dF _
The item with the index i denotes the inner changes in the system, whereas the index e
represents the exchange with the surroundings. If the energy conservation law is taken
into account, we find
-= - T P
where P = diSfdt 2 0 is the entropy production of the system.
The total flow of free energy Qr, may be described by the difference of the free
pumped into the system and the sum of the free energy flow @a carried
energy flow
away and the free energy flow @,n usable for application.
- (@a
+ On)*
QrP is often characterized as pump rate.
et al., Electrical Dissipative Structures
For the stationary state of the system we get :
QP = @a f @,, f TP.
The efficiency of an engine is defined as the quotient from the useful energy and the
difference of the input
and the energy @a carried away
The useful free energy Q,, is determined by the product of the potential difference Ag,
and the electric current I which flows through the resistance R,.
For the efficiency we obtain
Here the entropy production contains two parts: the chemical part and the diffusion
= TPchern
The external resistance R, is determined by the electric current I:
I = - -[Z+S+(C?
- c,) - Z-S-(C,+
- c,)]
The entropy production density for r chemical reactions is known from the theory of
irreversible thermodynamics [221
TGchem = 2 wrAr
where w,the chemical reaction rates and A, are the affinities of the chemical reactions,
which are estimgted by the chemical potentials of the reacting substances
2 vkrPk
vkr is the reaction number.
For our system of two boxes in which the dissociation reactions (2.12) takes place,
we get
,& X ( P B - pi+ - ~ i ) V~coto
= [ ( A - X,+X,) (PA -
+ ( B - (WG)(X,+Y
+ ( B - C X , + X , ) (X;i-)2
+ [ ( A- X,+X,)
- /d- K )
- Pi+ - P,)1 v2coto
where Vl .= V , = V are the volumes of the two boxes which are assumed to be equal.
The dlffusion part of the entropy production for the two compartment model is
determined by
Ann. Physik Leipzig 41 (1984) 415
where ,
:are the electrochemical potentials of the reacting substances. Assuming linear
Onsager relations between the fluxes and the forces the diffusion part is obtained from
L, are the Onsager coefficients.
Now all parts for the calculation of the efficiency of the electrical dissipative structure are available. For example we estimate the following set of parameters: C , =
10 mol m-3 ( A = 1); B = 0.9; D, = 2.5.
om2 s-l; L), = 2 . 5 . 10-6 cm2s-l; R, =
2.17 kn (8= 0.5); lo = lo-* s; C, = 10 mol m-3; I = 5 .
m; V = 6.5.
mola kW-l s - ~
H = 0 ; L+ = 3.1 . 10-l2; L- = 3.1 .
m-l C = 10”.
I n this case the arising potential difference will be 39.1 mV, the current which flows
through the resistance will be 18.0 PA. The useful free energy Dndetermined by (4.7)
has the value: Qn = 7.03 lO-’W. For the chemical part of the entropy production
calculated for a n ideal solution we get TPchem= 6.37 .
W. The obtained value for
the diffusion part of the entropy prQduction is very small TPdifr= 4.96. lo-% W,
therefore this contribution can be neglected. With all these data we find our final result :
= 1.1.10-4.
This analysis shows that the energy conversion based on usage of dissipative structures
is in principle possible but the efficiency of such “engines” is much lower than the efficiency of the present classical technical processes. By BARANOVSKI
[231 the efficiency
of such energy transformations is valued for another example.
5. Discussion
The ionic character of the particles influences the behaviour of reaction diffusion
systems with charged particles. The appearance of potential differences can only be
understood if the electric interactions are included in the theory. It is shown that nonlinear chemical reactions taking place in simple two compartment systems give rise to
potential differences. Such electrical dissipative structures are of interest for the generation of electric voltages and currents. Systems consisting of two charged species in which
only simple chemical reactions with polynomial kinetics of third order take place, are
unable to show oscillations, if the total number of cations and anions is assumed t o be
constant. Alternating voltages and currents may be observed if three substances participate in the chemical reactions. If an electrical dissipative structure is built up, a conversion of chemical energy into electrical energy can be observed. The electrical energy
stored in form of the potential difference may be used if a n external resistance will be
contacted to the system. It has been shown that for simple chemical kinetics always
a limiting load exists, where the dissipative structure breaks down.
The considerations of the efficiency of such “engines” which are based on electrical
dissipative structures show that the efficiency of the nonequilibrium batteries is much
lower than that for the classical technical processes of energy transformation. Whether
such processes are technical realizable and may work effectively has to be examined by
experimental investigations.
A c k n o w l e d g e m e n t . The authors thank G. VOSTAfor useful discussions.
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E. E.: Matematicheskaja fizika kletki. Moskau:
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Bei der Redaktion eingegangen am 21. Miin 1984.
Anschr. d. Verf.: Dipl. Phys. U. FEUDEL,
Sektion Physik 04, Humboldt-Universit zu Berlin
DDR-1040 Berlin, Invalidenstr. 42
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