Hydrodynamics of chiral liquids and suspensions

Abstract

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.

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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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