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INT-PUB-09-024

Hydrodynamics of chiral liquids and suspensions

A. V. Andreev,1D. T. Son,2and B. Spivak1

1Physics Department, University of Washington, Seattle, WA 98195

2Institute for Nuclear Theory, University of Washington, Seattle, WA 98195

(Dated: May 17, 2009)

We obtain hydrodynamic equations describing a fluid consisting of chiral molecules or a suspension

of chiral particles in a Newtonian fluid. The stresses arising in a flowing chiral liquid have a

component forbidden by symmetry in a Newtonian liquid. For example, a chiral liquid in a Poiseuille

flow between parallel plates exerts forces on the plates, which are perpendicular to the flow. A generic

flow results in spatial separation of particles of different chirality. Thus even a racemic suspension

will exhibit chiral properties in a generic flow. A suspension of particles of random shape in a

Newtonian liquid is described by equations which are similar to those describing a racemic mixture

of chiral particles in a liquid.

Equations of hydrodynamics express conservation of

mass, momentum and energy, and can be written as

∂tρ + ∂iJi= 0,

∂tPi+ ∂jΠij= 0,

∂tE + ∂iJE= 0.

(1a)

(1b)

(1c)

Here ∂tand ∂idenote time and spatial derivatives, ρ, P,

and E are correspondingly the densities of mass, momen-

tum and energy, and J, JEandˆΠ are the flux densities of

mass, energy and momentum (we indicate vector quan-

tities by bold face symbols and second rank tensors by

hats). The flux densities can be expressed in terms of

the hydrodynamic variables: the pressure p(r,t), tem-

perature T(r,t) and the hydrodynamic velocity v(r,t),

which we define via the equation

ρv = J ≡ P.

(2)

To lowest order in spatial derivatives we have [1]

Πij= ρvivj+ pδij− ηVij− ζδijdivv,

where Vij= ∂jvi+ ∂ivj−2

strain, and η and ζ are the first and the second viscosities.

This leads to the Navier-Stokes equations, which should

be supplemented by the equation of state of the fluid

and the expression for the energy current in terms of the

hydrodynamic variables.

For a dilute suspension of particles in a Newtonian liq-

uid, the basic hydrodynamic equations need to be sup-

plemented [1] by the conservation law for the current of

suspended particles,

(3)

3δijdivv is the rate of shear

∂tn + v ·∇n + divj = 0.

(4)

Here n(r,t) is the density of suspended particles, and

j(r,t) their flux density (relative to the fluid). To linear

order in the gradients of concentration, temperature and

pressure the latter can be written as

j = −D∇n − nλT∇T − nλp∇p.

(5)

Equation (2) remains unchanged and can be considered

as a definition of the hydrodynamic velocity v, which

is, generally speaking, different from the local velocity

u(r,t) near an individual particle of the suspension.

There are corrections to the flux densities of various

quantities, which are higher orders in spatial derivatives

of the hydrodynamic variables (for a review see, for ex-

ample, Refs. 2, 3). Moreover, there are nonlocal correc-

tions to the Navier-Stokes equations, which can not be

expressed in terms of higher order spatial derivatives of

hydrodynamic variables[4, 5, 6, 7].

Several studies focused on the effects of chirality on the

motion of suspended particles in hydrodynamic flows [8,

9, 10, 11, 12, 13, 14, 15, 16]. It was shown that non-chiral

magnetic colloidal particles can self-assemble into chiral

colloidal clusters [17].

In this article, we develop a hydrodynamic description

for the case of a suspension containing both right-handed

and left-handed chiral particles in a centrosymmetric liq-

uid. We show that in this case the corrections to the

Navier-Stokes equations contain new terms, which are

associated with the chirality of the particles. The sig-

nificance of these corrections is that they describe new

effects, which are absent in the case of centrosymmetric

liquid. Since certain types of hydrodynamic flow lead

to separation of particles with different chirality, these

corrections are important even in initially racemic sus-

pensions of chiral particles. For simplicity we consider

the case of incompressible fluids.

In a given flow an individual particle of the suspension

undergoes a complicated motion which depends on the

initial position and orientation of the particle. The hy-

drodynamic equations can be written for quantities which

are averaged over the characteristic spatial and temporal

scales of such motion.

In the presence of chirality the following contribution

to momentum flux density is allowed by symmetry:

Πch

ij= nchαη[∂iωj+∂jωi]+ηα1[ωi∂jnch+ωj∂inch], (6)

where ωi(r) =

nch= (n+− n−) is the chiral density, with n+ and

1

2?ijk∂jvk(r) is the flow vorticity, and

arXiv:0905.2783v1 [cond-mat.soft] 18 May 2009

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n−being the volume densities of right- and left-handed

particles respectively. Equations (1-6) should be supple-

mented by the expression for the chiral current, defined

as the difference between the currents of right- and left-

handed particles. Separating it into the convective part,

vnch, and the current relative to the fluid, jch, we write

the continuity equation as

∂tnch+ div(vnch) + divjch= 0.

(7)

Besides the conventional contribution given by Eq. (5)

with n replaced by nch, the chiral current contains a con-

tribution,˜jch, which depends on the flow vorticity:

˜jch

i

= n[β∇2ωi+ β1ωjVij],

where n = n++n−. The contributions to˜jch

only nωi are not allowed as there should be no chiral

current in rigidly rotating fluid.

In Eqs. (6) and (8) we keep only the leading terms in

the powers of ∂ivj, or in the order of spatial derivatives of

v. Although these terms are subleading in comparison to

those in the conventional hydrodynamic approximation,

they describe new effects which are absent in the latter.

Note that according to the Navier-Stokes equations

∇2curl(v) =ρ

(8)

i

containing

η{∂tcurl(v) + curl[(v∇)v]}.

(9)

Thus the first term in Eq. (8) arises either due to non-

stationary or non-linear in v nature of the flow. In par-

ticular, in stationary flows and to zeroth order in the

Reynolds number ∇2curl(v) = 0 and this term vanishes.

In spatially inhomogeneous flows the suspended parti-

cles rotate, generally speaking, relative to the surround-

ing fluid.This gives rise to separation of particles of

different chirality due to the propeller effect, and to the

chiral contribution to the momentum flux, Eq. (6).

The rotation of the particles relative to the fluid arises

due to two effects:

i) In the presence of the spatial dependence of vorticity,

ωi(r), the angular velocity of a particle is different from

ωi(r). This results in Eq. (6) and the first term in Eq. (8).

ii) A non-uniform hydrodynamic flow induces orien-

tational order in suspended particles similar to nematic

order in liquid crystals. In the presence of flow vorticity

orientation of particles induces their rotation with re-

spect to the surrounding fluid. This contributes both to

the chiral stress and the chiral flux. The latter contri-

bution is described by the second term in Eq. (8). The

contribution to the chiral part of the stress tensor asso-

ciated with orientational order was discussed in Ref. 8.

In most cases of practical importance the Reynolds

number corresponding to the particle size R is small. In

this regime the coefficients α, α1, β, and β1in Eqs. (6)

and (8) can be obtained by studying the particle motion

in the surrounding fluid in the creeping flow approxima-

tion [18, 19]. In this approximation the motion of parti-

cle immersed in the liquid is of purely geometrical nature

(see for example Ref. 20). Dimensional analysis gives an

estimate

α ∼ α1∼ χR4,β ∼ χR3,

(10)

where R is the characteristic size of the particles, and

the dimensionless parameter χ characterizes the degree

of chirality in shape of the particles.

The relative magnitudes of the different terms in

Eqs. (6) and (8) depend on the particle geometry. For

example, particles with the symmetry of the isotropic he-

licoid [18] can not be oriented in a shear flow. Therefore

the second term in Eq. (6) vanishes is this case.

The degree of orientation of the particles can be ob-

tained by balancing the characteristic directional relax-

ation rate due to the Brownian rotary motion, ∼ T/ηR3

with T being the temperature, with the rate of orienta-

tion due to the shear flow, ∼ Vij. Thus at small shear

rates the degree of particle orientation is ∼ VijηR3/T.

This leads to the estimate

β1∼ χηR4/T.

(11)

The second term in Eq. (8) is the leading term in the ex-

pansion in the rotational P´ eclet number Pe ∼ VijηR3/T.

At larger P´ eclet numbers terms of higher power in Vij

should be taken into account. For Pe ? 1 the particle

orientation becomes strong, and the corresponding con-

tribution to the chiral current can be estimated as

˜jch∼ χRω.

(12)

Equations (6) and (8) are written for the case when

there is no external force acting on the particles, e.g. for

a suspension of uncharged neutrally buoyant particles.

In the presence of an external force F, there will be ad-

ditional contributions to the chiral flux. The linear in

F contributions can be constructed by contracting the

antisymmetric tensor ?ijk with the velocity vi, force Fi

and two derivatives ∂i. For example, the following terms

exist when F is constant: (F · ∇)ω, F × ∇2v, ∇(F · ω).

These terms arise when the orientation of the suspended

particles can be characterized by a polar vector. In this

case the degree of particle orientation can be estimated

as ∼ RF/T. Thus the coefficients with which these terms

enter the chiral current jchare of order as χR3/T. In the

case when particles do not have a polar axis the degree

of their orientation, and the corresponding contribution

to the chiral flux are quadratic in F for small force.

The chiral contribution to the stress tensor Eq. (6)

leads to several new effects. Consider a Poiseuille flow

of a chiral liquid between parallel planes separated by a

distance d: vx= −∂xp(d2− 4y2)/8η, vz = vy = 0 (see

Fig. 1), with ∂xp being the pressure gradient along the

flow. If the chiral density is uniform the chiral part of the

stress tensor has only two non-vanishing elements, Πyz=

Πzy= −αnch∂xp/2. It describes a pair of opposing forces

2

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FIG. 1: A chiral liquid in a Poiseuille flow between parallel

plates exerts a pair of opposite forces on the plates F↑= −F↓,

which are directed into (⊗) and out of (?) the page. A flow of

a chiral liquid in a converging or diverging channel develops a

helicoidal component of velocity directed into and out of the

page, as shown at right. For a Newtonian fluid the flow lines

(dotted lines) lie in the plane of the figure.

per unit area exerted by the liquid on the top and bottom

planes. These forces are perpendicular to the flow, as

shown in Fig. 1. Assuming nchR3∼ 1 and using Eq. (10)

the magnitude of the chiral force per unit area of the

plane can be estimated as ∼ χR∂xp.

If nchis constant in space, then the volume force

density generated by the chiral part of the stress ten-

sor is fch

i

= −∂jΠch

from Eq. (9) that fch

i

is generated only in nonstation-

ary or nonlinear flows. In the special case of stationary

Poiseuille flow the chiral part of the stress tensor does

not generate a force density inside the fluid even at large

Reynolds numbers. Thus the flow pattern is not affected

by the fluid chirality. However, in a generic flow with

converging or diverging flow lines the fluid chirality does

affect the flow pattern. This is especially evident in flows,

which have a mirror symmetry in the absence of chiral

corrections. In these cases the chiral contribution to the

stress tensor results in mirror asymmetric corrections to

the flow velocity. For example a chiral liquid flowing be-

tween two surfaces with a varying distance between them,

see Fig. 1, will develop a helicoidal component of veloc-

ity with non-vanishing vorticity along the flow direction.

This can be checked explicitly for the exactly solvable

flow in a converging channel (§ 23 of Ref. 1).

Another consequence of Eq. (8) is a possibility of

separation of particles of different chirality in hydrody-

namic flows. It has been observed in numerical simula-

tions [10, 11, 12, 13] and recent experiments [15]. We

note that according to Eq. (9) in a stationary flow and in

the linear approximation in the shear rate ∂ivj, we have

∇2ωi= 0, and the first term in Eq. (8) vanishes. Thus

separation chiral isomers in the absence of external forces

acting on the particles is possible either in non-linear or

in non-stationary flows.

In the practically important case of a stationary Cou-

ette flow, the first term in Eq. (8) vanishes for arbitrary

Reynolds numbers, and the chiral current arises only due

to orientation of the particles. The latter increases with

ij= −nchαη∇2ωi. Then it is clear

the rotational P´ eclet number and saturates at Pe ? 1.

In this regime the chiral current becomes linear in the

flow vorticity ω, Eq. (12). The linear dependence of jch

on ω and saturation of the proportionality coefficient at

Pe → ∞ has been observed numerically in Refs. 10, 12.

Separation of particles by chirality can also be achieved

by subjecting the particles to an external circularly po-

larized electric or magnetic field. The orientation of the

particles along the field (e.g. due to the presence of a per-

manent electric or magnetic dipole moment or anisotropy

of the polarization matrix) will cause their rotation rel-

ative to the surrounding fluid. This will produce a chi-

ral flux along the circular polarization axis due to the

propeller effect [21, 22] (similarly, a stationary electric or

magnetic field will cause separation of particles by chiral-

ity in a rotating fluid). We also note that there are other

mechanisms of chiral current generation which do not in-

volve transfer of angular or linear momentum from the

ac-field to the particles [23]. Chiral separation by circu-

larly polarized magnetic field has been recently observed

in the experiments of Refs. 24, 25. The full quantitative

analysis of this effect is beyond the scope of this work.

Here we restrict the treatment to the experimentally rel-

evant regime of strong and slowly varying fields, where

thermal fluctuations can be neglected and the particles

are fully polarized along the instantaneous electric field.

In this case the problem is of purely geometric nature.

The chiral current becomes independent of the viscosity

of the fluid and can be expressed in terms of the Berry

adiabatic connection [20]. Below we express this adia-

batic connection in terms of the resistance matrix of the

particle [18]. The latter relates the external force F and

torque τ exerted on the particle to the linear velocity δv

and angular velocity δω relative to the fluid,

?F

Here we chose the origin of the reference frame at the

reaction center, so that the coupling tensorˆC is sym-

metric [18]. Since a uniform electric field exerts no force

on the particle we immediately obtain the relations,

τ

?

= −

?ˆK

ˆC

ˆΩˆC

??δv

δω

?

.

(13)

δv = −ˆK−1ˆCδω,

τ = −˜Ωδω,

(14a)

(14b)

where we introduced the notation˜Ω =ˆΩ −ˆCˆK−1ˆC.

The orientation of the particle relative to the lab frame

is described by the three Euler angles, φ, θ and ψ [26]. We

choose the axes of the body reference frame, x1,x2,x3, so

that x3points along the dipole moment of the particle.

For fully polarized particles the latter points along the in-

stantaneous direction of the electric field. Thus θ and φ

coincide with the polar angles of the electric field vector.

The value of ψ remains undetermined because the parti-

cle can be rotated by an arbitrary angle about x3without

changing its polarization energy. When the orientation

3

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of the electric field changes with time the particle ori-

entation angle about the instantaneous field direction,

ψ(t), also changes. Its time evolution can be determined

from the condition that projection of the torque onto the

x3 axis must vanish. This is clear because the torque

τ = d × E is perpendicular to the dipole moment d,

which points along x3. Writing Eq. (14b) in the body

frame,˜Ω3iωi = 0, and expressing ωi in terms the Eu-

ler angles, ω1 =˙φsinθsinψ +˙θcosψ, etc. we obtain

dψ = Aφdφ+Aθdθ where Aφand Aθplay the role of the

adiabatic connection components and are given by

Aφ= −cosθ −sinθ

1

˜Ω31

˜Ω33

?˜Ω31sinψ +˜Ω32cosψ

?˜Ω31cosψ +˜Ω32sinψ

?

,

(15a)

Aθ= −

?

.

(15b)

This defines˙ψ in terms of˙θ and˙φ. The displacement of

the particle can be obtained from Eq. (14a). Instead of

presenting the general formulae we focus on the practi-

cally relevant case of an ac-field of frequency ω0circularly

polarized in the xy plane: θ = π/2, φ = ω0t. In this case

it is easy to see that the average velocity along the x

and y axes vanishes. For the average z-component of the

velocity an elementary calculation gives,

δvz= χRω0=

ω0

2˜Ω33

Tr

ˆK−1ˆC

−˜Ω33

0

˜Ω31

00

−˜Ω33 0

˜Ω32

0

,

(16)

where the resistance tensor is expressed in the body

frame, and χ can be viewed as the dimensionless mea-

sure of the particle chirality. Since the coupling tensorˆC

changes sign under inversion it is clear that particles of

opposite chirality will move in opposite directions along

the z axis. By order of magnitude the chiral separation

velocity is vch

z

∼ Rω0, which is consistent with the recent

experimental findings [24, 25].

In the regime where the polarization energy in the elec-

tric field is smaller than the temperature the chiral cur-

rent is reduced compared to the above estimate. The

leading contribution at weak fields is proportional to the

intensity of the ac-radiation [22].

So far we discussed the case where the suspended par-

ticles consist of the opposite enantiomers of a single

species. However, the effects considered above exist even

in suspensions of particles of completely random shape in

a non-chiral liquid. In this case the definition of chiral-

ity requires clarification. For example one can define the

chirality of a particle by considering the direction of its

motion in a hydrodynamic shear flow or under the action

of an ac-electromagnetic field. Thus the same individ-

ual particle can exhibit different chirality with respect to

different external perturbations.

The set of Eqs. (1-8) still holds for a suspension of

random particles. In this case the auxiliary quantities

nch, and˜jchare defined in terms of the chiral component

of the stress tensor Eq. (6) and correspond to quantities

averaged over the random shape of the particles.

Finally, we note that symmetry allows contributions to

jchthat are proportional to the external magnetic field B,

for example˜jch∝ nch(∇T)2B. We believe that these ef-

fects are of fluctuational origin similar to those discussed

in Refs. 4, 5, 6, 7 and do not study them in this work.

We acknowledge useful discussions with D. Cobden, E.

L. Ivchenko and L. Sorensen. This work was supported

by the DOE grants DE-FG02-07ER46452 (A.V.A.), DE-

FG02-00ER41132 (D.T.S.) and by the NSF grant DMR-

0704151 (B.S.).

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