Colloids in Cholesterics: Size-Dependent Defects and Non-Stokesian Microrheology

0.43 MB·English

by  J. S. Lintuvuori

Colloids in cholesterics: Size-dependent defects and non-Stokesian microrheology
J. S. Lintuvuori1, K. Stratford2, M. E. Cates1 and D. Marenduzzo1
1SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, UK;
2EPCC, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, UK.
Wesimulateacolloidal particle(radiusR)inacholestericliquidcrystal(pitchp)withtangential
orderparameteralignmentattheparticlesurface. Thelocaldefectstructureevolvesfromadipolar
pairofsurfacedefects(boojums)atsmallR/ptoapairoftwisteddisclinationlineswrappingaround
the particle at larger values. On dragging the colloid with small velocity v through the medium
alongthecholesterichelixaxis(anactivemicrorheologymeasurement)wefindahydrodynamicdrag
force that scales linearly with v but superlinearly with R – in striking violation of Stokes’ law, as
1
generally used tointerpret such measurements.
1
0
PACSnumbers: 61.30.-v,83.80.Xz,61.30.Jf
2

n
a Understanding the properties of colloidal particles Wefocusonthecaseoftangentialanchoringinwhichthe
J moving in complex fluids is a challenging goal of soft directorfieldatthecolloid-fluidinterfacelieseverywhere

7 matter physics and fluid dynamics. The case of a New- parallel to the surface of the particle. We show that the

tonian fluid is very well understood, and it is accurately resulting defect structure depends crucially on the ratio
]
t described by Stokes’ law [1], which states that the vis- R/p. Onincreasingthisratioweobserveacrossoverfrom
f
o cousforcefeltbyaparticlemovingataconstantspeedv adipolarconfiguration,withtwodefectpatchesonoppo-
s islinearlyproportionaltothesolventviscosity,η,tov it- site sides of the particle, to a twisted set of disclination
.
t self and to the particle radius R. This fundamental and lines of opposite chiralities wrapping the colloid.
a
m elegant result is exploited routinely in active microrhe- This crossover could profoundly influence the many-
ology, a modern technique which consists of dragging a bodyself-assemblyofcolloidswithintheliquidcrystalto
-
d particlein afluid, andmeasuringthe force–velocityrela- form organized structures [7]: we defer this issue to fu-
n tion. From these data, and the knowledge of the probe turework. Hereweaddressamuchsimplerproblemthat
o
size, one can obtain an estimate of a fluid viscosity (ef- is nonetheless directly addressablein the laboratory. We
c
[ fective microviscosity) [2], η(active) = F , which is di- model an active microrheology experiment, in which we
rectly a property of the locaml eicnrovironm6eπnvtR. This is often draganisolatedcolloidthroughthe liquid crystal,defin-
1
preferrableto a bulk rheologyexperiment, e.g. when ad- ingtwoeffectiveviscositiesformotionalongandperpen-
v
7 dressing the jamming of dense suspensions [3]. dicularto the cholesterichelix. We find that, in contrast
8 Whathappenswhenaparticle,instead,movesinfluid tothenematiccase,theratiobetweenthesetwoeffective
4 with brokensymmetry? Here, much less is known. Sedi- viscosities,eveninthesmallErregime,dependsstrongly
1 mentation or “falling ball” experiments showed long ago on the size of the probe. This is because the effective
.
1 that the drag force in nematic liquid crystals (in which drag on a particle moving along the cholesteric helix is
0 molecular alignment is governed by a director field nˆ) is superlinearinitssizeR(withexponent∼1.7),instriking
1 anisotropic, as one might expect for a fluid of sponta- violationofStokes’law. Thispresentsaninstructivecase
1
neouslybrokenrotationalsymmetry. Morerecently,pas- in which microrheology delivers a probe-size dependent
:
v sive microrheology experiments and theories have quan- “viscosity”,essentiallyunrelatedtoanybulkmacroscopic
Xi tified the ratioof the viscositiesalongand perpendicular value,and onlyindirectly relatedto the materialparam-
to the local director field, and found these to differ by eters of the medium even as defined on the length scale
r
a about a factor of 2 [4]. These results were obtained in R. Finally, we address the case of intermediate Erick-
the linear regime where the Ericksen number Er, quan- sen number (Er ≈ 1), where the disclinations wrapping
tifying the ratio between viscous and elastic effects, is the dragged colloid are displaced downstream to form a
small: the particlemoves“slowly”. Aseriesofnumerical double twisting disclination wake.
simulations have also addressed the case of intermediate Methodology: The thermodynamics of the cholesteric
tohighEr,correspondingtononlinear(inv)microrheol- solvent is determined by the Landau–de Gennes free en-
ogy. It was found that the ratio between the viscosities ergyF. Itsdensityf isexpressedintermsofa(traceless
becomes much smaller in this case, and the defect struc- and symmetric) tensorial order parameter Q [8] as
ture changes, e.g. from a Saturn ring to a dipole [5, 6].
f = A0 1− γ Q2 − A0γQ Q Q + A0γ(Q2 )2
Hereweconsiderthehydrodynamicsofacolloidalpar- 2 (cid:0) 3(cid:1) αβ 3 αβ βγ γα 4 αβ
ticle (radius R) inside a cholesteric liquid crystal, which + K ∇ Q 2+ K ǫ ∇ Q +2q Q 2. (1)
2(cid:0) β αβ(cid:1) 2(cid:0) αγδ γ δβ 0 αβ(cid:1)
differs from the nematic by having a helical twist to
the director field with a pitch p. (This additional or- Here A is a constant (setting the energy scale), K is an
0
der breakstranslationalas well as rotationalsymmetry.) elastic constant, q = 2π/p, and γ is a temperature-like
0
2
control parameter governing proximity to the isotropic- (a) (b) (c)
to-cholesteric transition. In our notation Greek indices
denote Cartesian components and summation over re-
peatedindicesisimplied; ǫ isthe permutationtensor.
αγδ
We employ a 3D hybrid Lattice Boltzmann (LB) al-
gorithm [9] to solve the Beris-Edwards equations for the
evolution of the Q tensor [8]
FIG. 1: Defect structure close to the particle. The arrow
D Q=Γ −δF + 1tr δF I . (2) denotesthedirectionofthecholesterichelix. Forasmallpar-
t (cid:16) δQ 3 (cid:16)δQ(cid:17) (cid:17) ticle(a),R/p=1/4,therearetwodefectpatchesonopposite
Here Γ is a collective rotational diffusion constant and sides. These elongate to form twisted disclination pairs for
D is the material derivative for rod-like molecules [8]. particles with R/p≥1/2: (b) R/p=1/2 and (c) R/p=3/4.
t
The term in brackets is the molecular field, H, which
ensures that in the absence of flow Q evolves towards a
minimum ofthe freeenergy. The velocityfieldobeys the
defects of topological charge +1 (boojums) [13]. In the
continuity equationand a Navier-Stokesequation with a
case of a chiral nematic host, we find the behavior is de-
stress tensor, Π , generalised to describe liquid crystal
αβ
termined by the ratio between the colloidal size and the
hydrodynamics[9]. Forthehydrodynamics,thecolloidis
cholestericpitch. ForR/psignificantlysmallerthan1/2,
represented by the standard method of bounce-back on
we observe two defect patches on opposite sides of the
links [11], where the LB distributions which link solid to
particles (Fig. 1a). On the other hand, if the particle
fluid nodes in the underlying discretised space are used
size exceeds the pitch, then the minimum energy con-
to impose the appropriate boundary conditions at each
step. Tangential boundary conditions for nˆ are imposed figuration becomes a twisted pair of disclination lines of
on the particle surface (nˆ is free to rotate in the tangent opposite chirality,eachof whichwraps aroundthe parti-
cle (Fig. 1b and 1c). These can be viewed as elongated
plane). Order parameter variations create an additional
elastic force acting on the particle, Fel, which we com- boojums, in which a 3d disclination line of charge +1/2
lies adjacent to the surface terminating in a 2d defect of
puted by integrating the stress tensor Π over the par-
αβ
ticlesurface[12], Fel = dSΠ νˆ , whereνˆ isthelocal a charge +1/2 on the surface at each end. The length
α R αβ β β ofthese “chiralrings”is notconstant,ratherit increases
normal to the colloid surface. Furthermore, the particle
with the size of the particle, and for very large colloids
may rotate due to elastic and hydrodynamic torques.
(R=p) further disclination rings appear at the top and
The thermodynamics of chiral liquid crystals is deter-
bottom of the particle, leading to a more uniform cover-
mined by the chirality κ and the reduced temperature
age of the surface. Unlike the transition between Saturn
τ, which are givenin terms of previously defined quanti-
ties as: κ= 108Kq2/A γ and τ =27(1−γ/3)/γ. The rings and dipoles in standard nematic colloids [14], this
p 0 0 conformation change does not depend on the degree of
physicsofacolloidalparticlemovingintheliquidcrystal
surface anchoring. It should thus be more easily observ-
is controlled by the Ericksen number
able experimentally, purely by changing the particle size
Er=γ vR/K, with γ =2q2/Γ (3)
1 1 R or the chiral fraction (and hence p) in a mixed ne-
which measures the ratio between viscous and elastic matogen. Just as for the nematic case, the two defect
forces. In the expression for Er, γ is the rotational patternswhichwefoundhavesufficientlydifferentgeom-
1
viscosity of the liquid crystal, q is the degree of order- etry that they should mediate quite distinct two-body
ing in the system (for the uniaxial case with director nˆ, and many-body effective interactions, creating new av-
Q = q(nˆ nˆ −δ /3)). The Ericksen number should enuesfordirectingtheself-assemblyofcolloidsbytuning
αβ α β αβ
determine which dynamical regime we are in. As we properties of the surrounding matrix [7].
shall see, the presence of the cholesteric helix brings an-
Having seen the importance of the control parameter
other control parameter into play: R/p. This changes
R/p in determining the statics of colloids in cholester-
the physics completely with respect to the nematic case.
ics, it is natural to ask if this length scale ratio leads to
We give most of our results in simulationunits [9, 10].
different physics (from the nematic case, R/p = 0) for
To convert them into physical ones, we can assume an
their hydrodynamics as well. To address this question,
elastic constant of 6.5 pN, and a rotational viscosity of
we studied the dynamics of colloids of various radii in
1 poise. (These values hold for typical materials, and a
response to a constant pulling force, which is either par-
colloidal diameter of 1 µm.) In this way, one can show
allel or perpendicular to the axis of the cholesteric helix.
thatthesimulationunitsforforce,timeandvelocitymap
First we consider the Er≪1 regime.
onto 90 pN, 1 µs, and 0.03µm/s respectively.
Results: We first discuss the disclination structure Fig. 2 shows the average velocity–force relations ob-
around the colloidal particle. In the case of planar an- served, for dragging perpendicular and parallel to the
choring, a colloid in a nematic host leads to two surface cholesteric helix, at various R/p ratios. We fitted the
3
1.4 7
force 0001 ....14682 R = 24, ff⊥|| ∼f/v ln() 456 ...56555 αα p≈ e≈ r1 p1.6e.08np7 dS±a itr± co0au kl0.ll0ea.e16srl
0.2 R) 4
0 aln( 3 .35
0.5 R = 16, f 2.5
0.4 f|| 2
⊥
e 0.3 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
c
for 0.2 ln(R)
0.1
0 FIG. 3: ln(aR)–ln(R) relation for colloids dragged perpen-
dicular (closed squares) and along (open squares) the helix.
R = 8, ff|| Dashedandsolid linesarelinearfitsofln(aR)=β+αln(R).
e 0.1 ⊥ The dotted line is Stokes’ law f/v=6πηR,for η=0.1.
c
or
f
0 R dependence of the defect structure or to non-Stokes
0 0.0004 0.0008 0.0012 0.0016 featuresoftheflow. Butunlesstheseeffectsarefullyun-
velocity derstood,activemicrorheologyisuninformativeforchiral
nematics,inthesensethatitmeasuresacombinedprop-
FIG.2: (a)Averagevelocity–forcecharacteristicsforparticles erty of the probe and environment, rather than any ma-
with R/p equal to 3/4, 1/2 and 1/4 respectively. The fits terialproperty (localor otherwise)of the medium alone.
are with the formulas shown in the text. The averages were Remarkably, this breakdown arises at very low Er, in a
calculated from the last 2.5−5×104 simulation steps. At regime where the v dependence is still linear (Fig. 2).
steadystate,thevelocityalongthehelixshowsslightperiodic
Our results can be illuminated by considering the bulk
fluctuations around theaverage value.
shearrheologyofcholestericsundergoingpermeationflow
along the pitch direction. In this case, in any finite sam-
databymeansofthefollowingformulaforthedragforce: ple there is a regime of linear rheology, but the shear
viscosityincreases linearly with system size, divergingin
f(v)=fy+a(R)vR (4) the thermodynamic limit [18, 19]. Crudely treating the
colloid radius R as an effective sample size then gives
where f and a(R), with dimensions of force and viscos-
y α=2, not far from the observed value (α≈1.7).
ity respectively,areparametersofthe fit. We allowfor a
finite“yieldforce”f ,whichcanbeseenasthemicrorhe- Toexplorefurtherthephysicsinvolved,Fig.4showdi-
y
ological analogue of a yield stress. Our data show that rector profiles for a stationary colloid, and ones moving
theyieldforce,whilepossiblynon-zero,isverysmall[15], along the helical axis with Er ≈ 0.03, and Er ≈ 0.7. In
so that it does not appreciably change our analysis [16]. the stationary case the liquid crystal accommodates the
Much more striking than any yield force is the extended colloid by a local deformation involving the disclination
regimeinwhichtheforceislinearinvelocity. (Thiswould pattern of Fig. 1c. Interestingly, even at very small forc-
be the usualcriterionforacceptinganactivemicrorheol- ing the cholesteric layers slightly bend throughout the
ogymeasurementasmeasuringalength-scaledependent, whole simulation box: the elastic distortions induced by
but nonetheless linear, effective viscosity.) Fig. 3 shows the colloid motion appear long ranged (Fig. 4b). Here,
the dependence of f/v ∼ aR on R, for dragging both the disclination rotates around the helical axis and is
perpendicular and parallel to the cholesteric helix. We slightly displaced in the downstream direction, but still
fitted these data with a power law, f/v∼Rα. The data remains on the particle surface. At larger Er, (Fig. 4c),
corresponding to dragging perpendicular to the helical the bending oflayersis morepronounced,andthe discli-
axisare comparablewith those reportedfor nematic col- nation is fully displaced downstream to form a double
loids: a(R)∼Rα−1 isalmostconstantinagreementwith twisting disclination “wake” (Fig. 4d).
Stokes’law[17]. Thefitgivesα≈1.07±0.10;(notethat In summary, we have reported lattice Boltzmann sim-
the prefactor can depend on the in-plane angle between ulations of a colloidal particle embedded in a cholesteric
force and director). In contrast, pulling along the helix liquid crystal. Focussing on tangential anchoring of the
shows very different physics, with α≈1.7. liquid crystal at the colloidal surface, we have shown
One way to discuss these results is to insist on that by changing the ratio between particle size and
Stokes’lawasdefininganeffectivemicroviscosityη(R)= cholesteric pitch it is possible to control the topology
a(R)/6π,inwhichcaseformotionalongthepitchη(R)∼ of the local defects imposed by the presence of the col-
R0.7. This strong size effect could be attributable to the loid. For small particles, the equilibrium configuration
4
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FIG.4: SnapshotsofacolloidwithR/p=3/4inacholesteric:
[15] Seesupplementarymaterialatforatableoftheobtained
(a) stationary; (b) moving along the helical axis (Er≈0.03)
parameters.
and (c) likewise with Er ≈ 0.7. (d) Disclination structure
[16] Experiments on sheared cholesterics showed that flow
correspondingto(c). (a)-(c)wereproducedwithQMGA[20].
along thehelix does lead to alinear regime and noyield
Colors denotethemagnitude of nˆx (thehelix is along zˆ).
stress [18], which is also predicted by theory [19]. In the
microrheological case, the colloid induces a small stress
at zero flow, but this vanishes when integrated over its
has two defects at opposite poles of the sphere. Increas-
surface. Very small forces drive an oscillatory motion –
ing the particle size leads to a texture with two twisted whetherthereisayieldforceornotdependsonwhether
disclination lines of opposite polarity, wrapping around theparticlemotionoveralongtimeaveragesoutto0or
the colloid. Both these noveldefect structures should be exhibits a drift. Oursimulations favour thelatter.
detectable inexperimentswithcrossedpolarizers. When [17] Stokes’ law, when applicable, describes the limit of zero
volumefraction. Ourstudiesuseperiodicboundarycon-
pulling the particle along the cholesteric helix, we have
ditions and therefore effectively entail a finite volume
found the drag force is linear in v but depends superlin- fraction φ=4πR3/3L3 whereLis thebox size. Forpar-
early on particle size, in violation of Stokes’ law which
ticle size R = 8 we have investigated this φ dependence
is the usual basis for interpreting active microrheology and find that a 64-fold change in φ alters the f values
experiments [2]. Laser tweezers experiments (similar to bynomorethan30%(whichisconsistentwiththefinite
those in nematic [21] and in twisted nematic cells [22]) size corrections of Hasimoto [24] for a simple fluid). In
should be able to test our predicted superlinear scaling. Fig.3,theφvaluesfordifferentRvaryonly16-fold,from
1.0×10−3 and 1.6×10−2; moreover, this variation is
We hope that our results will stimulate further ex-
notcorrelatedwithR.ThereforethereportedRscalings
periments [23] and theoretical work on the microrheol-
cannotbedirectlyattributedtofiniteφeffects,although
ogy of colloids in cholesterics. Furthermore, similar phe-
these could contributeto theerror in fitted exponents.
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phasescharacterisedbyaspatiallyvariableorderparam- 3650 (2003).
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Acknowledgements: Work funded in part by EPSRC
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Grants EP/E030173 and EP/E045316. MEC is funded
ory Comput. 4, 468 (2008).
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Smaluykhet al. Phys. Rev. Lett. 95, 157801 (2005); I.I.
Smalyukh et al. Appl. Phys. Lett. 86, 021913 (2005).
[22] U. Tkalec et al.Phys. Rev. Lett. 103, 127801 (2009).
[23] These might include simultaneous 3D imaging and laser
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