Fractionalization of Interstitials in Curved Colloidal
Crystals
William T.M. Irvine1, Mark J. Bowick2 and Paul M. Chaikin31
James Franck Institute, University of Chicago, 929 E 57th street,
Chicago, IL 60637, USA
2 Physics Department, Syracuse University, Syracuse NY
13244-1130, USA3 Center for Soft Matter Research, New York
University,
4 Washington Place, New York, NY 10003, USA
Understanding the out-of equilibrium behaviour of point defects
in crystals, yields insights intothe nature and fragility of the
ordered state, as well as being of great practical importance.
Insome rare cases defects are spontaneously healed - a
one-dimensional crystal formed by a line ofidentical charged
particles, for example, can accommodate an interstitial (extra
particle) by a re-adjusting all particle positions to even out the
spacing. In sharp contrast, particles organized intoa perfect
hexagonal crystal in the plane (Fig. 1E) cannot accommodate an
interstitial by a simplere-adjustment of the particle spacing - the
interstitial remains instead trapped between lattice sitesand
diffuses by hopping[13–15], leaving the crystal permanently
defected. Here we report on thebehavior of interstitials in
colloidal crystals on curved surfaces (Fig. 1A,B). Using optical
tweezersoperated independently of three dimensional imaging, we
insert a colloidal interstitial in a latticeof similar particles on
flat and curved (positively and negatively) oil-glycerol interfaces
and imagethe ensuing dynamics. We find that, unlike in flat space,
the curved crystals self-heal througha collective rearrangement
that re-distributes the increased density associated with the
interstitial.The self-healing process can be interpreted in terms
of an out of equilibrium interaction of topologicaldefects with
each other and with the underlying curvature. Our observations
suggest the existenceof “particle fractionalization” on curved
surface crystals.
Much in the way that a sticker does not naturally fiton the
surface of a car bumper, crystals must deform tofit on curved
surfaces. Curvature changes the rules of ge-ometry. The interior
angles in a triangle, for example, nolonger add to 180◦ and
initially parallel lines can divergeor converge leading to
compression and stretching. Thisvery general geometry-induced
frustration applies to anyphase that possesses orientational order
in flat space suchas nematics, smectics and crystals.
A recent flurry of activity has investigated how a crys-tal can
undergo structural changes to relieve this frus-tration by
introducing topological defects. In hexago-nal crystals, two types
of topological defect are found:disclinations (Fig. 1F,G), that
correspond to a missing(inserted) 60◦ wedge of lattice and disrupt
orientationalorder, and dislocations (Fig. 1H), that correspond to
twoextra rows of particles that terminate at the core of thedefect
and disrupt translational order. Clearly both arenon-local in
origin and influence[3] and disrupt order inthe crystal. On
surfaces that are curved, however, topo-logical defects can play a
complementary role, relievingboth compressive and shear
stress[4–9]. For example ona sphere, on which the sum of the
interior angles in a tri-angle is increased from 180◦ by the
Gaussian curvature,5-fold coordinated disclinations can make up
this angularexcess (as seen on a soccer ball with its twelve
pentago-nal panels). When the lattice spacing is much less thanthe
radius of curvature of the surface, the isotropic andshear stress
created by curvature is relieved by groupsof defects that organize
into grain-boundary-like struc-tures that freely terminate within
the medium, character-ized by the excess disclination charge;
examples of struc-tures both neutral (‘pleats’[9]) and charged
(‘scars’[6])are shown in Figs 1C and 1D.
Our experimental system consists of positively chargedcolloidal
PMMA particles (∼ 2µm in diameter) coatedin a layer of
(poly)hydroxy stearic acid and suspended inoil (a CHB/dodecane
mixture). In the presence of an oil-glycerol interface, image
charge effects drive the bindingof the particles to the interface,
where, without wetting,they form a monolayer (Fig. 2A). The
repulsive screenedCoulomb interactions cause them to self-organize
into acrystalline lattice that conforms to the curved surface.By
index matching the mixture of CHB and dodecane tothe glycerol we
obtain a clean system that can be imagedfully using a confocal
microscope (Fig. 2C). The systemequilibrates into scarred and
pleated structures[9].
To study the behavior of interstitials we add a parti-cle to
this system using optical tweezers decoupled fromthe imaging (see
methods) and, by simultaneously imag-ing in three dimensions, we
follow the out-of-equilibriumdynamics of the defects on the curved
surface.
Interstitial defects have a local material origin, result-ing
from a single extra particle forced into an otherwiseordered
crystalline array. When a particle that is identi-cal to the other
particles is added to the crystal, however,its identity becomes
ambiguous; the crystal accommo-dates its presence by local
re-adjustments that leave twosignatures of the interstitial’s
presence. The first is a lo-calized spike in the compressional
strain (density) fieldthat corresponds to the additional particle’s
mass; thesecond is a bound triplet or doublet of
dislocations[18–20] as can be seen in Fig. 2B and supplementary
movie 2.In flat space these dislocations remain bound,
affectingneither translational nor orientational order.
Once added, an interstitial can move or diffuse by hop-ping
between lattice inter-sites[13–15] with the disloca-tions remaining
bound, preserving the local character of
arX
iv:1
310.
3000
v1 [
cond
-mat
.sof
t] 1
1 O
ct 2
013
2
A B
C D
E G HF
FIG. 1: Curved crystals and topological defects. (A) A
spher-ical crystal formed by self-assembled colloidal beads on a
liq-uid droplet (Diameter ∼60µm). (B) A negative curvaturecrystal
formed by the same colloidal beads on the surface ofa capillary
bridge (Bridge diameter ∼45µm). (C) A Delau-nay triangulation of a
typical equilibrated configuration forthe crystal (A) with
5-coordinated particles shown in red and7-coordinated particles
shown in yellow. (D) A Delaunay tri-angulation of a typical
equilibrated configuration for the crys-tal (B). (E) A regular
hexagonal lattice configuration. (F) A5-disclination in a planar
crystal. (G) A 7-disclination in aplanar crystal. (H) A dislocation
in a planar crystal.
the interstitial. Note that the original particle need
notdiffuse with the interstitial defect, but rather may remainin
the region it was added, as can be verified by trackingthe added
particle in this case.
The diffusive motion can be biased by stress fields[2].We
observe this in both flat and curved space: If theparticle is added
in close proximity to a grain boundary,the latter can absorb the
interstitial with little change instructure, while eliminating the
stress energy associatedwith the interstitial. The interstitial
hops towards thegrain boundary and is absorbed. This is the case in
flator curved space alike - as can be seen in Fig. 2D,E
andsupplementary movies 3 and 4.
Adding particles to crystals bound to positively andnegatively
curved surfaces, however, we observe an ad-ditional, strikingly
different, behavior: the addition ofa particle is followed by the
fissioning of the normallybound dislocations into pairs that
travel, gliding alongparallel Bragg planes in opposite directions,
leaving thecrystal region in which the particle was added
rotated(Fig. 3A for a spherical crystal and in Fig. 3B,D for a
A B
E
C D
Oil
PMMA
GlassWater
Holographic laser tweezer
Glass
Confocal Imaging
FIG. 2: Interstitials in flat space and interstitial
absorptionby grain boundaries. (A) Inserting a particle to create a
self-interstitial. (B) An interstitial in flat space typically
evolvesinto bound states of three or two dislocations with
vanishingnet Burgers’ vector (lattice spacting ∼ 3µm). (C)
Schematicof confocal imaging combined with laser tweezers. (D)
Aninterstitial in flat space close to a grain boundary is
absorbedby the latter (See supplementary movies). (E) An
interstitialcreated on a spherical crystal and subsequently
absorbed bya grain boundary scar.
negatively curved crystal, see also supplementary movies6 and
7).
We find that this intriguing mechanism, predicted the-oretically
and numerically in the special case of un-equilibrated arrangements
of disclinations on spheres[16,17], occurs generically in
experiments on equilibratedspherical droplets and equilibrated
negatively curvedcapillary bridges.
This non-local behavior raises the question - wheredoes the
particle go? By partially bleaching the parti-cles and subsequently
adding an unbleached, and there-fore brighter particle to the
crystal (Fig. 1B), we verifiedthat in both types of instability,
the specific particle thatwas added remains in the region in which
it was added,in agreement with the behavior in flat space.
Howeverin the case where the dislocations remain bound as
theydiffuse in flat space, or migrate towards a grain bound-ary as
seen before, the density increase associated withthe added particle
remains localized and has a contin-uous trajectory to the grain
boundary. In contrast, forthe case in which the interstitial
fractionalizes, the masstransport cannot be similarly localized -
though the massassociated with an extra particle clearly leaves the
region.
We investigated this transport through a numerical(Fig. 3C) and
experimental (Supplementary movie 3)investigation of the stress
fields associated with a frac-tionalization event. Fig. 3C shows
how the compressivepart of the stress field in the case of the
sphere extends,creating two branches that join up to the scars,
effec-tively delocalizing the increased density associated with
4
perimental run to match the refractive index of Glycerolas
measured by an Abbe refractometer. This avoids lens-ing by the
oil-glycerol interface, while allowing for a smallindex contrast
between the particles and the oil phase foroptical tweezing. The
Glycerol-Oil interfaces were pre-pared by emulsification (in the
case of spherical surfaces)and by deposition of glycerol droplets
in contact with airin capillary channels that were subsequently
filled by theparticle suspension (in the case of capillary bridges)
andsealed to avoid evaporation. The samples were imagedusing a
Yokogawa CSU-10 spinning disk confocal. Op-tical tweezing
independent of the confocal imaging wasachieved by substituting the
microscope condenser with asecond microscope objective. A
holographically shaped1064nm trapping laser was then projected
through theupper objective into the sample. Particle location
wasdetermined from the images using the IDL routines of
Ref.[26] and triangulation and defect identification usingcustom
codes written in Matlab.AcknowledgementsWe acknowledge discussions
with S. Sacanna,A.D. Hollingsworth, A. Grosberg, T. Witten andV.
Vitelli. This work was supported by Rhodia andthe English speaking
union (WTMI), the NationalScience Foundation grant DMR-0808812
(MJB), theMRSEC Program of the National Science Founda-tion under
Award Number DMR-0820341 and NASANNX08AK04G (PMC).
WTMI and MJB acknowledge hospitality fromthe Aspen Center for
Physics. Correspondenceand requests for materials should be
addressedto WTMI (email:[email protected]),
PMC(email:[email protected]) and MJB (email:
[email protected]).
[1] Chaikin, P. & Lubensky, T. Principles of Condensed
Mat-ter Physics, (Cambridge University Press, Cambridge,1995).
[2] Hirth, J.P. & Lothe, J. Theory of Dislocations,
(McGraw-Hill, New York, 1968).
[3] Nelson, D.R. Defects and Geometry in Condensed Mat-ter
Physics, (Cambridge University Press, Cambridge,2002).
[4] Nelson, D.R. & Peliti, L. Fluctuations in membranes
withcrystalline and hexatic order. J. Phys. (Paris) 48, 1085-1092
(1987).
[5] Pèrez-Garrido, A., Dodgson, M. & Moore, M. Influenceof
dislocations in Thomsons problem. Phys. Rev. B 56,3640-3643
(1997).
[6] Bowick, M.J., Nelson, D.R. & Travesset, A.
InteractingTopological Defects on Frozen Topographies. Phys. Rev.B
62, 8738-8751 (2000).
[7] Vitelli, V. Lucks, J.B. & Nelson, D.R. Crystallographyon
Curved Surfaces. Proc. Natl. Acad. Sci. USA 103,12323-12328
(2006).
[8] Bausch, A.R., et al. Grain Boundary Scars and
SphericalCrystallography. Science 299, 1716-1718 (2003).
[9] Irvine, W.T.M., Vitelli, V. & Chaikin, P.M. Pleats
incrystals on curved surfaces. Nature 468, 947-951 (2010).
[10] Lipowsky, P., Bowick, M.J., Meinke, J.H., Nelson, D.R.&
Bausch, A.R. Direct visualization of dislocation dy-namics in
grain-boundary scars. Nature Mater. 4, 407-411 (2005).
[11] Einert, T., Lipowsky, P., Schilling, J., Bowick, M.J.
&Bausch, A.R. Grain Boundary Scars on Spherical Crys-tals.
Langmuir 21, 12076-12079 (2005).
[12] Bowick, M.J., Giomi, L., Shin, H. & Thomas,
C.K.Bubble-raft model for a paraboloidal crystal. Phys. Rev.E 77,
021602-(1-4) (2008).
[13] Pertsinidis, A. & Ling, X.S. Diffusion of point defects
intwo-dimensional colloidal crystals . Nature 413,
147-150(2001).
[14] Pertsinidis, A. & Ling, X.S. Equilibrium
Configurationsand Energetics of Point Defects in Two-Dimensional
Col-
loidal Crystals. Phys. Rev. Lett. 87, 098303-(1-4) (2001).[15]
Pertsinidis, A. & Ling, X.S. Video microscopy and mi-
cromechanics studies of one- and two-dimensional col-loidal
crystals. New J. Phys. 7, 33 (1-27) (2005).
[16] Bowick, M.J., Shin, H. & Travesset, A. Dynamics
andinstabilities of defects in two-dimensional crystals oncurved
backgrounds. Phys. Rev. E 75, 021404-(1-8)(2007).
[17] Bowick, M.J., Nelson, D.R. & Shin, H. Inter-stitial
fractionalization and spherical
crystallography.Phys.Chem.Chem.Phys. 9, 6304-6312 (2007).
[18] Fisher, D.S., Halperin, B. & Morf, R. Defects in the
two-dimensional electron solid and implications for melting.Phys.
Rev. B 20, 4692-4712 (1979).
[19] Cockayne, E. & Elser, V. Energetics of point defects
inthe two-dimensional Wigner crystal. Phys. Rev. B 43,623-629
(1991).
[20] Jain, S. & Nelson, D.R. Statistical mechanics of
vacancyand interstitial strings in hexagonal columnar
crystals.Phys. Rev. E 61, 1599-1615 (2000).
[21] Bowick, M.J. & Giomi, L. Two-Dimensional Matter:Order,
Curvature and Defects. Adv. Phys. 58, 449-563(2009).
[22] DeVries, G.A. et al. Divalent Metal Nanoparticles. Sci-ence
315, 358-361 (2007).
[23] Bosma, G. et al. Preparation of Monodisperse, Fluo-rescent
PMMALatex Colloids by Dispersion Polymeriza-tion. J. Colloid
Interface Sci. 245, 292-300 (2002).
[24] Antl, L. et al. The Preparation of Poly(Methyl
Methacry-late) Latices in Non-Aqueous Media. Colloids Surf.
17,67-78 (1986).
[25] Leunissen, M.E., van Blaaderen, A., Hollingsworth,
A.D.,Sullivan, M.T. & Chaikin, P. M. Electrostatics at the
oil-water interface, stability, and order in emulsions and
col-loids. Proc. Natl. Acad. Sci. USA 104, 2585-2590 (2007).
[26] Crocker, J. & Grier, D.G. Methods of Digital Video
Mi-croscopy for Colloidal Studies. J. Colloid Interface Sci.179,
298-310 (1996).
References