Stokes-Einstein diffusion of colloids in nematics

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by  Frédéric Mondiot

Stokes-Einstein diffusion of colloids in nematics
Fr´ed´eric Mondiot, Jean-Christophe Loudet, Olivier Mondain-Monval, Patrick Snabre
Centre de Recherche Paul Pascal, Universit´e de Bordeaux & CNRS, 33600 Pessac, France
Alexandre Vilquin, Alois Wu¨rger
Laboratoire Ondes et Mati`ere d’Aquitaine, Universit´e de Bordeaux & CNRS, 33405 Talence, France
We report the first experimental observation of anisotropic diffusion of polystyrene particles im-
mersed in a lyotropic liquid crystal with two different anchoring conditions. Diffusion is shown to
obey the Stokes-Einstein law for particle diameters ranging from 190 nm up to 2 µm. In the case
of prolate micelles, the beads diffuse four times faster along the director than in perpendicular di-
rections,D /D ≈4. InthetheorypartwepresentaperturbativeapproachtotheLeslie-Ericksen
(cid:107) ⊥
4 equations and relate the diffusion coefficients to the Miesovicz viscosity parameters ηi. We provide
1 explicit formulae for the cases of uniform director field and planar anchoring conditions which are
0 then discussed in view of the data. As a general rule, we find that the inequalities ηb < ηa < ηc,
2 satisfied by various liquid crystals of rodlike molecules, imply D(cid:107) >D⊥.
PACS numbers: 05.40.Jc; 82.70.Dd; 83.80.Xz
n
a
J
Dispersions of colloids in nematic liquid crystals
9
(NLCs) show singular properties, that are related to the
2
anisotropy of the nematic phase and to the anchoring of

] thenematogensontheparticlesurface[1–4]. Thecolloid
t imposes on the neighboring LC molecules an orientation
f
o that locally breaks the uniform nematic alignment and
s gives rise to elastic interactions. In order to satisfy the
.
t global boundary conditions, each inclusion is accompa-
a
m niedbytopologicaldefectsthatdeterminethelong-range
deformation field and govern colloidal pattern formation
-
d [5–11].
n A Brownian particle in a NLC thus drags a nematic
o deformation along its random trajectory, and its diffu-
c
sion behavior constitutes a sensitive probe to the local
[
order parameter and surface anchoring. In an isotropic

1 medium, the Stokes-Einstein coefficient D = k T/3πηd
B
v is given by the particle diameter d and the scalar vis-
5
cosity η. A more complex situation arises in a nematic,
5 FIG.1: (a)Browniantrajectoryofa190nm-diameterfluores-
where viscosity is a tensor quantity and where the direc-
4 cent PS particle, consisting of 2800 time steps of 0.3 s. Dif-
tor and velocity fields exert forces on each other [12, 13].
7 fusion is faster parallel to the director (double arrow). Inset:
1. Thus the viscous stress of a diffusing particle on the sur- Schematic of the director field distortions around a sphere
rounding fluid, is not the same for motion parallel and
0 in planar anchoring conditions. The black dots symbolize
4 perpendiculartothedirector,resultingintwocoefficients “boojum” defects [3]. (b) Histogram of the measured parti-
1 D(cid:107) and D⊥. cle displacement δ parallel and perpendicular to the director
: So far anisotropic colloidal diffusion has been stud- during a time τ =0.3 s, as obtained from a sample of 20,000
v
ied in thermotropic NLCs, made of rod-like organic trajectory steps. The solid lines are Gaussian fits.
i
X molecules the anchoring of which is determined by the
r surface chemistry [14–18]. Rather generally, diffusion
a turns out to be faster along the director, with a ratio tance in nematic colloids, we found it worthy to inves-
D /D smaller than 2. A recent study on silica beads tigate particle mobility in lyotropic liquid crystals for
(cid:107) ⊥
dispersed in nematic 5CB with normal anchoring condi- which various anchoring conditions are easily achieved
tions[18], reportedthatthediffusioncoefficientsD and without altering the surface chemistry [19]. The latter
(cid:107)
D of large particles show the size dependence D ∝1/d does indeed influence the particle diffusion coefficients
⊥
expected from the Stokes-Einstein relation, yet surpris- as shown in Ref. [17]. Lyotropic LCs are water-based
ingly saturate for smaller particles at an effective hydro- surfactantmixtures,andinsuchsystems,anchoringcon-
dynamic diameter of about 300 nm. The role of surface ditions depend critically on the shape of the surfactant
chemistry, which controls the anchoring of the nemato- micelles(nematogens). Thelattercanbetuned,through
gens, was put forward in an attempt to rationalize the tiny changes of surfactant concentrations, from rodlike
observations. (CalamiticNematics,NCphase)todisklike(DiscoticNe-
Since boundary conditions are of paramount impor- matics, ND phase) [20, 21]. And for entropic reasons
2
probetheBrowniandiffusioninlyotropicphases. Atyp-
icaltrajectoryofa190nm-diameterparticlederivedfrom
2,800 snapshots is shown in Fig. (1a) for the NC case.
Itselongationalongthenematicdirectornindicatesthat
diffusion is faster in this direction. In the histogram of
Fig. (1c), we plot the displacements parallel and per-
pendicular to n, for a total of 20,000 trajectory steps.
The probability that the particle moves a distance δ in
time τ, P(δ,τ), is very well fitted by a Gaussian distri-
bution; its standard deviation is related to the diffusion
2
coefficient through δ2 − δ = 2Dτ [25]. In Fig. (2a),
we plot D and D as a function of the inverse particle
(cid:107) ⊥
radius in the NC phase whereas Fig. (2b) displays the
results for the ND phase. However, due to experimen-
tal limitations [26], only D could be determined in the
⊥
latter case. The straight lines confirm the linear depen-
FIG. 2: Stokes-Einstein evolution of the diffusion coefficients dence of the Stokes-Einstein relation whatever the an-
D(cid:107) (along the nematic director) and D⊥ (perpendicular to choring conditions. The friction coefficients are different
the nematic director) as a function of the particle diameter.
for motion parallel and perpendicular to n and the large
The ratio D /D is about equal to 4.
(cid:107) ⊥ anisotropy ratio D /D ≈ 4 indicates a strong nemato-
(cid:107) ⊥
hydrodynamic coupling in the LC matrix. Note also the
very close values measured for D in the ND phase and
⊥
[22], spontaneous planar (NC) and normal (ND) anchor- D intheNCphase. Ourresultsthendifferconsiderably
(cid:107)
ingconditionscanbeachievedatconstantsurfacechem- from the measurements of [18] in a themotropic NLC,
istry (and consequently constant anchoring strength), in where the diffusion coefficients become constant for par-
contrast with the thermotropic case where varying the ticles smaller than about 300 nm, with D /D ≈1.6.
(cid:107) ⊥
anchoring conditions requires a change in the particles In the remainder of this paper, we study how the dif-
surface chemistry. fusion anisotropy arises from the viscous properties of
InthisLetter,wereportonbothexperimentalandthe- a NLC. The fluctuation-dissipation theorem relates the
oretical work on anisotropic diffusion. We present data frictioncoefficienttothevelocityu=F/3πηdofaspher-
for particles with diameters d ranging from 190nm to icalparticledrivenbyanexternalforceF. Thuscalculat-
1.9µm, both in planar and normal anchoring conditions ing the Rayleigh function Ψ = Fu provides an effective
of the lyotropic LC at the particle surface. To the best viscosityintheformΨ=3πηdu2,whichtakestwovalues
of our knowledge, these are the first data obtained in η and η for motion parallel and perpendicular to the
(cid:107) ⊥
lyotropic systems and in both anchoring conditions. In director.
the theory part, we develop an original perturbative ap- ThefrictioncoefficientsarecalculatedfromtheLeslie-
proach which provides the effective viscosities to linear Ericksen equations of nemato-hydrodynamics for |n|=1
order in terms of the Leslie coefficients. The obtained and an incompressible fluid. Energy dissipation occurs
expressions enable a direct comparison with the exper- through two channels [12, 13],
imental relevant quantities and can account for the ob-
(cid:90)
served diffusion anisotropy. Ψ= dVψ, ψ =σ :A+h·N, (1)
The two NLC phases used are the NC and the ND
phases of the water/decanol/sodium dodecyl sulfate ly-
where A and N are thermodynamic fluxes, and σ and h
otropic system, which can be obtained at very close
the corresponding forces. The rate of strain tensor
experimental concentrations (NC 71/24.5/4.5 %; ND
73/23.5/3.5 %) [20, 21]. In the NC phase, the surfactant 1
A = (∂ v +∂ v ) (2)
moleculesformnanometer-sizedrodlike(prolate)micelles ij 2 i j j i
(withlongandshortaxesof9nmand3.5nm[23]),which,
is given by the symmetrized derivatives of the flow v(r)
in order to minimize their excluded volume, show pla-
in the vicinity of the particle moving at velocity u. The
naranchoringatthesurfaceofthedispersedpolystyrene
vector quantity
spheres (PS). In the ND phase, and for similar reasons,
the disklike (oblate) micelles (with diameter 8 nm and
N=((v−u)·∇)n−ω×n, (3)
thickness 3.5 nm [23]) anchor normally at the surface
(with the disk normal perpendicular to the surface) [19]. withthecurlω =1∇×v,expressestherateofchangeof
2
Unlike colloidal suspensions in thermotropic NLCs, this the director with respect to the background fluid. The
dispersion does not require surface functionalization. conjugate forces, that is the viscous stress tensor σ and
We used classical and fluorescence optical microscopy the molecular field h, are linear functions of the compo-
combined with standard video tracking routines [24] to nents of A and N [12, 13]. Inserting their steady-state
3
TABLE I: Effective viscosities for zero anchoring (uniform
director). The middle column is obtained from Eqs. (6) and
(7) with the Leslie coefficients of 5CB and MBBA [12, 13],
thelastoneisderivedfromStark’snumericalcalculations[3].
Uniform director present work numerically exact
5CB η (P) 0.429 0.381
(cid:107)
η (P) 0.724 0.754
⊥
MBBA η (P) 0.412 0.380
(cid:107)
η (P) 0.650 0.684
⊥ FIG.3: Schematicviewofacolloidalparticleinaliquidcrys-
tal. The particle moves along the z-axis; the shear in the
plane z = 0 is indicated by the decay of the fluid velocity
expressions in (1) one has field. In the left panel the director is parallel to the particle
velocity, with an effective shear viscosity η . The remainder
b
ψ =α (n·A·n)2+(α −α )N2 shows the perpendicular case n = e ; in the middle a view
1 3 2 0 x
of the x-z-plane with η , at right the y-z-plane with η . The
+(α +α +α −α )n·A·N c a
3 2 6 5
Miesovicz viscosities η ,η ,η are expressed in terms of the
a b c
+α A:A+(α +α )n·A·A·n. (4)
4 5 6 Leslie parameters.
Parodi’s relation α +α = α −α reduces the viscos-
3 2 6 5
ity tensor to five independent Leslie coefficients α . The Fig. 3; the effective viscosity for a particle moving along
i
various scalar products result in an intricate dependence the nematic order, u(cid:107)n , then reads
0
ontherelativeorientationofthemacroscopicdirectorn
0 8α 4η +η
and the particle velocity u. ηUD = 1 + b c . (6)
(cid:107) 70 5
In the absence of nematic ordering, n = 0, the power
density reduces to ψ = α (cid:80) A2 . The tensor (2) is Similarly, we find for the perpendicular case u⊥n
4 ij ij 0
readily calculated from the velocity field of a spherical
3α 5η +η +4η
particle moving in an isotropic liquid, ηUD = 1 + a b c . (7)
⊥ 70 10
(cid:18)3a a3 (cid:19) It is noteworthy that the five independent viscosities of
v= (1+ˆrˆr)+ (1−3ˆrˆr) ·u (5)
4r 4r3 (4) reduce to three or four terms. In Table I we compare
our formulae with Stark’s numerical calculations for the
where ˆr = r/r. The resulting Rayleigh function Ψ = liquid crystals 5CB and MBBA [3], and find that the
0
3πdα u2 defines the isotropic viscosity η = 1α . In a numbers differ by hardly 10%. Though it slightly un-
2 4 0 2 4
NLC,however,thevelocityanddirectorfieldsdependon derestimates the viscosity anisotropy, the linearization
eachotherthroughtheequationsforσ andh. ThenΨis approximation is therefore quantitatively correct.
acomplicatedfunctionoftheLesliecoefficients, anditis Becauseofthesmallweightofthefirsttermin(6)and
not possible to single out the dissipation due to a given (7), the anisotropy arises mainly from the Miesovicz vis-
term. Though the problem can be solved with consider- cosities. Its physical origin is illustrated in Fig. 3 for a
able numerical effort [1, 3, 27–30], the resulting numbers particlemovingintheverticaldirection.Thepoleregions
for the effective viscosities give no physical insight in the beingofminorimportance, wefocusontheshearflowin
underlying mechanism. theplanez =0,wheretheviscousstresssimplifiestothe
The present work relies on two approximations. First, in-plane derivatives of the vertical velocity component
weevaluate(2)and(3)withtheabovevelocityfieldv(r) ∂v /∂x and ∂v /∂y. The left panel shows the parallel
z z
ofaparticleinanisotropicliquid. Formallythisamounts case, and the middle and right panels the perpendicular
to linearize ψ with respect to the a . Second we use a one,withthecorrespondingshearviscosities. Thisquali-
i
simple parameterization for the director n(r) which is tativepictureisconfirmedbythelargeweightofη inthe
b
independent of the velocity field and which depends on parallel viscosity (6), and of η and η in (7). Data for
a c
the particle size through the reduced distance d/r only; commonNLCsmadeofrodlikemoleculessuggestthatα
1
in other words, the director has no intrinsic length scale. is small; more importantly, they satisfy the inequalities
With these assumptions, the Rayleigh function becomes η < η < η [13] and thus imply η < η , which is in
b a c (cid:107) ⊥
linear in the α and in the particle size d. line with our results.
i
Uniform director (UD). We start with the case where Planar anchoring (PA). A finite surface energy de-
theparticlesurfacedoesnotaffecttheliquidcrystalorder forms the nematic order parameter in the vicinity of the
parameter. Then the director is constant, n=n , and colloidalparticle. Hereweconsiderthecaseofplanaran-
0
withtheexplicitformofthetensor∂ v [31],thedissipa- choring,whichisillustratedintheinsetofFig. 1a. Asthe
i j
tion function can be calculated in closed form. It turns distance from the particle increases, n varies smoothly
outconvenienttorewritetheLeslieparametersα ,...,α toward the constant n . Even for the simplified one-
2 6 0
in terms of the Miesovicz viscosities η ,η ,η given in constant elastic energy, there is no general solution for
a b c
4
(C(cid:107)−C⊥)η . From experiments, ∆η < 0, which there-
TABLE II: Coefficients of the effective viscosity (9) for a d d d
fore implies η < η and suggests that η is small or
spherical particle with planar anchoring (PA), moving par- b a d
positive. The zero shear effective viscosity is given by
allel or perpendicular to the director. The measured values
η = m η +m η +m η , with m = 1 in a polycrys-
are obtained by fitting the straight lines in Fig. (2a) with S a a b b c c i 3
talline sample. Because of the planar anchoring condi-
D=k T/3πηd.
B
tions on the confining surfaces of the rheometer, we ex-
PA C C C C measured
1 b c d pect a smaller weight for η ; in addition, shear-induced
c
η(cid:107) 0.08 1.02 0.40 −0.48 0.31 Pa.s alignment would reduce ma. Indeed, we find that the
η⊥ 0.04 0.24 0.41 −0.15 1.24 Pa.s three equations for η(cid:107), η⊥, and ηS have solutions only if
m <0.15 and m >0.7, and strongly suggest η <η .
a,c b a c
TheseinequalitiesaresatisfiedbytheMiesoviczparame-
the spatially varying order parameter [3]. It is conve- tersthatarerequiredtofitourdata: Forexample,setting
niently parameterized by η =0andm =0.05,themeasuredviscositiesaremet
d a,c
with η =0.27 Pa.s, η =1.46 Pa.s, η =1.61 Pa.s. This
b a c
n=n0cosΘ−n⊥sinΘ, (8) discussion qualitatively agrees with that of η⊥ measured
for planar anchoring in the thermotropic NLC 5CB [17].
where n is a radial vector perpendicular to n . Be-
⊥ 0 Finally, we close with a remark on diffusion in dis-
cause of its rotational symmetry, the director is deter-
cotic nematics (ND phase) where the anchoring is nor-
mined by a single function Θ(r), which decays as 1/r3
mal and not planar anymore. As aforesaid, the data for
at large distances. Here we use the ansatz of Luben-
the perpendicular coefficient D in Fig. (2b) are almost
sky et al. [8], which for planar anchoring results in ⊥
Θ=(cid:80)∞ [sin(2kθ )/k](d/2r)1+2k,whereθ isthepolar identical to those for D(cid:107) in the NC phase. As discussed
k=1 n n above, the left panel of Fig. 3 implies η ∼ η for pro-
angle with respect to n , that is cosθ =ˆr·n . (cid:107) b
0 n 0 latemicelles; asimilarargumentforoblateonessuggests
With the director (8) and the velocity (5), the dis-
η ∼ η . Consequently, the important point is that,
sipation function (1) is calculated numerically for both ⊥ c
in the ND phase, one expects the Miesovicz viscosities
parallel and perpendicular alignement; it yields to the
to satisfy η < η < η . Thus, it does not come as
following viscosities c a b
a surprise that η (ND) and η (NC) take close values.
⊥ (cid:107)
ThisargumentimpliesmoreoverthatdiffusionintheND
η =C α +(1−C −C )η +C η +C η +C η , (9)
1 1 b c a b b c c d d
phase should be faster perpendicular to the director, i.e.
where we have defined a fifth independent parameter D(cid:107) <D⊥ [26].
η = 1(α −α )= 1(α +α ). Fromthenumericalcoef- Insummary,wehaveinvestigateddiffusioninlyotropic
d 2 6 3 2 5 2
ficientsinTableIIitisclearthattheviscosityanisotropy LCs with two different anchoring conditions. Our mea-
arisesessentiallyfromC andC . SinceC =0forauni- surementsconfirmStokesfrictionD ∝1/dinbothcases,
b d d
form director, finite values of C reflect distortions due unlike a previous study on thermotropic NLCs [18]. In
d
to anchoring. In the following, we discard the first term the NC phase, we found an unusually large viscosity ra-
becauseofthesmallcoefficientC . (Moreover, α isalso tio η /η ≈ 4 which can be accounted for thanks to
1 1 ⊥ (cid:107)
small for several thermotropic NLCs [12, 13]). our perturbative theoretical approach. Our analysis im-
Comparison with experiment. From the straight lines poses η < η < η on the Miesovicz parameters in the
b a c
in Fig. (2a) (NC phase, planar anchoring) and the NC phase (which is usually the case [32]), and suggests
Stokes-Einsteinrelation,wededucetheexperimentalval- η < η < η with D < D in the ND phase. As a
c a b (cid:107) ⊥
ues η = 0.31 Pa.s and η = 1.24 Pa.s. Using a plate- short-term follow-up work, we will evaluate Eq. (9) for
(cid:107) ⊥
cone rheometer, the zero shear effective viscosity was the case of normal anchoring and an independent mea-
found to be η (cid:39) 0.4 Pa.s at T = 25◦C. Though these surement of the Miesovicz viscosities together with the
S
numbersarenotsufficienttodetermineallLeslieparam- additionnal parameter η would be most desirable.
d
eters, they present several noteworthy constraints. In WeacknowledgefinancialsupportfromtheFrenchNa-
view of the coefficients listed in Table II and Eq. (9), tional Research Agency under grant # ANR-07-JCJC-
we deduce ∆η = η − η (cid:39) (C(cid:107) − C⊥)(η − η ) + 0023 and the Conseil R´egional d’Aquitaine.
(cid:107) ⊥ b b b a
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