Table Of ContentSelf-assembly and nonlinear dynamics of dimeric colloidal rotors in cholesterics
J. S. Lintuvuori1, K. Stratford2, M. E. Cates1, 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.
We study by simulation the physics of two colloidal particles in a cholesteric liquid crystal with
tangentialorderparameteralignmentattheparticlesurface. Theeffectiveforcebetweenthepairis
attractiveatshortrangeandfavoursassemblyofcolloiddimersatspecificorientationsrelativetothe
localdirectorfield. Whenpulledthroughthefluidbyaconstantforcealongthehelicalaxis,wefind
that such a dimer rotates, either continuously or stepwise with phase-slip events. These cases are
separatedbyasharp dynamicaltransition andlead respectivelytoaconstant oran ever-increasing
2
phaselag between thedimer orientation and thelocal nematic director.
1
0
PACSnumbers: 61.30.-v,83.80.Xz,61.30.Jf
2
n
a Self-assemblyisoneofthekeyaimsofpresent-daynan- leads to the formation of a dimer or longer chains. Here
J otechnology. The ideaunderpinning this conceptis that, we focus on the simplest case of a dimer, but progress
4 througha carefulcontrolof the interactions in a suspen- beyondthe purelystaticinvestigationof[6]toshowthat
sion of particles, it is possible to drive the spontaneous sucha dimer exhibits unexpected andintriguing dynam-
]
t formationofanorderedtargetstructureorpattern,start- icalproperties. When subjectedto anexternalforce(for
f
o ing from a disordered initial condition. The process is instancegravity)alongthecholesterichelix,thedimerro-
s spontaneousasitoftenentailstheminimisation,whether tates about this axis in a screw-like fashion. Depending
.
t globalor local, of the free energy of the system. An out- on the magnitude of the force, the dimer either rotates
a
m standing specific challenge in modern self-assembly is to continuouslyorexhibits phaseslippage,alternatingperi-
build a structure with a target dynamic feature, such as odsofsmoothrotationwithstaticspellsinwhichittrans-
-
d a synthetic microscopic rotor, walker or swimmer [1]. lates without rotation. This dynamical transition shows
n Colloidal dispersions in liquid crystals offer a useful similarnear-criticalbehaviourtothe depinningofdriven
o
system for the development of self-assembly strategies vorticesandofchargeddensitywavesinsuperconductors,
c
[ at the microscale. Even in nematics, the simplest liq- bothofwhichmaybestudiedwiththeFrenkel-Kontorova
uid crystalline phase, elastic distortions and topological model for transport in a periodic potential [8]. Another
1
defects mediate a variety of interactions resulting in the analogue is provided by the synchronization of coupled
v
formation of wires, colloidal crystals, cellular solids, and oscillatorsdescribedbythe Kuramotomodel[9]. Within
7
4 clusters entangled by disclinations [2–4]. This variety of our liquid crystal context, the phase slippage regime re-
9 self-assembledstructuresispossiblebecauseonecantune quires a specific free energy landscape which we discuss.
0 theliquidcrystalmediatedinteractionsbychanging,e.g., This provides potential for the design of self-assembled
.
1 the strength and nature of the liquid crystal ordering at systems with tunable dynamic properties.
0 the colloidal surface. These alter the local symmetry of Thesystemwestudyconsistsoftwosphericalcolloidal
2 thedirectorfieldneartheparticleandhencequalitatively particles of radius R moving in a cholesteric liquid crys-
1
change the effective interparticle forces. tal. To describe the thermodynamics of the chiral host,
:
v Here we show that a powerful way to extend the po- we employ a Landau–de Gennes free energy , whose
Xi tential for self-assembly of colloids in liquid crystals is density f may be expressed in terms of a tracFeless and
r to consider cholesterics, or chiral nematics. In cholester- symmetric tensor orderparameter Q [10] andis detailed
a ics the direction of the molecular order in the ground in[11]. Tangentialanchoringismodelledbyasurfacefree
state spontaneously twists. The spatial modulation of energy, fs = 21W(Qαβ−Q0αβ)2, where W is the strength
the twist is in the micron range, therefore of the order ofanchoringandQ0 isthepreferredorderparameterin
αβ
of typical colloidal sizes. We have previously shown [5] the tangent plane to the local spherical surface [12].
thatbyvaryingtheratiobetweenthesetwofundamental We employ a 3D hybrid lattice Boltzmann (LB) algo-
length scales, it is possible to change continuously the rithm[13]tosolvetheBeris-EdwardsequationsforQ[10]
topologyofthedefects,ordisclinationlines,surrounding
a single particle—morphological changes with no direct DtQ=Γ −δδQF + 31tr δδQF I . (1)
counterpartinnematics. Veryrecently,MackayandDen-
(cid:16) (cid:16) (cid:17) (cid:17)
niston[6]tookastepfurtherandstudiedtheinterparticle Here, Γ is a collective rotational diffusion constant and
elastic force felt in a cholesteric by two colloidal spheres D is the material derivative for rod-like molecules [10].
t
with tangential anchoring of the director field at their The term in brackets is known as the molecular field,
surface. Asinthenematiccase[7],thereisacomplexin- which in the absence of flow drives the system towardsa
terplaybetweenrepulsiveandattractivedirectionswhich free energy minimum. The boundary conditions for the
2
orderparameteronthecolloidalsurfacesaregivenby[4]:
1000
∂f ∂f (a) X
s
ν + =0 (2)
γ∂∂ Q ∂Q 500 Y
γ αβ αβ
Z
where ν is the local normal to the colloid surface.
γ T 0
The velocity field obeys the continuity and Navier- B 0.5
Ssctroikbeeslieqquuiadticorny,stwalithhydarostdryenssamteincsso[r10g]e.nWeriatlhisinedoutor hdye-- Uk/ -500 3kT0)B -0 .05
1 -1
bridscheme,wesolvetheNavier-StokesequationviaLB, U ( -1.5
-1000
and Eq. 1 via finite difference [5]. Colloids are repre- 0 60 120 180
sented by the standard method of bounce-back on links, θ (degrees)
-1500
whichleadstoano-slipboundaryconditionfortheveloc- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
ity field (see [5, 14] for details). Order parameter varia-
y
tions createanadditionalelastic forceactingonthe par- (b) n (d - 2R)/p d (c)
θ
ticle which is computed by integrating the stress tensor y
n
over the particle surface [4, 5]. d
φ
The dynamics is primarily controlled by the Ericksen
x
number, Er = γ1vR/K, where the rotational viscosity x z
γ =2q2/Γ, v is a velocitycharacteristicof the flow, and
1
q is the degree of orderingin the system. In the uniaxial
FIG. 1: [Color online.] (a) Interparticle elastic potential as a
case with director nˆ, Q =q(nˆ nˆ δ /3).
αβ α β − αβ function of reduced separation (d−2R)/p, along the helical
In what follows, we quote our results in simulation axis(z)andinthex−andy−directions. Theinsetshowsthe
units [11, 13]. To convert them into physical ones, we angular dependencein thex−y plane at (d−2R)/p≈0.05.
can specify an elastic constant of 28.6 pN, and a rota- Units are in kBT ≈ 4 pN nm. (b) Snapshots of the dimer
tional viscosity of 1 poise. (These values hold for typical showingthedisclination linesat(d−2R)/p≈0.05 whendis
materials,andacolloidaldiameterof1µm.) Inthisway, along y and along x. The Cartesian axes, the separation d,
the simulation units for force, time and velocity can be andthefar-fieldnematicdirector,nˆ areshown. (c)Definition
of theangles θ and φ used in the text. For error analysis see
mapped onto 440 pN, 1 µs, and 0.07µm/s respectively.
SupplementaryMaterial [11].
We first consider the interparticle elastic potential.
Two particles are placed centre-centre separation vector
d apart. Bothparticlesareheldfixedfor the durationof
These results confirm and extend those of [6] on the
a simulation in which the free energy is minimised. By
energetics of dimer formation in cholesterics. Our key
repeating simulations for different d, we map out the ef-
focus in the present work is dynamics.We place two par-
fective two-body potential as a function of the reduced
ticles initially near the weaker minimum of the potential
separation(d 2R)/pin the three coordinatedirections,
− inthedirectionperpendiculartonˆ (leftFig. 1b). Thisis
and as a function of angle in the x y plane (Fig. 1a).
− the first relatively deep local minimum two particles ap-
The potential is markedly anisotropic. While the poten-
proachingfromfarawaywouldencounter. Itis therefore
tialinzshowsstrongrepulsionatlargeseparations,there
a natural self-assembledconfigurationfor the dimer. We
is an attraction in the x y plane. The most favourable
− then pull each along the helical axis with force f.
configuration at small separations is along the director
field (x-direction in Fig. 1a). Here we estimate a max- At all force levels studied here, the moving dimer ro-
imum attractive force of 20 pN. Before this deep mini- tates about its centre of mass while d remains perpen-
mum is reachedthe dimer needs to overcomea repulsion dicular to the helical axis (inset Fig. 2e). The behaviour
(peak at (d 2R)/p 0.2), which is largest when the at low force (f 0.025) is illustrated in Fig. 2(a–d).
disclinations−attheop≈posingparticlesurfacesjoinup. In We quantify the r≤otation by measuring the angle, φ, be-
this bound state,the colloidssharetwodisclinationlines tween d and the x-axis and the angle, θ, between d and
which act as a glue between them (right Fig. 1b). For nˆ(z) (Fig. 1c), as a function of time. After an initial
separation vectors perpendicular to nˆ we find a stable transient, a smooth rotation is observed (Fig. 2e; open
minimum and no repulsive barrier (y-direction in Fig. symbols),butwithseparationdthatlagsbehindnˆ(z)by
1a). These results are far from the nematic limit stud- a constant phase angle θ(t) (Fig. 2f, open symbols) [17].
iedexperimentallyin[7]andtheoreticallyin[15], aswell We attribute this constant lag to a balance between the
as from results obtained in a twisted nematic cell [16]. viscousdragopposingtherotationofthepairinthex y
−
Most notably, the in-plane potential perpendicular to nˆ plane,andtheforcearisingfromtheangularvariationin
(here y) was always repulsive in the nematic [15]. The the rotating interparticle potential.
preferred configuration (here, along nˆ) is at an angle of At higher force (f 0.0275), the behaviour is man-
about 30◦ in the nematic [7, 15]. ifestly different. The ≥angle between d and x, shown in
3
(a) (b) for a single colloid moving along the helical axis [5].
z To understand the dynamical transition better, we
write down a set of phenomenological equations for the
evolutionof the dimer position, z(t), and for φ(t), which
y (c) (d) gives the direction of d in the lab frame (Fig. 1c). We
assumethatthebasicfeaturesofthecholestericordering
640
96 may be captured by an effective angular potential, also
560 (e)
es 480 90 periodic in z: V(φ,z). Our equations read as follows,
e
gr 400 84
e dφ 1 ∂V(φ,z)
) d 320 0 0.3 0.6 0.9 = (3)
φt( 240 dt −γφ ∂φ
160 dz 1 ∂V(φ,z)
80 =
dt −γ ∂z
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 z
V(φ,z) = Acos[B(φ q z)]+fz.
0
2160 − −
degrees 111048840000 fzfff=zzz===0000.0...00024325 (f) iγHtzievreaer,ceAonr(esultanaxnitatstsi,oofnfeanilsecrtoghnyes)teaaxnntdtesrBnrae(lldafitomerdecnintsogio,ntwhleehsisrl)eoatγarφetiopanonasd-l
(t) 720 and translational friction of the dimer, and whose exact
θ 360
values we will not need. Eqs. 3 may be solved by an
0
ansatz suggested by the behaviour in Fig. 2f. We write
0 0.2 0.4 0.6 0.8 1 1.2
6 φ(t)=Ωt+θ(f), where the director-dimer angle follows
simulation step/10
the most favourable orientation apart from a phase lag,
θ(f). This ansatzis asolutionprovidedthatf issmaller
FIG.2: [Color online.] (a-d) Snapshotsof a steadily rotating than a critical threshold f , i.e. in the rotor phase. By
c
dimer over a translation of half a pitch (a rotation of 180◦). estimating the average terminal velocity of the dimer as
Thedirectorfieldiscolourcodedaccordingtothelocaldirec- f/γ , we find that in the rotor phase the angular ve-
φ
tor (red along x, blue along y). The time evolution of (e) φ
locity is Ω 2πf/(γ p) (this is true within statistical
z
and (f) θ (defined in Fig. 1C), for the rotor (open symbols) ∼
error using data in Fig. 3), and that the critical force
andphaseslippage(closed symbols) motion. Theinset in(e)
also shows the angle between d and the helical-axis, which is fc = Aγzp/(2πγφ). Above this threshold, θ increases
remains close to 90◦. with time as in Fig. 2f (solid symbols). The asymptotic
velocity Ω (= dθ(t)/dt) f2 f2 [11] for f f
lag ∼ − c → c
provides a useful “order parameter” to characterise the
p
Fig.2e(closedsymbols),nowincreaseswithtimeinase- rotor-slippage transition. In contrast, the angular veloc-
riesofsteps. Thesestepscorrespondtointervalswithand ity Ω decreases as 1/f for large f.
withoutsignificantrotationofthedimerasitmovesalong Importantly, Eqs. 3 capture both the near-critical be-
the helix (see [11] for a movie and further discussion of haviourofΩ ,andthelargef behaviourofΩshownby
lag
the dynamics). Correspondingly, the phase lag θ(t) fails our full LB simulations (see Fig. 3a for an Ω 1/f fit
∼
to reacha steady value (Fig. 2f; closed symbols), but in- andFig. 3cforaΩ f2 f2 fit). Thenear-critical
lag ∼ − c
creasesindefinitely. We referto this as“phaseslippage”. behaviour of Ω in our transition is similar to that of
lag p
There is a clear transition between the smooth “rotor” the velocity of a chain of driven particles in a periodic
regime and the phase slippage regime. potential described by the Frenkel-Kontorova model [8],
Fig. 3a quantifies the dependence of the average rota- suggesting that our dimer provides a liquid crystal rep-
tional velocity of the dimer, Ω, on the applied force for resentative of a wider class of models [11]. Another ana-
simulations at a range of force values. It can be seen logueiswiththesynchronisationoftwodrivenoscillators
thatΩ is proportionalto f inthe lowforce rotorregime, describedbytheKuramotomodel[9,11],wheresynchro-
whileitdecreasesinthehighforcephaseslippageregime. nised and unsynchronised states correspond to the rotor
Fig.3bshowsthedependenceoftheaveragespeedofthe and slippage regimes respectively. At the same time, we
dimer along the helical axis with force, again showing a note that the physics of our cholesteric dimers is richer
transition. However, it is difficult to relate these transi- than that in Eqs.3. Whereas the symmetry of the prob-
tions to the equilibrium potential which is strictly valid lemsuggeststhatthereshouldalwaysbeaverylowforce
only at Er = 0. Although the largest Eriksen number regime in which the dimer behaves as a rotor, the exis-
remains low (Er 0.026), the potential is likely to be tence of the phase slippage regime at higher forces de-
≈
affectedby dynamicaleffects including the localbending pends on the form of the effective pair potential. The
of the cholesteric layers (visible in Fig. 2a-d [11]). The required conditions hold for tangential anchoring but,
localbending ofthe layersforEr 1,was alsoobserved according to our preliminary studies is violated in the
≪
4
curs as the forcing exceeds a critical threshold: its value
may be estimated via a simple theory considering the
interplay between a spatially periodic angular potential
andan externaldriving. The existence ofthe phase slip-
page regime is howeverhighly non-trivialand relies on a
delicate balance in the equilibrium and dynamic proper-
ties of our dimers: for example we have not observed it
for dimers with normal anchoring. An interesting possi-
bility for future researchwould be to study the effects of
FIG.3: Averagerotationalvelocity(Ω)(a)andsedimentation external electric field applied to our colloidal dimer, as
velocity(b)ofthecolloidaldimerasafunctionoftheforcing. done with platelets [20] and nematic colloids [21], where
Thenon-linearfitin(a)isΩ=c/f,withc>0aconstant. (c) unusual dynamics was triggered by the field.
showsΩlagasafunctionoff togetherwithafittodpf2−fc2,
with d>0 a constant and fc ≈0.0267±0.0005.
ThisworkwasfundedbyEPSRCGrantsEP/E030173
and EP/E045316. MEC is funded by the Royal Society.
normalanchoringcase,forwhichwehavenotobserveda
phase slip regime.
Ourstudy maybe viewedasa generalisationofaclas-
sical problem: the sedimentation of two spheres in a vis-
cous fluid. Intriguingly, Fig. 3b shows that the velocity– [1] C. E. Singa et al., Proc. Natl. Acad. Sci. USA 107, 535
force curvesarenot linear,eveninthe Stokes limit ofef- (2010); P. Tierno et al., Phys. Rev. Lett. 101, 218304
fectively zero Reynolds number. Rather, they appear to (2008).
[2] M. Ravnik et al., Phys. Rev. Lett. 99, 247801 (2007);
have different slopes (i.e. effective viscosities) in the ro-
J. C. Loudet, et al.,Langmuir 20, 11336 (2004).
tor andphase slippage regimes. This is differentto what
[3] T.Araki,H.Tanaka,Phys.Rev.Lett.97,127801(2006).
occurs for a single sedimenting particle in a cholesteric, [4] M. Skarabot et al. Phys. Rev. E 77, 061706 (2008);
whichleadstoalinearvelocity–forcerelationintheforce M. Conradi et al.,Soft Matter 5, 3905 (2009).
range simulated here [5]. This biphasic force-velocity [5] J. S. Lintuvuori et al., Phys. Rev. Lett. 105, 178302
curve is due to the dynamic transitionwe discussed, and (2010); J. Mat. Chem. 20, 10547 (2010).
has no counterpart in classical sedimentation, in either [6] F. E. Mackay and C. Denniston,EPL 94, 66003 (2011).
[7] I.I.Smalyukhet al.,Phys. Rev. Lett.95,157801(2005).
Newtonian or Maxwell fluids. In the Newtonian case,
[8] J. Frenkel and T. Kontorova, Zh. Eksp. Teor. Fiz. 8,
sinking side by side speeds up the particles, by up to
1340(1938);R.Besseling.R.NiggebruggeandP.H.Kes,
a factor of 2 [18], which is not true in our rotor phase Phys. Rev. Lett.82,3144(1999); R.Besselinget al.,Eu-
(wherewe findthatadimer sedimentsslower thanasin- rophys. Lett. 62, 419 (2003).
gle particle). In a Maxwell fluid, the repulsive or attrac- [9] J. A.Acebron et al.,Rev. Mod. Phys. 77, 137 (2005).
tiveinteractionbetweentwospheressedimentingside-by- [10] A.N.BerisandB.J.Edwards,ThermodynamicsofFlow-
side is controlledby normalstresses[19]. Inour case,we ing Systems, Oxford UniversityPress, Oxford, (1994).
[11] SeeonlineSupplementaryMaterialatXXXforadditional
find a novel velocity–dependent torque and a dramatic
details on our model and dynamical transition and for
dependence on the nature of the anchoring.
movies of thedynamics.
Inconclusion,wehavestudiedtheequilibriumanddy- [12] J.-B. Fournier, P. Galatola, Europhys. Lett. 72, 403
namic properties of two colloidal spheres in a cholesteric (2005).
liquidcrystal. We haveseenthatchiralityleadstoama- [13] D. Marenduzzo et al., Phys. Rev. E 76, 031921 (2007);
jorchangeinthe effective potentialfeltbythe pair,with M. E. Cates et al., Soft Matter 5, 3791 (2009).
respect to the nematic limit. The elastic forces we find [14] N.-Q. Nguyen and A. J. C. Ladd, Phys. Rev. E 66,
leadtothestabilisationofadimeratanangleofeither0◦ 046708 (2002).
or 90◦ degrees, as opposed to the 30◦ found in nematics. [15] M. R.Mozaffari et al.,Soft Matter 7, 1107 (2011).
[16] U. Tkalec et al.,Phys. Rev. Lett. 103, 127801 (2009).
These results, alongside those of Ref. [6], suggest that it [17] θ ≈ 348◦ and θ ≈ 330◦, for f = 0.25 and f = 0.2,
wouldbeinstructivetorepeattheexperimentsperformed respectively.Thesecorrespondtoθ=12◦ andθ=30◦ in
in Ref.[7]with a cholestericliquid crystal. Basedonour the inset of Fig. 1a.
results,onemayalsospeculatethatvariationsinparticle [18] M. Stimpson, G. B. Jeffery, Proc. Roy. Soc. A 111, 110
size (or cholesteric pitch) can affect the local free energy (1926).
[19] A. S. Khair and T. M. Squires, Phys. Rev. Lett. 105,
landscape of colloidal suspensions in liquid crystals, and
156001 (2010).
potentiallydrivetheself-assemblyofdifferentstructures.
[20] C.P.Lapointeetal.,Phys.Rev.Lett.105,178301(2010).
Ourmainresultisthat,whensubjectedtoanexternal [21] O. P. Pishnyak et al., Phys. Rev. Lett. 106, 047801
force, the dimer rotates, either smoothly as a corkscrew (2011);O.P.Lavrentovichetal.,Nature467,947(2011).
orintermittently,withphaseslippage. Thistransitionoc-
5
Supplementary information for: Self-assembly and nonlinear dynamics
of dimeric colloidal rotors in cholesterics
COMPUTATIONAL MODEL AND DETAILS
To describe the thermodynamics of the chiral liquid crystal host, we employ a Landau–de Gennes free energy ,
F
whose density f may be expressed in terms of a traceless and symmetric tensor order parameter Q [1] as
f = A0 1 γ Q2 A0γQ Q Q + A0γ(Q2 )2
2 − 3 αβ − 3 αβ βγ γα 4 αβ
+ K2(cid:0) ∇βQ(cid:1)αβ 2+ K2 ǫαγδ∇γQδβ +2q0Qαβ 2. (S1)
(cid:0) (cid:1) (cid:0) (cid:1)
Here, A sets the energy scale, K is an elastic constant, q = 2π/p where p is the cholesteric pitch, and γ is a
0 0
temperature-like control parameter governing proximity to the isotropic-cholesteric transition. Greek indices denote
Cartesian components and summation over repeated indices is implied; ǫ is the permutation tensor. The physics
αγδ
of our cholesteric dimers is determined by several dimensionless parameters. The chirality,
108Kq2
κ= 0, (S2)
s A0γ
and the reduced temperature,
27(1 γ/3)
τ = − , (S3)
γ
determine the equilibrium phase of the chiral nematic fluid [2]. We used the following parameters (in simulation
units): A =1.0, K =W 0.065, ξ =0.7, γ =3.0, p=32, q =1/2 and Γ=0.5. Note that in our units the density
0
isequaltounity,andsois≃therotationalviscosity,γ = 2q2. Ourparametersyieldτ =0,κ=0.3. Weusedaperiodic
1 Γ
cubic simulationbox ofvolume V = 1923 and particles ofradius R=7.25. Note that the exactforms ofEqs.S2 and
S3 follows the originalwork on Landau theory of blue phases by Grebel et al. [3]. The chirality, κ, differs by a factor
of 2, from that used by Wright and Mermin [4] as explained in the appendix A of [4]. For more detailed treatment
see eg. [5].
We considered a degenarate planar anchoring of the director nˆ at the particle surface. Following Fournier and
Galatola [6] , we constructed an orientational order tensor, Q0, planar to the surface normal ν as follows:
Q0 = δ ν ν Q˜ δ ν ν 1qδ , (S4)
αβ αγ − α γ γδ δβ − δ β − 3 αβ
(cid:0) (cid:1) (cid:0) (cid:1)
where
Q˜ =Q + 1qδ . (S5)
αβ αβ 3 αβ
CALCULATION OF THE ELASTIC PAIR POTENTIAL
The calculation of the elastic potential (Fig. 1a in the main text) was carried out by minimising the elastic free
energy of equation 1 by solving the Beris-Edwards equations [1] (Eq. (1) in the main text) in the absence of fluid
flow. This was done for pairs of particles held static at separation d. A first particle is placed with its centre at the
originwiththehelicalaxisinthez-direction. The far-fieldnematicdirector(nˆ(z))atz =0liesinthe x direction. A
−
secondparticleisplacedatcentre-centreseparationvectord fromthe first. By repeatingthe simulationsfordifferent
d an effective elastic two body potential can be mapped out.
Weuseadiscreterepresentationofthecolloidalparticlesurfaceinthelattice,whichisastandardmethodinlattice
Boltzmann models [7, 8]. This has a techical issue that the volume occupied of a particle can vary depending on
whereexactlythe centreofthe particlerislocatedwithrespecttothe lattice. Thiscanhaveasignificanteffectwhen
considering the elastic interaction energy of colloidal pairs embedded in liquid crystal host. To evaluate the error
6
associated with the discretisation, we carried out a series of simulations where we generated the particle coordinates
for the colloid pair as,
r =r +u, (S6)
1 0
and
r =r +d. (S7)
2 1
Here r denotes the originandthe components ofu areuniformly distributed randomnumbers u [0,1],giving the
0 α
∈
position of the centre of the particle relative to the lattice.
We carried out 8 independent scans consisting of 18 different separations along each of the three coordinate axis
x, y, z. Here it can be noted that for a constant u, the total volume occupied by the particles (V) is constant for all
the separations considered along the coordinate axes.
For the angular scan, we used 19 different angles, θ [0,180◦]. Here, the situation is manifestly different: V varies
with angle and is symmetric around θ = 90◦ only for∈u = 0 or 1. This gives a further uncertainty on the angular
α
scan in the x y plane at (d 2R)/p 0.05 shown in inset of Fig. 1a in the main text. From physical arguments
the elastic int−eraction energy−in the ch≈osen configuration, should be symmetric around θ = 90◦. We confirmed this
by simulation with u = 0. However, for u = 0, neither our individual angular scans nor the averaged curve were
α α
exactlysymmetric. So,wefurtheraveragedthe6 samplesaroundθ =90◦,using16independentsamplesforθ [0,90◦[.
∈
Fig. 1a in the main text shows the averaged interparticle elastic potential with descriptive error bars of 161k T,
B
161k T and 156k T for x, y and z,respectively,whereasfor the averagedangularscanallerrorbarsareshown. For
B B
both cases, these are estimated as the standard deviation of the mean.
DYNAMICS OF THE DIMER
To gain some deeper understanding of the dimer dynamics, we replot the time evolution of φ(t) and θ(t) (Fig. 2e
and 2f in the main text) against the scaled distance along the helical axis z/p, shown in Fig. S1. From the φ(z/p)
curve the rotor motion (Fig. S1a open symbols) is clear: after an initial transient, the colloid rotates following the
constant twist of the cholesteric host.
In the phase slippage regime, the colloidal dimer sediments along the helical axis with non-steady velocity: the
motion along z alternates between spells of slower and faster sedimentation. The motion is faster when the dimer
sinkswithoutrotatinginthe x y plane. Thiscanbe observedbycomparingthe timeevolutionoftheangleφ(t)and
−
phase lag θ(t) (solid symbols in Fig. 2e and 2f in the main text, respectively) with φ(z/p) and θ(z/p) (Fig. S1, solid
symbols). In Fig. S1, intervals of constant φ denote sedimentation without rotation, whereas intervals of constant θ
denote rotationlockedto the cholesteric helix. By comparing φ(t) (Fig. 2e) with φ(z/p) (Fig. S1a), one may observe
that intervals of rising φ, corresponding to a roto-translation of the dimer, last longer in time than the constant φ,
but covers a similar distance in space – therefore when rotating as a corkscrew the dimer takes longer to move the
same distance, hence is slower.
During the sedimentation in the phase slippage regime, the dimer goes through a repulsive barrier in the elastic
interactionpotentialatphaselagintervalsofθ(t)=180◦. Thismanifestsitselfinarepulsionbetweenparticlesleading
to a sharp increase of the surface-to-surface separation (r 2R), as shown in Fig. S2 (solid line). For the rotor
ij
−
motion,aftertheinitialtransientperiod,thedistancebetweenthecolloidpairstaysconstant(dashedlineinFig. S2).
THE ROTOR-PHASE SLIPPAGE TRANSITION
Finally,wecommentinmoredetailontheoriesrelatedtoourdynamicaltransitionbetweentherotorandthephase
slip regimes. Firstly, our simplified Eqs. (3) may be seenas a relativeof the Frenkel-Kontorovamodel, andof related
modelsforthedepinningofchargedensitywavesinsuperconductors. TheoriginalFrenkel-Kontorovamodelconsiders
the dynamics of a chain of forced coupled particles in a periodic potential (more realistic models for charge density
wave depinning consider impurities hence an additional random potential). The key result of the simplest model
is that there is a transition between pinned and moving particles. Just above the critical force, f , the asymptotic
c
velocity of the particles, v, scales as [9]:
f2 f2
v − c f f (S8)
c
∼ γ ∝ −
p
p
7
720
640 (a)
s
e 560
e
gr 480
e
d 400
p) 320
z/ 240
(
φ
160
80
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
1800
f =0.04
z
s 1440 f =0.03
e z (b)
e f =0.025
degr 1080 zfz=0.02
) 720
p
z/
( 360
θ
0
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
z/p
FIG. S1: Replot of the data in Fig. 2e and 2f in the main text, against z/p. For the rotor motion (open symbols) only every
5th point is plotted for clarity.
2
f = 0.025 (Rotor)
1.8 z
f = 0.03 (Slip)
z
1.6
R
2 1.4
-
rij
1.2
1
0.8
0 0.4 0.8 1.2 1.6 2
6
Simulation step/10
FIG. S2: Time evolution of the surface-to-surface gap, rij −2R, for the rotor (dashed line) and slip motion (solid line),
respectively.
wheref is the forcing,andγ isaneffective friction. The velocitythereforemaybe seenasanorderparameterfor the
dynamical depinning transition, which is a continuous transition in this framework. While in Eqs. (3) in the main
text there is a coupling between rotatory and translational motion, the analogue order parameter may be taken as
the asymptotic time derivative of the lag angle between the dimer orientationand the local director field, θ(t), which
we call Ω . Via a numerical solution of Eqs. (3) we find that Ω f2 f2 √f f , for f > f , further
lag lag ∼ − c ∝ − c c
validatingthe qualitativesimilarityhintedatabove. Inthisanalogy,ourrotorphasecorrespondstothe pinnedstate,
p
and the phase slip phase corresponds to the depinned one. The critical behaviour appears to be the same, provided
the appropriate order parameters are chosen.
Secondly, our model bears some similarities with the Kuramoto model [10], which is a paradigm to study the
synchronisation of a system of oscillators. In the particularly simple case of two coupled oscillators, the equations
8
50
40
30
g
a
Ωl
20
10
0
0 2 4 6 8 10
f
FIG. S3: Plot of Ωlag as a function of f, for a numerical solution of Eqs. (3) in the main text (γφ = γz = 1, q0 = 2π in
simulation units). The solid line is a fit to pf2−fc2, leading to fc ∼6.44.
describing the dynamics of the Kuramoto model are
dθ
1
= ω +K/2[sin(θ θ )] (S9)
1 2 1
dt −
dθ
2
= ω +K/2[sin(θ θ )], (S10)
2 1 2
dt −
where θ and θ describe the oscillators, ω and ω are the forcing for the two oscillators, and K >0 is the coupling
1 2 1 2
constant favouring synchronisation (θ = θ ). The relevant dynamical variable is ψ = θ θ , which obeys the
1 2 1 2
−
following evolution equation:
dψ
=∆ω K[sin(ψ)]. (S11)
dt −
where ∆ω =ω ω . If we define an (asymptotic) angular velocity of the phase difference between the oscillators as
1 2
Ξ = limt→∞ ddψt,−then Ξ may be estimated as 2π divided by the time needed for ψ to go from ψ0 to ψ0+2π (this is
independent of ψ ), namely
0
2π
Ξ = (S12)
∆t
2π
= (S13)
2π dψ
0 ∆ω−Ksin(ψ)
= R (∆ω2 K2) (S14)
−
where the final integral can be done with the methpod of residues, by changing variables to the complex number
z =eiψ and noting that sin(ψ)= eiψ−e−iψ. As in our simplified model, we find that the relevant “order parameter”,
2i
Ξ, behaves as ∆ω2 ∆ω2, with ∆ω =K, close to the transition.
− c c
Finally, the full lattice Boltzmann simulations of our cholesteric dimers lead to a similar scaling for Ω (see Fig.
p lag
3c in the main text). The critical force estimated through the fit is f 0.0267 0.0005,which is in agreement with
c
≈ ±
f ]0.025,0.0275[observedinsimulations. Thisshowsthatourminimaltheorycapturesthephysicsofthedynamical
c
∈
phase transition between the rotor and slip phases, and also suggests that this is in the same universality class of
oscillator synchronisationand of the depinning of charge density waves in a perfectly periodic medium.
Notethat,althoughthenear-criticalbehaviourofour“orderparameter”isgivenby √f f ,bothoursimulation
c
∼ −
data and the numerics corresponding to Eqs. (3) in the main text are better fitted by a function proportional to
f2 f2, as this better captures the behaviour of Ω further away from f .
− c lag c
p
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