RESEARCH ARTICLE

OPEN ACCESSPHYSICS

Scaling and spontaneous symmetry restoring of topologicaldefect dynamics in liquid crystal
Yohei Zushia

and Kazumasa A. Takeuchia,1Downloaded from https://www.pnas.org by University of Tokyo Library on October 3, 2022 from IP address 133.11.10.150.

Edited by Noel Clark, University of Colorado, Boulder, CO; received April 28, 2022; accepted August 24, 2022

Topological defects—locations of local mismatch of order—are a universal concept
playing important roles in diverse systems studied in physics and beyond, including theuniverse, various condensed matter systems, and recently, even life phenomena. Amongthese, liquid crystal has been a platform for studying topological defects via visualization,
yet it has been a challenge to resolve three-dimensional structures of dynamically
evolving singular topological defects. Here, we report a direct confocal observation ofnematic liquid crystalline defect lines, called disclinations, relaxing from an electricallydriven turbulent state. We focus in particular on reconnections, characteristic of suchline defects. We find a scaling law for in-plane reconnection events, by which the
distance between reconnecting disclinations decreases by the square root of time to thereconnection. Moreover, we show that apparently asymmetric dynamics of reconnecting
disclinations is actually symmetric in a comoving frame, in marked contrast to the twodimensional counterpart whose asymmetry is established. We argue, with experimental
supports, that this is because of energetically favorable symmetric twist configurations
that disclinations take spontaneously, thanks to the topology that allows for rotationof the winding axis. Our work illustrates a general mechanism of such spontaneous
symmetry restoring that may apply beyond liquid crystal, which can take place if
topologically distinct asymmetric defects in lower dimensions become homeomorphic
in higher dimensions and if the symmetric intermediate is energetically favorable.

SignificanceWhen order is formed, topologicaldefects may appear generically,
and as such, they are a key tocharacterize and control the orderformation observed in diﬀerent
ﬁelds of physics and related
disciplines. At the heart of suchapproaches is disentangling
universal and nonuniversal
aspects of topological defectdynamics. We do so by realizing
direct observation of
three-dimensional (3D) dynamics
of liquid crystalline defects.
Analyzing topologicalrearrangements called
reconnections, on one hand, weestablish the validity of the scaling
law known for quantum ﬂuid andtwo-dimensional (2D) liquid
crystal; on the other hand, we
reveal that the asymmetry
present in 2D defect dynamics
disappears in 3D. This ﬁnding isaccounted for by a generalmechanism that may hold beyond
liquid crystal.

liquid crystal | topological defect line | reconnection | scaling law

Topologically nontrivial configurations of order, called topological defects, may appear
generically and spontaneously when order is formed. As such, topological defects havebeen studied in diverse disciplines (1, 2), including cosmology (3), crystals and liquid
crystals (2), superconductivity and superfluid (4–9), and biology (10–19) to name but
a few. While there exist various kinds of defects characterized by different symmetries
and properties, defects may also enjoy common properties across different disciplines.
In this context, liquid crystal has the advantage that it is amenable to direct opticalobservations; various compounds and techniques exist; and as a soft matter system, itshows large response to external fields, being suitable for studying nonequilibrium andnonlinear effects (2, 20). This advantage has been recognized and used for decades, witha notable example of observing liquid crystal defects to test predictions for cosmic strings(21). Moreover, the scope of studies of liquid crystalline defects has been recently extended
remarkably, including the use of defects as templates for molecular self-assembly (22)and the recent surge of investigations of active nematic systems bearing relevance to life
phenomena (10–19).
Despite this history, resolving fully three-dimensional (3D) structures of liquid crystal
defects has not been straightforward, even for the simplest kind of defects, namely
nematic disclination lines. Well-known techniques for 3D observation of defects andother orientational structures are fluorescence confocal polarizing microscopy (23, 24)
and two- or three-photon excitation fluorescence polarizing microscopy (25–27). Bothtechniques allow one to reconstruct the 3D structure of the director field, by which one
can determine the position and structure of defects in principle. To do so, however, one
needs to reduce the effect of defocusing and polarization changes due to the birefringenceof liquid crystal. For singular defects, such as nematic disclinations, scattering at the core
gives another difficulty. The effect of birefringence can be significantly reduced by partial
polymerization of the medium (28), but this cannot be used to study dynamics of defects.
Here, we propose a method to capture dynamically evolving 3D structures of nematic
disclination lines by using confocal microscopy and a recently reported accumulation offluorescent dyes around the singular core of defects (29). This method allows us to visualize
the disclinations directly (Fig. 1), without reconstructing and analyzing the director field.
Using this technique, we observe reconnections of disclinations—a hallmark of such
topological defect lines—and characterize the reconnection dynamics in terms of scaling
and symmetry.
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Author aﬃliations: a Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan

Author contributions: K.A.T. designed research; Y.Z. performed research; Y.Z. and K.A.T. analyzed data; and Y.Z.
and K.A.T. wrote the paper.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
Copyright © 2022 the Author(s). Published by PNAS.
This open access article is distributed under Creative
Commons Attribution-NonCommercial-NoDerivatives
License 4.0 (CC BY-NC-ND).
1

To whom correspondence may be addressed. Email:kat@kaztake.org.
This article contains supporting information online athttps://www.pnas.org/lookup/suppl/doi:10.1073/pnas.
2207349119/-/DCSupplemental.
Published October 3, 2022.

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A−0.5 s

D

0.0 s

B

0.5 s

C
E

F

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0.0 s

Fig. 1. Reconnections and loop shrinkage. (A–C) Sketches of an in-plane
reconnection (A), an intersecting reconnection (B), and a loop shrinkage (C).
(D–F) Confocal observations of an in-plane reconnection (D), an intersecting
reconnection (E), and a loop shrinkage (F) (Movies S1–S4). E, Insets display sideviews of the event shown in E. (Scale bars: D–F, 50 μm; E, Insets, 20 μm.)

Observations of Disclination Dynamics
To study disclination dynamics, we add fluorescent dye to liquid
crystal and observe the fluorescence from the dye localized at the
3D disclinations by confocal microscopy. Using the previously
reported apparent length scale of dye accumulation, ≈0.33 μm(29), and the typical value of the diffusion coefficient of dye
molecules, ≈10−10 m2 /s (24, 30), we evaluate that dye can follow
the evolution of disclinations at the time resolution of roughly
1 ms. This is to compare with the timescale of the disclination
dynamics, which can be evaluated at γ1 2 /K with Frank constant K , rotational viscosity γ1 , and characteristic length scale
 of disclination lines (such as the radius of curvature) (20).
For typical mesogens (including the one used in this work),
we have K ≈ 101 pN and γ1 ≈ 102 mPa · s (20), so that, forexample, disclinations of length scale  ≈ 101 μm evolve overa timescale of roughly 1 s. Therefore, the disclination dynamics
can be faithfully captured by confocal images acquired at a timeinterval between 1 ms and 1 s (or longer for disclinations of larger
length scales). To fulfill this condition, we chose a laser-scanningconfocal microscopy equipped with a resonant scanner working at
8 kHz and a piezo objective scanner (M aterials and M ethods has
details).
A liquid crystal sample, MLC-2037 doped with fluorescentdye Coumarin 545T and electrolyte tetra-n-butylammonium
bromide, was filled in a cell that consists of parallel glass
plates with transparent electrodes (indium tin oxide) separated
by 130 μm-thick spacers (M aterials and M ethods). The
planer alignment condition was imposed on the cell surfaces.
We generated a large density of disclinations by using anelectrohydrodynamic turbulence (20), induced by an electricfield applied to the liquid crystal sample. The electric field wasthen removed, and disclinations started to undergo relaxation.
We indeed observed a large density of singular disclinations
upon removal of the electric field. The disclinations then
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exhibited coarsening dynamics, including reconnections andloop shrinkage (Fig. 1 and Movies S1–S4), similar to thoseobserved previously by bright-field microscopy (21, 31–34). We
also observed nonsingular disclinations terminating at singularones (SI Appendix, Fig. S1), as well as other kinds of defectstructure, as reported in past bright-field studies (31, 32).
Most disclination lines were found near the midplane between
the top and bottom surfaces and extended mostly horizontally
because of the homogeneous boundary condition we imposed. As
a result, most of the observed reconnections were classified into
the following two kinds: in-plane reconnections (Fig. 1 A and D
and Movie S2) and intersecting reconnections (Fig. 1 B and E andMovie S3). An in-plane reconnection consists of a pair of curved
disclinations in a nearly single horizontal plane, which approach inthat plane and reconnect (Fig. 1D and Movie S2). An intersecting
reconnection consists of a pair of disclinations crossing at differentz positions, which approach vertically and reconnect (Fig. 1E
and Movie S3). In this case, the upper disclination appeared dark
above the intersection (Fig. 1 E, Insets) and apparently bent when
the pair is close enough, presumably because of the lensing effectdue to the lower disclination. Since this prevented quantitative
analysis, in the following we focus on the in-plane reconnections
and study their reconnection dynamics. We analyzed a total of
40 in-plane reconnections without any noticeable nonsingulardisclinations in the field of view.
Scaling Law for In-Plane Reconnections
Using the confocal images of the in-plane reconnections, we
extracted the 3D positions of the two disclinations until the
moment of the reconnection (M aterials and M ethods). Measuring
how the minimum distance between the two disclinations, δ(t),decreases with time t (Fig. 2), we found the following scaling law
for all in-plane reconnections:δ(t)  C |t − t0 |1/2 ,[1]with a coefficient C . This power law is identical to that forannihilation of point disclinations in two-dimensional (2D) nematics (35, 36) as well as that for reconnections of quantum
vortices in quantum fluids (6, 7, 36). It is interesting to note thatthe interaction of disclinations in 3D nematics was theoretically~ −/Fig. 2. The scaling law for in-plane reconnections. Results for all 40 reconnection events are shown with diﬀerent colors. The error bars indicate
the uncertainty evaluated from the slicing and the Gaussian ﬁtting used todetermine the coordinates of the disclinations (Materials and Methods).

pnas.orgevaluated only very recently (37, 38) and that the power law in
Eq. 1 was derived in the case of straight disclinations. Although theexperimentally observed disclinations were not straight but curved
inward (Fig. 1D), we may argue that the time evolution of δ(t)is dominated by the interaction between the two closest points,
so that the disclination curvature did not affect the observed
power law significantly. The scaling law (Eq. 1) was also observed
numerically for curved disclinations in ref. 38. Of course, it isimportant to extend those theoretical approaches to the case of
curved disclinations and confirm the robustness of the power law
in Eq. 1.

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Apparent Asymmetry in the Laboratory FrameIn the case of 2D nematics, disclinations are point like and characterized by a topological invariant called the winding number.
Energetically stable are the disclinations of winding number ±1/2
(see the sketches in Fig. 4A), and disclinations of opposite signs
attract each other, approach, and annihilate. Here, it is well known
that such a pair of +1/2 and −1/2 disclinations approaches inan asymmetric manner, due to the different backflow generated
by the two defects (35, 39). It would be then natural to expect
analogous asymmetry to arise for line disclinations in 3D nematics. However, this is not so trivial from the viewpoint of topology
because +1/2 and −1/2 disclinations are topologically equivalent(homeomorphic) in 3D nematics (1, 2, 20). Besides, unlike point
disclinations, line disclinations have shapes and are deformable,
giving additional potential sources of asymmetry.
Here, we inspected this asymmetry experimentally. Insteadof the distance δ(t) between reconnecting disclinations, wemeasured the distance between each disclination line and the

A

C

B

D

E

Fig. 3. Apparent asymmetry of in-plane reconnections. (A) Deﬁnition of the
distances D1 (t) and D2 (t) from the reconnection point 
X0 . (B) Sketch of the
midpoint M(t)of the points on the disclinations closest to the reconnection
point X0 . (C) Distance Di (t) measured in the laboratory frame for an example
pair of reconnecting disclinations. (D) Histograms of the asymmetry parameter A (the square root of the ratio of the two slopes in C; see text) measured
in the laboratory frame (blue) and the comoving frame (red). Note that threeoutliers are not displayed in the blue histogram, taking A ≈ 3.2, A ≈ 5.5, andA ≈ 40 in the laboratory frame, but in the comoving frame, all data, including
those outliers, fell in the ﬁrst bin (max A = 1.03 ± 0.01). (E) Distance D̃i (t)measured in the comoving frame for the pair shown in C. E, Inset shows thedistance between the midpoint M(t)and the reconnection point 
X0 seen in thelaboratory frame. The error bars in C and E indicate the uncertainty evaluatedfrom the Gaussian ﬁtting and the search for the reconnection point used here
(Materials and Methods).

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reconnection point, D1 (t) and D2 (t) (Fig. 3A). Plotting Di (t)2
against t − t0 , we found a power law Di (t)  Ci |t − t0 |1/2 analogous to Eq. 1, with coefficients Ci that are typically asymmetricbetween the two disclinations (Fig. 3C has an example). The
asymmetry was also clear from the defect motion (SI Appendix,
Fig. S2A). Using the coefficients Ci , we define the asymmetry
parameter A byA≡

max{C1 , C2 }min{C1 , C2 }[2]and determined it for each reconnection event. By definition,
A = 1 for symmetric reconnections, and A > 1 for asymmetricones. The histogram of A (blue bars in Fig. 3D) shows that mostin-plane reconnections appear to be significantly asymmetric. We
suspected that the different curvature of the two disclination lines
may contribute to this asymmetry, but this effect turned out to beminuscule (SI Appendix).
Disappearance of Asymmetry in the Comoving
Frame
Let us now recall the fact that disclinations have extended line
structures and also, that the studied pairs were not the only defects
present in the system. It is, therefore, reasonable to consider thatthe reconnection dynamics may be affected by such extrinsic factors, which may induce flow and director changes superimposed tothe intrinsic reconnection dynamics. These effects are expected to
add a drift to the intrinsic motion of reconnecting disclinations.
To evaluate this drift, we located the point on each disclinationthat was closest to the reconnection point and inspected the (t) of the pair of the closest points
motion of the midpoint M(Fig. 3B). If the dynamics of the two disclinations is perfectly
symmetric, the motion of this midpoint is the drift itself andwill not show any singularity near the reconnection time. If the
dynamics is not symmetric, this midpoint will partly include thereconnection dynamics, showing the same singularity as Di (t) ∝|t − t0 |1/2 . This was indeed confirmed for the case of pair annihilation of 2D point disclinations reported by Tóth et al. (39)
(SI Appendix, Fig. S3). For 3D disclination lines, the behavior of (t) is shown in Fig. 3 E, Inset for the pair displayed in Fig. 3C.
M
This clearly shows linear dependence on time, suggesting that the
intrinsic dynamics of reconnection is actually symmetric. More evaluated by fitting dMover, using the drift velocity V
dt , we define
the comoving frame and measure the distance D̃i (t) between theclosest point and the reconnection point in this comoving frame.
The result shows that, remarkably, the reconnection dynamics
in this comoving frame is nearly perfectly symmetric (Fig. 3E
and SI Appendix, Fig. S2B). We carried out this analysis for allreconnection events, and for all cases, the asymmetry parameter
became very close to one (red bar in Fig. 3D), the largest deviation being A = 1.03 ± 0.01. Note that although the asymmetry
parameter A is expected to be independent of the choice ofreference frame in the limit t → t0 , it does depend in practicebecause the limit t → t0 is unreachable due to the finite time
resolution of the observation. Direct comparison of Di (t) andD̃i (t) (SI Appendix, Fig. S4; also compare Fig. 3 C and E) showsthat the scaling law Di (t), D̃i (t) ∝ |t − t0 |1/2 appears longer
in the comoving frame than in the laboratory frame, indicatingthat the results in the comoving frame are more reliable. This
will be supported in the next section on the basis of the director
configuration around the disclination pair.
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Spontaneous Symmetry Restoring
We have found that the asymmetry present in the 2D pair annihilation of ±1/2 point disclinations disappears for the in-plane
reconnections of 3D disclination lines. Obviously, if two disclination lines were straight and had ±1/2 director configurations
around (Fig. 4A), this pair would exhibit the same asymmetry as
its 2D counterpart. However, since +1/2 and −1/2 disclinations
are homeomorphic in 3D (1, 2, 20), the director can actuallytake an intermediate configuration that continuously transforms
between these two limiting structures (Fig. 4A). More precisely,
the winding of the director around a disclination line is notcharacterized by the winding number but by a unit vector that
 (e.g.,specifies the rotation axis of the director, denoted by Ω
refs. 37 and 38). With the unit tangent vector t whose head
and tail are set arbitrarily, the director rotates right handed by along a closed path that180◦ in the plane perpendicular to Ω, = t [or theturns right handed about the tangent vector t. If Ω
−1  
angle β ≡ cos (Ω · t) = 0], the director is essentially in theplane perpendicular to the disclination line, and the defect isequivalent to a +1/2 point disclination in that plane. Similarly, = −t (β = π), it is equivalent to a −1/2 point disclination.
if Ω
These two limiting structures, called the wedge disclinations, areinterpolated continuously by intermediate β. In particular, if ⊥ t (β = π/2), the director purely twists around the defect;Ω
hence, it is called a twist disclination.
Now, for a pair of reconnecting disclinations in plane, we have
two tangent vectors t1 and t2 , which are parallel near the reconnection point, so that we choose t1 = t2 . Then, it is reasonable to 1 = −Ω
 2 (β2 = π − β1 ) so that the disclinations may
assume Ω
attract each other most effectively, as inferred from the disclination
dynamics derived in the theoretical studies (37, 38). Indeed, 2 is expected in order to reduce the elastic energy cost
 1 = −Ω
Ω
due to the existence of the disclination pair. This leaves one free 1 ). If β1 = 0 or π, we have a pair of ±1/2
parameter, β1 (or Ωwedge disclinations (Fig. 4 B, Upper), which is equivalent to a pairof annihilating point disclinations in 2D nematics and therefore,
approaches asymmetrically (35, 39). By contrast, if β = π/2,
we have a pair of twist disclinations (Fig. 4 B, Lower) with πrotationally symmetric director field; in this case, the dynamics
of the two disclinations should also be symmetric.
Our experimental results of the vanishing asymmetry suggestthat all disclination pairs we observed spontaneously took thesymmetric twist configurations. This can be attributed to theanisotropic elasticity of liquid crystal; bulk deformation of thedirector can be decomposed into splay, twist, and bend deformations, characterized by different elastic constants denoted by
K1 , K2 , and K3 , respectively (20). For the mesogen used here,
MLC-2037, these are K1 = 11.6 pN, K2 = 6.1 ± 0.5 pN, K3 =13.2 pN (M aterials and M ethods and SI Appendix, Table S1).
Similarly to other typical mesogens, the elastic constant for twist
deformations is lower than that for splay and bend deformations.
Then, it follows that the twist configuration of the disclination
pair (Fig. 4 B, Lower) is energetically favored over the wedge
configuration (Fig. 4 B, Upper), which involves splay and bend
deformations of the director field. This explains why the twist configuration seemed to be exclusively observed in our experiments,
accounting for the disappearance of the asymmetry.
Moreover, we confirm the realization of the twist configuration
via the coefficient C of the power law in Eq. 1 as follows.
Balancing the drag force J γ1 (δ̇/2) according to Geurst et al. (40)with a dimensionless coefficient J ≈ 1.9, the rotational viscosity
γ1 , and the attractive force πK /2δ exerted to the pair with
 1 = −Ω
 2 under the one-constant approximation K1 = K2 =
Ω
K3 ≡ K (37), we obtainA+1/2
1/2 wedgedBttwisti t−1/2
1/2 wedgedC
Energytopologicallydifferent in 2D
homeomorphic
in 3D

Asymmetry
 (or β). (B) Sketches of the director ﬁeld around wedge
Fig. 4. Director conﬁgurations and asymmetry. (A) Director ﬁeld around a disclination for diﬀerent Ω(Upper) and twist (Lower) disclination pairs. The xy (Left) and yz (Right) cross-sections are shown. The “T” symbols indicate that the directors are not in the planeof the cross-section, with the T head above the paper. Note that the analyzed disclinations were not limited to those aligned in the x direction as sketched here.
(C) Spontaneous symmetry restoring.

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C2 =

2πK
.
J γ1

[3]

However, since the actual elastic constant is anisotropic, Eq. 3
is expected to hold with K ≈ K1 , K3 for the wedge configuration
and K ≈ K2 for the twist one. From our data (Fig. 2) and
γ1 = 132 mPa · s for our mesogen (SI Appendix, Table S1), we
obtain C 2 = 151 ± 27 μm2 /s (the range of error being the SD),
which is close to the value for the twist configuration, C 2 ≈1.5 × 102 μm2 /s, instead of that for the wedge one, C 2 ≈ 3.1 ×
102 μm2 /s. This supports the realization of the twist configuration in the disclination pairs we observed as well as the resulting
vanishing asymmetry we found in the comoving frame.
We also demonstrate the realization of the twist configuration
in a more direct but destructive manner through the pattern ofthe electroconvection induced in the sample. It is known thatnematic liquid crystal with negative dielectric anisotropy andpositive conductivity anisotropy, such as the one used in thepresent work, shows roll convection under a moderate applied
voltage (20). The direction of the rolls is determined by the
director near the midplane; it is normal to the director if thedirector is parallel to the cell surfaces, while patches of rolls ofdifferent directions are formed if the director is perpendicular
to the surfaces. Our observation reveals that the region between
disclinations shows convection rolls normal to those in the outerregion (SI Appendix, Fig. S5), indicating the twist director configuration as sketched in Fig. 4 B, Lower (SI Appendix has details).
While the twist configuration is expected from the energy
viewpoint, it is important to note that such a lowest-energy
configuration is to describe the equilibrium state, while our observations deal with relaxation to it. Upon quenching from theturbulent state, we expect that there exist various types of disclinations, from wedge to twist and in between. However, since allthese configurations are homeomorphic, disclinations are allowed
to change the configurations continuously toward the lowestenergy state (i.e., the twist configuration). This is not possiblefor 2D nematics, for which +1/2 and −1/2 disclinations are
topologically distinct. Mathematically, this is a consequence ofthe different homotopy groups between 2D and 3D nematics(1, 20). For 2D, it is π1 (RP 1 ) = Z, which distinguishes all
different winding numbers. In contrast, its 3D counterpart is
π1 (RP 2 ) = Z2 , which now distinguishes only the absence andthe presence of a nontrivial defect configuration. In particular,
wedge disclinations of winding number ±1/2 are now identified
through continuous transformation, with the symmetric twist
state found in the middle at the lowest energy (Fig. 4C). This
results in the realization of the symmetric reconnection dynamics,
as we observed experimentally.
At this point, it is not difficult to generalize the argument. If
the space of the order parameter field accommodating topologicaldefects of interest is related to real space, such as the nematic case(RP d−1 for d -dimensional space), the corresponding homotopy
group may also depend on the dimensionality. In the case where
there exist two asymmetric structures that are topologically distinct in a lower dimension but become homeomorphic in a higher
dimension, such as the case of ±1/2 nematic disclinations, thedefect in the higher dimension is allowed to take an intermediate structure that continuously interpolates the two asymmetricanalogs of those in the lower dimension. Then, it is likely thata symmetric intermediate structure exists. If it is energeticallyfavored, the asymmetry present in the lower dimension will tend
to disappear in the higher dimension spontaneously. In brief, if
two topologically distinguished and asymmetric configurations ina lower dimension become homeomorphic in a higher dimension
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and if the newly allowed symmetric configuration is energeticallyfavorable, symmetry is spontaneously restored. The symmetry inthe structure results in the symmetry in the dynamics. Our results
on reconnecting nematic disclinations constitute a clear example
of such spontaneous restoring of symmetry.
Concluding Remarks
We carried out a direct confocal observation of disclination dynamics in 3D nematics using the accumulation of fluorescent dyes
to disclinations. Our method successfully resolved characteristic
dynamics of disclination lines, such as reconnections and loop
shrinkage. Studying in-plane reconnection events in depth, wedemonstrated the distance-time scaling law (Eq. 1) predicted
for straight disclination pairs, despite the curved shape of theobserved disclinations. Moreover, we revealed that the dynamics
of reconnecting disclinations is only deceptively asymmetric in
the 3D case, being actually symmetric in the comoving frame.
This is explained by the spontaneous realization of symmetric
twist configurations, which are energetically favored because of
the lower twist elasticity. These observations led us to propose
a mechanism of such spontaneous symmetry restoring from ageneral viewpoint of topology and energy. In this regard, it is important to investigate the generality and limitation of this concept
in future studies. The first step would be to study intersecting
reconnections. Although we restricted the analysis to in-plane
reconnections in the present work, our argument suggests thatthe spontaneous symmetry restoring also holds for intersecting
reconnections. Simulations in ref. 38 showed that at least the samescaling law holds for intersecting reconnections. Furthermore, it isof prominent importance to test the fate of the symmetry restoringin the case where the condition K2 < K1 , K3 is not satisfied
and consequently, the twist configuration does not correspond to
the lowest-energy state. Such a situation may be realized by using
the divergence of K2 near the nematic–smectic transition (20),by the drop of K3 near the transition to the twist–bend nematic
phase (41), or by using nematic discotic liquid crystals (42).
Since the concept of topological defects is universal, it is important to think of similarities and dissimilarities in defect properties
across different disciplines of physics. For example, quantum
vortices in superfluid 4 He are known to have similar interactionenergy and show the same scaling law of δ(t) (Eq. 1) as observed
experimentally (6, 7, 36), while the corresponding homotogy
 if defined analogously,group is different, and the rotation axis Ω,
is fixed to Ω = t or −t. Further, it is tempting to seek examplesof the spontaneous symmetry restoring we proposed in this work.
Disgyration of superfluid 3 He is particularly interesting in thiscontext, for which the homotopy group is π1 (SO(2)) = Z for2D and π1 (SO(3)) = Z2 for 3D, and its asymmetric structures
as well as energy have been thoroughly discussed (5). We hopethat such approaches to general mechanisms will accelerate themultidisciplinary understanding of topological defects and thatthe visualization of nematic disclination dynamics reported here
will be a useful tool in this line.
Materials and Methods
Sample Preparation and Defect Generation. The experimental sample wasprepared as follows for the main results on the in-plane reconnections, whilechanges for other experiments are described in the end of this subsection. The
liquid crystal sample was nematic compound MLC-2037 (Merck; discontinued
product) doped with 0.5 wt % tetra-n-butylammonium bromide and 0.005 wt %ﬂuorescent dye Coumarin 545T. The mesogen MLC-2037 was chosen for its
low birefringence Δn = 0.0649 and negative dielectric anisotropy Δ < 0

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(SI Appendix, Table S1), the latter of which was used to induce electroconvection
to generate disclinations (20). The sample was introduced to a handmade cell,which consists of a coverslip and a glass plate both coated with indium tinoxide and 130 μm-thick polyimide tapes used as spacers. The inner surfaces
were coated with polyvinyl alcohol and rubbed to realize a homogeneous planaralignment.
To study disclination dynamics, we generated a large density of disclinations
by applying an alternating electric ﬁeld (rms amplitude 150 V, frequency 50 Hz)
to the sample, inducing an electrohydrodynamic turbulence called the dynamic
scattering mode 2 (20, 43). Then, we removed the electric ﬁeld and observed
relaxation of disclinations by a confocal laser scanning microscope (Leica SP8;
objective 20×, numerical aperture 0.75, oil immersion) equipped with a resonant scanner working at 8 kHz and a piezo objective scanner. The ﬂuorescent dyes
were excited at 488 nm by laser light polarized in the direction perpendicular to
the nematic easy axis, represented by x and y axes, respectively. The ﬂuorescencesignal in the range between 500 and 600 nm was confocally detected by aphotomultiplier tube detector (pinhole size 20 μm, roughly 0.35 times the Airyunit). The voxel size in the xy plane was 0.91 μm, and the spacing between z slices
was 1 μm. The numbers of voxels were 512, 128, and 21 in the x, y, z directions,
respectively. The time interval between consecutive confocal images was 0.255 s.
Compared with this, the time needed for ﬂuorescent dyes to follow the evolutionof disclination lines is expected to be much shorter, which we evaluate to beroughly 1 ms, using a length scale 0.33 μm reported as the apparent size of dye
accumulation in ref. 29 and the typical value of the diffusion coefﬁcient of dye
molecules, 10−10 m2 /s (24, 30).
Below, we describe the experimental conditions used for other observations.
Conditions and parameters that are not speciﬁed below were kept unchanged
from those for the in-plane reconnections. The intersecting reconnection displayed in Fig. 1E was observed in a sample that contained 0.25 wt % tetra-nbutylammonium bromide and 0.2 wt % Coumarin 545T. After an alternating
voltage of rms amplitude 150 V and frequency 100 Hz was removed, we observed
the intersecting reconnection in a manner similar to the case of the in-plane
reconnections, except that the number of voxels was 512 × 64 × 36 and thetime interval was 0.300 s. The loop shrinkage displayed in Fig. 1F was observedin a sample that contained 0.05 wt % tetra-n-butylammonium bromide and0.005 wt % Coumarin 545T.
Image Analysis. The data acquired by the confocal microscope were the ﬂuorescence intensity detected at the 3D position (x, y, z) and time t. Using the 3D
image at each time, we obtained cross-sections and extracted the coordinates ofthe disclinations as follows. First, we chose the cross-sections to use, either in thexz plane or in the yz plane, chosen so that the cross-sections become closer toperpendicular to the disclination lines. In each cross-section, the two disclinations
appear as bright spots. These bright spots were ﬁtted by a Gaussian function to
obtain the coordinates of the spot centers. Repeating this over all cross-sections,
we obtained a set of 3D coordinates along each disclination line. The closest
distance δ(t) between two disclinations (Fig. 2) was directly determined fromthese coordinates.
The time and the position of each reconnection event, as well as the distanceDi (t) (i = 1, 2) of a disclination from the reconnection point, were determined
as follows. For the reconnection time t0 , we determined it from the 2D image
constructed from the transmitted excitation laser to beneﬁt from the ﬁner time
resolution than that of the confocal images. For the position X0 = (X0 , Y0 , Z0 )of the reconnection point, we ﬁrst approximately located it from the series oftransmitted and confocal images (X0 and Y0 from the transmitted images, Z0 fromthe confocal images). Using this and the coordinates of the disclinations, we could
evaluate the distance Di (t) in the laboratory frame, but for the analysis presented
in the paper, we evaluated Di (t) more precisely in the following manner. First,
we ﬁtted the 3D coordinates of disclinations by smoothing splines to reduce thenoise and to interpolate the lines appropriately. In general, smoothing splines
s(x) for a dataset (xi , yi ) is such a function that minimizeswi (yi − s(xi ))2 + (1 − p)

  2 2
ds
dx,
dx2

without high wave number components, for the two coordinates that spanned
the cross-sections [i.e., for xz cross-sections, the obtained smoothing splines
were x(y) and z(y)]. Then, we also reﬁned the estimate of the reconnection
point X0 by using the coordinates of the disclinations before the moment of
the reconnection. Speciﬁcally, we determined X0 in such a way that the scaling
Di (t)  Ci |t − t0 |1/2 is satisﬁed most precisely in a time period before thereconnection under the constraint that X0 is not changed by more than 3 μm fromthe ﬁrst rough estimate. This was done by evaluating Di (t) for each candidate
position X0 in the range within 3 μm, ﬁtting it to Di (t)2 = ai |t − t0 | + bi , andchoosing the candidate X0 that minimizes b21 + b22 . The distance D̃i (t) in the
comoving frame was also determined analogously by using the position X0 thatdrifts with the velocity of the comoving frame.
The errors in the estimates of δ(t), Di (t), D̃i (t) were evaluated as follows. For
δ(t), the error bars in Fig. 2 indicate the square root of the sum of the squares of
the uncertainties in all coordinates of the two closest points. For the coordinates in
the cross-section, we used the 95% CI of the Gaussian ﬁtting as the uncertainty;
for the other coordinate, we used half the thickness of the cross-section (i.e., thevoxel size) as the uncertainty. For Di (t) and D̃i (t) (Fig. 3 C and E), the errors were
evaluated from the uncertainties in the coordinates of the reconnection point andthe closest point on the disclination line, again by the square root of the sum of
the squares. The uncertainties in the coordinates of the reconnection point wereconsidered to be half the size of the scanned region described above. For theuncertainties in the coordinates of the closest point on the disclination line, since
the closest point was located on the smoothing spline, we only considered theuncertainties (95% CI) for the coordinates in the cross-section that is closest to
the closest point on the spline.
Estimation of K2 . The twist elastic constant K2 of MLC-2037 was evaluatedby using the Fréedericksz transition under an external magnetic ﬁeld (20).
The Fréedericksz transition point Hic corresponding to the elastic constant Ki is
given by


Hic =

π
d

Ki
,
Δχ

[5]

where d is the cell thickness and Δχ is the magnetic anisotropy. For MLC2037, Δχ was unknown, but K1 is known (SI Appendix, Table S1). Therefore, wemeasured the Fréedericksz transition for both the splay and twist conﬁgurations,obtaining H1c and H2c , and used the ratio

H2c
K2
=
[6]
H1c
K1
to determine K2 from K1 .
We used a ready-made cell with homogeneous planar alignment (EHC, KSRO25/B107M6NTS; d = 25 μm) ﬁlled with MLC-2037. Using a superconducting
magnet, we applied a magnetic ﬁeld that was perpendicular to the cell surface forthe splay conﬁguration and parallel to the cell surface but that was perpendicular
to the easy axis for the twist conﬁguration. The Fréedericksz transition point
was determined by measuring the retardation change through the transmitted
light intensity that changed in a swept magnetic ﬂux density B under crossed
Nicols (SI Appendix, Fig. S6). The measurement for the twist conﬁguration was
performed at oblique incidence (5◦ ) to reduce the effect of polarization rotation(44, 45).
We measured the Fréedericksz transition eight times for the splay conﬁguration and three times for the twist conﬁguration. The light source was either ahalogen lamp or a light-emitting diode. For each measurement, we determined
the transition point twice: when the magnetic ﬁeld was increased and decreased.
As a result, we obtained a total of 16 estimates of the transition point Bc1 and6 estimates of Bc2 . By using all of them, we determined our ﬁnal estimates at
Bc1 = 4.4 ± 0.1 T and Bc2 = 3.2 ± 0.1 T. Then, it follows, by using Eq. 6 andK1 = 11.6 pN (SI Appendix, Table S1), that K2 = 6.1 ± 0.5 pN.

[4]Data, Materials, and Code Availability. Analysis results have been depositedin ﬁgshare (https://doi.org/10.6084/m9.ﬁgshare.21130301) (46).

with a smoothing parameter p and a weight wi , which is set to be one here. By
adjusting p, we obtained smoothing splines that reproduced the defect shape

ACKNOWLEDGMENTS. We thank F. Araoka at RIKEN Center for Emergent
Matter Science (CEMS) (Japan) for providing the experimental setup for thepi

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determination of K2 and for his help to measure that of MLC-2037; M. Tsubota forhis suggestion to study the coefﬁcient C of Eq. 1; M. Kobayashi for drawing ourattention to disgyration; and G.Tóth,C.Denniston,and J.M.Yeomans for allowing
us to use and analyze their data in ref. 39. We acknowledge the material data of
MLC-2037 in SI Appendix, Table S1 provided by Merck and their permission topresent them in this work. We also thank C. Denniston, J.-i. Fukuda, O. Ishikawa,
T. Ohzono, K. Katoh, J. V. Selinger, R. L. B. Selinger, and H. Watanabe for useful

discussions. This work was supported in part by Japan Science and Technology
Agency (JST) Precursory Research for Embryonic Science and Technology (PRESTO)
Grant JPMJPR18L6; by KAKENHI from Japan Society for the Promotion of Science
Grants JP19H05800, JP19H05144, JP20H01826, and JP22J12144; by a JSR
Fellowship (The University of Tokyo); and by the Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-leading Innovative Graduate
Study (WINGS) Program (The University of Tokyo).

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PNAS

2022

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https://doi.org/10.1073/pnas.2207349119

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