Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective
A potential dynamics approach is developed to determine the periodic standing and traveling wave patterns associated with self-propelling camphor objects floating on ring-shaped water channels. Exact solutions of the wave patterns are derived. The bifurcation diagram describing the transition betwee...
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irk-123456789-1528882019-06-14T01:25:26Z Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective Frank, T.D. A potential dynamics approach is developed to determine the periodic standing and traveling wave patterns associated with self-propelling camphor objects floating on ring-shaped water channels. Exact solutions of the wave patterns are derived. The bifurcation diagram describing the transition between the immobile and self-propelling modes of camphor objects is derived semi-analytically. The bifurcation is of a pitchfork type which is consistent with earlier theoretical work in which natural boundary conditions have been considered. Розвинуто метод потенцiальної динамiки для того, щоб вивчити картини перiодичних стоячих i бiжучих хвиль, пов’язаних iз самоурухомлювальними камфорними об’єктами, якi рухаються на кiльцеподiбних водяних каналах. Отримано точнi розв’язки хвильвових картин. Отримано напiваналiтично дiаграму бiфуркацiї, яка описує перехiд мiж нерухомою i самоурухомлювальною модами камфорних об’єктiв. Бiфуркацiя є вилоподiбна, що узгоджується з попередньою теоретичною роботою, в якiй розглянуто природнi граничнi умови. 2014 Article Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective / T.D. Frank // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43002: 1–12 — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 05.45.-a, 82.40.Ck DOI:10.5488/CMP.17.43002 arXiv:1501.03468 http://dspace.nbuv.gov.ua/handle/123456789/152888 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A potential dynamics approach is developed to determine the periodic standing and traveling wave patterns associated with self-propelling camphor objects floating on ring-shaped water channels. Exact solutions of the wave patterns are derived. The bifurcation diagram describing the transition between the immobile and self-propelling modes of camphor objects is derived semi-analytically. The bifurcation is of a pitchfork type which is consistent with earlier theoretical work in which natural boundary conditions have been considered. |
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Frank, T.D. |
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Frank, T.D. Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective Condensed Matter Physics |
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Frank, T.D. |
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Frank, T.D. |
| title |
Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective |
| title_short |
Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective |
| title_full |
Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective |
| title_fullStr |
Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective |
| title_full_unstemmed |
Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: A novel potential dynamics perspective |
| title_sort |
exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: a novel potential dynamics perspective |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/152888 |
| citation_txt |
Exact solutions for chemical concentration waves of
self-propelling camphor particles racing on a ring: A
novel potential dynamics perspective
/ T.D. Frank // Condensed Matter Physics. — 2014. — Т. 17, № 4. — С. 43002: 1–12 — Бібліогр.: 28 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT franktd exactsolutionsforchemicalconcentrationwavesofselfpropellingcamphorparticlesracingonaringanovelpotentialdynamicsperspective |
| first_indexed |
2025-07-14T04:21:27Z |
| last_indexed |
2025-07-14T04:21:27Z |
| _version_ |
1837594719712968704 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 4, 43002: 1–12
DOI: 10.5488/CMP.17.43002
http://www.icmp.lviv.ua/journal
Exact solutions for chemical concentration waves of
self-propelling camphor particles racing on a ring: A
novel potential dynamics perspective
T.D. Frank
CESPA, Department of Psychology, University of Connecticut, 406 Babbidge Road, Storrs, CT 06269, USA
Received October 8, 2014
A potential dynamics approach is developed to determine the periodic standing and traveling wave patterns
associated with self-propelling camphor objects floating on ring-shaped water channels. Exact solutions of the
wave patterns are derived. The bifurcation diagram describing the transition between the immobile and self-
propelling modes of camphor objects is derived semi-analytically. The bifurcation is of a pitchfork type which is
consistent with earlier theoretical work in which natural boundary conditions have been considered.
Key words: chemical concentration waves, self-motion, pitchfork bifurcation
PACS: 05.45.-a, 82.40.Ck
1. Introduction
A challenge inmodern-day research in biophysics and bioengineering is to create and understand bio-
chemical particle systems that mimic biological cell motion. In particular, at issue is to construct particles-
surface systems in which particles have the capability to move themselves over certain distances by con-
verting chemical energy into kinetical energy [1]. As pointed out in reference [1], a fundamental class
of such man-made self-moving systems is given by objects that are floating on a medium, are driven
by differences in the surface tension of that medium, and at the same time produce gradients of some
substance that affects the surface tension of the medium. Irrespective of engineering applications, self-
propelling chemical systems allow us to study principles that might be important for our understanding
of the physics of life, in general, and the so-called active Brownian particles [2], in particular. In fact, it
has been shown that under certain conditions self-propelling chemical “motors” exhibit negative friction
terms [1, 3] that are a hallmark of active Brownian systems [2, 4–8]. A variety of self-propelling chem-
ical motors have been studied [3, 9–14]. In particular, the self-motion of camphor particles moving on
water has been extensively studied, in particular, by Nakata and colleagues [15–20]. In this context, we
would like to point out that not only single camphor particles have been considered but also the inter-
action between two self-propelling camphor particles has been examined [21], the collective motion of a
small number (about 10) of camphor particles has been investigated [22, 23], and the spatial and velocity
distributions of many interacting camphor particles have been determined [24, 25].
A theoretical model for the self-motion of a single, solid camphor disc floating on water has been de-
veloped in terms of a Newtonian equation for the disc and a reaction-diffusion equation for the concen-
tration of the surface-tension active camphor molecules on the water surface [26]. In this context, analyt-
ical solutions of the concentration wave patterns under natural boundary conditions have been derived
and it has been shown that the self-propelling mode bifurcates from an immobile mode by means of a
pitchfork bifurcation. However, an experimentally very useful paradigm is the self-motion of a camphor
disc on a ring channel [15–20].
In view of the importance of ring-shaped designs for experimental research, the present study goes
beyond the case of natural boundary conditions. To the best of our knowledge, we will for the first time
© T.D. Frank, 2014 43002-1
http://dx.doi.org/10.5488/CMP.17.43002
http://www.icmp.lviv.ua/journal
T.D. Frank
present a theoretical analysis of the periodic case that can directly be applied to and compared with
the laboratory situation. In addition, while it is plausible to assume that the chemical wave patterns of
camphor particles on a ring exhibit a single peak, a clear explanation why this should be the case has not
been given so far. Such an explanation will be given below. To this end, a potential dynamic perspective
for the coupled particle-wave system will be developed.
Explicitly, the aim of the current study is threefold. First, we will work out a potential dynamics ap-
proach towave patterns of reaction-diffusion systems in order to qualitatively discuss the possible shapes
of the camphor concentration patterns associated with the self-motion of camphor discs floating on a ring
channel. Second, we will derive analytical solutions for the standing and traveling concentration waves
associated with immobile and self-propelling camphor discs, respectively, on a ring channel. Third, we
will numerically determine the bifurcation diagram for the case of periodic boundary conditions. In do-
ing so, we will show that the pitchfork bifurcation derived earlier for natural boundary conditions can
also be found in the case of periodic boundary conditions that is frequently used in experimental re-
search.
We consider a camphor disc traveling on a ring channel filled with water. Note that the disc releases
camphor to the water surface, which implies that in this system we need to distinguish between the cam-
phor disc and the camphor concentration on thewater surface. Thewidth of the ring channel is of the disc
diameter such that the disc can move only in one direction, along the ring. The ring has radius R and cir-
cumference 2πR. The disc position is described by the periodic variable y(t) ∈ [0,2πR], where t denotes
time. Moreover, the velocity of the disc is described by v(t). The concentration u of camphor molecules
on the water surface at time t and at a particular position x ∈ [0,2πR] along the ring is described by the
field variable u(x, t) Ê 0. The dynamics of the camphor disc is given by [26]
d
dt
y = v, (1.1a)
d
dt
v =−
aγ
(au+1)2
∂
∂x
u
∣
∣
∣
x=y(t )
−µv (1.1b)
with a,γ,µ > 0. Accordingly, the disc satisfies a Newtonian equation with a force generated by the cam-
phor concentration field u and a friction force proportional to the camphor disc velocity. From a mecha-
nistic point of view, the camphor concentration affects the surface tension, which in turn acts as a force
on the camphor disc (see introduction above). The effective force given as the first term on the right hand
side of equation (1.1b) can be regarded as a gradient force of a potential, where u is the potential. That
is, the camphor disc is driven away from regions of high camphor concentrations. The parameters γ and
a are related to certain details of the aforementioned mechanistic relationship between camphor con-
centration, surface tension, and the force acting on the camphor particle, see reference [26]. Finally, in
equation (1.1) the parameter µ denotes the friction coefficient. The camphor concentration field u(x, t)
satisfies the reaction-diffusion equation
∂
∂t
u =
∂2
∂x2
u−k u+F (x − y(t)) (1.2)
with k > 0. The term −k u describes the decay of the camphor concentration on the water surface due to
dissolution and sublimation. The function F describes the increase of camphor concentration on the wa-
ter surface due to the camphor disc depositing camphor molecules to the water surface. Mathematically
speaking, F corresponds to a source term. In this context, note that we consider not too long time periods
during which the mass loss of the camphor disc can be neglected. The source term F is defined by F = 1
for |∆| É r and ∆ = x − y and F = 0 otherwise, where r > 0 is the radius of the camphor disc. Note that
from a mathematical point of view for large r we may not think of the disc as a circular object. We may
imagine a solid object that has the shape of a segment of the ring with segment length 2r measured along
the curvilinear coordinate x. Then, we have r É πR. In the case r = πR, the object would be a full, solid
ring that covers the whole surface of the ring channel.
It is important to note that equations (1.1) and (1.2) actually represent rescaled equations such that
the variables x, y , u, and even the time variable t are given in dimensionless units. For example, the
(real world) laboratory time tlab measured in seconds is given by the dimensionless time t occurring in
43002-2
Chemical waves of camphor particles
equations (1.1) and (1.2) divided by a rate constant measured in 1/sec that describes the rate with which
camphor is released from the camphor disc to the water surface [26].
2. Potential dynamics perspective of the chemical wave pattern associ-
ated with a self-propelling camphor disc
2.1. Potential dynamics point of view
The objective is to study traveling wave solutions [27, Sec. 7.3] of the form
u(x, t) = g (x −c t) (2.1)
of the model defined by equations (1.1) and (1.2), where c is the velocity of the traveling wave. Let us
define z = x − c t as the phase coordinate of the wave. The reaction diffusion equation (1.1) describes a
standing or traveling wave pattern only if the source F depends on the phase coordinate z rather than
on x and y(t). Substituting z = x −ct into ∆= x − y(t), we obtain ∆= z +c t − y(t). The requirement that
∆ depends only on z implies that y(t) = c t + y0 with y0 ∈ [2πR]. This also leads to v(t) = c. The camphor
disc moves with the same velocity as the wave pattern. Substituting equation (2.1) into equation (1.2), we
obtain
∂2
∂z2
g =−c
∂
∂z
g −k g +F (z − y0) . (2.2)
Likewise, from equation (1.1) it follows that
∂
∂z
g
∣
∣
∣
z=y0
=−
µc
aγ
[ag (y0)+1]2 . (2.3)
The position y0 of the camphor disc is arbitrary. That is, if there is a wave pattern g (z, y0) with a particular
camphor disc position y0, then y0 +h has the solution g (z, y0 +h) = g (z −h, y0), which shifts the wave
pattern by h. Without loss of generality, we put y0 = 0 such that
d2
dz2
g =−c
∂
∂z
g −k g +F (z) (2.4)
and
d
dz
g
∣
∣
∣
z=0
=−
µc
aγ
[ag (0)+1]2 (2.5)
with F = 1 if z ∈ [0,r ]∪ [2πR−r,2πR] and F = 0 otherwise. The wave pattern is subjected to periodic
boundary conditions g (z) = g (z +2πR), which implies that the relations g (0) = g (2πR) and dg (0)/dz =
dg (2πR)/dz hold. The traveling wave equations (2.4) and (2.5) contain the standing wave equations as
special case. For the standing wave we put c = 0 such that
d2
dz2
g =−k g +F (z) (2.6)
and
d
dz
g
∣
∣
∣
z=0
= 0. (2.7)
These equations have a symmetric solution g (z)= g (−z).
Equations (2.4)–(2.7) can be solved using the potential dynamicsmethod for solving reaction-diffusion
equations (see e.g. reference [28, Chap. 9]). Accordingly, g is regarded as the position of a hypothetical
point particle that evolves in time t and moves with a particle velocity vg . In doing so, the phase coor-
dinate z is replaced by the time variable t . The particle motion is subjected to a potential force with a
potential V (g , t) that depends on time. Using these replacements [i.e., z → t and dg /dz → vg (t)], equa-
tion (2.4) becomes
d
dt
g = vg ,
d
dt
vg =−c vg −
∂
∂g
V (g , t) (2.8)
43002-3
T.D. Frank
and we are looking for periodic solutions g (t) with period T = 2πR. During the time intervals [0,r ] and
[T − r,T ] we have F = 1 and the potential V gives rise to the “force” −dV /dg = kg −1. By contrast, for
t ∈ (r,T − r ) we have F = 0 which implies −dV /dg = kg . In total, the potential is given by
V (g , t)=
{
−0.5k(g −1/k)2 for t ∈ {[0,r ]∪ [T − r,T ]}modulus T ,
−0.5kg 2 for t otherwise .
(2.9)
The two different potential forms described by equation (2.9) are illustrated in figure 1 and will be re-
ferred to as type I and II potentials, respectively. Both potentials are inverted parabolic potentials. For
t ∈ {[0,r ]∪ [T − r,T ]} the potential has a peak at g = 1/k (top panel), otherwise the potential has a peak
at g = 0 (bottom panel). Equations (2.8) and (2.9) have to be solved under initial conditions g (t = 0) and
vg (t = 0) that satisfy
vg (t = 0) =−
µc
aγ
[a g (t = 0)+1]2 , (2.10)
see equation (2.5). Moreover, the periodicity condition g (t +T ) = g (t) implies that the boundary condi-
tions g (t = 0) = g (t = T ) and vg (t = 0) = vg (t = T ) hold. Finally, we have the constraint g (t) Ê 0 because
g in the original context reflects the concentration of camphor molecules. The potential dynamics sub-
jected to these initial and boundary conditions is then determined by the two types of repulsive potentials
shown in figure 1.
0 5 10
−4
−2
0
V
0 5 10
−4
−2
0
g
V
Figure 1. Potentials of type I (top) and II (bottom) defined by equation (2.9). Here k = 0.1.
Let us discuss period solutions g (t) corresponding to standing wave patterns g (z) associated with
an immobile camphor disc. For c = 0, equation (2.9) describes a Newtonian equation without damping
that is solved under the initial conditions vg = 0 and g (0) = A > 0. If A > 1/k , then we have g → ∞
for t →∞ because both potentials decay monotonously for g > 1/k. For g (0) = 1/k , a periodic solution
is not possible for 0 < r < πR. However, in the limiting case r = πR, the dynamics of the hypothetical
particle is subjected only to the potential I force with Vmax = 1/k (top panel of figure 1) such that g (0) =
g (t) = 1/k is the solution of equation (2.9). That is, the wave pattern g (z) is a constant when the water
channel is completely filled with a solid, ring-shaped camphor object. In summary, for 0 < r <πR we have
g (0) = A ∈ (0,1/k). Due to the potential I force, for t ∈ [0,r ], the particle moves “downhill” with respect
to the type I potential V , that is, to the left towards g = 0, and reaches at time t = r a point g (r ) = B .
At t = r , the potential switches from type I to type II. Since we have vg (r ) < 0, the particle continues
to move towards g = 0. However, at this stage it moves “uphill” with respect to the type II potential
V and, consequently, is de-accelerated. At a time point t∗, the particle has zero velocity (vg (t∗) = 0).
At this instance, the particle has reached its minimal position g (t∗) = gmin. Due to the impact of the
potential II force, the particle is accelerated and starts to move “downhill” with respect to the type II
potential V , that is, it moves to the right. Due to the symmetry of the problem at hand, the time point t∗
is half of the period: t∗ = T /2 = πR. When the particle moves to the right, g increases and eventually, at
43002-4
Chemical waves of camphor particles
t = T −r , the particle reaches the position g = B again. Note that the “uphill”movement B → gmin and the
“downhill” movement gmin → B are described by a time-reversible Newtonian equation, which implies
that the trajectory g (t) is symmetric with respect to t = T /2. At t = T − r we have vg > 0. In particular,
we have vg (T − r ) = −vg (r ). Moreover, the potential switches from type II to type I. Due to the “initial”
velocity vg (T −r ) > 0, the particle goes “uphill” in the type I potential V , slows down, and finally reaches
the location g = A with vg = 0 at time point t = T . In short, the periodic solution follows the sequence
g (0)=gmax → g (r )=B → g (T /2)=gmin → g (T−r )=B → g (T )=g (0)
(2.11)
and exhibits a single maximum at g (0) and a single minimum at g (T /2). Moreover, the trajectory is
symmetric with respect to t = T /2 for t ∈ [0,T ] (and symmetric with respect to t = 0 for t ∈ [−T /2,T /2]),
which means that the periodic standing wave pattern g (z) has a symmetry axis.
Periodic solutions with vg (0), 0 related to travelingwave patterns g (z) induced by a self-propagating
camphor disc can be discussed in a similar way. In this context, we note that a necessary condition for a
period solution is g (0) < 1/k again. In order to see this, let us assume that g (0) > 1/k holds. For c < 0, this
implies vg (0) > 0 which implies g →∞ for t →∞ because both potentials I and II decaymonotonously for
g > 1/k. For c > 0 and g (0) > 1/k , we see that the particle has initial velocity vg (0) < 0 andmoves towards
g = 1/k , which is the peak of the type I potential V (figure 1, top panel). If |vg (0)| is not sufficiently large,
it will not reach the point g = 1/k. Rather, we will have vg (t ′) = 0 with g (t ′) > 1/k for t ′ ∈ [0,r ] which
implies g →∞ to t →∞. That is, a necessary condition for a periodic solution starting at g (0) > 1/k is
that |vg (0)| is sufficiently large such that the particle passes the point g = 1/k during the interval [0,r ].
However, the particle should return somehow to the initial position g (0) > 1/k. Once the particle has
passed the point g = 1/k at t ′ ∈ [0,r ], the only way to return to the subspace g = (1/k,∞) is to pass the
point g = 1/k at a later time point t ′′ > t ′ with velocity vg (t ′′) > 0. From g (t ′′) = 1/k and vg (t ′′) > 0 it
follows that g → ∞ for t → ∞. In summary, periodic solutions with vg (0) , 0 and g > 0 only exist for
g (0) ∈ (0,1/k). By analogy to the previous discussion for the case c = vg (0) = 0, for c > 0 (camphor disc
and wave pattern traveling to the right on the ring coordinate x) we obtain the sequence
g (0) = A → g (r )= B < A → g (t∗) = gmin < B → g (T − r ) =C > gmin
→ g (t∗∗) = D = gmax > A → g (T ) = g (0) = A (2.12)
with 0< r < t∗ < T − r < t∗∗ < T and vg = 0 at t∗ and t∗∗. As indicated, the maximum is reached at t∗∗ ∈
[T − r,T ] and does not correspond to the initial position g (0). In any case, there is a single minimum and
a single maximum, which means that a traveling wave concentration pattern g (z) is single-peaked and
exhibits a single minimum. The condition g (t∗∗) > g (T ) = g (0) implies for the original problem that the
camphor disc is located to the “right” of the peak of the traveling wave concentration pattern. For c < 0,
we conclude in a similar vein that the traveling wave pattern is single-peaked and has a single minimum.
As anticipated above, the potential dynamic picture reveals that both the standing and traveling wave
patterns exhibit a single peak only. However, while the standing wave patterns g (z) exhibit a symmetry
axis, traveling wave patterns g (z) are non-symmetric.
The next objective is to explicitly formulate the scenarios described by the sequences (2.11) and (2.12)
in order to obtain analytical solutions for g (t) and for the patterns g (z).
2.2. Standing waves
We will derive the next symmetric solutions of equation (2.8) with vg (0) = 0. In this case, it is suffi-
cient to determine the trajectory g (t) in the interval [0,T /2 = πR] because, as we will see below we can
take advantage of the constraint vg (T /2) = 0, see equation (2.11). For t ∈ [0,r ], the solution g (t) satisfies
equation (2.8) with c = 0, vg (0) = 0, and g (0) ∈ (0,1/k). The solution reads
g (t)=
[
g (0)−
1
k
]
cosh(
p
kt)+
1
k
. (2.13)
43002-5
T.D. Frank
In particular, we obtain
g (r )=
[
g (0)−
1
k
]
cosh(
p
kr )+
1
k
, vg (r ) =
p
k
[
g (0)−
1
k
]
sinh(
p
kr ) . (2.14)
For t ∈ (r,T /2] the solution g (t) satisfies equation (2.8) for c = 0 and the initial conditions g (r ) and vg (r )
given above. The solution reads
g (t) =
1
2
[
g (r )+
vg (r )
p
k
]
exp
[p
k(t−r )
]
+
1
2
[
g (r )−
vg (r )
p
k
]
exp
[
−
p
k(t−r )
]
= g (r )cosh
[p
k(t − r )
]
+
vg (r )
p
k
sinh
[p
k(t − r )
]
, (2.15)
which implies that
vg (t) =
p
kg (r )cosh
[p
k(t − r )
]
+ vg (r )cosh
[p
k(t − r )
]
. (2.16)
The functions (2.13)–(2.16) involve the unknown parameter g (0). Let us use the constraint vg (T /2) = 0
to determine g (0). Substituting t = πR with vg (t = πR) = 0 into equation (2.16) and substituting equa-
tion (2.14) into equation (2.16) as well, we obtain
ξw =−
1
k
sinh
[p
k(πR − r )
]
< 0 (2.17)
with ξ= g (0)−1/k and
w = sinh(
p
kr )cosh
[p
k(πR − r )
]
+cosh(
p
kr )sinh
[p
k(πR − r )
]
> 0.
(2.18)
Since the right hand side of equation (2.17) is negative and w > 0, we conclude that ξ< 0. In addition, we
see that
g (0) = ξ+
1
k
=
1
k
1−
sinh
[p
k(πR − r )
]
w
(2.19)
holds. From w > sinh[
p
k(πR − r )] it follows that the factor sinh[
p
k(πR − r )]/w > 0 is smaller than 1
such that g (0) ∈ (0,1/k). That is, g (0) satisfies the necessary condition for periodic solutions.
Equations (2.13)–(2.15) in combination with the initial condition (2.19) describe the analytical solution
g (t) in t ∈ [0,T /2]. As shown above, solutions of equation (2.8) with c = 0 and vg (0) = 0 are symmetric
with respect to t = 0 (and t = T /2). Consequently, solutions on a full period are given by g (t) defined by
equations (2.13)–(2.15) and (2.19) for t ∈ [0,T /2] and g (t) = g (−t) for t ∈ [−T /2,0]. Likewise the pattern
of the standing wave solution g (z) can be computed from equations (2.13)–(2.15) and (2.19) for z ∈ [0,πR]
and g (z)= g (−z) for z ∈ [−πR,0].
Figure 2 shows g (z) for a standing wave pattern. The solid line was obtained from equations (2.13)–
(2.15) and (2.19) and g (z) = g (−z). The circles were obtained by solving equation (2.8) numerically in
the interval t ∈ [0,T /2] with initial condition vg (0) = 0 by means of an Euler forward algorithm for the
dynamics of a Newtonian particle (single time step 10−5). Numerical and analytical solution methods
showed consistent results. Although the peak of the pattern shown in figure 2 looks kinky, from equa-
tions (2.13)–(2.15) it follows that in fact the functions g (t) and g (z) are smooth functions at t = z = 0 (i.e.,
continuously differentiable). This is illustrated in the insert of figure 2 that depicts the standing wave
pattern in a small region around z = 0.
Let us briefly address the case r → πR. For r = πR equation (2.13) yields g (t) = 1/k. Likewise, equa-
tion (2.19) reduces to g (0) = 1/k. This is consistent with the conclusion drawn above that for r = πR the
ring channel exhibits a homogeneous concentration pattern g = 1/k.
43002-6
Chemical waves of camphor particles
Figure 2. Standing wave solution g (solid line) computed from the analytical solution defined by equa-
tions (2.13)–(2.15) and (2.19). Circles correspond to the graph g determined by a numerical solution
method (see text). Insert shows a detail of the graph g (z) around z = 0. Parameters: k = 0.1, R = 5.0,
r = 0.2 (as in reference [26]).
2.3. Traveling waves
Our next objective is to find solutions of equations (2.8)–(2.10) for c , 0 which implies vg (0) , 0. To
this end, we will use equations (2.8) and (2.9) to determine the trajectory g (t) and the particle velocity
vg (t) in the full period [0,T ] as function of the unknown parameters g (0), vg (0) and c. We will then use
the three constraints given by equation (2.10) and by the two periodicity requirements g (0) = g (T ) and
vg (0) = vg (T ) to determine g (0), vg (0) and c.
For t ∈ [0,r ], equation (2.8) explicitly reads d2g /dt 2 = −cvg +kg −1. The solution involves the un-
known parameters c, vg (0), and g (0) and reads
g (t)= A1,0 exp{λ1t }+ A2,0 exp{λ2t }+
1
k
(2.20)
with eigenvalues
λ1(c) =−
c
2
+
√
c2
4
+k > 0, λ2(c) =−
c
2
−
√
c2
4
+k < 0 (2.21)
and amplitudes
A1,0 =
[
g (0)−
1
k
−
vg (0)
λ2
]
1
1−λ1/λ2
,
A2,0 =
[
g (0)−
1
k
−
vg (0)
λ1
]
1
1−λ2/λ1
. (2.22)
In particular, we obtain
g (r ) = A1,0 exp{λ1r }+ A2,0 exp{λ2r }+
1
k
,
vg (r ) = A1,0λ1 exp{λ1r }+ A2,0λ2 exp{λ2r } . (2.23)
For t ∈ (r,T − r ), equation (2.8) explicitly reads d2g /dt 2 = −cvg + kg . The solution involves the initial
conditions g (r ) and vg (r ) listed above and reads
g (t)= A1,r exp{λ1(t − r )}+ A2,r exp{λ2(t − r )} (2.24)
with the amplitudes
A1,r =
[
g (r )−
vg (r )
λ2
]
1
1−λ1/λ2
, A2,r =
[
g (r )−
vg (r )
λ1
]
1
1−λ2/λ1
.
(2.25)
43002-7
T.D. Frank
In particular, we obtain
g (T − r ) = A1,r exp{λ1(T −2r )}+ A2,r exp{λ2(T −2r )} ,
vg (T − r ) = A1,rλ1 exp{λ1(T −2r )}+ A2,r λ2 exp{λ2(T −2r )} . (2.26)
Finally, for t ∈ [T −r,T ] equation (2.8) explicitly reads d2g /dt 2 =−cvg +kg −1. The solution involves the
initial conditions g (T − r ) and vg (T − r ) (determined above) and reads
g (t)= A1,T−r exp{λ1[t − (T − r )]}+ A2,T−r exp{λ2[t − (T − r )]}+
1
k
(2.27)
with amplitudes
A1,T−r =
[
g (T − r )−
1
k
−
vg (T − r )
λ2
]
1
1−λ1/λ2
,
A2,T−r =
[
g (T − r )−
1
k
−
vg (T − r )
λ1
]
1
1−λ2/λ1
. (2.28)
In particular, we obtain
g (T ) = A1,T−r exp{λ1r }+ A2,T−r exp{λ2r }+
1
k
︸ ︷︷ ︸
f1
,
vg (T ) = A1,T−rλ1 exp{λ1r }+ A2,T−rλ2 exp{λ2r }
︸ ︷︷ ︸
f2
. (2.29)
As mentioned above, we put g (0)= g (T ) and vg (0) = vg (T ) such that
g (0) = f1
(
g (0), vg (0),c
)
, vg (0) = f2
(
g (0), vg (0),c
)
. (2.30)
The functions f1 and f2 occurring in equation (2.30) are defined by the expressions on the right hand
sides of equation (2.29) and by the amplitudes and initial conditions listed in table 1. These amplitudes
and initial conditions are explicitly defined by equations (2.22), (2.23), (2.25), (2.26) and (2.28) with the
eigenvalues λ1 and λ2 given by equation (2.21).
The two relations in equation (2.30) together with the constraint (2.10) provide three equations to
determine the parameters g (0), vg (0), and c. Once g (0), vg (0), and c have been determined, the trajectory
g (t) can be computed from equations (2.20), (2.24) and (2.27). In doing so, the analytical solution for the
traveling wave pattern g (z) can be found. It can be useful to eliminate c with the help of equation (2.10)
like c =−aγvg (0)/{µ[ag (0)+1]2}. Equation (2.30) then becomes
g (0)= h1
(
g (0), vg (0)
)
, vg (0) = h2
(
g (0), vg (0)
)
(2.31)
Table 1. Amplitude and “initial conditions” involved in the definition of the functions f1 and f2 occurring
in equations (2.29) and (2.30).
Amplitudes Initial conditions
A1,T−r [g (t − r ), vg (T − r ),c] g (T − r )[A1,r , A2,r ,c]
A2,T−r [g (t − r ), vg (T − r ),c] vg (T − r )[A1,r , A2,r ,c]
A1,r [g (r ), vg (r ),c] g (r )[A1,0, A2,0,c]
A2,r [g (r ), vg (r ),c] vg (r )[A1,0, A2,0,c]
A1,0[g (0), vg (0),c]
A2,0[g (0), vg (0),c]
43002-8
Chemical waves of camphor particles
Figure 3. Traveling wave solution g (solid line) computed from the analytical solution defined by equa-
tions (2.20), (2.24) and (2.27). Circles were obtained by the same numerical solution method as used in
figure 2. Parameters: a = 0.5, γ = 2.0, k = 0.1, R = 5.0, r = 0.2, µ = 0.1 (as in reference [26]). Estimated
parameters obtained both from MATLAB fsolve and the w1(t) and w2(t) dynamical system method:
g (0) = 0.3151, vg (0) = 0.1516, c = 1.1315.
with
h1 = f1
(
·,c =−aγvg (0)/
{
µ[ag (0)+1]2
})
, (2.32)
h2 = f2
(
·,c =−aγvg (0)/
{
µ[ag (0)+1]2
})
. (2.33)
We numerically solved equation (2.31) for a given set of parameters a, γ, k , R, r , µ in order to ob-
tain g (0) and vg (0). Subsequently, we calculated c from c = −aγvg (0)/{µ[ag (0)+1]2}. We solved equa-
tion (2.31) in two different ways. First, we used the MATLAB solver (fsolve) for coupled nonlinear func-
tions. Second, we numerically solved a dynamical system that exhibits a fixed point consistent with equa-
tion (2.31). More precisely, the two-variable dynamical system defined by dw1(t)/dt = w1 −h1(w1, w2)
and dw2(t)/dt = h2(w1, w2)−w2 was numerically solved (Euler forward with single time step 10−5) for
initial conditions w1(0) = 0.5/k and w2(0) = −0.1 (to obtain solutions with c > 0). For both methods, the
same numerical values for g (0), vg (0), and c were obtained up to 4 digits after the decimal point. Subse-
quently, the traveling wave pattern g (z) was computed from equations (2.20), (2.24) and (2.27). The result
is shown in figure 3. A detail of the pattern for values of z close to zero is shown in the insert. We see that
the camphor disc at z = 0 is located to the “right” of the peak of the concentration wave (i.e., the peak
position is to the “left” of z = 0).
Finally, we compute the bifurcation diagram for the same set of parameters a, γ, k , R, r as used
to calculate the wave pattern g (z) in figure 3. However, µ was not fixed. Rather, the parameter µ was
considered as control parameter and was varied in the range µ= [0,0.2]. The traveling wave velocity c as
function of µ was determined using the two aforementioned methods (MATLAB fsolve and dynamical
system w1(t), w2(t)). The propagation velocities c thus obtained are shown in figure 4. The bifurcation
diagram indicates that for periodic solutions g (z) there is a pitchfork bifurcation at a critical value µc
such that at µ = µc, the standing wave solution with c = 0 becomes unstable and two stable traveling
wave solutions emerge with either c > 0 or c < 0.
3. Discussion
A potential dynamics approach was used to determine the shape of chemical concentration patterns
on a ring-shaped water channel associated with a camphor disc floating on the water surface of the
channel either in an immobile or self-propelling fashion. The potential dynamics approach allowed us
43002-9
T.D. Frank
0 0.05 0.1 0.15 0.2
−5
0
5
µ
c
Figure 4. Pitchfork bifurcation describing the transition of a camphor disc floating on a water ring from
an immobile mode to a self-motion mode. Two methods were used: MATLAB fsolve (solid line) and
dynamical system w1(t), w2(t) (circles). See text for details.
to qualitatively determine the shape of the wave patterns and to show that the concentration patterns
are bounded from above. In rescaled dimensionless units, the maximal camphor concentration of stand-
ing or traveling wave patterns cannot exceed the inverse of the decay constant k. With the help of the
potential dynamics approach, analytical expressions for standing and traveling wave patterns were de-
rived. Finally, by means of the analytical expression for traveling wave patterns, the bifurcation diagram
describing the transition from standing to traveling waves (immobility to self-motion) when the friction
parameter µ is decreased was determined numerically.
The stable branches of the bifurcation diagram for values of µ smaller than the critical value µc but
close to µc (i.e., µ = µc − ǫ with ǫ > 0 small) may be described like µc −µ = L c2 +O(c4), where L > 0
holds and c corresponds both to the camphor disc velocity and the traveling wave velocity. Taking the
immobile-disc and standing-wave solution c = 0 into account, this relationship between µ and c becomes
(µc−µ)c = L c3 +O(c5) and describes a supercritical pitchfork bifurcation. This relationship was analyt-
ically derived in a previous study assuming natural boundary conditions [26]. Consequently, our consid-
erations indicate that a supercritical pitchfork bifurcation can also be observed in the case of periodic
boundary conditions.
In the aforementioned study it was also shown that under natural boundary conditions and when the
radius of the camphor disc becomes large, the velocity c and the control parameter µ can be related like
(µc−µ)c = L1c3 +L2c5 +O(c7) with L1 < 0 and L2 > 0. When the solution c = 0 is ignored, the simplified,
truncated relation µc−µ= L1c2+L2c4 describes the branches of self-propelling camphor discs associated
with traveling wave concentration patterns. It has the shape of a W when considering ǫ = µc −µ as a
function of c. Consequently, this second relationship describes a subcritical pitchfork bifurcation when
considering c as a function of ǫ=µc−µ. The subcritical bifurcation predicts that for appropriately chosen
values of µ, there are two different propagation velocities c > 0 (and likewise two velocities c < 0). We
were not able to find a counterpart of this observation for the case of periodic boundary conditions.
That is, the two numerical methods described in section 2.3 to determine the bifurcation diagram did not
indicate the existence of a W shaped relationship between µc−µ and c. In this context, it is important to
note that in reference [26] it was shown that the range of values µ for which two different positive (or
negative) velocities c exist is relatively small. Therefore, the issue of a subcritical pitchfork bifurcation
for solutions of the reaction-diffusion equation (1.2) subjected to periodic boundary conditions may be
addressed bymore sophisticated analytical or numerical methods that go beyond the scope of the present
study.
43002-10
Chemical waves of camphor particles
Acknowledgements
Preparation of this manuscript was supported in part by National Science Foundation under the IN-
SPIRE track, grant BCS-SBE-1344275.
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T.D. Frank
Точнi розв’язки для хвиль хiмiчної концентрацiї
самоурухомлювальних камфорних частинок, що рухаються
по кiльцю: нова перспектива потенцiальної динамiки
Т.Д. Франк
Вiддiл психологiї, Унiверситет м. Коннектикут, CT 06269, США
Розвинуто метод потенцiальної динамiки для того, щоб вивчити картини перiодичних стоячих i бiжучих
хвиль, пов’язаних iз самоурухомлювальними камфорними об’єктами, якi рухаються на кiльцеподiбних
водяних каналах. Отримано точнi розв’язки хвильвових картин. Отримано напiваналiтично дiаграму бi-
фуркацiї, яка описує перехiд мiж нерухомою i самоурухомлювальною модами камфорних об’єктiв. Бiфур-
кацiя є вилоподiбна, що узгоджується з попередньою теоретичною роботою, в якiй розглянуто природнi
граничнi умови.
Ключовi слова: хвилi хiмiчної концентрацiї, саморух, вилоподiбна бiфуркацiя
43002-12
Introduction
Potential dynamics perspective of the chemical wave pattern associated with a self-propelling camphor disc
Potential dynamics point of view
Standing waves
Traveling waves
Discussion
|