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Letters

Journal of the Physical Society of Japan 86, 043801 (2017)https://doi.org/10.7566/JPSJ.86.043801

Swimmer-Microrheology
Kento Yasuda, Ryuichi Okamoto, and Shigeyuki Komura+
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Hachioji, Tokyo 192-0397, Japan(Received February 9, 2017; accepted February 28, 2017; published online March 29, 2017)

We discuss the locomotion of a three-sphere microswimmer in a viscoelastic medium and propose a new type ofactive microrheology. We derive a relation that connects the average swimming velocity and the frequency-dependent
viscosity of the surrounding medium. In this relation, the viscous contribution can exist only when the time-reversal
symmetry is broken, whereas the elastic contribution is present only when the structural symmetry of the swimmer isbroken. Purcell’s scallop theorem breaks down for a three-sphere swimmer in a viscoelastic medium.

Microrheology is one of the most useful techniques formeasuring the rheological properties of soft matter andvarious biological materials including cells.1,2) There are two
diﬀerent methods: passive microrheology and active microrheology. In passive microrheology, both the local and bulk
mechanical properties of a medium can be extracted fromthe Brownian motion of a probe particle.3,4) In this method,
the generalized Stokes–Einstein relation (GSER) is used toanalyze thermal diﬀusive motion. In active microrheology,
on the other hand, the probe is actively pulled through theﬂuid with the aims of driving the medium out of equilibrium
and measuring mechanical responses.5,6) Within linear
response theory, the generalized Stokes relation (GSR) is
employed to obtain the frequency-dependent complex shear
modulus.
In this letter, we propose a new type of active microrheology using a microswimmer. Microswimmers are tiny
machines that swim in a ﬂuid such as sperm cells or motile
bacteria, and are expected to be applied to microﬂuidics andmicrosystems.7) As one of the simplest microswimmers, weconsider the Najaﬁ–Golestanian three-sphere swimmer model,8,9) where three in-line spheres are linked by two arms ofvarying length (see Fig. 1). Recently, such a swimmer has
been experimentally realized.10) We investigate its motion ina general viscoelastic medium, and obtain a relation thatconnects the average swimming velocity and the frequencydependent complex shear viscosity of the surrounding
viscoelastic medium. We show explicitly that the absenceof the time-reversal symmetry of the swimmer motion leads
to the real part of the viscosity, whereas the absence of thestructural symmetry of the swimmer is reﬂected in theimaginary part of the viscosity. Hence, we shall call the
proposed method the “swimmer-microrheology”. Our result
also indicates that Purcell’s scallop theorem,11,12) whichstates that time-reversible body motion cannot be used forlocomotion in a Newtonian ﬂuid, breaks down for a threesphere swimmer in viscoelastic media if the structuralsymmetry is violated.
The general equation that describes the hydrodynamics ofa low-Reynolds-number ﬂow in a viscoelastic medium isgiven by the following generalized Stokes equation:13)
Zt0¼
dt0 ðt  t0 Þr2 vðr; t0 Þ  rpðr; tÞ:ð1Þ1Here ðtÞ is the time-dependent shear viscosity, v is thevelocity ﬁeld, p is the pressure ﬁeld, and r stands for a three-

η[ω]

1

3

2

a

L1 (t)

L2 (t)

Fig. 1. (Color online) Najaﬁ–Golestanian three-sphere swimmer model.
Three identical spheres of radius a are connected by arms of lengths L1 ðtÞand L2 ðtÞ and undergo time-dependent cyclic motion. The swimmer isembedded in a viscoelastic medium characterized by a frequency-dependent
complex shear viscosity ½!.

dimensional positional vector. The above equation is further
subjected to the incompressibility condition, r  v ¼ 0. Fromthese equations, one can obtain a linear relation between thetime-dependent force FðtÞ acting on a hard sphere of radius aand its time-dependent velocity VðtÞ. In the Fourier domain,
this relation can be represented as
Vð!Þ ¼

1
Fð!Þ;6½!að2Þ

we use a bilateral Fourier transform for Vð!Þ ¼
Rwhere
1
i!t
dt
VðtÞe
Fð!Þ, while we employ a unilateral
1
Rand
1
one for ½! ¼ 0 dt ðtÞei!t . Equation (2) is the GSR,
which has been successfully used in active microrheology
experiments,5) and its mathematical validity has also been
discussed.6)
Next, we brieﬂy explain the three-sphere model for aminimum swimmer introduced by Najaﬁ and Golestanian.8,9)
As schematically shown in Fig. 1, this model consists ofthree spheres of the same radius a that are connected by
two arms of lengths L1 ðtÞ and L2 ðtÞ, which undergo timedependent motion. Their explicit time dependences will begiven later. If we deﬁne the velocity of each sphere along theswimmer axis as Vi ðtÞ with i ¼ 1; 2; 3, we have
L_ 1 ðtÞ ¼ V2 ðtÞ  V1 ðtÞ;ð3ÞL_ 2 ðtÞ ¼ V3 ðtÞ  V2 ðtÞ;
ð4Þ__
where L1 and L2 indicate time derivatives.
Owing to the hydrodynamic eﬀect, each sphere exerts aforce Fi on the viscoelastic medium and experiences a force
Fi from it. To relate the forces and the velocities in thefrequency domain, we use the GSR in Eq. (2) and the Oseen
tensor, in which the frequency-dependent viscosity ½! isused instead of a constant one.3,4) Assuming that a  L1 ; L2 ,we can write8,9)043801-1©2017 The Physical Society of Japan

J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by on 03/28/17

J. Phys. Soc. Jpn. 86, 043801 (2017)

V1 ð!Þ ¼

F1 ð!Þ
1 F2 ð!Þ  L1
1 ð!Þ
þ
6½!a 4½!
2
1 F3 ð!Þ  ðL1 þ L2 Þ1 ð!Þ
þ;
4½!
2

V2 ð!Þ ¼

1 F3 ð!Þ  L1
2 ð!Þ
;4½!
2

Mediumð5Þ

1 F2 ð!Þ  L1
F3 ð!Þ2 ð!Þ
þ;
4½!
2
6½!a

ð7Þ

ð8Þ

Since Eqs. (5)–(7) involve convolutions in the frequency
domain, we cannot solve these equations for arbitrary L1 ðtÞand L2 ðtÞ. Here we assume that the two arms undergo thefollowing periodic motion:
L1 ðtÞ ¼ ‘ þ d1 cosðtÞ;

ð9Þ

L2 ðtÞ ¼ ‘ þ d2 cosðt  Þ:ð10ÞIn the above, ‘ is the constant length, d1 and d2 are theamplitudes of the oscillatory motion, Ω is the common arm
frequency, and ϕ is the mismatch in the phases between the
two arms. In the following analysis, we generally assume that
d1 ; d2  ‘. The time-reversal symmetry of the arm motion ispresent when  ¼ 0 and π. Furthermore, we characterize thestructural symmetry of the swimmer by d1 and d2 , i.e., thestructure is symmetric when d1 ¼ d2 , while it is asymmetricwhen d1 ≠ d2 .
Since the arm frequency is Ω, we assume that the velocities
and the forces of the three spheres can generally be writtenasVi ð!Þ ¼ Vi;0 ð!Þþ

1
X

½Vi;n ð! þ nÞ þ Vi;n ð!  nÞ;

ð11Þ

n¼1

Fi ð!Þ ¼ Fi;0 ð!Þþ

1
X

½Fi;n ð! þ nÞ þ Fi;n ð!  nÞ;Y

Viscoelastic
N

Y

Y

N

Y

N

Y

N

Y

N
N

Swimmer motion
Rheological information

N
—

N
—

Y
N

Y
N

N
—

Y
00

Y
0

Y
0 ; 00

total swimming velocity V ¼ ðV1 þ V2 þ V3 Þ=3. Up to thelowest order terms in a, the average swimming velocity over
one cycle of motion becomes14)

where we Rhave used bilateral Fourier transforms such as1
1 i!tL1. Furthermore, the convolu1 ð!Þ ¼ 1 dt ½L1 ðtÞ e
tion
of
two
functionsisgenerallydeﬁned by g1 ð!Þ  g2 ð!Þ ¼
R1
0
0
0
d!
g
ð!

!
Þg
ð!
Þ
in
the
above equations.
12
1
As in the original study, we are interested in theautonomous net locomotion of the swimmer, and there are
no external forces acting on the spheres. If the inertia of thesurrounding ﬂuid can be neglected, we have the following
force balance condition:
F1 ðtÞ þ F2 ðtÞ þ F3 ðtÞ ¼ 0:

Viscous

Time-reversal symmetry
Structural symmetry

ð6Þ

1 F1 ð!Þ  ðL1 þ L2 Þ1 ð!Þ
4½!
2
þ

K. Yasuda et al.

Table I. Locomotion of a three-sphere swimmer in a viscoelastic medium
and the relevant rheological information.

1 F1 ð!Þ  L1
F2 ð!Þ
1 ð!Þ
þ
4½!
2
6½!aþ

V3 ð!Þ ¼

Lettersð12Þ

n¼1

where i ¼ 1; 2; 3 for the three spheres. Substituting Eqs. (11)and (12) into the six coupled Eqs. (3)–(8), we obtain a matrix
equation with inﬁnite dimensions.
Under the conditions d1 ; d2  ‘ and a  ‘, we areallowed to consider only n ¼ 1; 0; 1 and we further use theapproximation Fi;2  0. Then we can solve for the six
unknown functions Vi ð!Þ and Fi ð!Þ, and also calculate theV

7d1 d2 a 0 ½
5ðd12  d22 Þa 00 ½
sin  
;
2
0
0
24‘
48‘2

ð13Þ

where 0 ½ and 00 ½ are the real and imaginary parts of thecomplex shear viscosity, respectively, and 0 ¼ ½ ! 0 isthe constant zero-frequency viscosity.
The ﬁrst term in Eq. (13) can be regarded as the viscous
contribution and is present only if the time-reversal symmetryof the swimmer motion is broken, i.e.,  ≠ 0; . The second
term, on the other hand, corresponds to the elastic
contribution, and exists only when the structural symmetryof the swimmer is broken, i.e., d1 ≠ d2 . If we are able tocontrol d1 , d2 , and Ω of the swimmer, we can ﬁrst obtain0 ½ by measuring V as a function of Ω by setting d1 ¼ d2 .
Then we make a diﬀerence between d1 and d2 to examine thechange in V, which then yields 00 ½. The corresponding
complex shear modulus is simply obtained by G½ ¼
i½. This is a new type of active microrheology that we
propose in this letter.
For a purely Newtonian ﬂuid, namely, for a medium
characterized by a constant viscosity, the second term in
Eq. (13) vanishes, and the ﬁrst term coincides with theexpression obtained by Golestanian and Ajdari.9) It should be
noted here, however, that the velocity V in this case no longer
depends on the constant viscosity (because 0 ½=0 ¼ 1)
and we cannot measure it from V. Equation (13) also implies
that the swimmer cannot move in a purely elastic medium,for which we have 0 ! 1. Importantly, owing to thepresence of the second term, Purcell’s scallop theorem breaks
down for a three-sphere swimmer in a viscoelastic medium.
Namely, even if the time-reversal symmetry of the swimmer
motion is not broken, i.e.,  ¼ 0; , the present swimmer canstill move in a viscoelastic medium due to the second term aslong as its structural symmetry is broken, i.e., d1 ≠ d2 . On
the basis of Eq. (13), we have summarized in Table I the
motion of a three-sphere swimmer in a viscoelastic medium
and the relevant rheological information.
To further illustrate our result, we ﬁrst assume that thesurrounding viscoelastic medium is described by a simple
Maxwell model. In this case, the frequency-dependent
viscosity can be written as½! ¼ 0

1  i!M;
1 þ !2 2M

ð14Þ

where M is the characteristic time scale. Within this model,
the medium behaves as a viscous ﬂuid for !M  1, while it
becomes elastic for !M
1. Using Eq. (14), we can easily
obtain the average swimming velocity in Eq. (13) as

043801-2©2017 The Physical Society of Japan

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Letters

J. Phys. Soc. Jpn. 86, 043801 (2017)

K. Yasuda et al.

2

10viscous
Maxwellpower-lawτ /7d21 a

1

10

0

-1

10

-2

-1

10

10

0

1

10

10

Ωτ

2

10

1

48

2

τ /5(d12 − d22 )a

10

(b)

0

10

-1

10
10

-2

elastic
Maxwellpower-law

-3

10
10

-4
-2

-1

10

10

0

10

Ωτ

1

10

2

10

Fig. 2. (Color online) Average swimming velocity V as a function of ,
where Ω is the arm frequency and τ represents either M for a Maxwell ﬂuid
(red lines) or p for a power-law ﬂuid (green lines). In the power-law model,
we choose  ¼ 1=2. (a) Viscous contribution by setting  ¼ =2 and d1 ¼
d2 . Here V is scaled by 7d12 a=ð24‘2 Þ. The case for a viscous ﬂuid is plottedby the black line. (b) Elastic contribution by setting  ¼ 0 and d1 ≠ d2 . Here
V is scaled by 5ðd12  d22 Þa=ð48‘2 Þ. The case for an elastic medium isplotted by the black line.

V¼

7d1 d2 a1
sin 
24‘2 1 þ 2 2M

5ðd12  d22 Þa M:
ð15Þ
1 þ 2 2M
48‘2
The ﬁrst viscous term increases as V  for M  1,
while it decreases as V 1 for M
1. This is a uniquefeature of the viscoelasticity,7,15,16) but such a reduction
occurs simply because the medium responds elastically in thehigh-frequency regime. On the other hand, the second elastic
term increases as V 2 for M  1, and it approaches
a constant for M
1. In Fig. 2(a), we plot the averageswimming velocity V as a function of the dimensionless arm
frequency M when  ¼ =2 and d1 ¼ d2 . This plotcorresponds to the ﬁrst term in Eq. (15). As a reference,
the behavior of V  for a purely viscous ﬂuid is alsoplotted. Figure 2(b) is a similar plot when  ¼ 0 andd1 ≠ d2 , and corresponds to the second term in Eq. (15).
As a diﬀerent example, we next consider the case in which
the viscoelastic medium is described by a power-law modelsuch that13,17,18)
þ

5ðd12  d22 Þaðp Þ cosð=2Þ;ð17Þ
48‘2 pwhere p ¼ ð0 =G0 Þ1=ð1Þ . Here we have assumed that themedium behaves as a purely viscous ﬂuid in the lowfrequency limit characterized by a ﬁnite viscosity 0 .
According to the above expression, the swimming velocity
scales as V  in both the ﬁrst and second terms. For thepurely viscous case of  ¼ 1, the ﬁrst term reduces to theresult by Golestanian and Ajdari,9) while the second term
vanishes. For the purely elastic case of  ¼ 0, on the other
hand, the ﬁrst term vanishes and the second term remains,
although the latter no longer depends on the arm frequency
Ω. In Figs. 2(a) and 2(b), we have also plotted the averagevelocity V as a function of p when  ¼ 1=2. In both ofthese plots, the scaling behavior V 1=2 is seen.
Lauga considered the axisymmetric squirming motion of asphere (squirmer) embedded in an Oldroyd-B ﬂuid, which
represents a typical polymeric ﬂuid.19) He reported that thescallop theorem in a viscoelastic ﬂuid breaks down if thesquirmer has fore-aft asymmetry in its surface velocity
distribution. For a time-reversal deformation given by asimple sinusoidal gait, he showed that the average swimming velocity is given by V  De=ð1 þ De2 Þ, where theDeborah number is given by De ¼ O with a characteristic
relaxation time O in the Oldroyd-B model. Such a frequency
dependence of the swimming velocity is identical to thesecond term of Eq. (15) obtained for a Maxwell ﬂuid,
although Eq. (13) is more general. On the other hand, ourresult is diﬀerent from that by Curtis and Gaﬀney,20) because
they showed that the swimming velocity in a viscoelastic
medium is the same as that in a Newtonian ﬂuid.
To summarize, we have proposed a new active microrheology using the Najaﬁ–Golestanian three-sphereswimmer. The frequency dependence of the average swimming speed provides us with the complex shear viscosityof the surrounding viscoelastic medium. Here the viscous
contribution can exist only when the time-reversal symmetryof the swimmer is broken, whereas the elastic contribution ispresent only if its structural symmetry is broken.
Even though the argument in this Letter is restricted to theartiﬁcial three-sphere swimmer, we expect that our basic
concept can still be applied to more complex biologicalprocesses such as the motion of bacteria, ﬂagellated cellularswimming, and the beating of cilia. Since most of these
phenomena take place in a viscoelastic environment, we hopethat the concept of our new active microrheology will be usedin the future to reveal their mechanical and dynamical
properties.
þ

-2

10

ð16Þ

where the exponent can take values of 0  1. With thisexpression, the complex shear modulus also exhibits apower-law behavior, G½! ¼ G0 ði!Þ . The limits of  ¼ 0
and 1 correspond to the purely elastic and the purely viscouscases, respectively. In the case of a power-law ﬂuid, theaverage swimming velocity can be obtained from Eqs. (13)and (16) as
7d1 d2 aV¼ðp Þ sinð=2Þ sin 24‘2 p

10

24

2

½! ¼ G0 ði!Þ1 ;(a)043801-3©2017 The Physical Society of Japan

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J. Phys. Soc. Jpn. 86, 043801 (2017)

Letters

K. Yasuda et al.

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043801-4©2017 The Physical Society of Japan

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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