Spatio-temporal patternformation in reaction-diﬀusion
systems coupled with convection
and geometrical eﬀect

Hiroyuki KITAHATA

Abstract
To understand how living organisms maintain their lives, it is important toregard the living organisms as nonequilibrium open systems, since organisms
live through the dissipation of the chemical energy of the nutrition. So far, anumber of studies on physics under nonequilibrium open conditions have been
performed both experimentally and theoretically. Among these studies, spatiotemporal self-organization has been one of the most interesting topics. As atheoretical model of spatio-temporal pattern formation, reaction-diﬀusion systems are widely adopted. Under the framework of such a reaction-diﬀusion
system, spontaneous pattern formation can be generated theoretically. So far anumber of experimental, numerical, and theoretical studies have been performedon reaction-diﬀusion systems.
Reaction-diﬀusion equations can be adopted only when the ﬁeld is ﬁxed.
Thinking of the phenomena in the real world, however, the ﬁeld itself usually
moves. We introduce this eﬀect by adding the advection term to the reactiondiﬀusion equation. It can be said that this is the natural extension of thereaction-diﬀusion system. In this thesis, the mutual coupling between patternformation in reaction-diﬀusion systems and convective ﬂow is investigated. As
the scales of the ﬂow velocity and propagating-wave velocity are comparable,
the novel phenomena can be seen both experimentally and theoretically.
On the other hand, in the theoretical studies, the boundary eﬀects are rarely
considered. In order to discuss on the pattern formation such as a target pattern,a spiral pattern, etc., there is no need to think about the boundary. In thesestudies, the eﬀects of boundaries should be avoided and a lot of eﬀorts havebeen made to eliminate the boundary eﬀects. In the real world, however, theboundary of the ﬁeld often aﬀects the behavior of the whole system. If the typicalscale of the system is comparable to the characteristic length of the pattern, theboundary eﬀect is quite large.
Thus, the main topics in this thesis are the convective ﬂow coupled withreaction-diﬀusion systems and boundary eﬀects on reaction-diﬀusion systems.
These two eﬀects are important in order to understand the phenomena in thereal world. In this thesis some experimental results are exempliﬁed and theoretical discussion and/or numerical modeling are made for each system fromthe viewpoint of the reaction-diﬀusion systems with convective and/or boundaryeﬀects.

1

2

Abstract

This thesis is based on the following original papers. The chapters which
include the contents of the papers are indicated:
Published papers[1] H. Kitahata, R. Aihara, N. Magome, and K. Yoshikawa, Convective andperiodic motion driven by a chemical wave, J. Chem. Phys., 116, 5666 - 5672
(2002): Chapters 3 and 4.
[2] H. Kitahata, Spontaneous motion of a droplet coupled with a chemical
reaction, Prog. Theor. Phys. Suppl., in press: Chapter 4.
[3] H. Kitahata, S. Hiromatsu, Y. Doi, S. Nakata, and M. R. Islam, Self-motionof a camphor disk coupled with convection, Phys. Chem. Chem. Phys., 6, 2409
- 2414 (2004): Chapter 5.
[4] H. Kitahata and K. Yoshikawa, Chemo-mechanical energy transduction
through interfacial instability, Physica D, 205, 283 - 291 (2005): Chapter 6.
[5] H. Kitahata, R. Aihara, Y. Mori, and K. Yoshikawa, Slowing and stoppingof chemical waves in a narrowing canal, J. Phys. Chem. B, 108, 18956 - 18959(2004): Chapter 7.
[6] H. Kitahata, A. Yamada, S. Nakata, and T. Ichino, Propagation of photosensitive chemical waves on the circular routes, J. Phys. Chem. A, 109, 4973 4978 (2005): Chapter 9.
Reference papers[7] H. Kitahata, N. Magome, and K. Yoshikawa, Convective ﬂow driven bychemical reaction, Proc. Slow Dyn. Compl. Sys. (AIP Proc.), 708, 430 - 431(2004): Chapter 3.
[8] S. Nakata, Y. Doi, and H. Kitahata, Synchronized motion of a mobile
boundary driven by a camphor fragment, J. Colloid. Interface. Sci., 279, 503 508 (2004): Chapter 5.
[9] S. Nakata, S. Hiromatsu, and H. Kitahata, Multiple autonomous motions
synchronized with complex formation, J. Phys. Chem. B, 107, 10557 - 10559(2003): Chapter 6.
[10] K. Nagai, Y. Sumino, H. Kitahata, and K. Yoshikawa, Mode selection in
the spontaneous motion of an alcohol droplet, Phys. Rev. E, 71, 065301 (2005):
Chapter 6.
[11] Y. Sumino, H. Kitahata, K. Yoshikawa, M. Nagayama, S.-i. M. Nomura,
N. Magome, and Y. Mori, Chemo-Sensitive Running Droplet, Phys. Rev. E, 72,
041603 (2005): Chapter 6.
[12] H. Kitahata and K. Yoshikawa, Spatio-temporal pattern formation withoscillatory chemical reaction and continuous photon ﬂux on a micrometre scale,
J. Phys.: Condens. Mat., 17, S4239 - S4248 (2005): Chapter 6.

Contents
I

General Introduction

5

1 Introduction
1.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . .

77
10

2 Background
2.1 Reaction diﬀusion systems coupled with convection . . . . . . .
2.2 Belousov-Zhabotinsky reaction . . . . . . . . . . . . . . . . . . .
2.3 Marangoni eﬀect and interfacial tension . . . . . . . . . . . . . .

11
11
12
17

II Spontaneous Motion in Reaction Diﬀusion System
Coupled with Convection
21
3 Convective ﬂow coupled with BZ reaction
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
24
24
27
28

4 Spontaneous motion of a BZ droplet
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Spontaneous motion of a BZ droplet on oil phase . . . . . . . .
4.3 Convective ﬂow inside the spontaneously-moving droplet . . . .
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Deformation of a droplet coupled with BZ reaction . . . . . . .

32
32
33
33
35
36

5 Convective motion in water-camphor system
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Numerical calculation . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Eﬀects of convective ﬂow on the spontaneous motion . . . . . .

38
38
40
41
41
44

3

4

Contents

6 Spontaneous motion of a droplet in the other systems
6.1 Spontaneous motion in an alcohol-water system . . . . . . . . .
6.2 Spontaneous motion in an oil-water system . . . . . . . . . . . .
6.3 Spontaneous motion coupled with a chemical reaction . . . . . .

III

51
51
56
62

Boundary Eﬀects on Reaction-Diﬀusion Systems 65

7 Slowing and stopping of a chemical wave in a narrowing canal
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
68
69
69

8 Alternative stopping of a chemical wave in a photosensitive BZ
reaction ﬁeld
75
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
8.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
9 Synchronization of photosensitive chemical waves coupled in
two circular excitable ﬁelds
81
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
9.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
9.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85

IVGeneral Conclusion

10 Conclusion
10.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Future problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

93
95
95
96

Acknowledment

99

References

101

Part I
General Introduction

5Chapter 1
Introduction
In this chapter, the perspective of this thesis is shown. The thesis
includes mainly two topics: convective ﬂow coupled with reactiondiﬀusion systems, and the boundary eﬀects on reaction-diﬀusion systems. The importance of these two topics is depicted. Then, theoutline of this thesis is presented.

1.1

Perspective

To understand how living organisms maintain their lives, it is important toregard the living organisms as nonequilibrium open systems, since organisms
live through the dissipation of the chemical energy of chemical substances suchas adenosine triphosphate (ATP) and so on [13, 14]. So far, a number of studies
on physics under nonequilibrium open conditions have been performed bothexperimentally and theoretically.
Among these studies, spatio-temporal self-organization has been one of themost interesting topics. As a theoretical model of spatio-temporal pattern formation, reaction-diﬀusion systems are widely adopted. In the framework ofreaction-diﬀusion systems, it is assumed that the concentrations of chemicalcomponents or other physical quantities are deﬁned at every point, i.e., local
equilibrium is achieved, and the temporal changes in these physical quantities
are described as the combination of the local dynamics (reaction) and transportation from/to the neighborhood proportional to the gradient (diﬀusion).
Typically, a reaction-diﬀusion equation for a N-component system is writtenas∂ci= fi (c1 , c2 , · · · , ci , · · · , cN ) + Di ∇2 ci ,(1.1)∂twhere ci is the physical quantity such as concentration of the chemical component, Di is a diﬀusion constant and fi (c1 , c2 , · · · , ci , · · · , cN ) is the summation of
the local dynamics of the chemical reaction regarding the i-th component.
Under the framework of such a reaction-diﬀusion system, various spontaneous pattern formation can be predicted theoretically. A wave or wave packets
7

8

Chapter 1

Introduction

can propagate on the reaction ﬁeld and forms fascinating patterns such as a target pattern or a spiral pattern. These patterns have been analytically studied
based on the phase reduction, bifurcation theory, and so on [15,16]. On the other
hand, Turing proposed a novel framework of pattern formation called a “Turing
pattern”; i.e., a static pattern can be generated in a two-or-more-component
system when the diﬀusion constants are quite diﬀerent from each other [17].
There are also a large number of experimental systems which can be described
as reaction-diﬀusion systems. Belousov-Zhabotinsky (BZ) reaction is one ofthem [18, 19]. The detailed introduction of BZ reaction is in Chapter 2. The
oxidization pattern of carbon monoxide on platinum substrate [20], the patternof combustion wave [21] or some other systems have also been studied. In thesesystems, the chemical wave propagation and/or spatio-temporal patterns can beobserved. As for the Turing pattern, Ouyang et al. and de Kepper et al. havefound the Turing patterns using the chemical reaction in the gel systems [22,23].
So far, there are many reports on the novel spatio-temporal pattern formationsuch as inward spirals [24], multiarm spirals [25], etc.
In the living organisms, it was said that the spatio-temporal self-organization
is essential for the patterns on the animal skin, such as zebras, tigers and seashells[26]. Kondo et al. showed that the Turing pattern is seen in the pattern on theskin of a ﬁsh [27]. Today, many examples of such pattern formation in theliving organisms have been found [28]. One of the most famous one is the pulse
propagation on the nerve cells [29, 30]. It has been well established that, in
cable-like nerve cells, pulses on the membrane propagate at a constant velocity
and amplitude. With the physiological measurements, the phenomenologicalmodel called the Hodgkin-Huxley equation was established [29]. These equations
are reduced to FitzHugh-Nagumo equation by adiabatic approximation [31–33].
The FitzHugh-Nagumo equation with a diﬀusion term has been widely used asa theoretical model for nerve systems. Therefore, reaction-diﬀusion systems areoften used for a model of living organisms.
Reaction-diﬀusion equations, however, can be adopted only when the ﬁeld is
ﬁxed. In the other words, the spatial transport is allowed only through diﬀusion.
Thinking of the phenomena in the real world, the ﬁeld itself usually moves. For
example, the chemical reaction inside a cell should be aﬀected by the protoplasmic streaming, and the solution in the reactor is often stirred. Thus, it isimportant to consider the motion of the ﬁeld itself.
We introduce this eﬀect by adding the advection term to the reaction-diﬀusion
equation. It can be said that this is the natural extension of the reaction-diﬀusion
system.
If the ﬂow velocity and the typical scale of the change in the spatio-temporal
pattern, such as the velocity of propagating chemical waves, are quite diﬀerent,or if the typical length of the change in ﬂow direction and the typical lengthof the pattern are quite diﬀerent, there seems to be no interesting phenomena.
For example, if the ﬂow is much faster than the chemical wave propagation,
the chemical components or other physical quantities are stirred, and becomeChapter 1

Introduction9

uniform, which results in no pattern formation. If the ﬂow direction is almost
uniform throughout the system, there seems to be no eﬀects on pattern formation
but the pattern is transferred as a whole.
In this thesis, the mutual coupling between pattern formation in reactiondiﬀusion systems and convective ﬂow are investigated. As the scales of theﬂow velocities and the propagating wave velocity are comparable, the novel
phenomena can be seen both experimentally and theoretically. This coupling
between reaction-diﬀusion systems and convective ﬂow is the ﬁrst topic in thisthesis. The coupling between reaction-diﬀusion systems and convective ﬂow canbe regarded as an experimental model where chemical energy is transduced intomechanical motion.
On the other hand, in the theoretical studies, the boundary eﬀects are rarely
considered. In order to discuss on the pattern formation such as a target pattern,a spiral pattern, etc., there is no need to think about the boundary. In thesestudies, the eﬀects of boundaries should be avoided and a lot of eﬀorts have been
made to eliminate the boundary eﬀects.
In the real world, however, the boundary of the ﬁeld often aﬀects the behaviorof the whole system. If the typical scale of the system is comparable to thecharacteristic length of the pattern, the boundary eﬀect is quite important. In
the experiments using BZ reaction, the change in the behavior in a narrow
space was reported using a gel system [34, 35], a droplet system [36], a beadsystem [37, 38], etc.
As in the above paragraph, the nerve systems can be treated as reactiondiﬀusion systems. In recent years, based on the advances in experimental techniques, it is getting clearer that the manner of this pulse propagation criticallydepends on the thickness or width of the nerve cells. For example, it has beenreported that the dendritic shape of nerve cells strongly aﬀects the propagationof an excited wave [39–41].
In this thesis, thus, using BZ reaction as an experimental model, the boundary eﬀects on chemical wave propagation are exempliﬁed. The typical size ofthe system is around 100 µm to 1 mm, which corresponds to the typical widthof one chemical wave. This boundary eﬀects on reaction-diﬀusion systems aresecond topic in this thesis.
As a summary, the main topics in this thesis are the convective ﬂow coupledwith reaction-diﬀusion systems and boundary eﬀects on reaction-diﬀusion systems. These two eﬀects are important in order to understand the phenomena in
the real world as reaction-diﬀusion systems. In this thesis, some experimental
results are exempliﬁed and theoretical discussion and/or numerical modeling are
made for each system from the viewpoint of the reaction-diﬀusion systems withconvective and/or boundary eﬀects.

10

1.2

Chapter 1

Introduction

Outline of the thesis

In this section, the outline of this thesis is presented.
In Chapter 2, three important topics for discussion in this thesis are introduced: Reaction diﬀusion systems, BZ reaction, and Marangoni convection due
to the surface tension diﬀerence.
This thesis includes mainly two topics: convective ﬂow coupled with reactiondiﬀusion systems and boundary eﬀects on reaction-diﬀusion systems. These two
topics correspond to Parts II and III, respectively.
Part II includes four chapters. In Chapter 3, the convective ﬂow induced by
the repetitive change in interfacial tension in BZ reaction is exempliﬁed. The
characteristics of the convection are reproduced by numerical calculation basedon the reaction-diﬀusion equations coupled with convection. The contents ofthis chapter are published in References [1] and [7]. In Chapter 4, the spontaneous motion of a droplet of BZ reaction medium is shown. The mechanism
is discussed related to the convection inside the droplet. The contents of thischapter are published in References [1] and [2]. In Chapter 5, the spontaneous
motion of a camphor grain is ﬁrstly discussed. Then, the convective ﬂow induced by the surface tension gradient due to the nonuniform camphor layer on
the water surface is exempliﬁed, and numerically reproduced by the reactiondiﬀusion equation coupled with convection. At last, the eﬀects of the convection
to spontaneous motion are presented. The contents of this chapter are published
in References [3], [8], and [4]. Chapter 6 includes three topics: the spontaneous
motion of an alcohol droplet on the aqueous phase, the spontaneous motion ofan oil droplet on a glass substrate in surfactant solution, and the spontaneous
motion of a phenanthroline disk coupled with chemical reaction. These are theexamples of spontaneous motion induced by the interfacial tension diﬀerence.
The contents of this chapter are published in References [10], [11], [4], and [9].
Part III includes three chapters. In Chapter 7, the slowing and stoppingof the chemical wave in the narrowing glass capillary is exempliﬁed in BZ reaction. The mechanism is theoretically analyzed related to the surface-volume
ratio. The contents of this chapter are published in References [5] and [12]. In
Chapter 8, the narrowing reaction ﬁeld is achieved by the designed illumination
on photosensitive BZ reaction medium. The alternative disappearing of chemical waves is exhibited at a certain light intensity. In Chapter 9, the geometryof chemical wave propagating on a circular ring is studied. The two chemical
waves on two connected circular rings are also discussed from the viewpoint ofsynchronization. The contents of this chapter are published in Reference [6].
The contents in this thesis are summarized and the future problems areshown in Part IV.

Chapter 2
BackgroundIn this chapter, three topics are introduced, which are important forthe discussion in this thesis: Reaction-diﬀusion systems, BelousovZhabotinsky reaction, and Marangoni convection induced by the surface tension diﬀerence.

2.1

Reaction diﬀusion systems coupled with convection

In studies on spatio-temporal self-organization in nonequilibrium open systems,
reaction-diﬀusion systems are often used as representative model systems. The
advantage of a reaction-diﬀusion system is that spatio-temporal pattern formation can be described using simple equations regarding the dynamics of the local
concentrations of the chemicals. Thus, the dynamics of the system can be described as the combination of the local dynamics and transportation from/tothe neighborhood proportional to the gradient, or diﬀusion.
The diﬀusion term is derived as follows. Diﬀusive ﬂow J is assumed to
be proportional to the gradient of the concentration or some other physicalquantities, c,
J ∝ −∇c.
(2.1)
From the equation of continuity,∂c= −∇ · J,
∂t

(2.2)

the diﬀusion equation∂c= D∇2 c(2.3)
∂t
is obtained, where D is a positive constant called a “diﬀusion constant”.
On the other hand, the local dynamics of a chemical reaction can be describedusing ordinary diﬀerential equations based on the mass-action law, which states
1112

Chapter 2

Backgroundthat the reaction rate is proportional to the product of the concentrations of thereactants.
Taking these two terms into consideration, a reaction-diﬀusion equation isderived:∂c= f (c) + D∇2 c,(2.4)∂t
where f (c) is the dynamics of the chemical reaction. For an N-component
system, this equation can be easily extended:∂ci= fi (c1 , c2 , · · · , ci , · · · , cN ) + Di ∇2 ci .
∂t

(2.5)

Under the framework of the reaction-diﬀusion systems, the local concentrations of the chemical reagents can be described only on a ﬁxed ﬁeld. In the realworld, however, convective motion of the ﬁeld is often observed: for example,
as in the chemical reactions in a stirred solution and the biochemical reactions
in cytoplasm. To consider this convective motion, an advection term should be
introduced to the reaction-diﬀusion equations. The ﬂow J at the point with aﬁeld velocity of v is
J = −D∇c + cv,(2.6)
instead of eq. (2.1). Based on the conservation of the chemical components, thedynamics of c can be written as∂c= D∇2 c − ∇ · (cv).
∂t

(2.7)

Here, if the volume of the ﬁeld is incompressive (for example, the reaction in asolution),∇ · v = 0,
(2.8)
the advection term can be written as v · ∇c instead of ∇ · (cv).
Overall, the dynamics of c in the incompressive ﬁeld can be written∂c= D∇2 c − v · ∇c.
∂t

(2.9)

By adding the local dynamics of the chemical reaction, or the reaction term,
the reaction-diﬀusion-advection equations for the N-component system can be
written as∂ci= fi (c1 , c2 , · · · , ci , · · · , cN ) + Di ∇2 ci − v · ∇ci .
(2.10)∂tThis can be regarded as a reaction-diﬀusion-advection equation [12].

2.2Belousov-Zhabotinsky reaction

Belousov-Zhabotinsky reaction (BZ reaction) is famous as an experimental model
system for a reaction-diﬀusion system. In 1951, Belousov found the oscillatory

Chapter 2

Background

13

(a)

(b)

1 cm

1 cm

Figure 2.1: Rhythmic change in color in BZ reaction. When the solution isstirred, the solution oscillates between the (a) oxidized state (blue) and (b)reduced state (red).

behavior in color of the solution, when he made an experiments to make a model
chemical system for metabolic pathways (TCA cycle). Belousov used citric acidas a reactant and cerium as a catalyst. In this reaction, the solution is changed
between white and yellow. Unfortunately, Belousov’s observation was not widely
known, since people did not believe his experimental results. In 1964, Zhabotinsky used the catalyst of iron complex, i.e., ferroin, and reproduced Belousov’s
results. By using ferroin, the change in color is much easier to observe: thesolution changes between red and blue as shown in Fig. 2.1. In 1970, he observed the spontaneous pattern formation in the solution, as BZ solution wasallowed to stand [18]. For example, chemical waves propagate as concentricrings, or with a shape of spiral, which are called a target pattern, and a spiral pattern, respectively (Fig. 2.2) [19]. Then, Gáspár found that BZ reaction
can be sensitive to the light intensity by using ruthenium complex as a catalyst(Ru(bpy)3+3 ) [42–45].
BZ reaction is an oscillatory chemical reaction between the oxidized state
and reduced state. In the reaction process, an autocatalytic process is said
to be the most important. This autocatalysis induces the nonlinearity in thechemical reaction, which enables chemical oscillation as a limit cycle oscillationin nonlinear systems.
There were a lot of studies on the mechanism of BZ reaction by considering
all the elementary processes. In 1972, Field, Körös, and Noyes proposed thatten main processes are important for BZ reaction [46]. This model is called theFKN model. Based on this model, the kinetic equations on the concentrationsof the products, reactants, and intermediates are derived, though this is socomplicated. By adiabatic approximation on the fast processes, we can get a
simpler model with a smaller number of variables.

14

Chapter 2(a)Background

(b)

3 mm

3 mm

Figure 2.2: Spontaneous pattern formation in BZ reaction using ferroin as a catalyst. (a) Target pattern. (b) Spiral pattern. Red and blue regions correspondto the oxidized and reduced states, respectively.

Among the reduced models, the Oregonator has widely been used, which
was proposed by Field and Noyes. They assumed that the BZ solution takes
one of the two characteristic states: i.e., HBrO2 -rich state and HBrO2 -poorstate, where HBrO2 is an unstable intermediate. In this model, the following
ﬁve chemical processes are important:k

1
A + Y −→
X + P,k

2
X + Y −→
2P,k

3
2X + 2Z,
A + X −→k

4
A + P,
2X −→k5
Z + B −→
hY,

(2.11)(2.12)(2.13)(2.14)(2.15)

where A, B, P, X, Y, and Z correspond to BrO−
3 , CH2 (COOH)2 +BrCH(COOH)2 ,HOBr，HBrO2 , Br− and the oxidized catalyst, respectively. h is the parameter
which corresponds to the number of Br− ions produced by one catalyst moleculein the oxidized state.
Assuming that the temporal change in the concentrations of A and B is
negligible, the concentrations of A and B, i.e., A and B, can be set as constants.
Then, the kinetic diﬀerential equations for X, Y, and Z can be derived:dX= k1 AY − k2 XY + k3 AX − 2k4 X 2 ,
dt


dY
= −k1 AY − k2 XY + hk5 BZ,
dt
dZ= 2k3 AX − k5 BZ,
dt(2.16)(2.17)(2.18)

Chapter 2

Background15

where X ，Y ，and Z correspond to the concentrations of X，Y，and Z, respectively. ki (i = 1 · 5) are the reaction constants on each reaction (2.11) to (2.15)
and  and  are the positive small constants that correspond to the diﬀerencein the reaction rate. The above-mentioned diﬀerential equations are called the
3-variable Oregonator [47]．
Since the temporal change in Y is much faster than the changes in X and Z,
Y can also be reduced adiabatically, and the following model is obtained. Thismodel is called the 2-variable Oregonator (Tyson version) [48]. Here, it is notedthat X and Y is replaced by U and V , respectively.


U −q
dU
= U(1 − U) − f V≡ F (U, V ) ,dt
U +q(2.19)dV= U − V ≡ G (U, V ) ,(2.20)
dt
where U and V correspond to the concentrations of HBrO2 and the oxidized
catalyst, respectively. In other words, the region with larger V corresponds tothe oxidized state (blue if using ferroin) and that with smaller V corresponds
to the reduced state (red if using ferroin). f ，q ，and  are the parameters that
correspond to the threshold for excitation, excitability, and nondimensionalized
reaction constant, respectively. The two-dimensional dynamical system withthe equations (2.19) and (2.20) has one stable ﬁxed point (excited state) or one
unstable ﬁxed point and one stable limit cycle (oscillatory state) depending onthese parameters. For example, as the paremeter f is changed, a stable ﬁxed
point destabilizes and a limit cycle orbit comes out, i.e., the system exhibits
Hopf bifurcation. The nullclines of the dynamical systems are shown in Fig.
2.3.
On the contrary, Rovinsky and Zhabotinsky also proposed another way of
reduction [49]. This model is called the Rovinsky-Zhabotinsky model, which canbe written as follows:dU
V U −µ= U(1 − U) − 2qα,dt
1−V U +µ(2.21)

U
dV
=U −α
V,
(2.22)
dt
1−U
where U and V correspond to the concentrations of HBrO2 and [Fe(phen)3 ]3+ ,respectively.
The diﬀerence between Rovinsky-Zhabotinsky model and Tyson version 2U
variable Oregonator is the term of 1−U. It is said that the BZ reaction withferroin catalyst is reproduced better by the Rovinsky Zhabotinsky model, but
in the present article, Tyson version 2-variable Oregonator is adopted. Figure
2.4 shows the results of numerical calculation on the pattern formation using
Tyson version Oregonator.
So far, there have been a large number of studies on BZ reaction, since it
is easily achieved by mixing a few chemical compounds, and easily controlled.

16

Chapter 2(a) Oscillatory state(b) Excitable state
1.0

1.0

V

Background

0.5

V

0

0

0.5
U

0.5

0

1.0

0.01

0

0.5
U

1.0

0.01

Stablefixed point
V 0.005

V 0.005

0

Unstable
fixed point
0

0.005
U

0
0

0.01

0.005
U

0.01

Figure 2.3: Nullclines of the 2-variable Oregonator (Tyson version). (a) Oscillatory state. (b) Excitable state. The parameters are q = 0.0008 and (a) f = 1.5,(b) f = 4. The lower ﬁgures are the magniﬁcations near the intersections.

(a)

(b)

Figure 2.4: Numerical results on pattern formation using the Oregonator (Tyson
version). (a) A target pattern and (b) a spiral pattern are exhibited. The
parameters are f = 3, q = 0.0008, and  = 0.04.

Chapter 2

Background0

17

Surface tension: γ(x)axflow velocity: v = (u, v)bzFigure 2.5: Schematic illustration on the system considered here. At the upperboundary, the surface tension is deﬁned as γ(x). At the other boundaries, aﬁxed boundary condition is adopted.

The mechanism of BZ reaction has been almost solved, but it is widely used as amodel experimental systems for a reaction-diﬀusion system, or a nonequilibrium
open system.

2.3

Marangoni eﬀect and interfacial tension

It is known that spatial gradient of surface tension can cause convection. In thissection, some characteristics of surface-tension-driven convection, or Marangoni
eﬀect, are introduced.
There are two types of the Marangoni eﬀect: one is the thermal Marangoni
eﬀect and the other is the solutal Marangoni eﬀect. The former is the convection
driven by the surface tension gradient due to the temperature gradient, and thelatter is that due to the concentration gradient [50, 51]. As for the thermal
Marangoni eﬀect, there have been a number of reports, but there have been
only a few reports on the solutal Marangoni eﬀect. In this thesis, the solutal
Marangoni eﬀect is mainly concerned.
The Marangoni eﬀect can be understood in the framework of hydrodynamics.
Using the Navier-Stokes equation, the surface tension gradient is introduced asa shear at the surface. Here, we consider the system as shown in Fig. 2.5. The
x axis is set along the surface, and the z axis is perpendicular to the x axis.
Supposing the component of the ﬁeld velocity in x and z axis are u and v, thesurface tension gradient along x axis is proportional to the tangential gradient
of u:
∂u
∂γ
ν
=− ,
(2.23)∂z∂x
where ν is the kinetic viscosity of the medium. Here, it is to be noted that the18

Chapter 2

Backgroundsurface has no curvature. As the surface have a certain curvature, the pressure
diﬀerence between two phases should be considered, i.e., Laplace pressure [52].
In Chapters 3 and 5, we use another formulation, i.e., we introduce the surface tension as a volume force. We adopt the modiﬁed Navier-Stokes equations
for incompressive ﬂuid;∂v+ (v · ∇)v = η∇2 v − ∇p + Fi ,
ρ∂t

(2.24)∇ · v = 0.

(2.25)

andThe interfacial tension term Fi has the following shape:∂γδ(z)ex .
(2.26)Fi =∂x
In this formulation, the surface tension is introduced as a volume force. The
compatibility between eqs. (2.23) and (2.26) are conﬁrmed as follows: A steadystate is assumed and the pressure term is also assumed to be neglected. Then,eq. (2.24) becomesρv · ∇v = η∇2 v + Fi .
(2.27)
When ρ/η is small, i.e., Re = ρvl/η << 1, the nonlinear term can be neglected:
η∇2 v + Fi = 0.

(2.28)

Here, the system as shown in Fig. 2.5 is considered. In this discussion, wewant to know the features near the interface, where v is nearly 0. For the xcomponent of v, u, follows the equation:
η 2
∂γ
∇ u=
δ(z).
(2.29)
ρ∂x
By integrating eq. (2.29) with z from 0 to b,
η
ρ









∂u 
∂u 
∂γ


.
dz
+
−
=
2
∂z z=b ∂z z=0
∂x
0 ∂x
b ∂2u

(2.30)

When the eﬀect of the bottom interface is small enough, or the system isdeep enough, we can assume that


∂u 

= 0,
∂z z=b
at the bottom interface. The ﬁrst term of the left side is


 b 2
 b
∂ u
∂
∂v
dz =
−
dz
∂z
0 ∂x2
0 ∂x


 b
∂
∂v
−
dz
=
∂x
0 ∂z


∂v 
∂v 
=
−


∂x z=b ∂x z=0
= 0,(2.31)(2.32)

Chapter 2

Background

19

y
w
c1

c2
l

0

x

h
z
Figure 2.6: Schematic illustration on the theoretical system in three dimension. The competition between the transportation by diﬀusion and by ﬂow isimportant.

where the incompressive condition (eq. (2.25)) is used. Equation (2.30) can be
written as follows:

∂u 
∂γ

(2.33)ν=− ,
∂z z=0∂x
where ν = η/ρ.
Thus, the relationship between the surface tension and the tangential gradient of velocity (eq. (2.23)) is achieved.
By the way, the characteristics of the surface-tension driven convection canbe identiﬁed by the nondimensional number named Marangoni number. Here,the meanings of this nondimensional number are easily exempliﬁed.
We assume the system where two regions with the concentrations of c1 andc2 are at the distance of l. The width of the system is w and the depth is h.
The coordinate is set so that the x coordinate meets the horizontal surface asshown in Fig. 2.6. The origin is set at the position where the concentration isc1 .
Without convection, the proﬁle of the concentration should be linear:c(x) = c1 + (c2 − c1 )x ≡ c1 + ∆cx.

(2.34)

This concentration gradient causes the gradient in surface tension, which drives
the convective ﬂow. The driving force per unit area is(γ2 − γ1 ) w/l,

(2.35)

where γ1 and γ2 correspond to the surface tension for the regions with theconcentration of c1 and c2 , respectively. The driving force is assumed to be20

Chapter 2

Backgroundaveraged, and the driving force per unit volume can be written as(γ2 − γ1 ) w/lh = ∆γw/lh.

(2.36)On the other hand, if the Marangoni ﬂow with a velocity of v was induced,
the friction force per unit volume would be|η∇v| d ≈ηv
d.
h(2.37)

By assuming that the driving force is balanced to the friction force, i.e.,
ηv
∆γw
=
w
lh
h

(2.38)

we can derive the following equation,
v=

∆γ
.
η

(2.39)

When the concentration diﬀerence is small enough, the surface tension gradientis described as a linear relationship:
dγ
∆c.
dc

(2.40)

∆c dγ
.
η dc

(2.41)∆γ =Therefore, eq. (2.39) leadsv=

Considering the above discussion, the rate of transport by the ﬂow is(∆c)2 dγ.
Mﬂow = v∆c =η dc(2.42)On the other hand, the rate of transportation by diﬀusion is
Mdiﬀusion = D∆cl(2.43)

The critical value for whether convection is induced or not is the point where
the above two transport rates become on the same scale. By considering the ratio
between Mﬂow and Mdiﬀusion , we can discuss on whether convection is inducedby the interfacial tension gradient. The ratio is written as follows:
Mﬂow
∆c dγ
(∆c)2 dγ=
≡ Ma.
=
Mdiﬀusion
ηD∆c dc
ηD dc(2.44)

Thus, we can discuss on whether convection induced by surface tension happens or not using Marangoni number, Ma.

Part II
Spontaneous Motion in Reaction
Diﬀusion System Coupled with
Convection

21

Chapter 3
Convective ﬂow coupled with BZ
reactionIn this chapter, the convective ﬂow induced by interfacial tension
change in BZ reaction is exhibited. This system is a typical example of the reaction-diﬀusion-advection system. The convection isobserved in the experiments [1], and the properties of the convective
ﬂow can be reproduced in numerical calculations [7].

3.1Introduction

It was reported that the interfacial tension of BZ medium changes synchronizedwith the chemical oscillation, that is, BZ medium in the oxidized state hashigher interfacial tension than that in the reduced state [53]. The catalyst usedin the BZ reaction is ferroin, which is the complex of the ferrous ion and 1,10phenanthroline. Since the phenanthroline is hydrophobic, the catalyst tends tobe distributed near the surface and reduces the interfacial tension. A catalyst
with a higher valence (oxidized form) is even more hydrophilic than that with alower valence (reduced form). Therefore, in BZ solution, the surface distributionof the catalyst is less in the oxidized state than in the reduced state, which causes
the diﬀerence in surface tension.
Driven by the repetitive change in interfacial tension, Miike et al. reported
that Marangoni convection is induced at the free surface of BZ medium [54–56].
Though there have been some other reports on the convective ﬂow in BZ reaction
both experimentally and theoretically [57–64], the convection is induced on theperpendicular plane. Therefore the gravitational eﬀects are unavoidable.
In the present study, we adopted a system made of two bulk phases of BZ
reaction medium and oleic acid (oil), and observed the ﬂow near the interface ofthe two phases by adding small particles for visualization. Since the interface isperpendicular to the horizontal plane, the gravity does not aﬀect the convection.
Due to the change in interfacial tension, Marangoni convection was induced
synchronizing with the chemical wave propagation. Near the interface between
23

24

Chapter 3(i) side view

Convective ﬂow coupled with BZ reaction(ii) top view

BZ mediumoil (oleic acid)oil (oleic acid)spacerspacer(1 mm)
BZ medium

Figure 3.1: Schematic illustration on the experimental setup. Images are takenfrom below [1].

the both phases, the strong convective ﬂow was observed toward the oxidized
area, and two rolls were formed in the bulk phases. Numerical calculations basedon the reaction-diﬀusion-advection equations were also performed, and can well
reproduce the experimental results.

3.2

Experiments

All chemicals were analytical grade reagents and used without further puriﬁcation. An aqueous solution of ferroin, tris (1,10-phenanthroline) iron (II) sulfate, was prepared by mixing stoichiometric amounts of 1,10-phenanthroline(C12 H8 N2 ) and ferrous sulfate (FeSO4 ) in pure water. The water was puriﬁed
with a Millipore-Q system. BZ medium, in the excitable state, contains 0.15
M sodium bromate (NaBrO3 ), 0.30 M sulfuric acid (H2 SO4 ), 0.10 M malonicacid (CH2 (COOH)2 ), 0.03 M potassium bromide (KBr), and 5.0 mM ferroin([Fe(phen)3 ]2+ ). To visualize the convective ﬂow, polystyrene beads (General
Science Corporation, 10 µm in diameter) were dispersed throughout the medium.
BZ medium and oleic acid were situated between two glass plates (clearance :1.0 mm). A chemical wave was initiated using silver wire. The region near theinterface between BZ medium and oleic acid was observed from below by aninverted microscope (Nikon DIA-PHOT-TMD), as schematically shown in Fig.
3.1.

3.3

Results

Figure 3.2 shows the convective ﬂow that accompanied the propagation of achemical wave in BZ medium, where the ﬂow of the tracer beads was visualized
as the accumulation of video frames over 1.0 s, together with image-processing
to compensate for the nonuniformity of the illumination [65]. On the front ofChapter 3

Convective ﬂow coupled with BZ reaction

25

oleic acidtime (every 5.0 sec)

BZ solution
chemical wave

1 mm

Figure 3.2: Results of convective ﬂow on the horizontal plane. The proﬁles of
streamlines in 1.0 s are shown at the intervals of 5.0 s. The upper phase is oleic
acid and the lower phase is the BZ medium; the chemical wave propagated fromleft to right. Convective ﬂow was generated towards the wave front on bothsides (Left column) Schematic illustrations are also given to show the directionof convective ﬂow (Right column) [1].

Chapter 3

Convective ﬂow coupled with BZ reaction

time (every 10.0 s)

26

1 mm

Figure 3.3: Disappearance of convective ﬂow due to an inhibitor, iodine, of thechemical reaction. Pure oleic acid was replaced by oleic acid with iodine. The
chemical wave propagated from left to right in the bulk solution, but did notreach the interface, and no convective ﬂow was induced [1].

the chemical wave, large convective ﬂow, parallel to the interface and opposite
the direction of movement of the chemical wave, was induced in both the waterand oil phases. Parallel ﬂow in the same direction as the chemical wave wasobserved on the back of the chemical wave at the interface. In both the waterand oil phases near the intersection of the chemical wave with the interface, theﬂow sank into the bulk phases. Figure 3.3 shows the results of an experiment inwhich iodine was added to the oil phase. In this experiment, no convective ﬂow
was generated, despite the appearance of a chemical wave. Iodine is a potent
inhibitor of the BZ reaction [66, 67]. The iodine in the oil phase is expected
to inhibit wave generation near the interface, due to its diﬀusion from the oilto the aqueous phase through the interface. Thus, the absence of convective
ﬂow can be attributed to the failure of the chemical wave to propagate near theinterface. This convective ﬂow is clearly caused by the diﬀerence in interfacial
tension between the oxidized and the reduced states [53].

Chapter 3

Convective ﬂow coupled with BZ reaction

27(a)oleic acidx0
BZ solution
y

(b)
x
y

Figure 3.4: Results of numerical calculation. (a)The coordinate in the numericalcalculation. The results are for the lower half plane (y > 0). (b)Proﬁles of a
traveling chemical wave and the streamline in the bulk phase are shown. The
region with large V , i.e., the oxidized state, is drawn with blue. The experimental
trends have been thus represented in a qualitative manner. The parameters usedin this simulation are ρ = 1, η = 3, DU = DV = 0.5,  = 0.04, f = 3, q = 0.0008,
k = 0.01, and δ = 0.5 [7].

3.4

Numerical calculation

For a numerical simulation, we adopted the 2-variable Oregonator, which hasoften been used for reproducing BZ reaction. In many past studies, the patternin the settled BZ reaction medium is described as the reaction-diﬀusion equation
using Oregonator as a reaction term. However, in this system, the medium canmove. Therefore, we have to consider the dynamics of the medium, or ﬁeld,
itself. Consequently, we adopted Navier-Stokes equation for the movement of themedium. The medium is aﬀected by the interfacial tension due to the change inthe chemical components coupled with BZ reaction. Here, Cartesian coordinates
are set so that the interface meets the x axis as shown in Fig. 3.4(a). Thus, we
can write as follows:




∂
ρ
+ v · ∇ v = η∇2 v − ∇p + Fi ,
∂t

(3.1)

∇ · v = 0,

(3.2)

∂
+ v · ∇ U = F (U, V ) + DU ∇2 U,
∂t

(3.3)





28

Chapter 3Convective ﬂow coupled with BZ reaction

∂+ v · ∇ V = G (U, V ) + DV ∇2 V,
∂t

(3.4)

U −q
1
U(1 − U) − f V,
F (U, V ) =

U +q

(3.5)

G (U, V ) = U − V,

(3.6)





where ρ is the density, v is the velocity of the ﬂuid, p is the pressure, η is theviscosity, U and V are the concentrations of the represented chemical reagents,and f , , and q is the parameters corresponding to the nature of BZ medium.
Here, interfacial tension is assumed to be as a volume force, which has non-zerovalue in the area within the distance, δ, from the interface as is commented
in Chapter 2. The interfacial tension is proportional to the gradient of theconcentration of oxidized catalyst, V :


Fi =

k ∂V
e (0 < y < δ),∂x x
0(y > δ),(3.7)

where k is a positive constant and ex is the unit vector in the x direction.
In Fig. 3.4(b), the proﬁles in the bulk phase calculated using the above
equations are shown. In the numerical calculation, SMAC method (Simpliﬁed
marker and cell method) was used [68]. Near the interface, convective ﬂow is
induced toward the point where chemical wave touches the interface. The chemical wave near the interface propagates slower than that in the bulk phase. These
characteristics are qualitatively similar to those of the experimental results.

3.5

Discussion

Using the above-mentioned framework, the proﬁle of interfacial tension can be
estimated. The x and y components of ﬂuid velocity v are supposed to be u
and v, respectively. Here, the pressure term does not play an important role, itis neglected for simplicity. The x component of eq. (3.1) is


∂u
∂u
∂u
+u
+v
ρ
∂t
∂x
∂y







∂2u ∂2u=η++ Fi .
∂x2 ∂y 2(3.8)

As we approach the region near the interface, the velocity parallel to the interfacial plane, u, is much greater than that perpendicular to the plane, v. Thus we
can neglect the term ρv∂u/∂y. The term η∂ 2 u/∂y 2 represents the diﬀusion-likebehavior in the y direction. The characteristic time for passing of the chemical
wave, ∆t, is about 10 s based on the experimental results. Therefore, the characteristic depth of the region, ∆y, in which ﬂuid moves attached to the interface
is calculated asη∆t  0.3 cm.
(3.9)
∆y ∼
ρ

Chapter 3

Convective ﬂow coupled with BZ reaction(a)Fluid Velocity
u (mm s-1)

0.00
-0.10
-0.20

Interfacial Tension∆γ (mN m-1)

Gradient ofInterfacial Tension
dγ/dx (N m-2)

(b)

(c)

29

0.04
0.02
0.00
-0.02
0.008
0.004
0.000

-0.004(d)(a.u.)

3+[[Fe(phen)3] ]

140
120
100
-1.0 -0.5 0.0

0.5 1.0 1.5
Displacement x (mm)wave front

Wave Direction

Figure 3.5: Spatial proﬁle of various physics-chemical variables deduced from theexperimental results on convection generated by the BZ reaction, as in Fig. 3.2.
(a) Fluid velocity, u, parallel to the interface in the region near the interface. The
chemical wave propagates from left to right, and the origin is set at the positionof the wave front. (b) The gradient of interfacial tension, dγ/dx, calculated fromthe ﬂuid velocity proﬁle. (c) Interfacial tension, ∆γ, calculated by integrationof (b). (d) The chemical wave as represented by a change in the color of thesolution that corresponds to the concentration of ferriin ([Fe(phen)3 ]3+ ) [1].

30

Chapter 3

Convective ﬂow coupled with BZ reaction

From the experiments, the viscosity of the BZ medium was around 1.1 × 10−2
g cm−1 s−1 . In this calculation, the viscosity and density of BZ medium are
adopted. From now on, only the x component is considered, and the interfacial
tension is taken as the force working in the region within ∆y from the interface.
In this framework, the interfacial tension can be considered s a volume force,and can be combined with an interfacial tension constant γ as follows:∂γ= Fi ∆y.
∂x(3.10)

Since the chemical wave propagates at a constant velocity, c, the ﬂuid velocity
proﬁle, u, satisﬁes the diﬀerential equation:
∂u
∂u
=c .
∂t
∂x

(3.11)

From the experiment, the velocity of the chemical wave parallel to the interface is
c = 0.01 cms−1 . Thus, eq. (3.8) can be simpliﬁed to a one-dimensional ordinary
diﬀerential equation:


du
du
+u
ρ c
dx
dx



=η

d2 u
+ Fi .
dx2(3.12)

The ﬂuid velocity proﬁle near the interface was measured, and the results areshown in Fig. 3.5(a). Using these results, the ﬁrst-order derivative, the secondorder derivative, and the interfacial tension proﬁle are calculated. The proﬁle of
the gradient of the interfacial tension, dγ/dx, is shown in Fig. 3.5(b). Using thisproﬁle, the proﬁle of the interfacial tension, ∆γ, is calculated by integration, as
shown in Fig. 3.5(c). The proﬁle of the chemical wave is shown in Fig. 3.5(d),which was obtained from a quantitative analysis of the color in the reacting
solution. The proﬁle of the chemical wave corresponds to the proﬁle of theinterfacial tension, except for the slight diﬀerences in their peak positions andwidths. These diﬀerences are attributed to the eﬀect of the convective motionof the reacting solution. As in Fig. 3.2, the portion of the ﬂuid in the reduced
state (low [Fe(phen)3 ]3+ ) is pulled toward the front of the chemical wave. This
convective eﬀect causes retardation of the phase of the oscillatory reaction of
the wave front, together with narrowing of the chemical wave. The change
in interfacial tension between the oxidized state and the reduced state is on
the order of 0.01 mNm−1 , as in Fig. 3.5(c). This is about 1% of the value ina previous report [53], where the interfacial tension was measured under thecondition that the BZ medium was stirred and the medium was uniform. In
the present experiment, we estimated the change in interfacial tension under
the condition of propagation of a chemical wave without stirring the reaction
medium. Since the interfacial tension on the front of the wave is opposite thaton the back, the net velocity may be damped. In the analysis adopted above, we
assumed that the system was two-dimensional, however, the actual system wasChapter 3

Convective ﬂow coupled with BZ reaction

31

the space between two glass plates. The ﬂuid must be aﬀected by the friction
from the glass plates, which should disturb the motion of the ﬂuid. In addition,
there would be a rather large damping eﬀect, i.e., ﬂow in the aqueous phaseinevitably induces ﬂow in the oil phase. The discrepancy in the amplitude of thechange in interfacial tension can be attributed to these eﬀects. This estimationof the interfacial tension proﬁle from the ﬂuid velocity proﬁle might be useful inan analysis of ﬂuid dynamics.

Chapter 4
Spontaneous motion of a BZ
dropletIn this chapter, spontaneous motion of a small droplet of BZ reaction medium on an oil phase is exhibited [1]. This can be regarded
as an experimental system where chemical energy is transduced intomechanical energy. This spontaneous motion is related to the convection inside the droplet [2].

4.1

Introduction

In the previous chapter, Marangoni convection at the interface between BZ
medium and oil driven by a repetitive change in interfacial tension is presented.
Near the interface between these two phases, strong convective ﬂow is observed
toward the wave front, and two rolls are formed in both phases. Moreover,the experimental results can be reproduced by a numerical calculation using
the Navier-Stokes equation coupled with a reaction-diﬀusion-advection equation
based on the Oregonator.
The convection coupled with the BZ reaction can be regarded as chemomechanical energy transduction, but only the motion of the small particles forvisualization can be observed due to the convective ﬂow. Therefore, we designed
the system where the macroscopic motion is induced by the chemo-mechanicalenergy transduction in BZ reaction. In the past studies, Yoshida et al. have
exempliﬁed the spontaneous deformation of the gel synchronized with BZ reaction [69, 70], but the deformation is due to the characteristics of the gel. In thepresent study, only by making the system size small, a droplet of BZ medium
can be driven by the convection due to the interfacial tension. In this chapter,the experimental results and discussion on the mechanism are presented.
32

Chapter 4

Spontaneous motion of a BZ droplet

33

Digital Video Camera
BZ medium (droplet)

Oleic Acid
Figure 4.1: Schematic illustration of the experimental setup.
recorded from above [1].

4.2

Images were

Spontaneous motion of a BZ droplet on oil
phase

All chemicals were analytical grade reagents and used without further puriﬁcation. An aqueous solution of ferroin, tris (1,10-phenanthroline) iron (II) sulfate, was prepared by mixing stoichiometric amounts of 1,10-phenanthroline(C12 H8 N2 ) and ferrous sulfate (FeSO4 ) in pure water. The water was puriﬁed
with a Millipore Milli-Q system. BZ medium, in the oscillatory state, contains
0.15 M sodium bromate (NaBrO3 ), 0.60 M sulfuric acid (H2 SO4 ), 0.10 M malonic acid (CH2 (COOH)2 ), 0.03 M sodium bromide (NaBr), and 5.0 mM ferroin([Fe(phen)3 ]2+ ). To visualize convective ﬂow, polystyrene beads (General Science Corporation, 10 µm in diameter) were dispersed throughout the medium.
Using the experimental system shown schematically in Fig. 4.1, we observed the
motion of a 1.0-µl droplet of BZ medium ﬂoating on a layer of oleic acid.
Figure 4.2 shows the experimental results on the motion of a BZ droplet
ﬂoating on the surface of oleic acid. The spatio-temporal plot indicates a translational motion of the droplet accompanied by the propagation of a chemical
wave.

4.3

Convective ﬂow inside the spontaneouslymoving droplet

For clariﬁcation of the nature of this motion, the x axis is set in the directionof the spontaneous motion of the droplet, as shown in Fig. 4.3(a). The positive
direction is deﬁned as that in which the droplet moves ﬁrst. The chemical
wave propagated keeping the shape of a circle, and when the chemical wave
propagating in the negative x direction touched the interface, the droplet began
to move in the positive direction. After the entire droplet turned to the oxidized

34

Chapter 4

Spontaneous motion of a BZ droplet

space
0
1
2
3
4
5
time
(s)

0.5 mm

0.2 mmFigure 4.2: Spatio-temporal plot on the motion of a droplet of BZ medium withthe snapshots each 1.0 s. Each image on the dotted line is aligned with time toshow the translational motion of the droplet. The change in the shape of thedroplet was not observed [1].

θx

0.5 mmπ
0.1A
displacement

B

0.05

0

0

5

time (s)10π/2angle θ(rad)BAdisplacement (mm)

(b)

(a)

0

Figure 4.3: Experimental results regarding the convection inside a droplet of BZ
medium. (a) Two measured particles inside the droplet are shown as A and B.
(b) Time traces of the angles of the particles from the x axis (the direction of the
motion). The time trace of translational motion of the droplet is also shown [2].

Chapter 4

Spontaneous motion of a BZ dropleti)oxidized state(blue)

35

ii)

flow

motionreduced state
(red)

iv)

iii)

motionflow

Figure 4.4: Schematic illustration on the mechanism of the spontaneous motionof a BZ droplet [2].

state, it moved back in the negative direction as shown in Figs. 4.2 and 4.3(b).
To observe the characteristics of the convection inside the droplet, we measured the positions of the particles located near the interface of the droplet.
Figure 4.3(b) shows the time traces of the position of the center of the droplet
and the angles between the positions of the particles and x direction. The origin was set at the center of the droplet. As the droplet moved in the positive
direction, convective ﬂow was observed in the counterclockwise direction on theupper semicircle. In contrast, after the entire droplet reached the oxidized state,convective ﬂow was induced in the opposite direction, as schematically shown in
Fig. 4.4.

4.4

Discussion

We consider only the concentration of the catalyst near the surface. The BZ
reaction medium in the oxidized state has a higher interfacial tension than thatin the reduced state, since the distribution of the catalyst near the surface is
diﬀerent [53].
First, as the chemical wave touches the interface of the droplet, convective
ﬂow is induced by the diﬀerence in interfacial tension between the oxidized andreduced states. This convective ﬂow causes motion in the positive direction, since36

Chapter 4

Spontaneous motion of a BZ droplet

there is interaction between the aqueous and organic phases at the interface.
They exchange momentum, which results in the total translational motion [71].
Due to the convective ﬂow, the oxidized catalyst at the interface accumulates
in the region where the chemical wave propagates last. As the chemical wave
ﬁnishes propagation and the entire droplet turns to the oxidized state, the concentration gradient of the oxidized catalyst near the interface causes backward
convection. The interfacial tension at regions with less catalyst is higher thanthat in regions with more catalyst. Thus, the direction of the convective ﬂow
is reversed, which results in the back-and-forth spontaneous motion of the BZ
droplet. It is noted that we can neglect the return to the reduced state, since
the time scale of the spontaneous motion is much shorter than the period ofchemical oscillation.
By the way, the motive force of the droplet is the gradient of interfacial
tension, which is proportional to the length of where the chemical wave touches
the interface between the BZ medium and oleic acid. Hence, the interfacial
tension is proportional to r, the system size:|Fi | ∝ γr.

(4.1)

On the other hand, the mass, M, is proportional to the cube of the system size:
M = ρr3 .

(4.2)

The acceleration α is, thus, proportional to r −2 ,|Fi | = Mα,

(4.3)

γ 1
.
(4.4)
ρ r2
From the above discussion, the BZ droplet should move more as the size ofthe system gets smaller. However, the chemical reaction is hard to activate,
when the system is smaller. Moreover, the time lag from when chemical wave
touches one side to when it touches the other side becomes shorter, so the droplet
moves less. Considering these factors, a system size of ∼ 1 mm is suitable forobserving the spontaneous motion of a BZ droplet.
α∝

4.5

Deformation of a droplet coupled with BZ
reaction

When the BZ droplet sinks down and touches the bottom of the petri dish, itcannot move translationally, but rather becomes deformed. The experimental
results together with the schematic representation are shown in Fig. 4.5. In thissystem, the droplet is stuck to the bottom and chemical energy is transduced
into deformation of the droplet rather than translational motion. In this phenomena, the detailed mechanism has not yet been solved. Both experimental
and theoretical studies are needed to understand this phenomenon.

Chapter 4

Spontaneous motion of a BZ droplet

(a)

37

BZ medium (Aluminum powder at the interface)(1.0 µl)
oleic acidobservation(b)(every 3.0 s)

reduced state

1 mm

oxidized state

Figure 4.5: Spontaneous deformation of a droplet of BZ medium. (a) Experimental setup. The composition of the BZ medium was the same as that in
the above-mentioned experiment on a ﬂoating BZ droplet. We placed a small
amount of BZ medium (1.0 ml) at the bottom of the oleic acid phase, andobserved with an inverted microscope (Nikon DIAPHOT-TMD). Since the BZ
medium was in an oscillatory state, a chemical wave was initiated spontaneously.
(b) Schematic representation on the experimental results from the side and below. The arrows in the ﬁgures correspond to the motion of the interface observedby placing aluminum powder at the interface of the BZ droplet. The chemical
wave propagated from the center, and motion of the interface could be seen
toward the wave front. After the entire droplet reached the reduced state, deformation of the droplet was observed. Periodic change in shape was, thus,
generated [4].

Chapter 5
Convective motion inwater-camphor system
In this chapter, the spontaneous motion of a camphor disk is ﬁrstly
introduced. Then, convective ﬂow induced by the surface tension
gradient due to the camphor layer is investigated both experimentally and theoretically. Some systems are also exempliﬁed where thespontaneous motion of a camphor disk is aﬀected by the convective
ﬂow.

5.1

Introduction

In the 19th century, Rayleigh reported that camphor grains ﬂoating on water
exhibit irregular motion called a “camphor dance” [72]. When a disk of camphor
is placed on water, it moves in one direction almost at a constant velocity. Such
motion is induced by the gradient in the surface tension [73]. Since the camphor
disk is almost circular and is placed on the water gently, the system can beconsidered isotropic. The camphor molecule layer formed around the camphor
grain, which is axially symmetric. Due to the dissolution from the camphor
grain and sublimation from the camphor layer, the concentration of camphor
on the surface has a gradient. This gradient makes the resting state unstable,
and the disk begins to move at a constant velocity in a certain direction, thus
breaking axial symmetry [74].
When anisotropy is introduced to the shape of a camphor grain, its motion
can be easily controlled. For example, a camphor grain with the shape of a“hat (ˆ)”, as shown in Fig. 5.1 (b)-ii, exhibits unidirectional motion, while one
with the shape of a “comma (,)” as shown in Fig. 5.1 (b)-iii or iv, exhibits arotational motion [73].
This translational and rotational motion is described for the position of thecenter of gravity, r, and the rotational angle θ of the camphor grain:
m

d2 r
dr
= −kt + F,
2
dt
dt
38

(5.1)

Chapter 5(a)Convective motion in water-camphor system(b)observation

i)

ii)

iii)

iv)

39

water camphor grain

Figure 5.1: Spontaneous motion of a camphor grain on water. (a) Rough sketchof the experimental setup. (b) i) A circular camphor disk on the aqueous phase
does not stand still, but rather spontaneously moves in one direction. ii) A
camphor grain shaped like a “hat (ˆ)” exhibits translational motion at a constant velocity. iii and iv) A camphor grain shaped like a “comma (,)” exhibits
rotational motion [4].

d2 θ
dθ
(5.2)
= −kr + N,2dt
dt
where m is the mass, I is the moment of inertia of the camphor grain, andkt and kr are the viscosity resistance for translational and rotational motion,
respectively. F and N are the total force and torque working on the camphor
grain due to the surface tension gradient:
I



F=


N=

γndl,

(5.3)

γ (l − r) × ndl,

(5.4)

where the contour integral is executed along the boundary of the camphor grain,
γ is the surface tension, n is the normal vector, and (l − r) is the vector from thecenter of mass. The surface tension is dependent on the density of the camphor
layer, which can be approximately calculated as follows:γ(c) =

γ0.
c + c̃

(5.5)

The dynamics of the surface concentration of camphor, c, is described as areaction-diﬀusion equation:∂c= −kc + D∇2 c + G (r, ω) ,
∂t

(5.6)

where the terms, −kc and D∇2 c, correspond to the sublimation and surface
diﬀusion of the camphor layer, respectively. G (r, ω) is the term of development

40

Chapter 5

Convective motion in water-camphor system

(a)

time

(b)

Figure 5.2: Typical numerical results on the spontaneous motion of a camphor
grain shaped like (a) a “hat (ˆ)” and (b) a “comma (,)” based on eqs. (5.1) (5.7). The parameters, m and I, are calculated from the shape of the grain byletting the density as 0.1. The other parameters are kt = 103 , kr = 0.1, D = 10,
k = 0.01, α = 10, c0 = 1, γ0 = 1, and γ̃ = 1 [4].

of the camphor layer from the grain:G (r, ω) =α(c0 − c) if the point is inside the grain.
0
otherwise(5.7)

Numerical results based on the above equations are shown in Fig. 5.2.
When the shape of the grain is isotropic, N is cancelled and the grain exhibits
only translational motion. In contrast, when it is not isotropic, N has a ﬁnite
value, and it exhibits rotational motion as well as translational motion.
In the above discussion, convective ﬂow is not concerned; however, we canobserve strong convective ﬂow in the bulk phase. Therefore, in this chapter, we
will discuss the convective ﬂow due to the surface tension diﬀerence induced by
the development of camphor molecular layer from the disk. We also report on
the inﬂuence of the convection on the spontaneous motion of a camphor disk inthe last section of this chapter.

5.2

Experiments

Camphor was obtained from Wako Chemicals (Kyoto, Japan). Water was ﬁrst
distilled and then puriﬁed with a Millipore-Q system. A camphor disk (diameter: 3 mm, thickness: 1 mm) was prepared using a pellet die set for FTIR. 3 ml

Chapter 5

Convective motion in water-camphor system

41

Pt wirecamphor disk

Water with small particles

Figure 5.3: Schematic illustration on the experiments on the convective ﬂow
induced by the camphor disk [3].

of water was poured into a quasi-one-dimensional polystyrene chamber (innerlength: 90 mm, width: 5 mm, depth: 9 mm, water level: 7 mm). The temperature of the water chamber was adjusted to 293 ± 1 K with a thermoplate(TP-80, AS ONE Co. Ltd., Japan). Plastic beads (DIAION, HP20S, relative
density: 1.01, Mitsubishi Chemical Co. Ltd., Japan) were mixed in the aqueous
phase to visualize the convective ﬂow around the camphor disk. As shown in
Fig. 5.3, the camphor disk was ﬁxed with a platinum wire and the motion of
the plastic beads were observed from side with a digital video camera and thenanalyzed with an image-processing system.

5.3

Results

Figure 5.4 shows a streamline of the convective ﬂow observed by the dispersion ofplastic beads in the aqueous phases with diﬀerent viscosities. In this experiment,
the camphor disk was ﬁxed with a platinum wire at the air/aqueous interface.
The streamlines for 1/3 s are shown in the both ﬁgures. With an increase inviscosity, the velocity of convective ﬂow clearly decreased.

5.4

Numerical calculation

In order to discuss on the convective ﬂow in the water-camphor system, we madenumerical calculations using the following equations:

∂+ v · ∇ v = η∇2 v − ∇p + Fi ,
ρ
∂t

(5.8)

∇ · v = 0,

(5.9)

42

Chapter 5

Convective motion in water-camphor system

Fixed Camphor
(a)
Water
level

Fixed camphor disk

water
Convective flow

(b)
Water
level

3 mm

Figure 5.4: Trajectories of the convective ﬂow (analyzed time: 1/3 s), as visualized with plastic beads (diameter: 100-200 µm) in a side view when a camphor
disk was placed on (a) the surface of an aqueous phase with (a) 5 mol l−1 and(b) 10 mol l−1 glycerin. The viscosity is larger in the solution of (b) than thatof (a). The nature of convective ﬂow on the left side of the camphor disk (datanot shown) was similar to that on the right side [3].

Chapter 5

Convective motion in water-camphor system

43

(a)
c
1
0.5
0
0
5
10
y
-40

-20

0

20

40

x

(b)

flow velocity

v
0.1
0
5
0

10
-40

-20

0

20

40

y
x

Figure 5.5: Results of the numerical calculation. It is assumed that a camphor
disk is ﬁxed at the origin, i.e., xc = 0. (a) The surface concentration of thecamphor layer and the ﬂow velocity at representative points are shown. (b) To
clarify the magnitude of the ﬂow velocity, the absolute value of the ﬂow velocity
is shown. The parameters are ρ = 1, η = 1, D = 0.5, α = 0.1, c̃ = 1, c0 = 1,
γ0 = 1, δ = 0.5, r = 1, and b = 1 [3].

44

Chapter 5Convective motion in water-camphor system∂+ v · ∇ c = −αc + H(x, c) + D∇2 c,∂t(5.10)

where c is the surface concentration of the camphor, α is the rate constant of
sublimation, D is the surface diﬀusion constant of camphor, and H(x, c) is theterm of the supply of camphor from a grain. Letting that the position of thecamphor grain is x = xc and that the size is r, we can writeH(x, c) =b(c0 − c) (|x − xc | < r),0(|x − xc | ≥ r),(5.11)

where b is the rate constant of the supply of the camphor, and c0 is the saturated
surface concentration. The interfacial tension is


Fi =

∂γ
e
∂x x

0

(0 < y < δ),(y > δ),(5.12)

γ0,(5.13)
c + c̃
where γ0 /c̃ denotes the interfacial tension of the pure water surface without any
camphor.
In Fig. 5.5, the numerical results are shown, when a camphor grain is ﬁxed.
The numerical result qualitatively reproduces the experimental one, and we suggest the convective ﬂow may aﬀect the spontaneous motion of a camphor grain
on water. When changing the viscosity in the numerical calculation, the velocity of the convective ﬂow is decreased, which also reproduced the experimental
results (data not shown).
γ=

5.5Eﬀects of convective ﬂow on the spontaneous motion

Figure 5.6 shows the camphor motion on water in a chamber with a saw-toothshaped base. A chamber with a saw-tooth-shaped base made of polycarbonate
material was prepared by a fused deposition modeling process using TITAN(Stratasys Corporation) at U-tec Corporation. The camphor moved over threeteeth under the present conditions. The maximum velocity in the leftward motion (∼ 60 mm s−1 ) was greater than that in rightward motion (50 mm s−1 ).
The velocity decreased when the camphor disk passed over the front edge of thesaw tooth while moving to the right, as marked by leftward arrows. In contrast,
the velocity increased when the camphor disk passed over the edge while moving
to the left, as marked by rightward arrows.
Figure 5.7 shows the results of a numerical calculation under a saw-toothshaped boundary condition. Leftward ﬂow is stronger than rightward ﬂow near
the camphor disk (x = 0). Due to this stronger leftward ﬂow, leftward motion

Chapter 5

Convective motion in water-camphor system

45

Side View
Velocity (mm s-1)
50
0
-50

Top View

12Time (s)

14

16

18

50 mm

Figure 5.6: Results of the experiments using the chamber with a saw-toothshaped base. Right: Snapshots of camphor motion with a time interval of 0.2 s(top view) in the chamber shown schematically (side view). Left: Time variationof the velocity of a camphor disk. The analyzed data correspond to those in
the snapshots on the right. The positive (gray areas) and negative velocities
correspond to the motion to the right and left in the chamber. The rightward
and leftward arrows denote when the camphor disk passed over the front edgeof a saw-tooth blade from the right to the left and vice versa, respectively [3].

46

Chapter 5

Convective motion in water-camphor system

(a)
c
1
0.5
0
0
5
10
y
-40

-20

0

20

40

x

flow velocity

(b)
v
0.1
0
5
0
-40

10
-20

0

20

40

x

y

Figure 5.7: The numerical results under a saw-tooth-shaped boundary condition. (a) The surface concentration of the camphor layer and the ﬂow velocity
at the representative points are shown. (b) To clarify the magnitude of theﬂow velocity, the absolute value of the ﬂow velocity is shown. Leftward ﬂow
is stronger than rightward ﬂow near the camphor disk. Due to this stronger
leftward ﬂow, leftward motion of the camphor disk is accelerated and rightward
motion is decelerated near the edges of the saw-tooth blades. The experimental
trends shown in Fig. 5.6 can be explained by this eﬀect of the asymmetry in theboundary condition. The parameters are the same as those used in Fig. 5.5 [3].

Chapter 5

Convective motion in water-camphor system

47

(a)
1

2

3

4

5

4

5

30 mm
(b)
1

2

3

100 mmFigure 5.8: Snapshots of the spontaneous motion of a circular ring and an encircled camphor fragment under two diﬀerent initial conditions. (a) Counterclockwise rotation of the circular ring (ring α) and clockwise rotation of the camphor
fragment when the camphor fragment was placed onto the water surface before
the ring (ring α). The white line denotes the trajectory of a point marked on
the circular ring. The location of the point in 1 is indicated by an arrow in eachsnapshot. The trajectory of the camphor motion from the previous snapshot isshown by a gray line. (b) Intermittent motion of the circular ring with a camphor fragment when the circular ring was placed onto the water surface before
the camphor fragment. The time intervals between snapshots were (a) 3/2 s,and (b) 3 s in 1-2 and 1/3 s in 2-5 [8].

of the camphor disk is accelerated and the rightward motion is decelerated near
the edges of the saw-tooth blades. The experimental results shown in Fig. 5.6
can be explained by this inﬂuence of the asymmetric boundary condition.
Next, the motion aﬀected by the convection in two-dimensional system isconsidered [8].
Three mobile rings (circular ring: ring α, ratchet-shaped ring: ring β, mirror
image of ring β: ring γ) were prepared with a plastic plate (thickness: 0.1 mm).
A camphor grain with the shape of comma (,) and a mobile ring were put onto
the water surface and the motion of the camphor grain was observed.
Figure 5.8 shows the snapshots of the synchronized spontaneous motion of acircular mobile ring (ring α) and a camphor fragment depending on the initialconditions. In Fig. 5.8(a), the camphor fragment moved clockwise at 2π/3 rads−1 on the water surface as an initial condition, and then the circular ring α
was placed on the water surface to surround the camphor fragment. Under

48
(a)
1

Chapter 523Convective motion in water-camphor system

4

5

6

5

6

30 mm
(b)
1

2

3

4

30 mm

Figure 5.9: Snapshots of the spontaneous motion of an asymmetric ring and an
encircled camphor fragment (time interval: 2.0 s). The shapes of the ring in (a)(ring β) and (b) (ring γ) are mirror images of each other when they are placedon the water surface. To visualize the motion of the circular ring, a point wasmarked on the ring [8].

this condition, the camphor fragment continued to move clockwise at 2π/5 rads−1 and the circular ring moved counterclockwise at −π/30 rad s−1 . In Fig.
5.8(b), the circular ring α was placed on the water surface as another initialcondition, and the camphor fragment was placed on inside the ring settling on
the water surface. Under this condition, the camphor fragment settled after
slow movement for 10 min without rotation. The circular ring then maintainedintermittent motion together with the camphor fragment for 30 min, i.e., settlingof both the ring and the camphor fragment and rapid motion of them wererepeated at an interval of ∼ 30 s.
Figure 5.9 shows the snapshots of the synchronized motion between an asymmetric ring and a camphor fragment. In these experiments, a camphor fragment
was placed and then the ring was introduced to surround the fragment. In Fig.
5.9(a), the ring rotated in the direction opposite the rotation of the camphor
fragment. In contrast, the camphor fragment continued to rotate while the ringalmost did not move under the experimental condition in Fig. 5.9(b).
Figure 5.10(a) shows the trajectories of a camphor fragment, a circular ring,and two talc powders for visualization on a water surface. In this experiment,
the initial conditions are the same as those in Fig. 5.8(a). Convective ﬂow wasobserved along the inner side of the ring in the same rotational direction as thatof the ring. Figure 5.10(b) shows the trajectories of the talc powders during
the resting state of the circular ring and the camphor fragment when the initialconditions are the same as those in Fig. 5.8(b). Although the camphor fragment

Chapter 5

Convective motion in water-camphor system

(a)

49

(b)

10 mm

30 mm

CamphorFigure 5.10: (a)Trajectories of the camphor fragment, ring, and two talc powders on the water surface. In this experiment, the initial conditions are thesame as those in Fig. 5.8(a), and talc powders (diameter: 3-7 µm) were homogeneously added to the water surface to visualize the convective ﬂow driven
by the concentration diﬀerence of the camphor layer that developed from thesolid fragment. Convective ﬂow was observed along the inner side of the ringin the same rotational direction as that of the circular ring (time interval: 1/15s). (b)Trajectories of the movements of talc powders to visualize the convective
ﬂow generated around a camphor fragment in the resting state of the circularring and the camphor fragment on the water surface for a duration of 25 s. The
camphor grain exhibits intermittent motion as in Fig. 5.8(b) [8].

50

Chapter 5

Convective motion in water-camphor system

was settled in the circular ring, convective ﬂow circulated from the location of
the camphor fragment.
Figures 5.8(a) and 5.10(a) suggest that convective ﬂow, where the direction is
opposite that of camphor rotation, is maintained around the camphor fragment
even after the circular ring is placed, and therefore the convective ﬂow drives thecircular ring. By adding the ratchet to the circular ring, the eﬀects of the ﬂow
can increase or decrease depending on the direction of the ratchet as shown in
Fig. 5.9. In contrast, Figs. 5.8(b) and 5.10(b) suggest that the surface tension
around the camphor fragment reaches a balanced state because of the narrowsurface area within the circular ring and therefore the camphor fragment is at
rest immediately after the camphor fragment is placed inside the ﬂoating ring.
Next, convective ﬂow increases since the local settling of the camphor fragment
increases the concentration gradient of the camphor layer around the camphor
fragment, while circulating ﬂow around the camphor fragment may maintain thissettling, as seen in Fig. 5.10(b). When the concentration of the camphor layer
reaches a critical concentration, the excess camphor layer develops locally outof the circular ring and the balanced state is destroyed. Therefore, the circularring may move rapidly in a certain direction.
These spontaneous motions of a camphor grain and an encircled ring areclearly aﬀected by the convection induced by the camphor grain. So far, wecannot reproduce these phenomena by numerical calculation, because the system
is quite complicated, however, by reducing the eﬀects of the convection, I hopethat we can clearly describe the eﬀects of the convective ﬂow.

Chapter 6
Spontaneous motion of a droplet
in the other systems
In this chapter, three examples where chemical energy is transduced
into mechanical energy, or motion [4]. The ﬁrst example is the spontaneous motion of an alcohol droplet on the aqueous phase [10], thesecond one is the spontaneous motion of an oil droplet on a glass
substrate in the surfactant solution [11], and the last one is the spontaneous motion of a phenanthroline disk on the aqueous phase [9].
These motions can be explained as the energy transduction fromchemical energy into mechanical energy through interfacial tension
in reaction-diﬀusion systems. Although the convective ﬂow is not
explicitly considered in this chapter, the spontaneous motion canbe regarded as convective ﬂow in a wide sense. Therefore, they are
introduced in this chapter.

6.1

Spontaneous motion in an alcohol-water system

It is well known that alcohol has much lower surface tension than water, andthat mixtures of alcohol and water show many interesting phenomena, such as“tears of wine” [75, 76]. When alcohol with a rather long hydrocarbon chain isused, the oscillation of the surface tension is reported both experimentally andtheoretically [77–79].
A droplet of alcohol with a long hydrocarbon chain can be formed on water
for a rather long time. We adopted the pentanol as alcohol, and noticed theformation of the droplet on aqueous phase. In this section, the spontaneous
motion of such a droplet is presented.
We added pentanol to water in a petri dish and observed the phenomena
inside it as shown in Fig. 6.1(a). Initially, the pentanol immediately mixes withwater. After a near-saturating amount of pentanol is added, the pentanol does51

52

Chapter 6

Spontaneous motion of a droplet in the other systems

(a)

(b)

i) small droplet

observationii) large droplet
water

pentanol dropletiii) larger droplet

Figure 6.1: Spontaneous motion of a pentanol droplet on water. (a) Roughsketch of the experimental setup. Images are taken from above. (b) i) A small
droplet maintains a circular shape and moves irregularly. ii) A larger droplet
deforms into a “hat (ˆ)” and moves in one direction. iii) A much larger droplet
divides into smaller droplets and they move around [4, 10].

not mix any further with the solution, and a droplet is formed. The droplet does
not stand still but rather moves around, as shown schematically in Fig. 6.1 (b).
The feature of this spontaneous motion of a droplet is depending on the sizeof the droplet. A small droplet maintains a circular shape and moves irregularly.
A larger droplet deforms like a “hat (ˆ)” shape and moves in one direction. A
much larger droplet divides into smaller droplets and moves around as shown
schematically in Fig. 6.1 (b). The experimental results are exhibited in Fig.
6.2.
This mode change in spontaneous motion depending on the size of the droplet
can be understood by considering the instability for the perturbation. Here,the straight interface with a small perturbation is assumed and the following
two factors are considered. In order to simplify the treatment, we adopt theapproximation that the shape of the wave consists of parts of a circle as shown
schematically in Fig. 6.3.
The ﬁrst factor is the process of the minimizing the alcohol-water interface.
Here, the strength of the force per unit length is let as γ1 . In this case, theradius of the circle, R, and the length of the interface per wave length, L, shownin Fig. 6.3, are derived as
A2 + (π/2k)2
π2
R=
,
≈
2A
8Ak 2

(6.1)

Chapter 6

Spontaneous motion of a droplet in the other systems

(a)

53

(b)
8s

6s
4s
1s

5s
2s
0s

4s

3s

2s
0s
50 mm

50 mm

Figure 6.2: Spontaneous motion of an alcohol droplet on water-alcohol mixture.
(a) When a droplet is smaller, it moves irregularly. (b) When a droplet is larger,
it deforms and moves in one direction. The pentanol was mixed with ink inorder to visualize the droplet [10].
π/2k
L = 4R arcsin
R







2π
8A2 k 2
≈
1+
.
k
3π 2

(6.2)

Supposing that γ1 is proportional to ∂ [L/ (2π/k)] /∂A,∂
γ1 = a
∂A



L
2π/k



= aAk2 ,(6.3)

where a is a constant.
The second factor is the concentration gradient of the pentanol at the surface depending on the curvature of the water-alcohol interface. The pentanol
dissolves from the droplet to the surface of the aqueous phase, and then it issublimated to the air or dissolved to the aqueous phase. These processes can be
written as follows:∂c= D∇2 c − α (c − ca ) − β (c − cw ) ,
∂t

(6.4)

where c is the surface concentration of the pentanol, D is the diﬀusion coeﬃcient of pentanol on the surface, α is the evaporation rate of pentanol, β is thedissolution rate of pentanol, ca is the concentration at the gas-liquid equilibrium and cw is the concentration at the surface-bulk equilibrium in the aqueous
phase. By assuming the steady state with the Dirichlet boundary conditionat the pentanol-water interface, i.e., c = c0 at the interface, the concentration
gradient can be calculated using modiﬁed Bessel functions for a certain curvature. It is noted that the dissolution process of the pentanol from the droplet
corresponds to the adopted boundary condition in the steady state. In the convex region, the concentration gradient is larger than that in the concave region,

54

Chapter 6

Spontaneous motion of a droplet in the other systems

pentanol-water mixture
2π/k

R

L
interfacepentanolFigure 6.3: Schematic illustration of the tractable model. The perturbationfrom the straight interface is considered. The sine wave with the wave number
k is approximated as the combination of the parts of circles [10].

c
c0

R

dc
dr conv

pentanolphase
aqueous phasedc
dr convrRcdc
dr concc0

Raqueousphasepentanol phaserdc
dr concr
Rr

Figure 6.4: Schematic representation on the interfacial tension diﬀerence. As
the interface is assumed to be circular with the radius of R, the concentration
proﬁle is (c0 − Γ) K0 ( (α + β) /Dr)/K0(R̃) at the convex region (upper), and(c0 − Γ) I0 ( (α + β) /Dr)/I0 (R̃) at the concave region (lower), where K and I
are modiﬁed Bessel functions. The interfacial tension is nearly proportional to
the gradient of the concentration at the interface [10].

Chapter 6

Spontaneous motion of a droplet in the other systems

55

which means the perturbation of the interface tends to grow as schematicallyshown in Fig. 6.4. As the force strength per unit length is γ2 , it is described as
γ2 = b dc  dr 

conv


 
  dc 
 

−

  dr 

conc







,

(6.5)

where dc/dr|conv is the concentration gradient perpendicular to the interface inthe convex region, dc/dr|conc is that in the concave region, and b is a positive
constant. By the analytic calculation, the following representation is obtained:

γ2 = b (c0 − Γ) ⎛where Γ = (αca + βcw ) / (α + β), R̃ =modiﬁed Bessel functions.
Here, it is assumed that 0 < kA = δ(6.6)(α + β) /DR and I0 , I1 , K0 , K1 are
1. When kγ2 − γ1 ≈ −aAk2 +and when k

 ⎞

I1 R̃
α + β ⎝ K1 R̃ − ⎠ ,
D
K0 R̃
I0 R̃

1,

8bΓAk2
,
π2

(6.7)

1,
γ2 − γ1 ≈ −aAk2 +

8bΓAk2

π 2 ln 8Ak 2 / π 2 (α + β) /D

1, and γ2 − γ1 > 0 when kThis means γ2 − γ1 < 0 when k
γ2 − γ1 = 0 at k = kc , kc is obtained as follows:
π2
kc =
8δ



α+β
3b (c0 − Γ)
.
exp
D
2a

 .

(6.8)
1. By setting(6.9)

When the droplet size is smaller than 2π/kc , the droplet is stable and does
not deform. On the other hand, when it is larger than 2π/kc , the ﬂuctuation ofa certain mode grows and the droplet can separate into several smaller droplets.
Just when the droplet size is around 2π/kc , only one mode is destabilized. Insuch a case, a droplet is deformed into the shape of a “hat(ˆ)” and moves in one
direction.
The mode change in spontaneous motion depending on the size of the droplet
can be understood based on the reaction-diﬀusion equations. The results canpredict the experimental trends when the temperature or concentration of theaqueous phase is changed. In fact, such experiments have been performed, andthe theoretical prediction has been conﬁrmed, though the experimental data are
not shown in this thesis.
Thus, in this framework, the convective ﬂow is not considered, but the strong
convective ﬂow is conﬁrmed in the experiments. By considering the convective
ﬂow, various modes of spontaneous motion may be achieved.

56

Chapter 6

Spontaneous motion of a droplet in the other systems(a)

Oil droplet 35 µl
15 mm
Aqueous phase
Glass substrate
40 mm

3.4 mm10 mm

1/3 s per frame(b)Figure 6.5: Spontaneous motion of an oil droplet on a glass substrate. (a)Schematic illustration on the experimental setup. (b) Snapshots of an oil droplet
from side every 1/3 s. The droplet moves spontaneously almost at a constantvelocity in one direction [4, 84].

6.2

Spontaneous motion in an oil-water system

There have been some reports on reactive droplets on a glass plate [80, 81].
However, such reactive droplets cannot cross their own trails; i.e., they show aself-avoiding trajectory. On the other hand, as a droplet of nitrobenzene including iodine (I2 ) and saturated potassium iodide (KI) is put onto a glass substrate
in an aqueous solution of surfactant such as stearyltrimethylammonium chloride(STAC) solution, the oil droplet exhibits ameba-like motion [82, 83]. When theoil droplet is put onto the long rectangular glass substrate, the droplet goes backand forth as shown in Fig. 6.5 [84].
We have also experimentally observed that the chemo-sensitive motion ofan oil droplet on a glass substrate treated by acid. It was found that an oil
droplet can show turning back motion, stopping motion, and slowing motion
when the substrate was partially treated with acid as shown in Fig. 6.6. Withdetailed observations, however, it turned out that the motion of an oil droplet
severely depends on the condition of the substrate, which was hard to control

Chapter 6

Spontaneous motion of a droplet in the other systems

57

experimentally.
Here, the mechanism of the droplet motion is considered. The oil droplet
moves dissolving adhered STA+ ions on the glass substrate. Since STA+ and I−
3
in the organic phase make an ion-pair, a diﬀerence in interfacial energy arises
from the diﬀerence in the concentration of STA+ ions on the glass surface between the front and back of an oil droplet. Thus, the condition of the glass
surface on which STA+ ions aggregate plays an important role in the motion ofan oil droplet. The stopping and slowing of an oil droplet can be attributed tothe eﬀect of inertia, which moves the oil droplet to the area where there are notenough STA+ ions on the glass surface to produce a driving force. Furthermore,
the dependence of the droplet motion on the condition of the acid-treated region
can be attributed to the diﬀerence in the number of adsorption sites on the glass
surface.
Based on the above discussion, we make a simple one-dimensional model.
The x axis is set along the surface of the glass substrate. The droplet proceeds
on a line with stretching and shrinking. The positions at the both ends of thedroplet are set as x1 (t) and x2 (t), where x1 < x2 . The concentration of STA+
ions on the glass surface is set as u(x, t). The dynamics of x1 and x2 are written
as follows:


∂E 
d2 x1
dx1

−
ρ 2 = −µ0
− β(r0 )(x1 (t) − x2 (t) − r0 ),dt
dt∂x x=x1(6.10)



∂E 
dx2
d2 x2

−
ρ 2 = −µ0
− β(r0 )(x2 (t) − x1 (t) + r0 ),dt
dt∂x x=x2(6.11)

where ρ, µ0 , and r correspond to the surface density, viscosity, and the diameterof the oil droplet, respectively. β(r) is the elastic constant of the oil droplet withthe diameter r0 in the equilibrium state [11]:β(r) =

2γw.
r(6.12)

E(u) is the surface energy of the glass surface modulated by the STA+ ionsadsorbed on it:e0
E(u) =,(6.13)
1 + aum
where e0 , a and m are the positive constants. On the other hand, the dynamics
of u is written as follows:∂2u∂u= d 2 + F (u, x, x1 (t), x2 (t); 0 ),∂t∂x

(6.14)

where d is the diﬀusion constant of the STA+ ions molecules on the glass surface. F (u, x, x1 (t), x2 (t); 0 ) corresponds to the desorption of the STA+ ions

58

Chapter 6

Spontaneous motion of a droplet in the other systems

(a)

1cm

1s
Acid

Acid

0.1 sec per frame
1cm

Acid

Acid

0.1 sec per frame

(b)

1s
(c)

1cm

1s
Acid

Acid

(d)

0.1 sec per frame
1cm

1s
Acid

Acid
0.1 sec per frame

Figure 6.6: Experiments on spontaneous droplet motions. A 15-µl oil droplet
was put onto a narrow acid-treated glass substrate. Images from the experimentand a spatio-temporal image of oil-droplet motion are shown. An oil droplet
showed (a) shuttling motion, (b) intermittent shuttling motion, (c) stopping,and (d) slowing [11].

Chapter 6

Spontaneous motion of a droplet in the other systems
u1

u0

0

l0

u1

u0

β (r0)

σ
x2

σ
x1

59

x
l-l0

l

Figure 6.7: Schematic diagram of the model of oil-droplet motion on a glass
surface. The lower arrow represents the spatial coordinate, where 0 to 0 and− 0 to are treated with acid. u0 and u1 are the saturated concentration of
the STA+ ions on the glass substrate on bare and acid-treated glass substrate,
respectively. The oil droplet is modeled as two lines (perpendicular to x-axis,
whose mass density is σ, connected to a spring with an elastic constant of β(r0 )[11].

from the glass surface to the oil droplet. For this term, we assume the following
description:⎧⎪⎨ −k1 u,(x2 (t) ≤ x ≤ x1 (t))
F (u, x, x1 (t), x2 (t); 0 ) = k2 (u0 − u), ( 0 < x < x2 (t), x1 (t) < x < − 0 )⎪⎩k3 (u1 − u), (0 < x ≤ 0 , − 0 ≤ x < )(6.15)
where k1 , k2 , u0 , u1 , and 0 are the positive constants that correspond to
the desorption rate, adsorption rate, saturated concentration of the STA+ ionson the bare glass surface, that on the acid-treated glass surface, the length of
the glass substrate, and the length of the glass substrate treated with acid,respectively (see Fig. 6.7). The initial and boundary conditions are set as

u(0, x) =∂u∂u(t, 0) =(t, ) = 0,∂x∂x(6.16)

u0 , ( 0 < x < − 0 )u1 , (0 < x ≤ 0 , − 0 ≤ x < )(6.17)

x1 (0) = x10 , x2 (0) = x20 (< x10 ),
dx2
dx1
(0) =
(0) = 0.
(6.18)dt
dt
To normalize eqs. (6.10) - (6.18), we introduce the following dimensionlessvariables:
u
x
x1
x2
U = , X1 = , X2 = , X = , T = k1 t.
(6.19)
u0
We then derive the following dimensionless equations from eqs. (6.10) - (6.18):


∂E 
d2 X1
dX1
−
=
−µ
− B(R0 )(X1 (T ) − X2 (T ) − R0 ),

dT 2
dT
∂X X=X1(6.20)

60

Chapter 6

Spontaneous motion of a droplet in the other systems


dX2
d2 X2
∂E 

=
−µ
− B(R0 )(X2 (T ) − X1 (T ) + R0 ),
−
dT 2
dT
∂X X=X2
E(U) =

E0,
1 + AU n(6.21)(6.22)∂2U∂U=D+ F (U, X, X1(T ), X2 (T ); L0 ),(6.23)∂T∂X 2⎧(X2 (T ) ≤ X ≤ X1 (T ))
⎪
⎪ −U,
⎪
⎨ K (U − U), (L < X < X (T ),2002
F (U, X, X1 (T ), X2 (T ); L0 ) =
⎪
X
(T
)
<
X
<
1 − L0 )⎪1⎪⎩
K3 (U1 − U), (0 < X ≤ L0 , 1 − L0 ≤ X < 1)
(6.24)
with the following initial and boundary conditions:



∂U
∂U
(T, 0) =(T, 1) = 0,∂X
∂X

(6.25)

1, (L0 < X < 1 − L0 )U1 , (0 < X ≤ L0 , 1 − L0 ≤ X < 1)

(6.26)u(0, x) =X1 (0) = X10 , X2 (0) = X20 (< X10 ),
dX1
dX2
(0) =
(0) = 0,
dT
dT
where

(6.27)µ0
e0
r0,, E0 =
,
R
=
0
σk1
σk12

µ=
2

W
2γw
β(r0 )=, W =, A = aun0 ,
2
σk1
R0
σk12

B(R0 ) =k2
k3
u1, K3 = , U1 = ,
k1
k1
k1
u0
x10
x20
0
, X20 =.
L0 = (< 1), X10 =

D=

d

2

, K2 =

We performed a numerical simulation using eqs. (6.20) - (6.24) under the
initial and boundary conditions shown in eqs. (6.25) - (6.27).
We only changed the parameter U1 , while all of the other parameters were
ﬁxed. It is noted that U1 corresponds to the saturated concentration at the region
treated by acid. Spatio-temporal plots of the droplet given by the numericalcalculations are shown in Fig. 6.8.
When U1 is small enough, i.e., it is diﬃcult for STA+ ions to adsorb to the
acid-treated region, the oil droplet moves back and forth inside the untreatedregion, as shown in Fig. 6.8(a). As U1 increases slightly, the oil droplet does
not turn back quickly but rather rests at the border between the acid-treatedregion and the untreated region. After a brief stop, it begins to move backward,
and this intermittent shuttling motion is repeated as shown in Fig. 6.8(b). For

Chapter 6(a)Spontaneous motion of a droplet in the other systems

(c)

0
200

t

200
400

400

600

t

600

800

800

1000
0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

(d)

0
200

t

0.6

0.8

1

0.6

0.8

1

x

x
(b)

61

0
40

t

400

80

600

120

800
0

0.2

0.4

0.6
x

0.8

1

0

0.2

0.4
x

Figure 6.8: Spatio-temporal plots of the results by the numerical calculations
using eqs. (6.20) - (6.27). Only the parameter U1 was changed as (a) U1 = 0.65,(b) U1 = 0.658, (c) U1 = 0.68 and (d) U1 = 0.75. The other parameters areµ = 0.2, R = 0.05, E0 = 5 × 10−5 , A = 1, m = 4, B(R) = 2, D = 2.5 × 10−7 ,L0 = 0.25, U0 = 1, K2 = 1, K3 = 1, X10 = 0.23, and X20 = 0.27. The thicklines along the x axis correspond to the regions treated with acid [11].

62

Chapter 6

Spontaneous motion of a droplet in the other systems

much larger U1 , the oil droplet does not go back any more, as shown in Figs.
6.8(c) and (d). The oil droplet goes into the acid-treated region and it moves
on very slowly or stops according to the value of U1 .
In this system, the diﬀerence in the chemical energy of the dissolution of thesurfactants between the water phase and oil phase is transduced into motion.
We hope that we can discuss the eﬃciency of the chemo-mechanical energy
transduction using the developed model.

6.3

Spontaneous motion coupled with a chemical reaction

1,10-phenanthroline (C12 H8 N2 ) is used for making the ferroin, which is often
used as a catalyst of BZ reaction. Putting phenanthroline and ferrous sulfate(FeSO4 ) into pure water, they make a complex at a ratio of 3:1. A phenanthroline
molecule has hydrophobic parts and hydrophilic parts, and it has the surface
activity like camphor. Thus, as a disk of phenanthroline is put onto the water,
it moves spontaneously. In this section, we report the spontaneous motion ofa phenanthroline disk on pure water or ferrous sulfate solution, and the modechange of the spontaneous motion according to the change in the concentrationof FeSO4 .
When the concentration of FeSO4 is low enough, the uniform motion of thegrain was observed. With the increase in the concentration of FeSO4 , the uniform motion changed to intermittent motion (periodic change between motion
and rest) and its period and resting time increased. However, this intermittent
motion reverted to uniform motion and the velocity increased with a further
increase in [FeSO4 ]. The mechanism of this characteristic motion depending on[FeSO4 ] is discussed in relation to the surface tension, which depends on theconcentration of a surface-active layer composed of 1,10-phenanthroline and atris-(1,10-phenanthroline) iron complex ([Fe(phen)3 ]2+ , ferroin) as the driving
force and the solubility of ferroin in the aqueous phase.
Figure 6.9 shows the snapshots for (a) uniform motion ([FeSO4 ]= 0 mol−1l ), (b) intermittent motion ([FeSO4 = 5 mmol l−1 ), and (c) uniform motion([FeSO4 = 100 mmol l−1 ) of the spontaneous motion of the phenanthroline disk
together with the time variation of its velocity. As for the uniform motion, a redcolored layer, which was composed of ferroin, was remained on the trajectoryof the motion. On the intermittent motion, the density of the red colored layer
around the disk increased at the resting state, but the resting state was changed
to the rapid motion when the density of the red colored layer reached to a criticalvalue, as seen in Fig. 6.9(b).
Figure 6.10 shows the mode diagram of the spontaneous motion of thephenanthroline disk depending on [FeSO4 ]. The velocity of uniform motion
decreased with the increase in [FeSO4 ], and changed to intermittent motion(periodic change between motion and rest) at [FeSO4 ] =3 mmol l−1 . Here, weChapter 6

Spontaneous motion of a droplet in the other systems

(a)
1

2

3

4

63

5 mm

time interval: 0.2 s
V (mm s-1)

10
5
0
10 s

(b)
1

2

5

3

6

4

5 mm

time interval in 1-4 : 4 s
87

V (mm s-1)

time interval in 4-8 : 0.1 s
40
20
0
10 s
(c)
1

2

3

5 mm 4V (mm s-1)

time interval: 0.2 s
10
5
0
10 s

Figure 6.9: Snapshots of the spontaneous motion of a phenanthroline disk on
the FeSO4 aqueous solution at diﬀerent concentrations ((a) 0, (b) 5, and (c)100 mmol l−1 ) (top view) together with the time variation of the velocity of thephenanthroline disk [9].

64Chapter 6

Spontaneous motion of a droplet in the other systems

unifrom
motionintermittent
motionambiguous
region

8
6
20

4

10

2
0

0

20

40

60
80
Concentration of FeSO4 (mmol l-1)

100

Period (s) ( )V (mm s-1) ( )300Figure 6.10: The mode of the spontaneous motion of a phenanthroline disk
depending on [FeSO4 ]. The velocity of uniform motion (0 ≤ [FeSO4 ] ≤ 2, 30 ≤[FeSO4 ] ≤ 100 mmol l−1 ) including the ambiguous region (11 ≤ [FeSO4 ] ≤ 20mmol l−1 ), and the frequency of intermittent motion (3 ≤ [FeSO4 ] ≤ 10 mmoll−1 ) are plotted [9].

determined as the ambiguous motion if the velocity of the motion was lower than
1 mm s−1 and the resting time of the intermittent motion was higher than 1 min.
The frequency of intermittent motion decreased and the resting time increased
with the further increase in [FeSO4 ], and the phenanthroline disk almost settled
at [FeSO4 ] = 10 mmol l−1 . However, the uniform motion reverted again over
[FeSO4 ] = 30 mmol l−1 , and the velocity increased with the further increase in[FeSO4 ].
The mechanism of this spontaneous motion can be described as a reactiondiﬀusion system with convection. The phenomenon of the spontaneous motionof a phenanthroline disk is interesting because the formation of the reaction
product and the convective ﬂow can be easily observed. We hope that thesystem will be designed where the spontaneous motion of a phenanthroline disk
and oscillation or spatio-temporal pattern formation in BZ reaction can interact
with each other, and create a novel spatio-temporal pattern.

Part III
Boundary Eﬀects on
Reaction-Diﬀusion Systems

65

Chapter 7
Slowing and stopping of achemical wave in a narrowing
canal
In Part III, the geometrical eﬀects of boundary in reaction-diﬀusion
systems are exempliﬁed. In this chapter, the slowing and stoppingof the chemical wave in the narrowing glass capillary is presented in
BZ reaction. The mechanism is theoretically analyzed related to thesurface-volume ratio.

7.1

Introduction

It has been claimed that the essence of the neuronal system can be understoodin terms of a simple diﬀerential equation as a kind of reaction-diﬀusion systems[31–33]. Thus, a neuronal pulse is a manifestation of a dissipative structure
generated in nonequilibrium open systems. Several experimental and theoreticalstudies have been undertaken on the reaction-diﬀusion systems as a model ofthe neuronal system. It is known that the FitzHugh-Nagumo equation can well
reproduce the features.
In such studies, the Belousov-Zhabotinsky (BZ) reaction is often used as anexperimental model of the reaction-diﬀusion systems. In this system, chemical
waves propagate at a constant velocity and amplitude, and thus exhibit thecharacteristics similar to those of pulses in nerve cells. The characteristics ofthe BZ reaction can be well described using a numerical model, the Oregonator.
The Oregonator model has a mathematical structure similar to the FitzHughNagumo equation. Therefore, to better understand information processing in
the neuronal system, it may be useful to compare it with the chemical waves in
the BZ reaction, although there exists a fundamental diﬀerence between them:a neuronal pulse propagates on the two-dimensional tubular membrane and achemical wave propagates in the three-dimensional bulk phase.
67

68

Chapter 7

Slowing and stopping of a chemical wave

CCD Camera
BZ solution
Chemical wave
Ag wire0.036
1Capillary0x (mm)x0 = 5.5

Figure 7.1: Schematic representation of the experimental system. The capillary
was arranged horizontally and observed from above by a CCD camera [5].

Over the past decade, the behavior of the chemical waves in a narrow space
has been studied using a gel system [34,35], a droplet system [36], and a bead system [37, 38]. Tóth and co-workers reported that a chemical wave does not propagate when it expands from a thin capillary to the bulk solution. They claimed
that a chemical wave can perform information processing [85, 86]. Masere et al.
performed experiments on the propagation of a chemical wave in a vertical capillary. They discussed their experimental results with special emphasis on thegravitational eﬀect [87]. In these previous studies, the features of the chemicalwave propagation in a narrow straight capillary with a constant diameter were
examined.
In this chapter, we show that a chemical wave slows, stops, and then disappears in a narrowing glass capillary. We discuss this phenomenon in term of thesurface-volume ratio for the reaction ﬁeld, i.e., the surface can seriously aﬀect
chemical wave propagation.

7.2

Experiments

All chemicals were analytical-grade reagents and used without further puriﬁcation. An aqueous solution of ferroin, or tris (1,10-phenanthroline) iron (II)sulfate, was prepared by mixing stoichiometric amounts of 1,10-phenanthroline(C12 H8 N2 ) and ferrous sulfate (FeSO4 ) in pure water. The water was puriﬁed with a Millipore-Q system. BZ medium contained 0.35 M sodium bromate(NaBrO3 ), 0.39 M sulfuric acid (H2 SO4 ), 0.12 M malonic acid (CH2 (COOH)2 ),0.05 M sodium bromide (NaBr), and 4.0 mM ferroin (Fe(phen)2+3 ) (excitablecondition). We left the BZ solution for about 10 min and then stirred it todiminish the formation of bubbles in the solution.
We prepared a glass capillary (borosilicated glass, length: 100 mm, inner
radius: 0.75 mm, Sutter Instrument Co.) with a decreasing width by pulling

Chapter 7

Slowing and stopping of a chemical wave

69

a straight capillary with a Micropipette Puller (Model P-97/IVF, Sutter Instrument Co.). The both ends of the capillary were open and the gradient of
the inner radius was 0.036. In order to make easier to show the experimental
results, the x axis is set as shown in Fig. 7.1. The capillary was immersedin the BZ medium, fulﬁlled with the BZ medium inside it, and then situatedon a horizontal plate for observation with a CCD camera (Digital Microscope
Unit, Keyence). As soon as bubbles formed inside the capillary, we stopped theobservation: the data reported in the present study are only from the experimental series without bubble formation. All the experiments were performed at
the room temperature (20 ± 3 ˚C). Images were recorded on a videotape andanalyzed by an image processing system.

7.3

Results

A chemical wave was initiated at the wider end by a silver wire without disturbance. A chemical wave propagating toward the narrower end slowed, stopped,and eventually disappeared. The oxidized solution in the capillary then reverted
to the reduced state. The spatio-temporal plot of images of the capillary along
the long axis is shown in Fig. 7.2(a), and the snapshots in every 5.0 s are shownin Fig. 7.2(b). They clearly show that the velocity of the chemical wave gradually decreases before it stops. This feature was seen in every chemical wave
initiated. With each subsequent wave, the position at which the chemical wave
stopped tended to shift slightly toward the narrower end. Figure 7.3 shows the
temporal change in the position (a) and velocity (b) of the chemical wave, andthe velocity depending on the position (c). From the observation, the point
where the chemical wave stops propagation, xs , was 2.7 ± 0.2 mm. We have
conﬁrmed the reproducibility of the experimental trends through a number of
repeated experiments.

7.4

Discussion

We discuss here the characteristic change in the manner of wave propagation ina glass capillary. According to an analysis with the Oregonator, the velocity of
the chemical wave in the BZ reaction can be written as [88]:v≈

1(k5 HAD)1/2 ,
2

(7.1)

where H is the concentration of hydrogen ion, A is the concentration of bromate
ion, D is the diﬀusion constant of the activator, and k5 is the rate constant of
the component reaction, which is introduced as an autocatalytic process by summation of the two elemental processes in the Oregonator model [88]. Equation(7.1) is valid for a plane and steady traveling wave but does not apply to a wavethat shows a change in velocity. By assuming that the concentration dependence

70

Chapter 7(a)Slowing and stopping of a chemical wave

position x (mm)
0
2
4

(b)

0

100

200

300
1 mm
time (s)1 mm(every 5.0 s)

Figure 7.2: Experimental results on the propagation failure in a narrowing glass
capillary. The chemical wave propagated from the left (wider end) to the right(narrower end). (a) Spatio-temporal plot of the chemical wave propagation. (b)Snapshots with an interval of 5.0 s. When the wave propagated to the narrower
region, it slowed, stopped, and then disappeared. The darker and brighter areas
correspond to the reduced and oxidized states, respectively [5].

Chapter 7

Slowing and stopping of a chemical wave(a)position x (mm)

3
2
1
0

(b)
velocity (mm s-1)

71

0.03
0.02
0.01

(c)

0

0

100

time (s)200300

velocity (mm s-1)

0.04
1st
2nd
3rd
4th
5th

0.03
0.02
0.01
00

1

2
3
position x (mm)4Figure 7.3: Features of the propagation failure of chemical waves in the experiments as shown in Fig. 7.2. (a) Temporal changes in the position and (b) velocityof chemical waves. (c) The relationship between position and velocity [5].

72

Chapter 7

Slowing and stopping of a chemical wave

0.5

∼
b
a (x0 − x)gradient : 450

1−

1st
2nd
3rd
4th
5th

0
0

0.001
v2 (mm2 s-2)

Figure 7.4: Plot of v 2 vs 1 − b̃/{a(x0 − x)}. Five chemical waves were measured
and the diﬀerent marks correspond to the diﬀerent waves. By linear ﬁtting, thegradient was calculated to be 450 mm−2 s2 [5].

of the local velocity along the capillary is also given in a similar way, we simply adapt eq.(7.1) to interpret the experimental trends in a semi-quantitative
manner.
The x axis is set in the direction of wave propagation. We describe the inner
radius of the glass capillary, r, by assuming it as a part of a cone:r = a(x0 − x),(7.2)

where a corresponds to the gradient of the inner radius of the glass capillary,and x0 is the position that would be the top of the cone.
The concentration of hydrogen ion, H, can be decreased by the eﬀect of theglass surface of the capillary. It is noted that H is the average concentration
over the cross section, and that the eﬀect on the hydrogen concentration by
the glass surface is eﬀective only near the surface. Amemiya et al. reported
that the eﬀective hydrogen ion concentration is not homogeneous in the porous
glass system [89]. The sodium ions connected to the SiO−
2 groups could be,thus, replaced by hydrogen ions, which causes a decrease in the concentration ofhydrogen ions near the glass surface. We assume that this eﬀect is proportional
to the surface-volume ratio:

2πr
Heﬀ = H 1 − b 2
πr







b̃
≡H 1−
,
r

(7.3)

where b is a positive constant that reﬂects the eﬀect of the glass surface. Although, at present, it seems diﬃcult to evaluate b from an experiment, as aChapter 7

Slowing and stopping of a chemical wave

73

velocity (mm s-1)

0.04

0.03

0.02

0.01
xs = 2.70102position x (mm)34Figure 7.5: Relationship between position and velocity of a chemical wave basedon the theoretical discussion by eq.(7.4). The parameters used here are determined from the experiments [5].

future subject, it may be of value to try to measure the change in the chemical environment near the surface. b̃ (= 2b) is a positive constant introduced tosimplify the equation.
From the above equations, we can derive the relationship between the position and velocity:v = v0b̃1−a(x0 − x)

1/2

,

(7.4)

where v0 is the velocity of the chemical wave without any eﬀects from the glass
surface. By letting v = 0 in eq.(7.4), we can determine the position, xs , where
the chemical wave stops:b̃(7.5)
xs = x0 − .
aEquation (7.4) can also be written as:


v
v0

2

=1−

b̃
.
a(x0 − x)(7.6)

From the experimental results, the parameters were determined to be xs = 2.7± 0.2 mm, x0 = 5.5 mm, and a = 0.036. Using these values, we plotted therelationship between v 2 and 1 − b̃/{a(x0 − x)} in Fig. 7.4, where b̃ is calculated
as b̃ = 0.20±0.01 using eq. (7.5). The results show a linear relationship between
these two values. By linear ﬁtting, the gradient was calculated to be 450 ± 50
mm−2 s2 . This value corresponds to 1/v02 , indicating that v0 = 0.047 ± 0.003
mm s−1 , which roughly corresponds to the wave velocity obtained through the74

Chapter 7

Slowing and stopping of a chemical wave

experiment using a wide capillary (0.06 ± 0.01 mm s−1 ). Using this value, theplot between v and x is shown in Fig. 7.5. The relationship between v and x in
the experiments, as shown in Fig. 7.3(c), was well reproduced in a quantitative
manner. The velocity proﬁle near the point where a chemical wave stops can bethus explained.
In the above discussion, the decrease in the eﬀective concentration of thehydrogen ion, Heﬀ , near the glass surface seems to be induced by the adsorption
by the SiO−
2 groups on the glass surface. With time, this substitution continues,
and the eﬀect of the glass surface on the concentration of the hydrogen ion
decreases. This may be the cause of the slight shift in the position at which thechemical wave stops.
As we performed the experiments under the oxygen atmosphere, it is possible that the oxygen disturbs chemical-wave propagation. On this problem, we
carried out the observation using a straight capillary with open ends, and conﬁrmed that absorption of the oxygen from an open end has negligible eﬀect on
the manner of wave propagation. Therefore, it is most probable to expect thatthe signiﬁcant factor is the absorption of the hydrogen ion to the inner wall of
the glass capillary.

Chapter 8
Alternative stopping of achemical wave in aphotosensitive BZ reaction ﬁeld
In Chapter 7, the chemical waves of BZ reaction in a narrowing glass
capillary was investigated, and it is suggested that the eﬀects of theglass surface is important. In this chapter, therefore, the narrowing
reaction ﬁeld is achieved using light illumination in photosensitive
BZ reaction in order to avoid the eﬀects of the glass. Then, it isobserved that the chemical wave disappears alternatively at a certain
light intensity.

8.1

Introduction

In the previous chapter, the natures of the chemical wave in the narrowing glass
capillary were investigated. When a chemical wave goes to the narrower area,
it propagates more and more slowly, and ﬁnally it stops and disappears. The
mechanism of the slowing and stopping of the chemical wave can be explained
as the eﬀects of the glass surface, however, the physical chemistry near the glass
surface is so diﬃcult that the characteristics of the reaction diﬀusion system are
hidden.
In this chapter, we made the narrowing reaction ﬁeld using photosensitive
BZ reaction. As written in Chapter 2, using Ru-catalyst, the BZ reaction has
photosensitivity [42–44]. The mechanism of photosensitivity is considered asfollows: Ruthenium catalyst absorbs light and is excited. When the excited
catalyst is relaxed, it produces the bromide ion, which is the inhibitor of the BZ
reaction. Therefore, in the region where light is illuminated, the inhibitor of BZ
reaction increases.
Using a slide projector and a ﬁlter with a certain shape, we can achieve thereaction ﬁeld with an arbitrary shape. In this chapter, chemical wave propaga75

76

Chapter 8

Alternative stopping of a chemical wave

Digital Video
CameraBlue Filter

Silicone oil
Membrane filter
Glass plate

Slide projector
Miirror(a)

OHP film with a shape(b)

Figure 8.1: Schematic representation of experimental setup.

tion on the V- and X-shaped reaction ﬁelds as shown in Fig. 8.1 are investigated.
On such reaction ﬁelds, the interesting phenomena on chemical wave propagation can be observed by changing the brightness of light irradiation. At a certain
brightness, the positions at which chemical waves stop propagation change every
two waves. In this chapter, the results of experiments and simple discussion arepresented.

8.2

Experiments

All chemicals were analytical-grade reagents and used without further puriﬁcation. The water was puriﬁed with a Millipore-Q system. The BZ medium contained 0.45 M sodium bromate (NaBrO3 ), 0.30 M sulfuric acid (H2 SO4 ), 0.20
M malonic acid (CH2 (COOH)2 ), 0.05 M sodium bromide (NaBr), and 1.7 mM
ruthenium bipyridyl chloride (Ru(bpy)3 Cl2 ) (excitable condition). We mixed
the chemical compounds except the ruthenium catalyst, and then the ruthenium catalyst solution was added to the solution waiting for around 4 minutes.
The solution was heated till the ﬁrst chemical wave was observed. Cellulosenitrate membrane ﬁlters (Advantec, A100A025A with a pore size of 1 µm) were
soaked in the BZ solution. The soaked membrane was gently wiped with a pieceof ﬁlter paper to remove excess solution and placed on a piece of slide glass.
The surface of the membrane ﬁlter was completely covered with around 1 mlof silicone oil to prevent it from drying and to protect it from the inﬂuence ofChapter 8

Alternative stopping of a chemical wave
(a)

77

(b)

A

B

A

B

B

A

B

Time

1 min

A

3 mm

Figure 8.2: Experimental results on the reaction ﬁeld with the triangular shape
as in Fig. 8.1(a). Spatio-temporal plots achieved by aligning the images on thewhite broken line on the upper picture with time are shown.

oxygen. The experiments were carried out at room temperature. Using a slide
projector and the OHP ﬁlms with a certain shape printed, we could obtain theclear image on the membrane ﬁlter. Chemical waves were prevented from propagating in the brighter area, and they can propagate only in the darker area. We
made the series of the experiments by changing the light intensity in the brighter
region. The features of chemical wave propagation were recorded from above
through a blue ﬁlter (Asahi Techno Glass, V-42) using a digital video camera.
The schematic representation of the experimental setup is shown in Fig. 8.1.

8.3

Results

When the shape of the reaction ﬁeld was triangular as shown in Fig. 8.1(a),the results of the chemical wave propagation are shown in Fig. 8.2. The chemical wave propagating from the broader region disappeared as they came to thenarrower region. When the light intensity in the brighter area was smaller, thechemical waves disappeared at the same point as shown in Fig. 8.2(a). When it
was larger, on the other hand, the points at which chemical waves disappearedchanged every two wave as in Fig. 8.2(b). It was noted that the disappearing points sometimes changed every one in three or in more complicated way;
however these phenomena were ambiguous and could not be observed stably.

78

Chapter 8(a)Alternative stopping of a chemical wave

(b)

(c)

Time

1 min

A
A

B

B

A

B
2 mm

Figure 8.3: Experimental results on the reaction ﬁeld with the combination of
two triangles as in Fig. 8.1(b). Spatio-temporal plots achieved by aligning theimages on the white broken line on the upper picture with time are shown.

When the shape of the reaction ﬁeld was the combination of two triangles asshown in Fig. 8.1(b), the results of the chemical wave propagation are shown in
Fig. 8.3. When the light intensity in the brighter area was smaller, all chemical
waves propagated through the narrowest region as shown in Fig. 8.3(a). As thelight intensity was slightly larger, every second wave could propagate throughthe narrowest region as in Fig. 8.3(b). When it was even larger, all chemical
waves disappeared and failed propagation through the narrowest region as inFig. 8.3(c).

8.4Discussion

The Oregonator model is modiﬁed for photosensitive BZ reaction [90–92]:


U −q
1
∂U
U(1 − U) − (f V + A)
=
∂t

U +q



+ DU ∇2 U,

(8.1)

∂V
= U − v + DV ∇2 V,(8.2)∂twhere U and V are the dimensionless variable that correspond to the concentrations of the activator (HBrO2 ) and the oxidized catalyst ([Ru(bpy)3 ]3+ ), respectively. f ,  and q are the positive parameters that determine the nature of the
BZ reaction, DU and DV are the diﬀusion constants for the activator and theoxidized catalyst, and A is the variable that is proportional to the light intensity.
For simplicity, in this chapter, we adopt a one-dimensional model. In theexperiments, the light intensity outside the reaction ﬁeld was changed. It isChapter 8

Alternative stopping of a chemical wave

79

x

Time

Space

x = xcFigure 8.4: Numerical results based on eqs. (8.3) to (8.5). The spatio temporalplots are shown. The parameters are f = 3,  = 0.04, q = 0.0008, A = 100,
a = 8 × 10−9 , and DU = DV = 1.

assumed that when the reaction ﬁeld becomes thin, the diﬀusion from/to theilluminated region becomes large. The activator diﬀuses from the reaction ﬁeld
to the illuminated region, and the inhibitor (bromide ion) diﬀuses from theilluminated area to the reaction ﬁeld. This transfer can be described as follows:




1
U −q
∂U
=
+ F (U, x) + DU ∇2 U,
U(1 − U) − f V
∂t

U +q

(8.3)

∂V
= U − v + DV ∇2 V,
∂t

(8.4)

a(x − xc )2 −A UU −q− U , (x ≥ xc )+q0,
(x < xc )

(8.5)



F (U, x) =where the a(x − xc )2 is the factor that correspond to the amount of the transfer
between the reaction ﬁeld and illuminated region. A is the variable which is
proportional to the light intensity in the illuminated region. This function is

80

Chapter 8

Alternative stopping of a chemical wave

adopted in order to change the parameter smoothly. A UU −q− U corresponds+qto the diﬀusion of the activator, U, from the reaction ﬁeld to the illuminatedarea and the eﬀects of the illumination on the activator, −A UU −q.
+q
Based on the eqs. (8.3) to (8.5), numerical calculation was performed. The
results of the calculation are shown in Fig. 8.4. The positions at which chemical
waves stop are changed alternatively. It is said that this simple model canrepresent some essences of the experimental results.

Chapter 9
Synchronization ofphotosensitive chemical waves
coupled in two circular excitableﬁelds
In this chapter, the geometry of chemical wave propagating on a circular ring is presented. The two chemical waves on the two contacted
circular rings are also discussed from the viewpoint of synchronization [6].

9.1Introduction

In Chapters 7 and 8, the anomalous chemical wave propagation on the reaction
ﬁeld with a small scale is reported. In this chapter, on the other hand, thechemical wave propagation on the excitable ﬁeld with a shape of circular rings
is investigated.
Experimental and theoretical studies on wave propagation on an excitable
media may help us to not only understand signal processing in biological systems[93] such as nerve impulses [26] but also to create novel methods for artiﬁcial
processing such as image processing [43, 44, 94, 95] and logic operations [85, 92,
96–99]. The Belousov-Zhabotinsky (BZ) reaction has been widely investigatedas an excitable or oscillatory chemical system. The BZ reaction on a membrane,e.g., ﬁlter paper, [100, 101] naﬁon membrane, [102–104], gel [34, 69] or glassﬁlter, [105] has been widely studied because the spatio-temporal pattern of wave
propagation can be regulated by the geometry of the excitable ﬁeld, which iseasily prepared by cutting or printing. In such systems, however, it can betechnically diﬃcult to cut a membrane ﬁlter with a completely similar shapeand to regulate the number of chemical waves and their intervals.
A photosensitive experimental set-up of the BZ reaction makes it easy to cre81

82

Chapter 9

Synchronization of chemical waves

ate excitable ﬁelds of various geometries, which are drawn by computer softwareand then projected on a membrane ﬁlter soaked with BZ solution using a liquidcrystal projector. In this case, light illumination produces bromide ion which
inhibits the oscillatory reaction, i.e., the degree of excitability can be adjusted
by changing the intensity of illumination. Therefore, the number of chemical
waves and their intervals can be spatio-temporally regulated by illumination.
In this chapter, the propagation of chemical waves in the photosensitive
BZ reaction on an excitable ﬁeld (a circular ring or a ﬁgure “8” composed of
two equivalent circular rings) illuminated with a liquid-crystal projector wasinvestigated. The nature of wave propagation depending on the ratio of theinner and outer radii of the circular rings was well ﬁtted as the involute ofa circle [106–109]. When the two equivalent circular rings slightly in contact
with one another, two chemical waves collided at a location apart from theintersection. When two circular rings completely overlapped at the intersection,
the location of wave collision alternated with time. The essential feature of wave
propagation was qualitatively reproduced by a numerical calculation based on
the reaction-diﬀusion equations using the modiﬁed Oregonator.

9.2

Experiments

Ru(bpy)3 Cl2 was used as a catalyst for the photosensitive BZ reaction. The BZ
solution consisted of [NaBrO3 ] = 0.45 M, [H2 SO4 ] = 0.3 M, [CH2 (COOH)2 ] = 0.2
M, [KBr] = 0.05 M, and [Ru(bpy)3 Cl2 ] = 1.7 mM. Cellulose-nitrate membraneﬁlters (Advantec, A100A025A) with a pore size of 1 µm were soaked in BZ
solution (5 ml) for about 1 min. The soaked membrane was gently wiped withﬁlter paper to remove excess solution and placed on a petri dish (diameter: 100mm). The surface of the membrane ﬁlter was completely covered with 1 ml
silicone oil (Wako, WF-30) to prevent it from drying and to protect it fromthe inﬂuence of oxygen. The experiments were carried out in an air-conditioned
room at 298 K, at which the reaction medium showed no spontaneous excitationand no change in behavior for approximately 30 min.
The medium was illuminated from below as schematically shown in Fig. 9.1.
The high-pressure mercury bulb of a liquid-crystal projector (MITSUBISHI,LVP-XL8) was used as a light source and the spatial intensity distribution was
controlled with a personal computer. A black and white picture on the liquidcrystal projector served as an illumination mask to create the appropriate boundary, and the light intensities were 4.0×102 lx and 1.7×104 lx for black and white
regions, respectively. The experiments were monitored from above with a digital
video camera (SONY, DCR-VX700) and recorded on videotape. A blue opticalﬁlter (Asahi Techno Glass, V-42) with a maximum transparency at 410 nm wasused to enhance the image of the chemical waves. The light intensity at theilluminated part was measured with a light intensity meter (ASONE, LX-100).

Chapter 9

Synchronization of chemical waves

83

Digital Video Camera

Personal Computar
Blue Filter
Silicone Oil
Petri dishMembrane Filter
MirrorLiquid-crystal
Projector

Figure 9.1: Schematic representation of experimental setup [6].

9.3

Results

9.3.1

Propagation of a single wave on a circular ringFirst, we examined the radius-dependent propagation of a single wave on a reaction ﬁeld in the shape of a circular ring. In this experiment, two chemical waves
initially propagated in opposite directions. By the local illumination around
one wave within 1 s, the illuminated wave was disappeared, but another one
continued to propagate. Therefore the uni-directional chemical wave on the ring
was achieved. Figure 9.2(a) shows the velocity of wave propagation on a circularring depending on rout /rin (rin : inner radius, rout : outer radius of the circularroute). In this experiment, rin was varied under constant rout (= 6 mm). The
velocity of the chemical wave at the inner boundary of the circle (vin ) was almost independent of rin . On the other hand, that at the outer boundary (vout )was linearly increased with 1/rin , which indicates that the angular velocity of
the chemical wave (ω) was determined by the wave propagation near the inner
radius, i.e. ω = (vin /2π)/rin and vout = 2πrout ω. The phase diﬀerence (∆θ),
which is schematically deﬁned in Fig. 9.2, increased with rout /rin , as shown in
Fig. 9.2(b), i.e., the deformation of the chemical wave increased with the widthof the circular ring.

9.3.2

Collision of two chemical waves on two equivalentcircular rings

Next, we examined the interaction between two chemical waves on a reaction
ﬁeld composed of two equivalent circular rings at rin = 2.1 mm and rout /rin = 2.0.
In this experiment, one chemical wave was initially generated in each ring in the

84

Chapter 9

Synchronization of chemical waves

10 mm
0.2

0

1

2
rout/rin3vin (mm/s) ( )

0.1

rin

vin
rout

0

Figure 9.2: Wave propagation on a circular ring depending on rout /rin for vin(empty circle) and ∆θ (ﬁlled square). Corresponding snapshots are indicated in
the upper part. ∆θ, rout , rin , and vin are schematically deﬁned at the right sideof the ﬁgure [6].

same direction (clockwise), and the chemical wave in the right ring reached theintersection before that in the left ring.
When the distance between the centers of the two rings was nearly equal
to 2rout , i.e., the two rings were just slightly in contact with one another, thelocation of the collision of the two waves was determined to be in the left ringnear the intersection, as shown in Figs. 9.3 (a-1) and (b-1). Here, θL and θR are
the angles corresponding to the positions of pulses on the left and right innerrings, and they are plotted at the moment two pulses collide in Fig. 9.3 (b). The
location of the collision converged to 1.93π rad with time, and this convergent
value did not change even when the diﬀerence in the initial phases was larger
than 1.93π rad (data not shown) [110].
When the distance between the centers of the two rings was equal to rin +rout ,i.e., the two rings completely overlapped one another, the location of the collisionof the two waves changed alternatively, as shown in Figs. 9.3 (a-2) and (b-2),
i.e., the chemical wave from the right ring reached the intersection ahead ofthat from the left ring, as seen in snapshots 1 and 3, or vice versa, as seen insnapshots 2 and 4.

Chapter 9

Synchronization of chemical waves

85

(a-1)
1

2

3

4

1

2

3

4

(a-2)

(b-1)

(b-2)

0

10 mm

0

1

2

3

4

number of collision

1

2

3

4

Figure 9.3: Interactive wave propagation between two chemical waves in two
equivalent circular rings that (1) were slightly in contact and (2) completely
overlapped one another. (a) Top view of the snapshots of the wave propagation
with the time intervals of (1) 300 sec and (2) 250 sec. (b) θR (ﬁlled circle) andθL (empty circle) when the two waves collided for (b-1) slightly connecting ringsand (b-2) completely overlapping rings. θL and θR were schematically deﬁned
on the right side. The numbers of collision (1, 2, 3, 4) correspond to those in(a) [6].

9.4

Discussion

9.4.1

The nature of chemical wave propagation in a single
circular ring.

Based on the experimental results and those in the related papers [106–109],
we now discuss the characteristics of the BZ wave propagation in a reaction
ﬁeld designed by the light illumination. First, we analyze the shape of wave
propagation depending on rout /rin of the circular ring. If we suppose that thewave propagates on the reaction ﬁeld without any eﬀects from the boundary,and that the wave cannot move into the brighter area, the distance from a pointon the inner circle (point α) to another point (point β), which is one of thelocations at which the chemical wave propagates along the inner circle, is thesame as that to the intersection (point γ) between the chemical wave and thetangent line at the noted point (point α), as schematically indicated in Fig. 9.4(a), since the chemical wave propagates in a direction perpendicular to the wave

Chapter 9
(a)
rout

(b)

π/2

∆θ (rad)

86

π/4

β

rin
α

Synchronization of chemical waves

0

γ

1

2

chemical wave

3

rout/rin

Figure 9.4: (a) Schematic representation on the chemical wave with the shapeof the “involute” of a circle. (b)Comparison between the experimental results
and the analytical prediction. The solid squares are the experimental results
shown in Fig. 9.3, and the solid curve is the analytical prediction based on eq.
(9.6) [6].

front with a uniform velocity. Therefore, the shape of the chemical wave may
be regarded as the involute of a circle [106–109].
From the geometrical perspective, when the location of the chemical wave
on the inner circle of the ring (point β) is expressed as x = rin cos θin andy = rin sin θin , that on the chemical wave is expressed using parameter θ,


x
y





= rin



cos θ
sin θ



+ rin (θin − θ)− sin θ
cos θ



,

(9.1)

where θ is the phase of the point α. We calculate the location of the intersection
between the chemical wave and the outer circle with a radius of rout , (point γ).
Here, the phase of the intersection is set as θout . From




2
2
= rin
1 + (θin − θ)2 ,
rout

and
tan θout =(9.2)

rin sin θ + rin (θin − θ) cos θtan θ + (θin − θ)=,rin cos θ − rin (θin − θ) sin θ
1 − (θin − θ) tan θ(9.3)

we can gettan θout = tan (θin + α) ,where tan α =

(9.4)

2
2
rout
− rin
/rin = θin − θ. From eq. (9.4),

θout = θin + α = θin −

2
2
rout
− rin
/rin + arctan



2
2
rout
− rin
/rin



.

(9.5)

(rout /rin ) − 1 .

(9.6)

Therefore, we can derive∆θ = θin − θout =(rout /rin ) − 1 − arctan
2



2



Chapter 9

Synchronization of chemical waves

87

Figure 9.4 (b) shows the theoretical results obtained based on eq.(9.6). These
theoretical results suggest that the experimental results regarding the nature of
the chemical wave propagation shown in Fig. 9.4 can be quantitatively ﬁtted bya theoretical consideration based on the geometrical discussion. In the previous
study, Müller et al. and Noszticzius et al. reported that the shape of the chemical
wave is the involute of a circle and that it can be regarded as an Archimedes
spiral far enough from the tip of the wave [106–109]. In this study, the chemical
wave is well ﬁt to the involute of a circle but not a spiral, since the core of the
spiral is much larger than that of the spiral pattern formed spontaneously.

9.4.2

The nature of chemical wave propagation on two
equivalent circular rings.

Following the above discussion, we calculated the nature of chemical wave propagation in a reaction ﬁeld composed of two equivalent circular rings. We derived
a characteristic phase diﬀerences ∆θ0 that depended on the shape of the reaction
ﬁeld. First, it should be noted that there are two types of interaction between
the chemical waves on the two rings: in one type, the chemical wave on one ring
(ring A) propagates and touches the inner radius of the other ring (ring B), andin the other type, the chemical wave does not touch it. In the former case, thephase of the chemical wave on the ring A aﬀects the phase of the chemical wave
on ring B. Therefore, the phase diﬀerence between the chemical waves on thetwo rings is determined only by the shape of the reaction ﬁeld. This diﬀerencein phase on the two circular rings is set as ∆θ0 . In the latter case, in contrast,
the chemical wave on the ring A does not aﬀect the chemical wave propagation
on ring B. These two cases are selected by the initial phase diﬀerence between
θL and θR when they reach a stationary propagation without collision. This
characteristic phase diﬀerence ∆θ0 is analytically derived as follows:
We consider chemical wave propagation on a ﬁeld composed of two circularrings with overlap d, as shown in Fig. 9.5(a). Here, we derive the characteristicphase diﬀerence ∆θ0 between the phase of the chemical wave in one ring (ring A)
and that of the chemical wave on the other ring (ring B) after the chemical wave
on the ring A aﬀects chemical wave propagation on ring B. The time required
for a chemical wave to reach a certain point can be calculated from the shortest
path. Therefore, we have to consider the two cases as shown in Fig. 9.5.
In the case in Fig. 9.5 (b-1), i.e.,2rout− (rout − d/2)2 > rin ,(9.7)

the shortest path from one ring to the other ring is a straight line, and thelength, L, is
L = 2rout − d.
(9.8)
If we suppose that the chemical wave on the ring A aﬀects the chemical wave
propagation on ring B, the phases of the chemical waves on rings A and B, θA

88

Chapter 9

Synchronization of chemical waves

(a)
rout
rin

d (b-2)

(b-1)

rout
rin

L

L

Figure 9.5: Schematic representation of (a) the shape of the reaction ﬁeld and(b) the shortest path when the chemical wave in one ring propagates to the otherring. (b-1) When the shortest path is a straight line. (b-2) When the shortest
path is a V-shaped line [6].

and θB , can be written as
θA = −
and

π
v
+ t,
2 rin


(9.9)


v
π
L
θB = +
t−,(9.10)
2 rinv
where t is time. Therefore, the characteristic phase diﬀerence ∆θ0 is derived as∆θ0 = θB − θA = π −2rout − d.
rin

(9.11)On the other hand, in the case of Fig. 9.5 (b-2), i.e.,2rout− (rout − d/2)2 < rin ,(9.12)

the shortest path is a V-shaped line, and L is derived as
2
2
− rin
.
L = 2 rout(9.13)

If we suppose that the chemical wave on ring A aﬀects the chemical wave propagation on ring B, θA and θB can be written as
θA = −Θ0 +andθB = Θ0 +

v
t,
rin

(9.14)


v
L
t−
,
rin
v

(9.15)

Chapter 9

Synchronization of chemical waves

89

2

D = d / rin

Not defined
1

I
II
0

1

2

3

R = rout / rin

Figure 9.6: Phase map of the synchronization of chemical wave propagation ina reaction ﬁeld composed of two equivalent circular rings depending on R =rout /rin and D = d/rin . Region I corresponds to collision at one point, andregion II corresponds to alternative collisions [6].

where









rin
rout − d/2+ arccos.
Θ0 = arccos
rout
rout

(9.16)

Therefore, the characteristic phase diﬀerence ∆θ0 is derived as
2
2
2 rout− rin.
∆θ0 = θB − θA = 2Θ0 −rin(9.17)

In summary, the characteristic phase diﬀerence ∆θ0 is written as∆θ0 = θB − θA =and

π − 2R(R > D/4 + 1/D)√+ D,
2
2Θ0 − 2 R − 1 (R < D/4 + 1/D)








(9.18)

D1+ arccos 1 −.
(9.19)
Θ0 = arccos
R
2R
These equations are written in the nondimensional form using R = rout /rin and
D = d/rin . The phase map shown in Fig. 9.6 is drawn according to the sign of∆θ0 .
According to the above discussion, when the initial phase diﬀerence is smaller
than |∆θ0 |, the phase diﬀerence is maintained. Though we do not show the experimental data corresponding to this case, but this features are conﬁrmed in theexperiments [110]. On the other hand, when the initial phase diﬀerence is larger
than |∆θ0 |, the chemical waves on the two rings aﬀect each other: If ∆θ0 < 0(Region I), the phase diﬀerence is ﬁxed as |∆θ0 |. If ∆θ0 > 0 (Region II), thephase diﬀerence changes alternatively. Thus, the location of the collision of the

90

Chapter 9(a)rout/rin = 1.5

rout/rin = 1.8

Synchronization of chemical waves

rout/rin = 3.0

(b)
0.1

0

vin (a.u.) ( )

0.05

0
1

3
rout/rin

5

Figure 9.7: Results of the numerical calculation for a reaction ﬁeld in the shapeof a circular ring based on the modiﬁed Oregonator model shown in eqs. (9.20)
and (9.21). (a) Snapshots for the various values of rout /rin . (b) Wave velocity
along the inner boundary vin (empty circle) and the phase diﬀerence between thelocations of the chemical wave along the inner and outer boundaries ∆θ (ﬁlled
square) depending on rout /rin . The curve derived from the analytical discussionis also described as a solid curve. The parameters used in the calculations are
f = 1,  = 0.05, q = 0.00015, and DU = DV = 1. The chemical wave is initiatedby setting U = 0.5 at a certain point. Some time later, the two chemical waves
propagate in opposite directions. By increasing the variable for light intensity,
A, in a certain area for some length of time, we can make one wave disappear,
and a unidirectional chemical wave is achieved [6].

two waves changes alternatively. Figure 9.6 shows a phase map that depends on
R (the ratio between the inner and outer radii, rout /rin ) and D (the ratio between
the overlap and the inner radius, d/rin ). The characteristics of the chemical wave
propagation shown in 1 and 2 in Fig. 9.3 in the experiments correspond to regions I and II, respectively. The properties of chemical wave propagation on two
equivalent circular rings can be understood analytically within this framework.

Chapter 99.4.3

Synchronization of chemical waves

91

Numerical calculation using the modiﬁed Oregonator model

We performed numerical calculations based on the modiﬁed Oregonator [90–92]:


1
U −q
∂U
=
U(1 − U) − (f V + A)
∂t

U +q



+ DU ∇2 U,(9.20)∂V(9.21)= U − v + DV ∇2 V,∂twhere U and V are the dimensionless variable that correspond to the concentrations of the activator (HBrO2 ) and the oxidized catalyst ([Ru(bpy)3 ]3+ ), respectively. f ,  and q are the positive parameters that determine the nature of the
BZ reaction, DU and DV are the diﬀusion constants for the activator and theoxidized catalyst, and A is the variable that is proportional to the light intensity.
We set A = 0.03 in the brighter area and 0.005 in the darker area, respectively.
The results of the numerical calculation for a reaction ﬁeld in the shape of acircular ring are shown in Fig. 9.7. The relationship between ∆θ and rout /rin is
described together with the analytical prediction by eq. (9.6). This dependency
is well described by the analytical discussion, and the experimental, theoreticaland numerical results are all consistent with one another.
Figure 9.8 shows the numerical results for a reaction ﬁeld in the shape of aﬁgure “8”, which is composed of two equivalent circular rings. When the twocircular rings slightly in contact with one another, the phase diﬀerence is ﬁxed
at a constant value, as shown in Figs. 9.8 (a-1) and (b-1). On the other hand,when they completely overlap, the phase diﬀerence changes alternatively; i.e.,
the location of the collision changes alternatively, as shown in Figs. 9.8 (a-2)
and (b-2). These experimental and analytical results are well reproduced by thenumerical calculations.

92

Chapter 9

Synchronization of chemical waves(a-1)

time step: 48(a-2)

time step: 44

θR( ) or θL( ) (rad)

(b-1)

(b-2)
π/4

π/4

0

0

−π/4

0

3

6

−π/4
0
9
number of collision

3π/2

π

3

6

π/2
θR

0(2π)
θL

0(2π)

π/2

3π/2

π

9

Figure 9.8: Results of a numerical calculation for a reaction ﬁeld in the shapeof a ﬁgure “8” composed of two equivalent circular rings based on the modiﬁed
Oregonator model shown in eqs. (9.20) and (9.21). (a) Snapshots of a typical nature of wave propagation for (a-1) slightly connected rings and (a-2) completely
overlapping rings. (b) Phases of the locations of collision versus the numberof collision for (b-1) slightly connecting rings and (b-2) completely overlapping
rings. The parameters and the initial conditions are the same as those in Fig.
9.7. The shape of the ﬁeld is rout /rin = 2 for both (1) and (2), d/rin = 2/9 for(1), and d/rin = 1 for (2) [6].

Part IV
General Conclusion

93

Chapter 10
Conclusion
10.1

Conclusion

In this thesis, the experimental and theoretical results on the reaction diﬀusion
systems coupled with convection and geometrical eﬀects are demonstrated.
As for the reaction-diﬀusion systems coupled with convection, the spontaneous motion can be exhibited induced by the spatio-temporal pattern formationin reaction-diﬀusion systems. At the same time, the convective ﬂow of the ﬁeld
can cause the spatio-pattern formation:
In the BZ reaction system, convective ﬂow in the bulk phase can be observed
induced by the repetitive change in interfacial tension of BZ medium (Chapter
3). This convection can cause the spontaneous motion of a small droplet of BZ
reaction medium (Chapter 4). In the water-camphor system, convective ﬂow
coupled with the surface diﬀusion of the camphor layer can be observed. This
convective ﬂow can aﬀect the spontaneous motion of a camphor disk (Chapter
5). These two systems can be understood as a reaction-diﬀusion systems coupled with convection through interfacial tension. Some other systems, i.e.: thespontaneous motions of an alcohol droplet in the water-alcohol system, an oil
droplet on the glass substrate in the surfactant solution, and a phenanthroline
disk in the ferrous sulfate solution are also exhibited (Chapter 6).
As for the boundary eﬀects in reaction-diﬀusion systems, the characteristicsof the chemical wave changes when the system size is comparable to the chemical wave itself. In this part, the anomalous behavior of chemical waves are
exempliﬁed:
In Chapter 7, the slowing and stopping of the chemical wave in the narrowing
glass capillary is exempliﬁed in BZ reaction. The mechanism is theoretically analyzed related to the surface-volume ratio. The scaling analysis was performed,
and the experimental results can be reproduced. In Chapter 8, the narrowing
reaction ﬁeld is achieved using light illumination in photosensitive BZ reaction.
The alternative disappearing of the propagating chemical waves are observed.
In Chapter 9, the geometry of chemical wave propagating on a circular ringis studied. The two chemical waves on two connected circular rings are also95

96

Chapter 10

Conclusion

discussed from the viewpoint of synchronization.
Considering the phenomena in the real world, especially in the living organisms, the rhythmicity, pattern formation, and directional motion should becoupled with each other. The system has a ﬁnite size and the boundary of thesystem should aﬀect seriously. In order to discuss on the mechanism of thenonequilibrium systems, these convective eﬀects and boundary eﬀects cannot be
avoided. The discussions in this thesis may help to understand the mechanism
on how the convective ﬂow and geometrical eﬀects of the boundary aﬀect thenonequilibrium systems.

10.2

Future problems

In proceeding the studies on the reaction diﬀusion systems coupled with convective ﬂow or geometrical eﬀects of the boundary, a lot of problems or diﬃculties
have come out and some remained to be unsolved so far. The followings aresome of the future problems:
In Part II, several numerical calculations based on the reaction-diﬀusionadvection equations coupled through the interfacial tension gradient were performed. The numerical results reproduce the experimental results to some extent, but some characteristics cannot be reproduced. For example, in the experiments of the convective ﬂow induced by BZ reaction (Chapter 3), the strong
convective ﬂow can be observed before the wave front. On the other hand, inthe numerical calculation, strong ﬂow can be observed only near the wave front.
I guess that these diﬀerences are due to the nonequilibrium condition. Theinterfacial tension is deﬁned in the equilibrium state, but in this case, the concentration of the surface-active compounds is not uniform, which means that thesystem is under nonequilibrium condition. Thus, the interfacial tension cannot
be deﬁned in the similar way to that in the equilibrium state. The interfacial
tension under nonequilibrium condition should be studied more from both themicroscopic and macroscopic points of view.
In Part III, some characteristic behaviors of the chemical wave propagation
are presented in the reaction ﬁeld with the size of 10 µm to 1 mm. In thisscale, the singularity is due to the surface-volume ratio. In other words, thecondition near the surface is diﬀerent from that in the bulk phase, and thetotal behavior changes because the ratio of the region aﬀected from the surface
relatively increases.
In a much smaller system, another singularity may happen: the ﬂuctuationof the concentration. In the reaction with an autocatalytic process, such as
the BZ reaction, has a state with a little concentration of some chemical compounds. For example, in the BZ reaction, the concentration of the activator isvery little during a certain span in the oscillation. The ﬂuctuation of the number of molecules is supposed to be proportional to N 1/2 , where N is the numberof molecules. Thus, in a very small system, the ﬂuctuation of the number ofChapter 10

Conclusion

97

molecule can appear as the ﬂuctuation of the period of the chemical oscillation.
This is an interesting problem from the viewpoints of not only physics but also
life sciences, since the behavior of the molecules inside the cell should be aﬀectedby this ﬂuctuation.

Acknowledgment
I would like to thank Professor Kenichi Yoshikawa for advising me for six and ahalf years. I have been impressed in his ideas and the way of thinking. I would
not study on the nonequilibrium physics without his kind advice. I also thankProfessor Hideki Seto for his advice. His research area is rather far from mine,
but his opinion is very useful for me.
I am grateful to Professor Satoshi Nakata (Nara University of Education).
I have felt very interesting to collaborate with the researchers with the otherbackground. I also appreciate the member of his laboratory, Ms. Yukie Doi,
Ms. Akane Terada, Mr. Shin-ichi Hiromatsu, Ms. Akiko Yamada, Ms. Sayaka
Morishima for their kind help.
I would like to thank Professor Yoshihito Mori (Ochanomizu University) for
his kind advice on the chemical reaction. Discussion with him has inspired me.
I am also grateful to thank Professor Masaharu Nagayama (Kanazawa University) for his kind advice on theoretical analysis and numerical simulation.
I thank very much for Dr. Nobuyuki Magome (Nagoya-Bunri College) for
teaching me the experimental technique, especially BZ reaction. I also thankDr. Ryoichi Aihara (West Virginia University) for a lot of discussion. He gaveme the chance to proceed the ﬁrst study on the convective ﬂow in BZ reaction.
I am also grateful to Dr. Takatoshi Ichino (Kinki University). He taught me on
the experiment of BZ reaction, and the discussions with him are helpful for me.
I am grateful to Professor Jerzy Gorecki (Polish Academy of Sciences). He
has kindly advised me on the boundary eﬀects of BZ reaction, and he proposed
the experiments using photosensitive BZ reaction.
I would also like to thank Professor Hidehiro Oana (Tokyo University), Professor Tatsuo Akitaya (Meijo University), Professor Satoru Kidoaki (Kyusyu
University), Dr. Hiroyuki Mayama (Hokkaido University), Mr. Koji Kubo (Kyoto University) for their kind advice on how to proceed the study. I am also
grateful to Dr. Ikuko N. Motoike (Future University Hakodate), Dr. Shin-ichiro
M. Nomura (Tokyo Medical and Dental University), Mr. Masahiro I. Kohira(Chuo University), and Dr. Sada-atsu Mukai (JAMSTEC) for their kind advice
and discussion. With Dr. Takahiro Sakaue (Kyoto University), Dr. Masatoshi
Ichikawa (Kyusyu University), Dr. Takahiro Harada (Kyoto University), Mr.
Tsutomu Hamada, and Mr. Natsuhiko Yoshinaga, I have made a lot of discussion on various topics. These discussions have inspired me a lot. With Mr.
Yutaka Sumino and Mr. Ken Nagai, I have studied on the spontaneous motion.
99

100
I appreciate the discussions with them so much. I cannot write down the nameof every member, but I have spent a great time with the colleagues. Active
discussions with them, talks on coﬀee or alcohol, and kind advices from them
are the greatest help for me.
I am grateful to Ms. Yoshimi Mayama, Ms. Tomoko Kawakami, Ms. Mami
Harada, Ms. Yuka Yamafuji, and Ms. Naoko Yamawaki for their kind support.
They have given us the great environment to study in the laboratory.
I was supported by JSPS Research Fellowships for Young Scientists for one
year. The present work is supported in part by Grant-in-Aid for Scientiﬁc Research in Priority Areas (No. 17049017) from the Ministry of Education, Culture, Sports, Science and Technology of Japan and by Kurata Grants from theKurata Memorial Hitachi Science and Technology Foundation.
At last, I am very grateful for all the kind supports from my friends, and myfamily.

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