Table Of ContentHydrodynamics of topological defects in nematic liquid crystals
G´eza T´oth1, Colin Denniston2, and J. M. Yeomans1
1 Dept. of Physics, Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP
2 Dept. of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218.
(February 1, 2008)
Weshowthatback-flow,thecouplingbetweentheorderparameterandthevelocityfields,hasa
2 significant effect on the motion of defects in nematic liquid crystals. In particular the defect speed
0 can depend strongly on the topological strength in two dimensions and on the sense of rotation of
0
the director about thecore in threedimensions.
2
n 61.30.Jf,83.80.Xz,61.30.-v
a
J
Topological defects arise in all areas of physics from of constant magnitude. Thus they are inadequate to ex-
1
2 cosmicstrings[1]tovorticesinsuperfluidhelium[2]. Al- plorethehydrodynamicsoftopologicaldefectswherethe
though the physical systems are very different, many as- magnitude of the order parameter has a steep gradient
] pects of the observed phenomena match even quantita- and becomes biaxial within the core region [9]. Here we
t
f tively,makingitpossibletotestcosmologicalpredictions considerthemoregeneralBeris-Edwards[10]formulation
o
experimentally in condensed matter systems [1]. De- ofnematohydrodynamics,wheretheequationsofmotion
s
. fectsareclassifiedaccordingtotheirtopologicalstrength, are written in terms of a tensor order parameter Q.
t
a and in many cases,the symmetries of the field equations Theequilibriumpropertiesoftheliquidcrystalarede-
m
lead to dynamics where defects of opposite topological scribed by a Landau-de Gennes free energy density [3].
- strength can be mapped into each other. In this Letter This comprises a bulk term
d
n we show an example, of topological line defects in liquid A γ Aγ Aγ
o crystals [3], where this symmetry is broken, due to the fb = 2(1− 3)Q2αβ − 3 QαβQβγQγα+ 4 (Q2αβ)2,
c couplingtoanadditionalfield,theflow-field,ortoelastic
[ (1)
constants that are not equal.
1 In liquid crystals the topological defects are moving whichdescribesafirstordertransitionfromtheisotropic
v
within a liquid and therefore one must expect hydrody- to the nematic phase atγ =2.7,togetherwith anelastic
8
namics to play an important role in their dynamics. In contribution
7
13 pchaartnigciunlgar,dairsecthtoerdfiefeelcdtsamndovteh,ethveelcoocuitpylinfigelbdet(wsoe-ecnaltlhede fd = L1(∂αQβγ)2+ L2(∂αQαγ)(∂βQβγ)
2 2
20 bpaercikm-fleonwta)lmevaiydepnlacye[a4]sisghnoiwficsatnhtaptatrhtisinisthinedmeeodtitohne.cEaxse- + L3Qαβ(∂αQγǫ)(∂βQγǫ), (2)
0 2
when a nucleated domainwhere the director field is hor-
/
at izontal grows in a twist or vertical environment due to wheretheL’sarematerialspecificelasticconstants. The
Frank expression for the elastic energy, written in terms
m the influence of the surfaces or an external electric field.
of the derivatives of the director, can be simply mapped
The speed of the domain boundary is found to depend
-
to (2) [10]. A controls the relative magnitude of f and
d strongly on the local defect configuration. b
n fd. TheGreekindiceslabelthe Cartesiancomponentsof
Previous investigations [5–8] of defect dynamics have
o Q, with the usual sum over repeated indices.
c either ignoredthe flow fieldor takenaccountofits effect The dynamics of the order parameter is described by
: phenomenologically. Hereweaimtogeneralizethis work
v the equation
i by treating the full hydrodynamic equations of motion
X for a nematic liquid crystal. We consider the annihila- (∂t+~u·∇)Q−S(W,Q)=ΓH, (3)
r tion of a pair of defects of strength s = ±1/2. We find
a where~uisthe bulk fluidvelocityandΓis acollectivero-
that back-flow can change the speed of defects by up to
tational diffusion constant. The term on the right-hand
∼100%. Defects of different strength couple to the flow
side of equation (3) describes the relaxationof the order
field in different ways. This leads to a dependence of
parameter towards the minimum of the free energy F
speed on strength which can occur either as a result of
the back-flow or if the elastic energy is treated beyond a δF δF
H=− +(I/3)Tr . (4)
one-elastic constant approximation. δQ (cid:26)δQ(cid:27)
The hydrodynamics of liquid crystals is often well de- The term on the left-hand side is
scribedbytheEriksen-Leslie-Parodiequationsofmotion,
S(W,Q)=(ξD+Ω)(Q+I/3)+(Q+I/3)(ξD−Ω)
which are written in terms of the director field. How-
ever these are restricted to an uniaxial order parameter −2ξ(Q+I/3)Tr(QW), (5)
1
whereD=(W+WT)/2andΩ=(W−WT)/2arethe the director on the x axis (where we define x as the axis
symmetricpartandtheanti-symmetricpartrespectively connecting the two defects cores). This corresponds to
of the velocity gradient tensor W =∂ u . ξ is related the transformation
αβ β α
to the aspect ratio of the molecules.
Q →−Q , Q →−Q . (8)
The velocity field ~u obeys the continuity equation and xy xy yx yx
aNavier-Stokesequationwithastresstensorgeneralized The order parameter fields of the two defects with topo-
to describe the flow of nematic liquid crystals logicalchargess=±1/2transformintoeachother. Thus
approachesbasedonasimpleGinzburg-Landauequation
1
predict that when the defects move they follow symmet-
σ =−ρTδ −(ξ−1)H (Q + δ )
αβ αβ αγ γβ γβ
3 ric dynamical trajectories.
1
Fig. 2(a) shows the position of two annihilating topo-
−(ξ+1)(Q + δ )H
αγ αγ γβ
3 logical defects, with topological charge s=+1/2 (upper
1 δF curve)ands=−1/2(lowercurve),asafunctionoftime.
+2ξ(Q + δ )Q H −∂ Q , (6)
αβ αβ γǫ γǫ β γν
3 δ∂αQγν Fig. 2(b) shows the velocities of the defects as the func-
tion of their separation D. (See footnote [13] for the
where ρ and T are the density and temperature. Notice
simulation parameters.)
that the stress (6) depends on the molecularfield H and
on Q. This is the origin of back-flow. Details of the
equations of motion can be found in reference [10]. The
equations (1)–(6) were solvednumerically using a lattice
Boltzmann algorithm described in [11].
Consider a pair of defects of topological strength s =
±1/2 situated a distance D apart in a nematic liquid
crystal, as shown in Fig. 1(a). We consider a two-
dimensional cross section of the two line defects, assum-
ing thatthe orderparameterdoesnotchangeinthe per-
pendiculardirection(althoughthedirectormaypointout
of this plane). The two defects are topologically distinct
onlyintwodimensions,buteveninthreedimensionsthey
are separatedby an energy barrier for the typical elastic (a)
constants we study here.
Aphenomenologicalequationofmotioncanbewritten
down by assuming that the attractive force between the
two defects [3] is counterbalancedby a friction force [7]
dD
D =µ0ln−1(D/Rc), (7)
dt
where D is the defect separation, R is the defect core
c
size and µ0 is a constant.
In Ref. [8] the director field and the trajectory of the
defects were obtained analytically and the defect veloc-
ity was determined as a function of parameters of the
medium. (A review of the earlier development of the
theory of defect dynamics is also given in [8].) However
this and, as far as we are aware,all other analyses of de-
(b)
fect dynamics have ignored back-flow. This means that FIG.1. (a) The director field of two annihilating topolog-
the approach to equilibrium is relaxational, determined ical defects with strength s=±1/2. (b) Velocity field of the
entirely by the derivative of the free energy with respect two defects.
to the order parameter, with the flow playing no role. A
further simplification in previous work is that the Frank Consider first the dashed trajectories. These were
elastic constants were assumed to be equal. obtained with the flow field switched off, the case for
We can examine relaxational dynamics using a whichequation(3)reducestoaGinzburg-Landaumodel.
Ginzburg-Landauequation for the director field [12], i.e. As expected the two defects move with the same speed
Eqn. (3) with the velocity set to zero. The Ginzburg- and annihilate halfway between their initial positions
Landau equation with a single elastic constant is invari- (markedbyahorizontallineinFig.2(a)). Theresultsfor
ant under a local coordinate transformation mirroring theGinzburg-Landaumodelfitthesimpleformula(7)for
2
µ0 =124µm2/sandRc =0.0233µmforD &Rc. Around This flow depends onthe signof the spatialderivativeof
D ∼R the formula overestimates the defect speed. the director orientation. As a result, these two sources
c
of flow reinforce in one case and partially cancel in the
0.5 other, thus giving the anisotropy. The flow field around
the defects is depicted in Fig. 1(b). A strong velocity
)
m vortex pair is formed around the +1/2 defect, with the
µ flowpointing in the directionofdefect propagation. The
(
n flow around the −1/2 defect is much weaker, and points
o
siti 0 opposite to the direction of defect motion.
o
p The solid line in Fig. 2(a) corresponds to a simulation
ct of the full hydrodynamic equations of motion (1)–(6).
e
ef There is a marked decrease in the time to coalescence.
D
This is primarily because the speed of the s = +1/2 de-
−0.5 fect is increased by ∼ 100% compared to the case with-
0 5 10
out flow for D & 0.25µm. The s = −1/2 defect is only
Time (ms)
slightly affected by the back-flow; the change in veloc-
0.4
ity is less than 20%. Due to the speed anisotropy the
(b)
defects do not meet halfway between their initial posi-
0.3 tions. For defects initially 1µm apart the displacement
s) of the coalescence point (∆x1 = 0.149µm) is smaller
m than might be expected from the substantial speed-up
m/0.2
ofthe s=+1/2defect. Thisis becausethe relativeflow-
µ
v ( induced increase in velocity drops dramatically near the
0.1 defect core (D . 0.25µm) where the defects are moving
the fastest. At these short separations the relaxational
dynamics dominates the hydrodynamics.
0
0.2 0.4 0.6 0.8 Changing the various material parameters of the sam-
D (µm) ple will affect the velocity of the defects and their speed
FIG. 2. (a) Positions of the defects as a function of time. anisotropy. We find, as expected, that as the viscosity
Upper (lower) curves are for s = +1/2(−1/2). The different
in the Navier-Stokes equation increases the motion ap-
lines correspond to a Ginzburg-Landau model(dashed), the
proaches that of the Ginzburg-Landau model. Increas-
hydrodynamic equations of motion with Γ = 6.25 Pa−1s−1
(solid) and with Γ = 7.76 Pa−1s−1(dotted). (b) Defect ing Γ in (3) increasesthe speed of relaxationto the min-
imum of the free energy, thus increasing the speed of
speed as a function of separation. Upper and lower solid
curves: the s = +1/2 and s = −1/2 defect trajecto- the defects. The speed anisotropy howeverdecreases be-
ries with hydrodynamics and Γ = 6.25; Dashed curve: cause the weight of the free energy relaxation process is
Ginzburg-Landau model. The +1/2 defect is considerably increased relative to the hydrodynamics. For example
(100%) accelerated for D > 0.25µm compared to the results the dotted curve in Fig. 2(a) corresponds to Γ = 7.76
oftheGinzburg-Landaumodel. Thespeedofthe−1/2defect Pa−1 s−1 (compared to Γ = 6.25 Pa−1 s−1 for the solid
is only slightly affected bythe back-flow. curve). The displacement of the coalescence point is
∆x2 =0.128µm<∆x1. Decreasing A in the free energy
Consider the effect of the back-flow under the trans- (1)increasesthe defectsize. The defects movefasterbut
formationofEq.(8). Examiningthestresstensoronecan thevelocitydisparitydecreases,againbecausetheimpor-
see that the last term in Eq.(6) does not change under tanceoftherelaxationaldynamicsisincreasedrelativeto
the transformation but the off-diagonal elements of the the hydrodynamics.
other terms have their sign inverted. This reflects the The results in Fig. 2 are for a single elastic constant
two sources of back-flow in this problem. The first has (equal Frank elastic constants or, equivalently, L2 =
to do with the defect core. Order is suppressed in the L3 = 0). If a more general model for the elasticity is
core, which results in an increase in viscosity at the core considered, allowing L2 6=0 or L3 6= 0, the invariance of
(The isotropic viscosity α4, is proportional to (1 −q)2 theGinzburg-Landauequationunderthetransformation
where q is the magnitude of the order [10,11]). As a re- (8)is broken. Thereforeone mightexpect a difference in
sult,themovementofthecoreinducessimilarvorticesto the flow velocities of s = ±1/2 defects even if back-flow
those produced when a solid cylinder is moved through is not considered.
a fluid. This flow is independent of the sign of the de- Thisisindeedthecase. IfL2 =0andL3 <0(L3 >0)
fect, and points into the direction of defect propagation the s = +1/2 (s = −1/2) defect moves faster. For ex-
at the core. The second source of back-flow comes from ample, if L1 = 8.73 pN, L2 = 0, and L3 = 15.88 pN
thereorientationofthedirectorfieldawayfromthecore. [14] a comparison of the velocities v+1/2, v−1/2 of the
3
s=1/2 and s=−1/2 defects respectively gives a speed [2] D.V.Osborne,Proc.Phys.Soc.LondonA63,909(1950).
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and S.F. Edwards, The Theory of Polymer Dynamics,
anisotropy caused by the back-flow is ∼+68%. The dis-
Clarendon Press, Oxford, (1989).
placementofthecoalescencepointis∆x=0.029µm. For
[4] E. J. Acosta, M. J. Towler and H. G. Walton, Liquid
most materials L3 > 0 (corresponding to K11 < K33), Cryst. 27, 977 (2000).
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52, 718 (1995).
have been observedin experiments on pi-cell liquid crys-
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wallsisimportantinmediatingtheformationoftheoper- and Ref. therein.
ating(bend)statefromtheground(splay)state. Defects [11] C. Denniston, E. Orlandini and J. M. Yeomans, Euro-
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gate the role of defect motion in many other new gener- [13] Simulations were performed on a 350×350 lattice with
ation liquid crystal devices. For example multi-domain A = 1.0, L1 = 0.55, L2 = L3 = 0, Γ = 0.625,
ξ = 0.59, ρ = 2.0, T = 0.5 and γ = 3. Periodic bound-
nematic modes improveviewing angles atthe expense of
ary conditions were used in the x direction and free
introducing defects into the director profile and under-
boundaries were used in the y direction. Given suitable
standing the behavior ofsuch defects as the electric field
pressure, length and time scales these parameters can
isvariedwillhelpcontroldeviceperformance. Inzenithal be mapped to a lattice size 4.4µm×4.4µm, L1 = 8.73
bistablenematicdevicesswitchingisbetweentwo(meta) pN, Γ = 6.25 Pa−1s−1. The Frank elastic constants are
stable zero-field states. Switching between the states is K11 = K22 = K33 = 2L1q2 = 4.37pN and Miesowicz
mediated by the movement of topological defects [18]. viscosities 0.02 Pasto0.1 Pas.Thematerial parameter
To conclude, we have used a formulation of nemato- values chosen are close to those of 5CB.
dynamics based on the tensor order parameter to study [14] L1 = 8.73 pN, L2 = 0 and L3 = 15.88 pN correspond
the hydrodynamics of topological defects in nematic liq- to Frank elastic constants K11 = K22 = 3.04 pN, and
uidcrystals. Wefindthatthecouplingbetweentheorder K33 =7.01 pN.
parameterfieldandtheflowhasasignificanteffectonde- [15] If L3 = 0 and the director is confined to the xy plane,
the Frank elastic free energy density (assuming uniaxi-
fect motion: in particular it introduces a substantial dif-
ality and constant magnitude of order) can be written
ferencebetweenthevelocitiesofdefectsofdifferenttopo- as f = κ(∇θ)2, where θ is the angle of the director to
2
logical charge. Similar but smaller velocity anisotropies thehorizontalaxisandκ=(2L1+L2)q2.Thedynamics
can result from changing the elastic constants. of the medium, ∂tθ = −ΓδF/δθ = Γκ∇2θ, is invariant
We would like to thank E. J. Acosta, C. M. Care, J. under mirroring on the x axis. Moreover a given L1 and
Dziarmaga,S. Elston, K. Good, O. Kuksenok, S. Lahiri, L2 correspond to thesame dynamicsas L′1 =L1+L2/2
′
N.J.MottramandT.Sluckinforhelpfuldiscussions. We and L2 = 0. (Note that there is no speed anisotropy in
acknowledgethesupportofSharpLaboratoriesofEurope the latter case with the Ginzburg-Landau model for the
at Oxford. C.D. acknowledges funding from NSF Grant tensor order parameter.) For defects this equivalence is
onlyapproximateduetothebiaxialityandthechangein
No. 0083286.
the magnitudeof order in the core.
[16] L1 = 8.73 pN, L2 = 15.88 pN and L3 = 0 corre-
spond to Frank elastic constants K11 = K33 =8.34 pN,
K22 =4.37 pN.
[17] C.Denniston,G.T´othandJ.M.Yeomans,J.Stat.Phys.
107, 187 (2002).
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4
