Motivated by recent experiments [Science {\bf 299}, 1042 (2003)] reporting
that carbon nanotubes immersed in a flowing fluid displayed an electric current
and voltage, we numerically study the behaviour of a collection of Brownian
particles in a channel, in the presence of a flow field applied on similar but
slower particles in a wide chamber in contact with the channel. For a suitable
range of shear rates, we find that the flow field induces a unidirectional
drift in the confined particles, and is stronger for narrower channels. The
average drift velocity initially rises with increasing shear rate, then shows
saturation for a while, thereafter starts decreasing, in qualitative agreement
with recent theoretical studies [Phys. Rev. B {\bf 70}, 205423 (2004)] based on
Brownian drag and ``loss of grip''. Interestingly, if the sign of the
interspecies interaction is reversed, the direction of the induced drift
remains the same, but the flow-rate at which loss of grip occurs is lower, and
the level of fluctuations is higher.Comment: 7 pages, 9 figure

Ananthakrishna, G.

Das, Moumita

Ramaswamy, Sriram

Sood, A. K.

English

arXiv

Motivated by recent experiments [1] reporting that carbon nanotubes immersed in a flowing fluid displayed an electric current and voltage, we numerically study the behaviour of a collection of Brownian particles in a channel, in the presence of a flow field applied on similar but slower particles in a wide chamber in contact with the channel. For a suitable range of shear rates, we find that the flow field induces a unidirectional drift in the confined particles, and is stronger for narrower channels. The average drift velocity initially rises with increasing shear rate, then shows saturation for a while, thereafter starts decreasing, in qualitative agreement with recent theoretical studies [2] based on Brownian drag and "loss of grip". Interestingly, if the sign of the interspecies interaction is reversed, the direction of the induced drift remains the same, but the flow-rate at which loss of grip occurs is lower, and the level of fluctuations is higher

Indian Academy of Sciences

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Europhysics Letters PREPRINT
Flow-induced currents in nanotubes: a Brownian dynam-
ics approach
Moumita DAS1,2, Sriram RAMASWAMY1, A. K. SOOD1 and G. ANANTHAKR-
ISHNA3
1 Department of Physics, Indian Institute of Science, Bangalore 560012, India
2 Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA-
02138,USA.
3 Materials Research Centre, IISc, Bangalore 560012, India.
PACS. 02.70.Ns – Molecular dynamics and particle methods.
PACS. 05.10.Gg – Stochastic analysis methods(Fokker-Planck,Langevin, etc).
PACS. 61.46.+w – Nanoscale materials.
Abstract. – Motivated by recent experiments [1] reporting that carbon nanotubes immersed
in a flowing fluid displayed an electric current and voltage, we numerically study the behaviour
of a collection of Brownian particles in a channel, in the presence of a flow field applied on similar
but slower particles in a wide chamber in contact with the channel. For a suitable range of shear
rates, we find that the flow field induces a unidirectional drift in the confined particles, and is
stronger for narrower channels. The average drift velocity initially rises with increasing shear
rate, then shows saturation for a while, thereafter starts decreasing, in qualitative agreement
with recent theoretical studies [2] based on Brownian drag and “loss of grip”. Interestingly, if
the sign of the interspecies interaction is reversed, the direction of the induced drift remains
the same, but the flow-rate at which loss of grip occurs is lower, and the level of fluctuations is
higher.
Recent experiments show that liquid flow past single walled carbon nanotube (SWNT)
bundles generates voltage [1] and electric current [2] in the nanotube, along the direction of
flow. The current and voltage are found [1,2] to be highly sublinear functions of the fluid flow
rate. The direction of flow-induced current relative to fluid flow is determined [1, 2] by the
nature of the ions in the vicinity of the nanotubes. The experiments in [1,2] also showed that
the one-dimensional nature of the SWNTs was essential for the generation of a net electrical
signal in the sample: experiments on graphite did not generate a measurable signal and those
on multiwalled carbon nanotubes produced a signal approximately 10 times weaker than that
for the single walled nanotubes, the dimensions of the sensing element and flow speeds of the
ionic liquid remaining same. Electrokinetic [3] and phonon-wind [4] based explanations give
a linear current versus flow rate dependence. There is also a recent proposal [5] involving
stick-slip and barrier hopping of ions giving a sublinear dependence. Ghosh et al. [2] explain
the phenomenon as follows: Thermal fluctuations in the ionic charge density in the fluid near
the nanotube produce a stochastic Coulomb field on the carriers in the nanotube. At thermal
equilibrium, i.e., if there is no mean fluid flow, the fluctuation-dissipation theorem tells us
c© EDP Sciences
2 EUROPHYSICS LETTERS
that the power spectral density of these fluctuations contributes to the frictional drag on
the carriers. The total friction or resistivity of the carriers is thus the sum of contributions
from the ambient ions and from phonons or other mechanisms intrinsic to the nanotube. An
imposed fluid flow advects the ions. The carriers in the nanotube then have to move at a
speed determined by balancing the drag forces due to the ions and the nanotube. However,
increasing the fluid velocity also Doppler-shifts the autocorrelation function of the ionic charge
density, hence speeding up its time-decay as seen by the carriers and reducing the drag due to
the ions. This results in a saturation of the flow-induced voltage and current. The calculation
of [2] was done in a simplified framework, where the ions were assumed to move with a single
mean velocity. In the experiment the velocity of fluid (and ions) increases with distance
from the nanotube surface. Ions nearest the nanotube contribute the strongest Coulomb field
but move the slowest. It would be interesting, first, to see if the form of the dependence of
current on flow-speed remains intact when these competing tendencies are taken into account.
Secondly, it remains to be seen if the theory of [2] produces a weaker effect when extended to
flow past a higher dimensional conductor. Lastly, only the two-point correlations of the ionic
Coulomb field enter the (Gaussian) treatment of [2], which means that changing the sign of
the ionic charge can have no effect. Testing the importance of non-Gaussian fluctuations is
important, but hard to do analytically. All three questions are easily resolved in a numerical
simulation.
Motivated by these considerations, we carry out a Brownian dynamics study on a model
system (Figs. 1) consisting of two types of particles, (i) particles ‘A’ of low diffusivity in a
chamber (ions) and (ii) particles ‘B’ of comparatively high diffusivity (carriers) confined in an
adjoining channel (the nanotube) of depth WB much smaller than that of the chamber. Both
‘A’ and ‘B’ particles are assumed to be charged particles with pairwise screened Coulomb
interactions. Our simulations implement the theoretical ideas of [2] with an important differ-
ence: the ions flowing past the nanotube have a strong velocity gradient in the vicinity of the
nanotube, as shown in Fig. 1. For a suitable range of shear rates (magnitude of velocity gra-
dient), we find that an imposed flow of the ‘A’ particles indeed induces a unidirectional drift
in the ‘B’ particles in the channel. The average drift velocity initially rises with increasing
shear rate, then shows saturation for a while, and thereafter starts decreasing, in qualita-
tive agreement with the theoretical arguments of [2]. We further observe that the induced
drift decreases substantially when WB is increased, underlining as in the experiments of [1],
the central role of reduced dimensionality. Interestingly, if the sign of the A-B interaction
is reversed, the direction of the induced drift remains the same, in agreement with [2], but
the flow-rate at which crossover to saturation occurs is lower, and the level of fluctuations is
higher.
Model and Results. – Let us now present our study and its results in more detail. We
consider for simplicity a two dimensional (x-y) system consisting of two species of particles,
say, A representing the ions in the chamber (of dimension 10ℓ × 10ℓ, ) and B, representing
the carriers in the narrow channel (of dimension 10ℓ × ǫℓ, ǫ ≤2), with ℓ = (ρA)
−1/2, where
ρA is the mean number density of ions. We work with periodic boundary conditions in the
x direction, and hard walls at y = 10ℓ, y = 0 and y = −ǫℓ. The flow field imposed on A
particles is plane Couette, with velocity in the x direction and gradient in the y direction till
a certain separation between the walls (5ℓ), beyond which the velocity is uniform (Fig. 1).
This flow geometry effectively models the flow field near a small obstacle in bulk flow, such
that gradients are concentrated in the boundary layer on the surface of the obstacle.
There is no flow field imposed on B particles. Both A and B particles are overdamped and
the pairwise interactions are screened Coulomb in nature. Further B particles are 100 times
M. Das et al.: Flow-induced currents in nanotubes: Brownian dynamics 3
Carriers
Ions
W
B
5
5
10
y=0
l
l
l
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Fig. 1 – A schematic diagram of the model.
more diffusive than their A counterparts. The positions R(t) (= x(t), y(t)), evolve according
to overdamped Langevin equations, with independent Gaussian, zero-mean, thermal white
noise sources hA (or hB), interparticle forces ∇V from the pair potentials, and for the ions
an additional force due to the flow field γ̇x̂. Let us nondimensionalise our variables as follows:
scale all lengths by ℓ, energy by Boltzmann’s constant kB times temperature T (and hence
force by kBT/ℓ), and time by the time τ taken by a carrier to traverse a distance ℓ. Then our
nondimensional discretised Langevin equations are for a given type (A or B) of particles,
RA(t + δt) = RA(t) + DA[γ̇x̂ − ∇VAA − ∇VAB]δt +
√
2DAδt hA(t), (1)
RB(t + δt) = RB(t) − DB∇VBAδt +
√
2DAδt hB(t) (2)
where DA(= 1) and DB(= 100) are the nondimensionalised Brownian diffusivities for the
ions and carriers respectively and obey the fluctuation dissipation relation, i.e., for example
for A particles, 〈hA
i(0)hA
j(t)〉 = 2Iδijδ(t), where I is the unit tensor and i, j label particle
indices. The pair potentials have the screened Coulomb form V (r) = (U/r) exp(−κr) (where
U is a coupling coefficient of the DLVO form (1).) at interparticle separation r, with different
κ’s for AA and AB interaction, while there is no BB interaction. Also both the A and B
particles carry charges of identical sign and magnitude. The dimensions of the box containing
A particles is L = 10 and W = 10, whereas for B particles L = 10, and W = ǫ(ǫ ≤ 2). Keeping
WA=10 and κABℓ=2 fixed, we monitor the behavior of the system by changing: (1) Width of
the channel containing the carriers WB, (2) Ion-ion interaction strength, κAAℓ and (3) sign
of the A − B i.e. ion-carrier interaction. We also explore the effect of changing the screening
length of the ion-carrier (A − B) interaction, keeping other parameters constant. We have
studied the behavior of this model for a system with NA = 100 particles and ρA/ρB = 1 and
ρA/ρB = 1/2, ρA and ρB being the number densities for the ions and carriers respectively.
The results presented here pertain to the latter case, the observed behavior being qualitatively
same for both cases.
(1)This DLVO coefficient is given by ZiZje
2exp(κ(σi + σj))/(ǫ(1 + κσi)(1 + κσj)), where ǫ is the dielectric
constant of the solvent, Zie is the charge on and σi is the radius of the i
th particle. In our simulations, the
dimensionless value of the κ independent part of U is ≈ 15.3, σB/σA = 0.1 and 2σA/ℓ ≈ 0.455.
4 EUROPHYSICS LETTERS
0 5 10
0
0.5
1
1.5
γ τ
<
V
d>
0.5 1 1.5 2
0
0.5
1
W
B
<
V
d>
. 
Fig. 2 – The induced drift of the carriers as a function of the ionic flow speed for different WB (stars,
diamonds and squares correspond to WB = 0.5,WB = 1.0 and WB = 2.0 respectively) for κAAℓ=1.
The inset shows the average drift velocity 〈vd〉 as a function of WB, for γ̇τ = 4.0 (circles) and 2.0
(triangles) respectively.
For WB = 2, we observe that the carriers do have a unidirectional drift in the direction
of the flow. The average drift velocity of the carriers initially increases with the flow rate,
reaches a maximum and then decreases at large values of the flow rate as seen in Fig. 2. This
is broadly in agreement with the ideas of [2]. On further decreasing WB to first 1.0 and then
0.5 (Fig.2), we find that the drift velocity induced is progressively larger than for WB = 2.0,
and the saturation occurs at a lower value of the flow rate.
γ τ
<
V
d>
0 25 50
0
1
2
. 
Fig. 3
γ τ
<
v d
>
0  2.5 5  7.5 10 
0
0.5
1
. 
Fig. 4
Fig. 3 – Induced drift as function of flow speed for κAAℓ=0.2, WB=0.5. The vertical dashed lines
show the standard deviation from the mean drift velocity.
Fig. 4 – Induced drift as function of flow speed for κAAℓ=4, WB=0.5. Note that the velocity
weakening takes place at a much smaller shear rate than in Fig.3. The vertical dashed lines show the
standard deviation from the mean drift velocity.
M. Das et al.: Flow-induced currents in nanotubes: Brownian dynamics 5
0 2 4
0
3
6
γ τ
<
V
d>
. 
 
Fig. 5
0 250 500
0
i
V
d/
<
V
d>
Fig. 6
Fig. 5 – Induced drift as a function of the ionic flow speed for κAAℓ=1, WB=1, with attractive
ion-carrier interaction.
Fig. 6 – Drift velocities Vd of the carriers at different times (represented by the index ’i’) scaled
by the average value 〈Vd〉 at κAAℓ=1, WB=1, for repulsive (lines) and attractive (circles) ion-carrier
interactions.
Now keeping the wall to wall distance fixed at WB = 0.5, we study the behavior of the
system at very small (κAAℓ=0.2) (Fig. 3)and large (κAAℓ=4) ion-ion interaction screening
strength (Fig. 4). A smaller value of κℓ corresponds to stronger ion-ion interactions and it is
observed that in such a case the induced effect on the carriers is stronger and is sustained till
a larger value of the ionic flow speed at which the weakening sets in.
We now look at the scenario where the two species A (ions) and B (carriers) have opposite
charges (Fig. 5). The wall to wall distance is kept fixed at WB = 1. We observe that the
induced drift is stronger than when both A and B particles carry like charges, at the same WB.
The fluctuations in the drift velocity are however much larger than for the case of like charges
(Fig. 6): the ratio of standard deviation to mean is ∼ 4 times that for the repulsive case. This
can be explained as follows: The attraction of the carriers for the ions brings both close to the
wall separating them, from time to time, enhancing the flow induced current. Subsequently the
presence of the walls pushes them apart, reducing the induced current again. This implies large
fluctuations in the current. For like charged ions and carriers both the Coulomb repulsion and
the walls act in tandem to keep them apart, thus the intermittent enhancement and reduction
in current does not occur.
In order to understand the dependence of the effect on the strength of A-B interaction,
we monitored the system for different values of the screening length corresonding to the A-B
interaction (Fig. 7) keeping the channel width WB and A-A interaction strength constant.
We find that the effect increases on increasing the screening length κ−1AB and is maximum
where κ−1AB is comparable to the width of the channel WB ; therafter it saturates and doesn’t
show a marked increase. This shows that the contribution to the flow induced effect indeed
chiefly comes from the A particles (or ions) present in a boundary layer of order of the width
of the channel at the interface of the chamber and the channel.
We observe that our simulations (Fig. 8) agree well with the theoretical predictions of a
saturating dependence of the induced velocity of the carriers on the imposed flow speeds (Fig.
9). We further find the current decreases when γ̇ passes a threshold value γ̇c, and that γ̇c is
6 EUROPHYSICS LETTERS
2 6 10 14
0
0.5
1
1.5
2
γ τ
<
V
d>
. 
Fig. 7 – The flow induced drift velocity of B particles as a function of the flow speed of A particles,
for κ−1AB = 0.1ℓ (circles), 0.5ℓ (diamonds), 5ℓ (cross) and 10ℓ (plus), for fixed WB=1.0 and κAAℓ = 1.
smaller for smaller WB . This observation demands a theoretical explanation which at present
we do not have. We hope this will motivate further experiments to check this prediction. It is
significant that we are able to observe this type of “Brownian drag” and transfer of momentum
even in the absence of momentum conservation and in the extreme limit of no inertia. Also,
though strictly speaking fluctuation dissipation relations are valid only in equilibrium, a naive
extension of FDT as in [2] to the flowing case would imply a decrease in the rate of relaxation
with increasing flow rate, and hence a “loss of grip” and a velocity weakening, which is what
we find.
10
−2
10
−1
10
0
10
1
0
0.2
0.4
0.6
γ τ
<
V
d>
. 
Fig. 8
1E-5 1E-4 1E-3 0.01 0.1 1
0.0
0.2
0.4
 
 
I (
µA
)
uL (cm/s)
Fig. 9
Fig. 8 – Induced velocity with flow speed for parameter values as in Fig. 3 on a log scale for comparison
with the behavior predicted by [2] as shown in Fig. 9.
Fig. 9 – Current I as a function of flow speed uL. Figure provided by [2].
M. Das et al.: Flow-induced currents in nanotubes: Brownian dynamics 7
∗ ∗ ∗
We thank S. Ghosh for useful discussions and providing figure 9. MD acknowledges CSIR,
India for support, and SR and GA thank the DST, India for support through the Centre for
Condensed Matter Theory. AKS thanks DST for financial support of his research on carbon
nanotubes.
REFERENCES
[1] S. Ghosh, A.K. Sood and N. Kumar, Science, 299 (2003) 1042.
[2] S. Ghosh, A.K. Sood, Sriram Ramaswamy and N. Kumar, Phys. Rev. B, 70 (2004) 205423
[3] A. E. Cohen (comment); S. Ghosh, A. K. Sood, and N. Kumar (reply), Science, 300
(2003) 1235.
[4] P. Kral and M.Shapiro, Phys. Rev. Lett., 86 (2001) 131
[5] B. N. J. Persson, U. Tartaglino, E. Tosatti, and H. Ueba, Phys. Rev. B, 69 (2004) 235410


Flow-induced currents in nanotubes: a brownian dynamics approach

Publications of the IAS Fellows

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v1
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  1
 M
ay
 20
05
Europhysics Letters PREPRINT
Flow-induced currents in nanotubes: a Brownian dynam-
ics approach
Moumita DAS1,2, Sriram RAMASWAMY1, A. K. SOOD1 and G. ANANTHAKR-
ISHNA3
1 Department of Physics, Indian Institute of Science, Bangalore 560012, India
2 Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA-
02138,USA.
3 Materials Research Centre, IISc, Bangalore 560012, India.
PACS. 02.70.Ns – Molecular dynamics and particle methods.
PACS. 05.10.Gg – Stochastic analysis methods(Fokker-Planck,Langevin, etc).
PACS. 61.46.+w – Nanoscale materials.
Abstract. – Motivated by recent experiments [1] reporting that carbon nanotubes immersed
in a flowing fluid displayed an electric current and voltage, we numerically study the behaviour
of a collection of Brownian particles in a channel, in the presence of a flow field applied on similar
but slower particles in a wide chamber in contact with the channel. For a suitable range of shear
rates, we find that the flow field induces a unidirectional drift in the confined particles, and is
stronger for narrower channels. The average drift velocity initially rises with increasing shear
rate, then shows saturation for a while, thereafter starts decreasing, in qualitative agreement
with recent theoretical studies [2] based on Brownian drag and “loss of grip”. Interestingly, if
the sign of the interspecies interaction is reversed, the direction of the induced drift remains
the same, but the flow-rate at which loss of grip occurs is lower, and the level of fluctuations is
higher.
Recent experiments show that liquid flow past single walled carbon nanotube (SWNT)
bundles generates voltage [1] and electric current [2] in the nanotube, along the direction of
flow. The current and voltage are found [1,2] to be highly sublinear functions of the fluid flow
rate. The direction of flow-induced current relative to fluid flow is determined [1, 2] by the
nature of the ions in the vicinity of the nanotubes. The experiments in [1,2] also showed that
the one-dimensional nature of the SWNTs was essential for the generation of a net electrical
signal in the sample: experiments on graphite did not generate a measurable signal and those
on multiwalled carbon nanotubes produced a signal approximately 10 times weaker than that
for the single walled nanotubes, the dimensions of the sensing element and flow speeds of the
ionic liquid remaining same. Electrokinetic [3] and phonon-wind [4] based explanations give
a linear current versus flow rate dependence. There is also a recent proposal [5] involving
stick-slip and barrier hopping of ions giving a sublinear dependence. Ghosh et al. [2] explain
the phenomenon as follows: Thermal fluctuations in the ionic charge density in the fluid near
the nanotube produce a stochastic Coulomb field on the carriers in the nanotube. At thermal
equilibrium, i.e., if there is no mean fluid flow, the fluctuation-dissipation theorem tells us
c© EDP Sciences
2 EUROPHYSICS LETTERS
that the power spectral density of these fluctuations contributes to the frictional drag on
the carriers. The total friction or resistivity of the carriers is thus the sum of contributions
from the ambient ions and from phonons or other mechanisms intrinsic to the nanotube. An
imposed fluid flow advects the ions. The carriers in the nanotube then have to move at a
speed determined by balancing the drag forces due to the ions and the nanotube. However,
increasing the fluid velocity also Doppler-shifts the autocorrelation function of the ionic charge
density, hence speeding up its time-decay as seen by the carriers and reducing the drag due to
the ions. This results in a saturation of the flow-induced voltage and current. The calculation
of [2] was done in a simplified framework, where the ions were assumed to move with a single
mean velocity. In the experiment the velocity of fluid (and ions) increases with distance
from the nanotube surface. Ions nearest the nanotube contribute the strongest Coulomb field
but move the slowest. It would be interesting, first, to see if the form of the dependence of
current on flow-speed remains intact when these competing tendencies are taken into account.
Secondly, it remains to be seen if the theory of [2] produces a weaker effect when extended to
flow past a higher dimensional conductor. Lastly, only the two-point correlations of the ionic
Coulomb field enter the (Gaussian) treatment of [2], which means that changing the sign of
the ionic charge can have no effect. Testing the importance of non-Gaussian fluctuations is
important, but hard to do analytically. All three questions are easily resolved in a numerical
simulation.
Motivated by these considerations, we carry out a Brownian dynamics study on a model
system (Figs. 1) consisting of two types of particles, (i) particles ‘A’ of low diffusivity in a
chamber (ions) and (ii) particles ‘B’ of comparatively high diffusivity (carriers) confined in an
adjoining channel (the nanotube) of depth WB much smaller than that of the chamber. Both
‘A’ and ‘B’ particles are assumed to be charged particles with pairwise screened Coulomb
interactions. Our simulations implement the theoretical ideas of [2] with an important differ-
ence: the ions flowing past the nanotube have a strong velocity gradient in the vicinity of the
nanotube, as shown in Fig. 1. For a suitable range of shear rates (magnitude of velocity gra-
dient), we find that an imposed flow of the ‘A’ particles indeed induces a unidirectional drift
in the ‘B’ particles in the channel. The average drift velocity initially rises with increasing
shear rate, then shows saturation for a while, and thereafter starts decreasing, in qualita-
tive agreement with the theoretical arguments of [2]. We further observe that the induced
drift decreases substantially when WB is increased, underlining as in the experiments of [1],
the central role of reduced dimensionality. Interestingly, if the sign of the A-B interaction
is reversed, the direction of the induced drift remains the same, in agreement with [2], but
the flow-rate at which crossover to saturation occurs is lower, and the level of fluctuations is
higher.
Model and Results. – Let us now present our study and its results in more detail. We
consider for simplicity a two dimensional (x-y) system consisting of two species of particles,
say, A representing the ions in the chamber (of dimension 10ℓ × 10ℓ, ) and B, representing
the carriers in the narrow channel (of dimension 10ℓ × ǫℓ, ǫ ≤2), with ℓ = (ρA)
−1/2, where
ρA is the mean number density of ions. We work with periodic boundary conditions in the
x direction, and hard walls at y = 10ℓ, y = 0 and y = −ǫℓ. The flow field imposed on A
particles is plane Couette, with velocity in the x direction and gradient in the y direction till
a certain separation between the walls (5ℓ), beyond which the velocity is uniform (Fig. 1).
This flow geometry effectively models the flow field near a small obstacle in bulk flow, such
that gradients are concentrated in the boundary layer on the surface of the obstacle.
There is no flow field imposed on B particles. Both A and B particles are overdamped and
the pairwise interactions are screened Coulomb in nature. Further B particles are 100 times
M. Das et al.: Flow-induced currents in nanotubes: Brownian dynamics 3
Carriers
Ions
W
B
5
5
10
y=0
l
l
l
                               
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Fig. 1 – A schematic diagram of the model.
more diffusive than their A counterparts. The positions R(t) (= x(t), y(t)), evolve according
to overdamped Langevin equations, with independent Gaussian, zero-mean, thermal white
noise sources hA (or hB), interparticle forces ∇V from the pair potentials, and for the ions
an additional force due to the flow field γ˙xˆ. Let us nondimensionalise our variables as follows:
scale all lengths by ℓ, energy by Boltzmann’s constant kB times temperature T (and hence
force by kBT/ℓ), and time by the time τ taken by a carrier to traverse a distance ℓ. Then our
nondimensional discretised Langevin equations are for a given type (A or B) of particles,
RA(t+ δt) = RA(t) +DA[γ˙xˆ−∇VAA −∇VAB]δt+
√
2DAδt hA(t), (1)
RB(t+ δt) = RB(t)−DB∇VBAδt+
√
2DAδt hB(t) (2)
where DA(= 1) and DB(= 100) are the nondimensionalised Brownian diffusivities for the
ions and carriers respectively and obey the fluctuation dissipation relation, i.e., for example
for A particles, 〈hA
i(0)hA
j(t)〉 = 2Iδijδ(t), where I is the unit tensor and i, j label particle
indices. The pair potentials have the screened Coulomb form V (r) = (U/r) exp(−κr) (where
U is a coupling coefficient of the DLVO form (1).) at interparticle separation r, with different
κ’s for AA and AB interaction, while there is no BB interaction. Also both the A and B
particles carry charges of identical sign and magnitude. The dimensions of the box containing
A particles is L = 10 andW = 10, whereas for B particles L = 10, andW = ǫ(ǫ ≤ 2). Keeping
WA=10 and κABℓ=2 fixed, we monitor the behavior of the system by changing: (1) Width of
the channel containing the carriers WB, (2) Ion-ion interaction strength, κAAℓ and (3) sign
of the A−B i.e. ion-carrier interaction. We also explore the effect of changing the screening
length of the ion-carrier (A − B) interaction, keeping other parameters constant. We have
studied the behavior of this model for a system with NA = 100 particles and ρA/ρB = 1 and
ρA/ρB = 1/2, ρA and ρB being the number densities for the ions and carriers respectively.
The results presented here pertain to the latter case, the observed behavior being qualitatively
same for both cases.
(1)This DLVO coefficient is given by ZiZje
2exp(κ(σi + σj))/(ǫ(1 + κσi)(1 + κσj)), where ǫ is the dielectric
constant of the solvent, Zie is the charge on and σi is the radius of the i
th particle. In our simulations, the
dimensionless value of the κ independent part of U is ≈ 15.3, σB/σA = 0.1 and 2σA/ℓ ≈ 0.455.
4 EUROPHYSICS LETTERS
0 5 100
0.5
1
1.5
γ τ
<
V d
>
0.5 1 1.5 2
0
0.5
1
WB
<
V d
>
. 
Fig. 2 – The induced drift of the carriers as a function of the ionic flow speed for different WB (stars,
diamonds and squares correspond to WB = 0.5,WB = 1.0 and WB = 2.0 respectively) for κAAℓ=1.
The inset shows the average drift velocity 〈vd〉 as a function of WB, for γ˙τ = 4.0 (circles) and 2.0
(triangles) respectively.
For WB = 2, we observe that the carriers do have a unidirectional drift in the direction
of the flow. The average drift velocity of the carriers initially increases with the flow rate,
reaches a maximum and then decreases at large values of the flow rate as seen in Fig. 2. This
is broadly in agreement with the ideas of [2]. On further decreasing WB to first 1.0 and then
0.5 (Fig.2), we find that the drift velocity induced is progressively larger than for WB = 2.0,
and the saturation occurs at a lower value of the flow rate.
γ τ
<
V d
>
0 25 50
0
1
2
. 
Fig. 3
γ τ
<
v d
>
0  2.5 5  7.5 10 
0
0.5
1
. 
Fig. 4
Fig. 3 – Induced drift as function of flow speed for κAAℓ=0.2, WB=0.5. The vertical dashed lines
show the standard deviation from the mean drift velocity.
Fig. 4 – Induced drift as function of flow speed for κAAℓ=4, WB=0.5. Note that the velocity
weakening takes place at a much smaller shear rate than in Fig.3. The vertical dashed lines show the
standard deviation from the mean drift velocity.
M. Das et al.: Flow-induced currents in nanotubes: Brownian dynamics 5
0 2 40
3
6
γ τ
<
V d
>
. 
 
Fig. 5
0 250 500
0
i
V d
/<
V d
>
Fig. 6
Fig. 5 – Induced drift as a function of the ionic flow speed for κAAℓ=1, WB=1, with attractive
ion-carrier interaction.
Fig. 6 – Drift velocities Vd of the carriers at different times (represented by the index ’i’) scaled
by the average value 〈Vd〉 at κAAℓ=1, WB=1, for repulsive (lines) and attractive (circles) ion-carrier
interactions.
Now keeping the wall to wall distance fixed at WB = 0.5, we study the behavior of the
system at very small (κAAℓ=0.2) (Fig. 3)and large (κAAℓ=4) ion-ion interaction screening
strength (Fig. 4). A smaller value of κℓ corresponds to stronger ion-ion interactions and it is
observed that in such a case the induced effect on the carriers is stronger and is sustained till
a larger value of the ionic flow speed at which the weakening sets in.
We now look at the scenario where the two species A (ions) and B (carriers) have opposite
charges (Fig. 5). The wall to wall distance is kept fixed at WB = 1. We observe that the
induced drift is stronger than when both A and B particles carry like charges, at the sameWB.
The fluctuations in the drift velocity are however much larger than for the case of like charges
(Fig. 6): the ratio of standard deviation to mean is ∼ 4 times that for the repulsive case. This
can be explained as follows: The attraction of the carriers for the ions brings both close to the
wall separating them, from time to time, enhancing the flow induced current. Subsequently the
presence of the walls pushes them apart, reducing the induced current again. This implies large
fluctuations in the current. For like charged ions and carriers both the Coulomb repulsion and
the walls act in tandem to keep them apart, thus the intermittent enhancement and reduction
in current does not occur.
In order to understand the dependence of the effect on the strength of A-B interaction,
we monitored the system for different values of the screening length corresonding to the A-B
interaction (Fig. 7) keeping the channel width WB and A-A interaction strength constant.
We find that the effect increases on increasing the screening length κ−1AB and is maximum
where κ−1AB is comparable to the width of the channel WB ; therafter it saturates and doesn’t
show a marked increase. This shows that the contribution to the flow induced effect indeed
chiefly comes from the A particles (or ions) present in a boundary layer of order of the width
of the channel at the interface of the chamber and the channel.
We observe that our simulations (Fig. 8) agree well with the theoretical predictions of a
saturating dependence of the induced velocity of the carriers on the imposed flow speeds (Fig.
9). We further find the current decreases when γ˙ passes a threshold value γ˙c, and that γ˙c is
6 EUROPHYSICS LETTERS
2 6 10 140
0.5
1
1.5
2
γ τ
<
V d
>
. 
Fig. 7 – The flow induced drift velocity of B particles as a function of the flow speed of A particles,
for κ−1AB = 0.1ℓ (circles), 0.5ℓ (diamonds), 5ℓ (cross) and 10ℓ (plus), for fixed WB=1.0 and κAAℓ = 1.
smaller for smaller WB . This observation demands a theoretical explanation which at present
we do not have. We hope this will motivate further experiments to check this prediction. It is
significant that we are able to observe this type of “Brownian drag” and transfer of momentum
even in the absence of momentum conservation and in the extreme limit of no inertia. Also,
though strictly speaking fluctuation dissipation relations are valid only in equilibrium, a naive
extension of FDT as in [2] to the flowing case would imply a decrease in the rate of relaxation
with increasing flow rate, and hence a “loss of grip” and a velocity weakening, which is what
we find.
10−2 10−1 100 101
0
0.2
0.4
0.6
γ τ
<
V d
>
. 
Fig. 8
1E-5 1E-4 1E-3 0.01 0.1 1
0.0
0.2
0.4
 
 
I (
µA
)
uL (cm/s)
Fig. 9
Fig. 8 – Induced velocity with flow speed for parameter values as in Fig. 3 on a log scale for comparison
with the behavior predicted by [2] as shown in Fig. 9.
Fig. 9 – Current I as a function of flow speed uL. Figure provided by [2].
M. Das et al.: Flow-induced currents in nanotubes: Brownian dynamics 7
∗ ∗ ∗
We thank S. Ghosh for useful discussions and providing figure 9. MD acknowledges CSIR,
India for support, and SR and GA thank the DST, India for support through the Centre for
Condensed Matter Theory. AKS thanks DST for financial support of his research on carbon
nanotubes.
REFERENCES
[1] S. Ghosh, A.K. Sood and N. Kumar, Science, 299 (2003) 1042.
[2] S. Ghosh, A.K. Sood, Sriram Ramaswamy and N. Kumar, Phys. Rev. B, 70 (2004) 205423
[3] A. E. Cohen (comment); S. Ghosh, A. K. Sood, and N. Kumar (reply), Science, 300
(2003) 1235.
[4] P. Kral and M.Shapiro, Phys. Rev. Lett., 86 (2001) 131
[5] B. N. J. Persson, U. Tartaglino, E. Tosatti, and H. Ueba, Phys. Rev. B, 69 (2004) 235410


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Flow-induced currents in nanotubes: a Brownian dynamics approach

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