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Letters

Journal of the Physical Society of Japan 86, 113801 (2017)https://doi.org/10.7566/JPSJ.86.113801

Thermally Driven Elastic Micromachines
Yuto Hosaka1, Kento Yasuda1, Isamu Sou1, Ryuichi Okamoto2, and Shigeyuki Komura1+
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan
2
Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan(Received August 28, 2017; accepted September 19, 2017; published online October 5, 2017)

We discuss the directional motion of an elastic three-sphere micromachine in which the spheres are in equilibriumwith independent heat baths having diﬀerent temperatures. Even in the absence of prescribed motion of springs, such a
micromachine can gain net motion purely because of thermal ﬂuctuations. A relation connecting the average velocity
and the temperatures of the spheres is analytically obtained. This velocity can also be expressed in terms of the averageheat ﬂows in the steady state. Our model suggests a new mechanism for the locomotion of micromachines innonequilibrium biological systems.

Microswimmers are tiny machines that swim in a ﬂuid,
such as sperm cells or motile bacteria, and are expected to be
applied to microﬂuidics and microsystems.1) By transforming
chemical energy into mechanical work, these objects change
their shape and move in viscous environments. Over thelength scale of micromachines, the ﬂuid forces acting onthem are governed by viscous dissipation. According toPurcell’s scallop theorem,2) time-reversal body motion
cannot be used for locomotion in a Newtonian ﬂuid. As
one of the simplest models exhibiting broken time-reversal
symmetry, Najaﬁ and Golestanian proposed a three-sphereswimmer,3,4) in which three in-line spheres are linked by twoarms of varying length. Recently, Pande et al. and the present
authors independently proposed a generalized three-spheremicroswimmer in which the spheres are connected by twoelastic springs.5,6)
In the previous three-sphere microswimmer models, either
the arm lengths or the natural lengths of the springs wereassumed to undergo prescribed cyclic motions.3–6) Such
active motions can lead to net locomotion if the swimmingstrokes are nonreciprocal. From a practical point of view,
however, it is not a simple task to implement these motions at
micron length scales. Another approach for extending the
Najaﬁ–Golestanian model is to consider the arm motions as
occurring stochastically.7–10) Although proteins or enzymes
are naturally designed to include such sophisticated molecular mechanisms, it is still a substantial challenge to constructthem artiﬁcially. It should also be noted that thermal
agitations due to surrounding ﬂuids become more signiﬁcantat these small scales.
In this letter, by using an elastic three-sphere micromachine,5,6) we suggest a new mechanism for locomotionthat is purely induced by thermal ﬂuctuations. To highlight
this eﬀect, we do not consider any prescribed motion of thenatural lengths.6) On the other hand, the key assumption in
our model is that the three spheres are in equilibrium withindependent heat baths having diﬀerent temperatures. In thiscase, heat transfer occurs from a hotter sphere to a colder one,
driving the whole system out of equilibrium. We show that a
combination of heat transfer and hydrodynamic interactions
among the spheres can lead to directional locomotion in thesteady state. We analytically obtain the expression for the
average velocity in terms of the sphere temperatures. Ourﬁnding is further conﬁrmed by numerical simulations. Since
our model has a similarity to a class of thermal ratchet models

η

x1
T1
a

KA

x2
T2

KB

x3
T3

Fig. 1. (Color online) Thermally driven elastic three-sphere micromachine
in a ﬂuid of viscosity η. Three spheres of radius a are connected by two
harmonic springs with elastic constants KA and KB . The time-dependent
positions of the spheres are denoted by xi ðtÞ (i ¼ 1; 2; 3) in a one-dimensional
coordinate system. Importantly, the three spheres are in equilibrium withindependent heat baths at temperatures Ti .

that have been intensively studied before,11–13) the suggested
mechanism is relevant to nonequilibrium dynamics ofproteins and enzymes in biological systems.
As schematically shown in Fig. 1, we consider a threesphere micromachine and take into account the elasticity inthe internal spring motions.5,6) This model consists of three
hard spheres of radius a connected by two harmonic springs
A and B with spring constants KA and KB , respectively. The
natural length of the springs, ‘, is assumed to be constant.
The total energy is given by
E¼

KA
KB
ðx2  x1  ‘Þ2 þ
ðx3  x2  ‘Þ2 ;
2
2

ð1Þ

where xi ðtÞ (i ¼ 1; 2; 3) are the positions of the three spheresin a one-dimensional coordinate system and we assume x1 <x2 < x3 without loss of generality. Owing to the hydrodynamic interactions, each sphere exerts a force on theviscous ﬂuid of shear viscosity η and experiences an opposite
force from it. In general, the surrounding medium can beviscoelastic,14) but such an eﬀect is not included in this letter.
We consider a situation in which the three spheres are in
equilibrium with independent heat baths at temperatures Ti .
When these temperatures are diﬀerent, the system is drivenout of equilibrium because a heat ﬂux is generated from ahotter sphere to a colder one. Denoting the velocity of eachsphere by x_i , we can write the equations of motion of thethree spheres as
x_1 ¼

113801-1

KA
KA ðx2  x1  ‘Þ
ðx2  x1  ‘Þ 
x2  x1
6a4
KB ðx3  x2  ‘Þ KB ðx3  x2  ‘Þþþ 1 ;
x2  x1
x3  x1
4
4

ð2Þ

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Letters

J. Phys. Soc. Jpn. 86, 113801 (2017)

KA ðx2  x1  ‘ÞKAðx2  x1  ‘Þ
x2  x1
4
6a
KB
KB ðx3  x2  ‘Þ
ðx3  x2  ‘Þ þ 2 ;
þx3  x2
6a4
KA ðx2  x1  ‘Þ KA ðx2  x1  ‘Þx_3 ¼
x3  x1
x3  x2
4
4
KB ðx3  x2  ‘ÞKB
ðx3  x2  ‘Þ þ 3 ;þ
x3  x2
4
6a

x_2 ¼

ð3Þ

ð4Þ

ð5Þ

where Dij is the mutual diﬀusion coeﬃcient. When i ¼ j, Dii
is simply given by the Stokes–Einstein relation, i.e.,
kB Ti
;
Dii ¼
6a

ð6Þ

where kB is the Boltzmann constant. When i ≠ j, on the other
hand, we assume the following general relation:kB ðTi ; Tj Þ
Dij ¼;
4jxi  xj j

ð7Þwhere ðTi ; Tj Þ is a function of Ti and Tj . For example, therelevant eﬀective temperature can be the mobility-weighted
average,16) which in the present case is given by ðTi ; Tj Þ ¼
ðTi þ Tj Þ=2 because all the spheres have the same size.
However, its explicit functional form is not needed here,
and it only needs to satisfy an appropriate ﬂuctuation
dissipation theorem in thermal equilibrium, i.e., Ti ¼ Tj .
This is because we only consider the limit of a  ‘ in thepresent study.
It is convenient to introduce a characteristic time scale
given by  ¼ 6a=KA . We further deﬁne the ratio between
the two spring constants as  ¼ KB =KA . We shall denote thetwo spring extensions byuA ðtÞ ¼ x2  x1  ‘;

uB ðtÞ ¼ x3  x2  ‘:

ð8Þ

Notice that these quantities are related to the sphere velocities
in Eqs. (2)–(4) as u_A ¼ x_2  x_1 and u_B ¼ x_3  x_2 , respectively. In the following analysis, we generally assume thatuA , uB  ‘ as well as a  ‘, and focus only on the leadingorder contribution.
To present the essential outcome of the model, we ﬁrst
consider the simplest symmetric case, i.e., KA ¼ KB ( ¼ 1).
We introduce theR 1bilateral Fourier transform for any functionfðtÞ asRfð!Þ ¼ 1 dt fðtÞei!t and the inverse transform as
1fðtÞ ¼ 1 ðd!=2Þfð!Þei!t . Solving the time derivative ofEq. (8) with the aid of Eqs. (2)–(4) in the frequency domain,
we obtainuA ð!Þ 

ð2 þ i!Þ1 ð!Þ  ð1 þ i!Þ2 ð!Þ  3 ð!Þ
3  4i! þ ð!Þ2
þ OðaÞ;uB ð!Þ 1ð1 þ 2 þ 3 Þ þ Oðu3A ; u3B Þ:
3
By taking its statistical average, we further obtainhVðtÞi  uA ð!  !0 ÞuA ð!0 Þi þ Oða2 ; u3A ; u3B Þ:

ð12Þ

ð13Þ

Next, we substitute Eqs. (9) and (10) into Eq. (13) and use
the relation hi ð!Þj ð!0 Þi ¼ ð2Þð2Dij Þð! þ !0 Þ, as directly
obtained from Eq. (5). After some calculation, we havehVð!Þi ¼

a
2iðD33  D11 Þð!Þ:2
4i  !2
4‘

ð14Þ

Notice that the cross correlations for i ≠ j can be neglected
here because these are higher-order contributions of Oða2 Þ.
Transforming back to the time domain, we obtain the averagevelocity as
hVi ¼

akB ðT3  T1 Þ kB ðT3  T1 Þ¼;
16KA ‘296‘2

ð15Þ

where we have used Eq. (6).
The above expression is an important result of this letterand deserves further discussion. The average velocity is
proportional to the temperature diﬀerence T3  T1 . Since wehave assumed x1 < x2 < x3 , the swimming direction is froma colder sphere to a hotter one, i.e., hVi > 0 when T3 > T1
and vice versa. It is also remarkable that Eq. (15) does notdepend on the temperature T2 of the middle sphere. HencehVi ¼ 0 when T1 ¼ T3 even though T1 and T3 can be
diﬀerent from T2 . However, the presence of the middle
sphere is essential for directional locomotion because thehydrodynamic interactions among the three spheres are
responsible for it. Notice that a two-sphere micromachine
cannot move even if the temperatures are diﬀerent. This isbecause, if we keep only the ﬁrst two terms in Eqs. (2) and(3) plus the noise terms 1 and 2 for the two spheres, we canimmediately see that hx_1 þ x_2 i ¼ h1 i þ h2 i ¼ 0.
Having discussed the simplest symmetric case, we now
present the result for general asymmetric cases when KA ≠KB ( ≠ 1). By repeating the same calculation as before, thetwo spring extensions in Eq. (8) now becomeð9Þð2 þ i!Þ1 ð!Þ  ð þ i!Þ2 ð!Þ  3 ð!Þ
3  2ð1 þ Þi! þ ð!Þ2
þ OðaÞ;uB ð!Þ 
ð10Þ

ahu2 ðtÞ  u2A ðtÞi þ Oða2 ; u3A ; u3B Þ;
8‘2  B

ð11Þ

where we have used huA ðtÞi ¼ huB ðtÞi ¼ 0 in the lowestorder of a. In the Fourier domain, Eq. (12) can be written interms of convolution as
Z1
ad!0
huB ð!  !0 ÞuB ð!0 Þ
hVð!Þi  2
8‘  1 2

uA ð!Þ 1 ð!Þ þ ð1 þ i!Þ2 ð!Þ  ð2 þ i!Þ3 ð!Þ
3  4i! þ ð!Þ2

þ OðaÞ;where OðaÞ indicates the terms of the order of a.

The velocity of a three-sphere micromachine is generally
given by VðtÞ ¼ ðx_1 þ x_2 þ x_3 Þ=3, which now becomes


a
u2 u2
uB  uA þ B  AVðtÞ 
4‘
2‘ 2‘
þwhere we have used the Stokes’ law for a sphere and theOseen tensor in a three-dimensional viscous ﬂuid.
Furthermore, the white-noise sources i ðtÞ have zero mean,hi ðtÞi ¼ 0, and their correlations satisfy15)hi ðtÞj ðt0 Þi ¼ 2Dij ðt  t0 Þ;

Y. Hosaka et al.

1 ð!Þ þ ð1 þ i!Þ2 ð!Þ  ð2 þ i!Þ3 ð!Þ
3  2ð1 þ Þi! þ ð!Þ2

þ OðaÞ:Then, the average velocity is113801-2

ð16Þ

ð17Þ
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J. Phys. Soc. Jpn. 86, 113801 (2017)hVðtÞi 

Letters

Y. Hosaka et al.

ahu2B ðtÞ  u2A ðtÞ þ 3ð1  ÞuA ðtÞuB ðtÞi
8‘2 
ð18Þ
þ Oða2 ; u3A ; u3B Þ:

100

The substitution of Eqs. (16) and (17) into the Fouriertransformed expression of Eq. (18) yields the averagevelocity hVð!Þi similar to Eq. (14). By performing theinverse Fourier transform, we ﬁnally obtain the general
expression for the average velocity:kB½ð2  5ÞT1
hVi ¼
144‘2 ð1 þ Þ
 ð7  7ÞT2 þ ð5  2ÞT3 :

50

0

ð19Þ

When  ¼ 1, Eq. (19) reduces to Eq. (15), as expected.
When the three temperatures are identical, i.e., T1 ¼ T2 ¼ T3 ,one can also show that the velocity vanishes, hVi ¼ 0. This
indicates that an elastic three-sphere micromachine can attaina ﬁnite velocity owing to the temperature diﬀerence among
the spheres, rather than its structural asymmetry.
To conﬁrm our analytical prediction, we performed
numerical simulations of the coupled stochastic equations
in Eqs. (2)–(4) when  ¼ 1. The equations can be discretizedaccording to Storatonovich interpretation.13) Then, thestrength of thermal noise acting on each sphere is determinedby a dimensionless parameter Si ¼ ½2kB Ti =ðKA ‘2 Þ1=2 . We
performed simulations for (i) thermally asymmetric nonequilibrium cases in which S1 ¼ S2 ¼ 0 and S3 ¼ 0:2 or S3 ¼
0:141, and (ii) a thermally symmetric equilibrium casein which S1 ¼ S2 ¼ S3 ¼ 0:067. For comparison, we also
performed a simulation for (iii) a thermally asymmetricnonequilibrium two-sphere case in which S1 ¼ 0 andS2 ¼ 0:133 (sphere 3 does not exist). The average over 100
independent runs has been taken for each case.
In Fig. 2, we plot the obtained center-of-mass positionXðtÞ ¼ ðx1 þ x2 þ x3 Þ=3 as a function of time t. For case (i),we clearly see that a micromachine migrates towards the
positive direction with well-deﬁned ﬁnite velocities. The
dashed lines correspond to the analytical result in Eq. (15),which is in good agreement with the numerical simulations.
However, a micromachine cannot gain any net displacement
for cases (ii) and (iii). Our simulation result clearly
demonstrates that a three-sphere micromachine can acquire
directional motion because of thermal ﬂuctuations only when
the three spheres have diﬀerent temperatures.
In the above numerical simulations, a three-sphere micromachine undergoes not only ballistic motion but also
diﬀusive motion due to the presence of thermal ﬂuctuations.
The crossover time separating these two diﬀerent regimes canbe roughly estimated by the condition 2Dt  hVi2 t2 , where
the total diﬀusion coeﬃcient is approximately given byD  kB T=ð18aÞ with T ¼ ðT1 þ T2 þ T3 Þ=3. By denoting
the temperature diﬀerence in Eq. (15) as T ¼ T3  T1 ,the crossover time is roughly obtained as t  D=hVi2 
‘4 T=½akB ðTÞ2 . When S1 ¼ S2 ¼ 0 and S3 ¼ 0:2, forexample, we estimate t  300, which is much smaller than
the total simulation time in Fig. 2.
Next, we argue that the analytically obtained velocity inEqs. (15) or (19) can be related to the ensemble averageof heat ﬂows in the steady state. Within the framework of“stochastic energetics” proposed by Sekimoto,13) the heat
gained by the i-th sphere per unit time is expressed as

0

2

4

6

8

10

Fig. 2. (Color online) Simulations of scaled center-of-mass position X=‘of an elastic micromachine as a function of scaled time t= when a=‘ ¼ 0:1.
The strength of thermal noise is Si ¼ ½2kB Ti =ðKA ‘2 Þ1=2 . Case (i): thermallyasymmetric nonequilibrium three-sphere micromachine with S1 ¼ S2 ¼ 0
and S3 ¼ 0:2 (red) or S3 ¼ 0:141 (green). Case (ii): thermally symmetricequilibrium three-sphere micromachine with S1 ¼ S2 ¼ S3 ¼ 0:067 (black).
Case (iii): thermally asymmetric nonequilibrium two-sphere micromachinewith S1 ¼ 0 and S2 ¼ 0:133 (cyan). The dashed lines are the plots ofEq. (15) with the respective parameters.

dQi¼ 6aðx_i þ i Þx_i ;ð20Þ
dt
where x_i and i are given by Eqs. (2)–(4). In the calculationof average heat ﬂows, we also consider terms up to theleading-order contribution of a=‘. For example, only the ﬁrst
term on the r.h.s. of Eq. (2), KA uA =ð6aÞ, and the noise
term, 1 , are taken into account when we eliminate x_1 in
Eq. (20). We further use the statistical properties of quantities
such as hu2A i and huA 1 i, which can be estimated according
to Eqs. (16) and (5).
Then, the lowest-order average heat ﬂows are obtained
as


dQ1
kB
½ð3 þ 2ÞT1  ð3 þ ÞT2  T3 ; ð21Þ¼
dt 0 6ð1 þ Þ


dQ2
kB
½ð3 þ ÞT1 þ ð3 þ 2 þ 32 ÞT2¼
dt 0 6ð1 þ Þ ð þ 32 ÞT3 ;

ð22Þ

dQ3
kB
½T1  ð þ 32 ÞT2¼
dt 0 6ð1 þ Þþ ð2 þ 32 ÞT3 ;ð23Þwhich all vanish when T1 ¼ T2 ¼ T3 . It is also remarkablethat the above lowest-order heat ﬂows satisfy12,13)






dQ1
dQ2
dQ3
þ
þ
¼ 0:
ð24Þ
dt 0
dt 0
dt 0
Assuming a linear relation between the velocity in Eq. (19)and the heat ﬂows in Eqs. (21)–(23), we obtain an alternative
expression for the velocity:




 
a
3  5 dQ1
5  3 dQ3
hVi ¼
þ: ð25Þ
8KA ‘2 1 þ  dt 0 ð1 þ Þ dt 0
For the symmetric case of  ¼ 1 corresponding to Eq. (15),the above expression reduces to113801-3©2017 The Physical Society of Japan

J. Phys. Soc. Jpn.
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J. Phys. Soc. Jpn. 86, 113801 (2017)

hVi ¼

a
8KA ‘2





 
dQ3
dQ1

:
dt 0
dt 0

Letters

ð26Þ

This relation indicates that the average velocity is determinedby the net heat ﬂow between spheres 1 and 3.
Finally, we brieﬂy comment on previous relevant works.
Using coupled Langevin equations, Dunkel and Zaid
investigated the interplay between the diﬀusive and selfdriven behaviors of an elastic three-sphere swimmer.17) In
this work, however, the temperature of the system wasassumed to be uniform. In addition, hydrodynamic simulations of a self-thermophoretic Janus particle were reported
in Ref. 18 to reproduce the experimental result.19) Again, ourmodel diﬀers from this model because thermal ﬂuctuationsof internal degrees of freedom cause the locomotion ofan elastic micromachine. We also note from Eq. (19) thathVi ≠ 0 for symmetric temperatures T1 ¼ T3 ≠ T2 as long as
the structural asymmetry exists.
In summary, we have shown that an elastic three-spheremicromachine in a viscous ﬂuid can acquire directionalmotion because of thermal ﬂuctuations when the spheres
have diﬀerent temperatures. We have obtained an expression
for the average velocity that is related to the temperatures andaverage heat ﬂows. Such a mechanism for the locomotion of
micromachines is expected to play important roles innonequilibrium biological systems. In the future, we shallgeneralize our calculation to the case in which the spheres
have diﬀerent sizes.4) In such a calculation, one needs to take
into account higher-order contributions in a and uA , uB . It
would be interesting to investigate how these nonlinear
contributions aﬀect the nonequilibrium dynamics of thermally driven micromachines.

Y. Hosaka et al.

Acknowledgments S.K. and R.O. acknowledge support from a Grant-inAid for Scientiﬁc Research on Innovative Areas “Fluctuation and Structure”(Grant No. 25103010) from the Ministry of Education, Culture, Sports, Scienceand Technology of Japan and from a Grant-in-Aid for Scientiﬁc Research (C)(Grant No. 15K05250) from the Japan Society for the Promotion of Science(JSPS).

+

komura@tmu.ac.jp
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113801-4©2017 The Physical Society of Japan