PHYSICAL REVIEW E 81, 041807 共2010兲

Tumbling motion of a single chain in shear flow: A crossover from Brownian
to non-Brownian behavior
Hideki Kobayashi* and Ryoichi Yamamoto†Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japanand CREST, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan
共Received 13 October 2009; published 26 April 2010兲
We present the numerical results for the dynamics of a single chain in steady shear flow. The chain isrepresented by a bead-spring model and the smoothed profile method is used to accurately account for theeffects of thermal fluctuations and hydrodynamic interactions acting on beads due to host fluids. It wasobserved that the chain undergoes tumbling motions and that its dimensionless frequency F = 6␲␩␴3␯ / kBT
depends only on the Peclet number Pe with a power law F ⬀ Pe␣, where kB is the Boltzmann constant, T is thetemperature, and ␴ is the diameter of the beads. The exponent ␣ clearly changes from 2/3 to 1 around thecritical Peclet number, Pec, indicating that the crossover reflects the competition of thermal fluctuation andshear flow. The presented numerical results agree well with our theoretical analysis based on Jeffrey’s work.
DOI: 10.1103/PhysRevE.81.041807

PACS number共s兲: 83.80.Rs, 47.57.Ng, 82.20.Wt

I. INTRODUCTION

The dynamics of solid particles dispersed in host fluids is
an important problem in many different fields of science andengineering. The macroscopic properties of such dispersions
共the elastic modulus, viscosity, and thermal and electric conductivities兲 greatly depend on the dynamics of the particles
in the host fluids. In equilibrium states, the dynamics ofsmall dispersed particles are strongly affected by the thermal
fluctuations of their host fluids. When flow is imposed, thedynamics are also affected by the flow of the host fluids.
Because it is difficult to experimentally analyze these complex particle dynamics, which are coupled both to thermal
fluctuations and to fluid flow, numerical simulations are particularly important for understanding the properties of particle dispersions in detail 关1兴.
For a single Brownian chain fluctuating in the shear flowof a Newtonian fluid, it has been suggested that the tumbling
frequency ␯ is proportional to the shear rate ␥˙ 2/3 关2–9兴. Thishas been experimentally confirmed by Schroeder, Teixeira,Shaqfeh, and Chu 关2兴 for the dynamics of individual DNA
molecules in a linear shear flow. Primarily, Smith, Babcock,and Chu 关10兴 measured the power spectral density 共PSD兲 andthe probability distribution function 共PDF兲 of the extension
length of each DNA molecule for various Weissenberg numbers Wi= ␥˙ ␶, where ␥˙ is the shear rate and ␶ is the relaxation
time of the chain orientation. The PSD of polymer extension
exhibits no peaks. In subsequent experiments 关2,3兴, however,the focus has been put on the PSD of the orientation angle ␸,
where ␸ = 0 when the DNA molecule lies perfectly in theflow direction. These experimental results support a simple
power law, ␯␶ ⬀ Wi2/3, where ␯ is the peak frequency of the
PSD. The relaxation time ␶ is considered to be a constant if
the temperature is constant. This leads to ␯ ⬀ ␥˙ 2/3. Similar
results have also been obtained in other experiments 关3,4兴,
numerical simulations 关5,6兴, and theoretical analyses 关7–9兴.

*hidekb@cheme.kyoto-u.ac.jp†

ryoichi@cheme.kyoto-u.ac.jp1539-3755/2010/81共4兲/041807共8兲

Although the DNA molecules mentioned above can be considered flexible chains, a quite similar power law was obtained using numerical simulations for a Brownian linear
rigid rod as well 关2兴.
For a single non-Brownian 共thermally nonfluctuating兲
flexible chain in shear flow, the tumbling frequency ␯ is expected to be proportional to the shear rate ␥˙ 关11兴. A single
non-Brownian rigid rod with a finite aspect ratio is known to
exhibit a cyclic tumbling motion in shear flow, as describedby Jeffrey’s equation, ␯ ⬀ ␥˙ 关12–14兴.
From the above experimental findings, one would expect
to observe a crossover from Brownian 共␯ ⬀ ␥˙ 2/3兲 to nonBrownian 共␯ ⬀ ␥˙ 兲 behavior with increasing shear rate; however, such a clear crossover has not yet been reported. Thiscrossover has not yet been successfully predicted by previous numerical 关5,6兴 and theoretical 关7–9兴 studies where adispersed chain is treated as an end-to-end vector, namely, as
an infinitely thin line. Therefore, the rotational motion of thechain cannot be sustained across ␸ = 0 without thermal fluctuations. When a thermal fluctuation exists, the orientation of
the thin line can fluctuate around ␸ = 0. This leads to tumbling motions even for a thin line; however, the frequency of
the tumbling motion ␯ is always proportional to ␥˙ 2/3, regardless of the shear rate 关9兴.
We thus aim to analyze this crossover by use of a direct
numerical simulation approach. In the present study, we
simulated the tumbling motion of a chain using a smoothed
profile method 共SPM兲 that accurately takes into account thermal fluctuations and hydrodynamic interactions 关15–18兴. The
chain is represented by a bead-spring model, wherein eachbead is modeled as a spherical object with a finite radius aand undergoes free rotation. Rigid rods or flexible chains are
represented with or without a constraint force on bond bending. Apart from the previous numerical models, the presented
bead-spring model naturally takes into account the finite
thickness of the experimentally used chains or rods. A theoretical analysis has also been developed to understand themechanisms underlying the crossover.

041807-1©2010 The American Physical Society

PHYSICAL REVIEW E 81, 041807 共2010兲

HIDEKI KOBAYASHI AND RYOICHI YAMAMOTO
II. METHODS
A. MODEL

We solve the dynamics of a single chain in a Newtonian
solvent using SPM 关15–18兴. In this method, boundaries between solid particles and solvents are replaced with a continuous interface by assuming a smoothed profile. This enables us to calculate hydrodynamic interactions both
efficiently and accurately, without neglecting many-body interactions. The equation governing a solvent with a density
␳f and a shear viscosity ␩ is a modified Navier-Stokes equation,

␳f

再

冎

⳵ vជ
ជ 兲vជ = − ⵜជ p + ␩ⵜជ 2vជ + ␳f␾ជ
f p + ជf shear ,+ 共vជ · ⵜ⳵t共1兲

ជ · vជ = 0, where vជ 共rជ , t兲 andwith the incompressible condition ⵜ
p共rជ , t兲 are the velocity and pressure fields of the solvent,respectively. A smoothed profile function 0 ⱕ ␾共rជ , t兲 ⱕ 1 distinguishes between the fluid and particle domains, yielding␾ = 1 in the particle domain and ␾ = 0 in the fluid domain.
These domains are separated by thin interstitial regions, thethicknesses of which are characterized by ␰. The body force␾ជf p is introduced to ensure the rigidity of the particles andthe appropriate nonslip boundary condition at the fluid/particle interface. The mathematical expressions for ␾ and␾ជ
f p are detailed in previous papers 关15,16兴. The external
force ជf shear is introduced to maintain a linear shear 关19兴, expressed by

vx =

冦

− ␥˙ y冉Ly− ␥˙ − y +2− ␥˙ 共y − Ly兲

冊

共0 ⬍ y ⬍ Ly/4兲,冧

共Ly/4 ⬍ y ⬍ 3Ly/4兲,
共3Ly/4 ⬍ y ⬍ Ly兲,d
ជ i = Fជ Hi + Fជ Pi + Fជ Ci + G
ជ Vi ,
V
dt

ជIi ·

d
ជi=N
ជ Hi + G
ជ⍀
⍀
i ,
dt

d
ជ i = Vជ i ,R
dt

ULJ共rij兲 =冦 再冉 冊 冉 冊 冎
4⑀

0

共2兲

共3兲

共4兲

ជ i are the position, translational velocity,ជ i, Vជ i, and ⍀where Rand rotational velocity of the beads, respectively. M i and ជIi
ជ Hi and N
ជ Hi are theare the mass and moment of inertia, and Fhydrodynamic force and torque exerted by the solvent on theជ Vi and Gជ⍀beads, respectively 关15,16兴. G
i are the random force
and torque, respectively, due to thermal fluctuations. The
temperature of the system is defined such that the long-time

冧

␴ 12
␴ 6
−
+ ⑀ 共rij ⬍ 21/6␴兲,
rij
rij
共5兲
1/6
共rij ⬎ 2 ␴兲,ជ i − Rជ j兩. The parameter ⑀ characterizes thewhere rij = 兩R
strength of the interactions, and ␴ represents the diameter ofthe beads. Consecutive beads on a chain are connected by a
FENE potential of the form

再 冉 冊冎

1
r 2,
UFENE共r兲 = − kcR20 ln 1 −
2
R0

共6兲

ជ i兩, kc = 30⑀ / ␴2, and R0 = 1.5␴. Fជ Ci is the conwhere r = 兩Rជ i+1 − Rstraint force acting on the ith bead due to the bond-angle
constraints that cause the chain to form a straight line, and it
is used only for the rigid rod case. This is given byជ Ci =
F⳵⳵ Rជ i

冉兺
N

␣=3

冊

ជ␣ ,
␮
ជ␣ · ⌿ជ ␣ = 共␣ − 2兲Rជ 1 − 共␣ − 1兲Rជ 2 + Rជ ␣,⌿

where ␥˙ is the shear rate and Ly is the system size in the y
direction.
In the present study, the chain is represented as either arigid rod or a flexible chain. We use a bead-spring model
consisting of N beads in a single chain. The bead size issufficient to fit several mesh units. Therefore, it is necessary
to consider the torque exerted on the bead. The motion of theith bead is governed by the following Newton’s and Euler’s
equations of motion with stochastic forces:
Mi

diffusive motion of dispersed particles reproduces correct behavior 关17,18兴.
ជ Pi represents the potential force due to direct interbead
Finteractions such as Coulombic and Lennard-Jones potentials. We use a bead-spring model as a model of a polymericchain with a truncated Lennard-Jones potential and a finitely
extensible nonlinear elastic 共FENE兲 potential. The truncatedLennard-Jones interaction is expressed in terms of ULJ

共7兲

共␣ = 3, . . . ,N兲, 共8兲ជ ␣ = 0 is the constraint condition to be satisfied. ␮
where ⌿
ជ ␣ isa Lagrange multiplier associated with the intramolecularforces of the constraints chosen such that the constraint conជ ␣ = 0 is satisfied at a time t + h, where h is the time
dition ⌿increment of a single simulation step.
B. SIMULATION

Numerical simulations have been performed in three dimensions with periodic boundary conditions. The latticespacing ⌬ is taken to be the unit of length. The unit of time
is given by ␳f⌬2 / ␩, where ␩ = 1 and ␳f = 1. The system size is
Lx ⫻ Ly ⫻ Lz = 32⫻ 16⫻ 64. The other parameters include: ␴= 4, ␰ = 2, ⑀ = 1, ␩ = 1, M i = 4␲a3 / 3, N = 5, and h = 0.067.
In the presented simulations, the Navier-Stokes equation
is discretized with a dealiased Fourier spectral scheme inspace and with a second-order Runge-Kutta scheme in time.
To follow bead motions, the position, velocity and angularvelocity of the beads are integrated with the AdamsBashforth scheme.
At t = 0, the chain aligns along the x axis, which is theshear direction. The run-time of our simulations is about
3520/ ␥˙ . The range of kBT is 5.0⫻ 10−4 ⬍ kBT ⬍ 1.0 and thatof ␥˙ is 1.0⫻ 10−3 ⬍ ␥˙ ⬍ 4.0⫻ 10−2.
From the symmetry of the system, to analyze the tumbling
motion of a chain, we only have to consider the projected
tumbling motion on the x-y plane. We introduce the chain

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TUMBLING MOTION OF A SINGLE CHAIN IN SHEAR …

PHYSICAL REVIEW E 81, 041807 共2010兲

10

-1

10

-2

10

-3

10

-4

10

-2

10-5

νp

PSD

100

10-6
10-7

10

-3

10

-4

10-8
10-9
10-10
10

-11

10-1

100

101

102

103

104

105

ν/νp

10

FIG. 1. 共Color online兲 The behavior of PSD as a functionof ␯ / ␯ p for rigid rods at kBT = 0.0005 and ␥˙ = 0.001 共blue 䊐兲,
kBT = 0.006 and ␥˙ = 0.002 共red 䊊兲, and kBT = 0.01 and ␥˙ = 0.008
共green 䉭兲.

orientation angle ␸, which is the angle between the x axis
and the projected end-to-end vector on the x-y plane.
Evidence of periodic tumbling motion should appear in
the PSD per unit time 关2,3兴. Therefore, to investigate thespectral properties of the orientation, we calculate the PSD
using a fast Fourier transform. The PSD is expressed as
PSD共␻兲 =冏冕冏␸共t兲exp共i␻t兲dt .

共9兲

III. RESULTSIn Figs. 1 and 2, the PSDs of the chain orientation angle ␸show a peak at a specific frequency ␯ p, as shown in previous
studies 关2,3兴. Furthermore, the PSD data obtained at different
conditions of shear rate and temperature tend to lie on asingle master curve if a normalized frequency ␯ / ␯ p is used.
This is true for both rigid rods and flexible chains. This result
implies that the tumbling motion of chains is fully characterized by ␯ p.
10

-2

10

-1

10

Shear rate

FIG. 3. 共Color online兲 The behavior of ␯ p as a function of ␥˙ at
kBT = 0.0005. Rigid rod 共green closed circle兲 and flexible chain 共red
open circle兲. The blue solid line corresponds to the law ␯ p ⬀ ␥˙ . Error
bars are derived from the half band width of PSD.

In Fig. 3, we find that ␯ p follows the law ␯ p ⬀ ␥˙ for ␥˙⬍ 0.02 at kBT = 5.0⫻ 10−4, for both the rigid rod and the flexible chain cases. Meanwhile, Fig. 4 shows ␯ p ⬀ ␥˙ 0.68 for ␥˙⬍ 0.02 at kBT = 1.0 for both.
Figure 3 also shows that, for ␥˙ ⬎ 0.02, ␯ p is lower than thefrequency expected from the law ␯ p ⬀ ␥˙ . P. Bagchi and S.
Balachandar have reported that, at a finite Reynolds number
approximately equal to 1, the sphere rotation frequency in a
linear shear flow decreases at a much slower rate than ␥˙ / 2
关20兴. The Reynolds number, Re, is given by Re= ␳ f ␥˙ ␴2N / ␩.
In our paper, Re is equal to 1.6 at ␥˙ = 0.02. We only consider
␯ p in the region of ␥˙ ⱕ 0.02, so the effects of finite Reynolds
numbers do not influence the results.
We arrange data sets using the Peclet number to consider
the effect of competition between shear and fluctuation. The
Peclet number is the dimensionless number that relates therate of shear flow to the rate of thermal fluctuation. In ourwork, the Peclet number, Pe, and dimensionless frequency,F, are expressed as
10

10

-3

-2

0

-1

10

-3

10

-4

νp

PSD

10-2

10-5
10

10

-3

10

-4

-6

10-7
10

-8

10-9
10

-10

10

-1

0

10

10

1

10

2

10

3

4

10

5

10

ν/νp

10

-3

-2

10

-1

10

Shear rate

FIG. 2. 共Color online兲 The behavior of PSD as a function of␯ / ␯ p for flexible chains at kBT = 0.0005 and ␥˙ = 0.001 共blue 䊐兲,
kBT = 0.006 and ␥˙ = 0.002 共red 䊊兲, and kBT = 0.01 and ␥˙ = 0.008
共green 䉭兲.

FIG. 4. 共Color online兲 The behavior of ␯ p as a function of ␥˙ at
kBT = 1.00. Rigid rod 共green closed circle兲 and flexible chain 共red
open circle兲. The blue solid line corresponds to the law ␯ p ⬀ ␥˙ 2/3.
Error bars are derived from the half band width of PSD.

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PHYSICAL REVIEW E 81, 041807 共2010兲

HIDEKI KOBAYASHI AND RYOICHI YAMAMOTO
104

F = 0.16 Pe
103

F

102

F = 0.82 Pe0.65

101

0

10

-1

10

0

10

1

10

10

2

3

10

10

4

10

5

Pe

FIG. 5. 共Color online兲 The behavior of F as a function of Pe fora rigid rod. The orange solid and blue dotted lines were calculated
from a least-squares fit of the data points. The orange solid line
corresponds to Pe0.65. The blue dotted line corresponds to Pe.

6␲␩␴3␥˙
,
k BT

共10兲

6 ␲ ␩ ␴ 3␯ p
.
k BT

共11兲

Pe =

F=

We plotted the behavior of F as a function of Pe for the rigid
rod and flexible chain cases in Figs. 5 and 6. F was found todepend only on Pe because the data sets have the same valueof F with the same value of Pe, even when the shear rates
and temperatures are different. In the rigid rod case, F⬀ Pe0.65 for Pe⬍ 106 and F ⬀ Pe for Pe⬎ 106. In the flexible
chain case, F ⬀ Pe0.68 for Pe⬍ 156 and F ⬀ Pe for Pe⬎ 156.
The behaviors of F for the rigid rod case and the flexible
chain case are roughly equal, although the values of the

FIG. 7. 共Color online兲 The behavior of ␯ p as a function of ␥˙ at
kBT = 0. Rigid rod 共red +兲, flexible chain 共green ⫻兲. The data sets
for the rigid rods correspond to the law ␯ p = 0.178␥˙ . The data sets
for the flexible chains correspond to the law ␯ p = 0.190␥˙ .

Peclet numbers are different at the boundary where the exponent of Pe changes.
We define the critical value at which the exponent of Pe
drastically changes from almost 2/3 to 1 as the critical Peclet
number, Pec. F ⬀ Pe2/3 for Pe⬍ Pec; otherwise, F ⬀ Pe. Whenfluctuations dominate the system, the exponent is nearly
equal to 2/3. On the other hand, when shear flow dominates,
the exponent is exactly equal to 1.
In order to understand the behavior of F in the limit ofPe→ ⬁, we examine the behavior of ␯ p in the limit of kBT→ 0 共Figure 7兲. As shown in Figs. 5 and 6, the behavior of F
in the region of Pec ⬍ Pe is roughly equal to the behavior of␯ p in the limit of kBT → 0. Therefore, we believe that the
proportional relation F ⬀ Pe can be true across the entire region of Pec ⬍ Pe.
IV. DISCUSSION

104

A. Comparison with other rfaceesults
F = 0.18 Pe

103

F

102

101

F = 0.94 Pe0.68

100

10-1
100

101

102

103

104

105

Pe

FIG. 6. 共Color online兲 The behavior of F as a function of Pe fora flexible chain. The orange solid and blue dotted lines were calculated from a least-squares fit of the data points. The orange solidline corresponds to Pe0.68. The blue dotted line corresponds to Pe.

In our work, we calculate the tumbling motion of a single
chain for 0.0005⬍ kBT ⬍ 1.00 and 0.001⬍ ␥˙ ⬍ 0.04. As reported in Jeffrey’s paper 关12兴, which treated non-Brownian
particles, ␯ p follows the law ␯ p ⬀ ␥˙ at kBT = 5.0⫻ 10−4, asshown in Fig. 3. In previous papers 关5–8兴 that treated Brownian particles, ␯ p follows the law ␯ p ⬀ ␥˙ 2/3 at kBT = 1.0, as
shown in Fig. 4.
The dimensionless frequency, F, depends only on Pe. ThisPe dependence on F can be described by a power law, F⬀ Pe␣. The exponent ␣ drastically changes from 2/3 to 1 at
Pec; ␣ equals 2/3 for Pe⬍ Pec, while ␣ equals 1 for Pec⬍ Pe. In the case of a rigid rod with N = 5, Pec ⬇ 106, andPec ⬇ 156 in the flexible chain case. When fluctuations dominate the system, F follows the law of F ⬀ Pe2/3. On the other
hand, when shear flow dominates, F follows the law ofF ⬀ Pe.
Gerashchenko and Steinberg 关4兴 claim that there are two
dynamical regimes of polymer motion at WiⰇ 1, depending

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PHYSICAL REVIEW E 81, 041807 共2010兲

on the polymer extension R. When R Ⰶ Rmax, where Rmax isthe maximum polymer extension, the tumbling frequency ␯
is constant and independent of Wi. On the other hand, whenR ⬃ Rmax, ␯ is proportional to Wi2/3. In Fig. 4, we did notobserve ␯ p to be independent of ␥˙ . We consider the chain
length to be too short in our work. The chain cannot keep thecoil state along the shear direction for long times. The chain
is fully stretched at short notice and is always rotated, although Pe is small.
Previously, Szymczak and Cieplak 关21兴 discussed the conformational dynamics of a single long protein in shear flow
and found two characteristic tumbling frequencies, ␯1 and ␯2.
They showed that the higher frequency ␯1 follows the law of␯1 ⬀ ␥˙ ; however, the lower frequency ␯2 follows the law of␯2 ⬀ ␥˙ 2/3. When the protein is tightly packed, it essentiallyshows a spherical rotation in shear flow. As a result, ␯1 is
proportional to ␥˙ . The lower frequency, ␯2, corresponds tothe stretching-collapse cycle; hence, ␯2 is proportional to␥˙ 2/3. Although our results are similar to theirs, the phenomena in our system are essentially different from those in theirworks because rigid rods cannot fold.
Davoudi and Schumacher 关22兴 analyzed the stretching ofpolymers in a turbulent flow. It is known that the polymers
undergo a coil-stretch transition at Wi⬇ 1 / 2 in this system.
For Wi⬍ 1 / 2, polymers are in the coiled state, and their size
distribution is stationary. In contrast, for Wi⬎ 1 / 2, the polymers are in the stretched state. Their stretching carries on
until their lengths reach the finite extensibility limit or until
turbulence stops the growth of the polymers. They found the
maximum Lyapunov exponent to be ␭ ⬃ ␥˙ 3/2. However,Chertkov et al. reported ␭ ⬃ ␥˙ 2/3 in their work, where ␭−1 isexpressed as the mean stretching time scale. Davoudi and
Schumacher claimed also that their study could not be compared with the analytic results of Chertkov et al. They defined the shear time scale as Ts = ␥˙ −1 and the fluctuation timescale as T f = D−1, where D is the strength of Gaussian fluctuation. Chertkov analyzed the polymer dynamics in the region of Ts Ⰶ T f . Davoudi analyzed the polymer dynamics inthe region of ␶␩ ⬍ Ts, where ␶␩ is the Kolmogorov time. Notably, Davoudi’s work studies a different regime of polymer
stretching than the analytic model of Chertkov’s work.
Chertkov et al. studied in the shear-dominated regime,
whereas Davoudi and Schumacher studied in the turbulencedominated regime.
Our work analyzed polymer dynamics in the region ofTs Ⰶ T f . We do not consider the effect of turbulent flow. We
only considered the region with particle Reynolds numberRe⬍ 1.6, so the effects of finite Reynolds numbers do not
influence the results. In Davoudi’s work, by contrast, theregion of particle Reynolds number in the stretched state isestimated to be Re⬘ ⬎ 15, where Re⬘ = vL / ␯, ␯ is kinetic viscosity, v is the root-mean square of the turbulent velocity
fluctuation, and L is the mean length of polymer. Therefore,
our study cannot be compared with Davoudi’s work.

FIG. 8. The geometry of the present simulations.

rods with finite aspect ratios is governed by the equationr2 − 1 2
1dsin ␸ − ␥˙ 2,
␸ = − ␥˙ 2
r +1
r +1
dt

where the aspect ratio is r = L / ␴ and L is the length of thechain.
In our work, we consider the equation that governs theangle ␸ of a Brownian rigid rod with a finite aspect ratio. To
consider the diffusion of thermal fluctuation, we introduce
white noise ␨共t兲 into Eq. 共12兲. We can then write down thefollowing equation for a Brownian rigid rod:r2 − 1 2
1dsin ␸ − ␥˙ 2+2
␸ = − ␥˙ 2
r +1
r +1
dt

冑

Dr
␨,
cos2 ␪

具␨共t兲␨共t⬘兲典 = ␦共t − t⬘兲,

共13兲
共14兲

where Dr is the rotational diffusion constant. On the basis of
the shell model 关23,24兴, the rotational diffusion constant Dr
for a rigid rod is calculated as
Dr =3关ln r + d共r兲兴kBT,
␲␩L3

d共r兲 = − 0.662 +

0.917 0.05
− 2 .
r
r

共15兲

共16兲

In the shell model mentioned above, the contour of the macromolecules of arbitrary shape is represented by a shell composed of many identical small beads. The shell model can beadequately modeled by decreasing the size of the beads.
In the case of Dr Ⰶ ␥˙ , the dynamics of the angle ␸ become
decoupled from the angle ␪ between the end-to-end vector
and the x-y plane because the angle ␪ is approximately zero.
We can then write down the following equation:r2 − 1 2
1dsin ␸ − ␥˙ 2
+ 2冑Dr␨ .
␸ = − ␥˙ 2
r +1
r +1
dt

共17兲

For short times, the mean square displacement of ␸共t兲 in time
t is written as
具兩␸共t兲 − ␸共0兲兩2典 = 4Drt

B. Theoretical analysis

The geometry of the chain in our paper is depicted in Fig.
8. In Jeffrey’s work 关12兴, the angle ␸ of non-Brownian rigid

共12兲

共for Drt Ⰶ 1兲.

共18兲

In order to analyze the tumbling motion of a single chain,
we only have to consider the dynamics in the stochastic area,

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PHYSICAL REVIEW E 81, 041807 共2010兲

HIDEKI KOBAYASHI AND RYOICHI YAMAMOTO⳵⳵J␸ = 0,
P共t, ␸兲 +
⳵t
⳵␸

再

共19兲

冎

r2 − 1 2
1⳵
sin ␸ + ␥˙ 2
+ 2Dr
P共t, ␸兲,J␸ = − ␥˙ 2
r +1
r +1
⳵␸

FIG. 9. 共Color online兲 Schematic diagram of two areas and theangle ␸t. The angle ␸t is the boundary angle of the area dominatedby the first term and the area dominated by the second and thirdterms on the right-hand side of Eq. 共17兲.

where the effect of thermal fluctuation is not negligible. In
the case of Dr Ⰶ ␥˙ , the time required to pass through thestochastic area is sufficiently larger than the time required topass through the other area. The tumbling motion can only
be understood by considering the stochastic area. The stochastic area is dominated by the second and third terms on
the right-hand side of Eq. 共17兲. The remaining area is dominated by the first term on the right-hand side of Eq. 共17兲.
Figure 9 shows the two areas and the angle ␸t, which is theboundary angle between the two areas. We call the second
and third terms on the right-hand side of Eq. 共17兲 the shear
and the fluctuation terms, respectively.

where P共t , ␸兲 is the PDF of the angle ␸ and J␸ is the probability flow. Each term in the braces of Eq. 共20兲 corresponds
to a respective term on the right-hand side of Eq. 共17兲.
Next, we focus on the stationary PDF Pst共␸兲 ⬅ P共t = 0 , ␸兲.
Because ⳵ Pst共␸兲 / ⳵t = 0, namely, ⳵J␸ / ⳵␸ = 0 from Eq. 共19兲, J␸is ␸ independent and constant: J␸ = −␥˙ 关共r2 − 1兲sin2 ␸p Pst共␸p兲
+ Pst共␸p兲兴 / 共r2 + 1兲, where ␸p is the angle at which Pst共␸兲 hasa peak. We surmise that the angle ␸t satisfies the equation
given by共r2 − 1兲sin2 ␸ Pst共␸兲⳵ Pst共␸兲
Pst共␸兲
J␸
= − ␥˙ 2− 2Dr= − ␥˙
.
2
r +1
r +1
⳵␸
2
共21兲
By substituting the J␸ expressed by ␸p into Eq. 共21兲, therelation of ␸t to ␸p is given as− ␥˙

共r2 − 1兲sin2 ␸p Pst共␸p兲 + Pst共␸p兲
r2 + 1

1. Infinite aspect ratio caseFirst, we consider Eq. 共17兲 in the limit where r → ⬁. This
limit is consistent with treating a chain as an end-to-end
vector. We can rewrite Eq. 共17兲 as ␸˙ = −␥˙ sin2 ␸ + 2冑Dr␨, inwhich the shear term does not exist. This equation is identical to the equation used in previous works 关5–8兴.
Previous works 关4–6兴 have reported ␸t ⬃ 共Dr / ␥˙ 兲1/3. In theregion of ␸t ⬍ ␸ ⬍ ␲, the shear flow rapidly rotates the orientation of the chain from ␲ to ␸t in a time tl ⬇ ␥˙ −1. In theregion of 0 ⬍ ␸ ⬍ ␸t, the chain orientation almost aligns
along the shear direction. Because the effect of shear becomes sufficiently small in this region, the effect of thermal
fluctuations only contributes to rotate the chain orientationfrom ␸t to 0 in time tr = ␸2t / 4Dr ⬀ ␥˙ −2/3, as calculated with Eq.
共18兲. In the case of tr ⬎ tl, tr dominates the chain tumbling
time. We can consider ␯ p ⬀ tr−1. Therefore, we conclude from
Eqs. 共10兲 and 共11兲 that F ⬀ Pe2/3.

= − 2␥˙

共r2 − 1兲sin2 ␸t Pst共␸t兲
.
r2 + 1

共22兲

The angle ␸t is calculated by solving Eq. 共22兲. Pst共␸t兲 isexpanded in powers ⌬␸ = ␸t − ␸p. When we neglect the second order of ⌬␸ and higher, we can relate ␸t to ␸p as

␸t =

冑冉

冊

1
1 2
␸p + 2
.
2
r −1

共23兲

It should be noted that ␸t , ␸p Ⰶ 1. We can estimate ␸t withthe angle ␸p.
Next, we attempt to calculate the analytical form of ␸p. In
the case of Dr / ␥˙ Ⰶ 1, the formal solution for Pst共␸兲 is givenby
Pst共␸兲 = C1

冕

␲

0

2. Finite aspect ratio case

Next, we consider Eq. 共17兲 with a finite aspect ratio, r. We
can expect F to be proportional to Pe2/3 when the fluctuation
term dominates in the region 0 ⬍ ␸ ⬍ ␸t, as this case agrees
with the limit of r → ⬁. Additionally, we can expect F to be
proportional to Pe when the shear term dominates in theregion 0 ⬍ ␸ ⬍ ␸t because this case corresponds to the nonBrownian rigid rod. Pec is defined as the Peclet number atwhich the dominating term in the stochastic area changes
from the fluctuation term to the shear term with increasing
shear rate.
From Eq. 共17兲, we can write down the corresponding
Fokker-Planck equation as

共20兲

冉

f共␺, ␸兲 = ␺ − 1 −冉d␺ exp −冊冊

␥˙f共␺, ␸兲 ,4Dr

2
sin ␺ cos共␺ − 2␸兲,
r2 + 1

共24兲

共25兲

where C1 is determined from the normalization condition,
兰0␲ Pst共␸兲d␸ = 1. In the limit of r → ⬁, it is known that ␸p= 共Dr / ␥˙ 兲1/3 关6兴. When r is finite, ␸p is different from
共Dr / ␥˙ 兲1/3 because the shear term influences Pst共␸兲.
To estimate the effect of the shear term, we introduce thetimes t2⬘ and t3⬘, where t2⬘ and t3⬘ represent the times required topass through the region 0 ⬍ ␸ ⬍ 共Dr / ␥˙ 兲1/3, depending only on
the shear term and the fluctuation term, respectively. t2⬘ isestimated by dividing 共Dr / ␥˙ 兲1/3 by the shear term as

041807-6

TUMBLING MOTION OF A SINGLE CHAIN IN SHEAR …

PHYSICAL REVIEW E 81, 041807 共2010兲

冑再 冉 冊 冎
1

C共r,Pe兲 =r + 1 1 − C⬘2

C⬘ + 422

r2 − 1

C⬘

.

共31兲

We consider that F is proportional to Pe2/3 when the effect ofthermal fluctuation is more significant than the effect ofshear flow in the region of 0 ⬍ ␸ ⬍ ␸t and that F is proportional to Pe in the opposite case. t2 is the time required topass through the region 0 ⬍ ␸ ⬍ ␸t by the shear term,
t2 =共r2 + 1兲CDr1/3.
␥˙ 4/3共32兲

t3 is the time required to pass through the region 0 ⬍ ␸ ⬍ ␸tby the fluctuation term,
FIG. 10. 共Color online兲 The behavior of ␸t as a function of Pe.
␸t⬘ is ␸t calculated with ␸p, obtained from numerical integration
共red +兲. The green solid line corresponds to the behavior of ␸t⬙,where ␸t⬙ is ␸t, obtained from ␸p = 关t⬘2 / 共t⬘2 + t⬘3兲兴 ⫻ 共Dr / ␥˙ 兲1/3.

t2⬘ =共r2 + 1兲Dr1/3,
␥˙ 4/3共26兲

and t3⬘ is estimated by Eq. 共18兲 as
t3⬘ =

1

.
4Dr1/3␥˙ 2/3

共27兲

Because the Brownian chain is rotated by both the shear term
and the fluctuation terms, we expect that ␸p is given byC⬘共r,Pe兲 ⬅

t2⬘
t2⬘ + t3⬘

␸p = C⬘共r,Pe兲

冉 冊

再

再

= 1+

Dr 1/3
,
␥˙

共28兲

冎冎

2/3 −1
1
r3
Pe.
4r2 + 1 18关ln r + d共r兲兴

共29兲
For large ␥˙ / Dr where the present theoretical analysis isvalid, C⬘共r , Pe兲 ⬃ t2⬘ / t3⬘ finally approaches zero. We confirmed, however, that t2⬘ / t3⬘ still remains finite 共⬃1兲 aroundPec ⬇ 100, where ␥˙ / Dr ⬇ 1000.
Figure 10 shows the behaviors of ␸t⬘ and ␸t⬙ as a functionof Pe, where ␸t⬘ is ␸t calculated with Eq. 共23兲 and ␸p, which
is obtained from the numerical integration of Eq. 共24兲, and␸t⬙ is ␸t calculated with Eq. 共23兲 and ␸p, which is expressedas Eq. 共28兲. In this figure, it is shown that Eq. 共28兲 is established because the behavior of ␸t⬘ agrees well with ␸t⬙ forPeⱕ 100. We can obtain ␸t by substituting Eq. 共28兲 into Eq.
共23兲. Therefore, ␸t is given by␸t = C共r,Pe兲冉 冊

Dr 1/3
,
␥˙

共30兲

t3 =

C2

.
4Dr1/3␥˙ 2/3

共33兲

It is thought that Pec is the Peclet number that satisfies t2= t3. If we obtain the value of r, we know the value of Pec
because both t2 and t3 are functions of Pe and r. Our results
show that Pec ⬇ 115 at r = 5. In the numerical results obtained
from our work, Pec = 106. The analytical result agrees wellwith our numerical result. Moreover, the numerical condition
Dr Ⰶ ␥˙ is satisfied because Dr / ␥˙ ⬇ 10−3 Ⰶ 1 at Pe⬇ 100.
Therefore, the considerations in this section are reasonable in
the region near Pec.
The considerations in this section agree well with those ofprevious experimental results 关2–4,10,14兴. In experimental
works that measured the frequencies of DNA rotation
关2–4,10兴, the DNA molecules contained roughly 400 persistence lengths. The persistence length is thought to correspond to r. Thus, we conclude that Pec ⬇ 516. These experiments were carried out in the region of PeⰆ Pec, and F is
proportional to Pe2/3. In experimental work that measured thefrequencies of freely rotating rigid dumbbells 关14兴, the aspectratio r of the rigid dumbbell corresponds to 2 and Pec ⬇ 60.
The experiments were carried out in the region of Pec Ⰶ Pe,
and F was proportional to Pe. From these results, the considerations in this section are reasonable.
V. CONCLUSION

In our work, we calculated the tumbling motion of asingle chain using an SPM that takes into account thermal
fluctuations and hydrodynamic interactions for 0.0005⬍ kBT ⬍ 1.00 and 0.001⬍ ␥˙ ⬍ 0.02. We conclude that the dimensionless frequency, F, depends only on Pe. The dependence of F can be described by a power law F ⬀ Pe␣. The
exponent ␣ sharply changes from 2/3 to 1 on Pec. In the caseof a rigid rod with N = 5, Pec ⬇ 106, and in the case of aflexible chain with N = 5, Pec ⬇ 156. The behavior of F for
both cases is similar, while only the values of Pec are different from each other.
We have presented F to be proportional to Pe2/3 when thethird term on the right-hand side of Eq. 共17兲 dominates in theregion 0 ⬍ ␸ ⬍ ␸t, and F is proportional to Pe when the041807-7

PHYSICAL REVIEW E 81, 041807 共2010兲

HIDEKI KOBAYASHI AND RYOICHI YAMAMOTO

second term of Eq. 共17兲 dominates in the region 0 ⬍ ␸ ⬍ ␸t.
We have estimated the angle ␸t at which the fist term of J␸,expressed as Eq. 共20兲, is comparable to the sum of the second and third terms of J␸, expressed as Eq. 共20兲.
A proposed mechanism for this exponent change is thatthe effect of thermal fluctuation is more significant than theeffect of shear flow only for 0 ⬍ ␸ ⬍ ␸t, whereas in the othercase, the effect of thermal fluctuation is negligible. The

former contribution leads to F ⬀ Pe2/3, and the latter contribution leads to F ⬀ Pe.

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ACKNOWLEDGMENT

The authors would like to express their gratitude to Dr. T.
Murashima, Dr. Y. Nakayama, Dr. K. Kim, and Dr. T.
Iwashita for useful comments and discussions.

041807-8