PHYSICAL REVIEW E 90, 052501 (2014)
Creating arbitrary arrays of two-dimensional topological defects
Bryce S. Murray,1 Robert A. Pelcovits,2 and Charles Rosenblatt1
1
Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106, USA
2
Department of Physics, Brown University, Providence, Rhode Island 02912, USA
(Received 17 September 2014; published 14 November 2014)
An atomic force microscope was used to scribe a polyimide-coated substrate with complex patterns that serve
as an alignment template for a nematic liquid crystal. By employing a sufficiently large density of scribe lines,
two-dimensional topological defect arrays of arbitrary defect strength were patterned on the substrate. When used
as the master surface of a liquid crystal cell, in which the opposing slave surface is treated for planar degenerate
alignment, the liquid crystal adopts the pattern’s alignment with a disclination line emanating at the defect core
on one surface and terminating at the other surface.
DOI: 10.1103/PhysRevE.90.052501
PACS number(s): 61.30.Jf
I. INTRODUCTION
The study of high curvature liquid crystal patterns, especially topological defects, has been of interest for over a
century [1–3]. Defects generally are categorized in terms of
the defect strength s ≡ ψ/2π , where ψ is the nematic
director rotation angle when one circumnavigates the defect
line. Here the quantity s often is referred to as the “defect
charge.” Studies of defects can be used to obtain ratios
of elastic constants [4], albeit with limited accuracy [1].
Defects are central in establishing the director profile in
geometrically confined liquid crystals [5], and serve as an
important signature when determining the symmetry of phases,
for example, distinguishing between a biaxial nematic phase
and a smectic-C phase [6]. Defect dynamics provide another
important field of study [7,8], as defect motion is extremely
sensitive to boundary effects and provides information about
surfaces and impurities. Of recent interest are defects as traps
for micro- and nanoparticles [9–16], as the presence of the
particles reduces or eliminates the energy associated with the
defect core. Moreover, by decorating the nanoparticles with
appropriate ligands, in principal one can establish liquid crystal
anchoring conditions at the particles’ surfaces.
Until now it has been relatively difficult to manufacture
and study defect pairs or a regular array of defects whose
director fields are determined by elasticity and that have
certain desired features, such as specific value(s) of s. This
is especially true for closely spaced defect cores. Eakin
et al. [17], Crawford et al. [18], and Gorkhali [19] have
used polarized holographic techniques to create one- and
two-dimensional (2D) arrays, where the 2D arrays contained
a regular pattern of s = ±1 topological defects and their
associated line disclinations. Fleury, Pires, and Galerne rubbed
a substrate with a poly(tetrafluoroethylene) rod to create an
array of topological defects, where the resulting disclination
lines ultimately were used to trap nanoparticles that formed
a three-dimensional (3D) architecture of microwires [11].
Despite the power of these approaches, both tend to be
limited to relatively large spacings between defect cores
and have not been shown to produce arrays of arbitrary,
but topologically allowed, combinations of charges s. A
recent optical technique pioneered by Ackerman, Qi, and
Smalyukh [20] has resulted in layer-by-layer creation of 3D
patterns of electrically erasable “torons,” with spacings as
1539-3755/2014/90(5)/052501(6)
small as 10–20 μm. The patterns demonstrated, however, tend
to exhibit an identical director profile around each core, not
resembling the elastically controlled profile associated with a
schlierenlike texture. Here we report the technique of atomic
force microscope (AFM) scribing of a polymer alignment layer
to create complex and arbitrary director patterns, especially
2D arrays of topological defects. We demonstrate defect core
spacings as close as 20 μm, although in principle the spacings
can be made as small as a few multiples of the liquid crystal
ϕ
extrapolation length L = K/W2 , where K is an appropriate
combination of splay and bend elastic constants K11 and
ϕ
K33 , and W2 is the azimuthal anchoring strength coefficient
of the liquid crystal at the substrate. Typically L ranges
between 100 and 500 nm [21] and can depend on temperature
and surface preparation. For defect core separations much
smaller than 2 or 3 μm the liquid crystal director is unable
to follow the pattern and tends to adopt a relatively uniform
orientation [22,23], thus establishing an effective lower limit
for the core separation. The advantage of AFM is its simplicity
and versatility, facilitating placement of defect cores of any
combination of charge(s) s and at any position. By scribing the
substrate with lines corresponding to the 2D director profile
calculated on the basis of elasticity alone, we are able to
stabilize an arbitrary director pattern in a thin liquid crystal
cell.
Consider a planar-oriented nematic liquid crystal where the
director n̂ is given by nx = sin θ cos ψ, ny = sin θ sin ψ, and
nz = cos θ , where ψ corresponds to the director’s azimuthal
orientation in the xyz frame and θ to the polar angle. In
cylindrical coordinates r, φ, and z, where z is perpendicular
to the substrate plane, Nehring and Saupe used the equal
elastic constant approximation and assumed planar anchoring
(θ = π /2) to obtain the solution to Laplace’s equation,
viz., ψ = sϕ + c [1], where c is a constant. (The solution
for unequal elastic constants, e.g., when the bend modulus
K33 diverges near the nematic–smectic-A transition while the
splay modulus K11 remains regular [24], has been worked
out by Dzyaloshinskii [25] but is beyond the scope of the
present work.) Because n̂ = −n̂, a 2π rotation of the azimuthal
coordinate φ about the z axis at r = 0 corresponds to a
director rotation ψ by mπ , where m is an integer. This
condition allows only certain values of s, the strength or
“charge” of the defect, s = ± 12 ,±1,± 32 ,±2, . . ., as well as
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©2014 American Physical Society
MURRAY, PELCOVITS, AND ROSENBLATT
PHYSICAL REVIEW E 90, 052501 (2014)
a limit on constant c, viz., 0 c < 2π . The occurrence of
half-integer singularities necessitates that the alignment of
the director is planar [1]. For multiple charges the director
orientation over the entire two-dimensional surface can be
obtained by summing ψi for all defects i centered at positions
ri = ai x̂ + bi ŷ. Finally, to generate working patterns, ψ must
be expressed in terms of Cartesian coordinates:
y − bi
ψ=
si tan−1
+ ci .
(1)
x − ai
i
II. PATTERN GENERATION
Let us first discuss the scribing algorithm. Each pattern consists of a set of pregenerated paths, with each path containing
a set of locations for the AFM stylus to visit in sequence. This
information is generated prior to the scribing process using an
algorithm that was adapted from a plotting technique to allow
reasonably strict control of line density [26]. The maximum
and minimum acceptable distances between lines are input
as parameters. The user chooses an initial seed location to
start the algorithm. Starting points for all subsequent lines are
algorithm generated by selecting candidate locations around
existing lines. The script checks through all of these candidate
seeds until a location that is further than the minimum line
spacing from all other lines is found. It then solves the vector
field, given by Eq. (1), in each direction to follow the streamline
until it meets some stopping criterion. The pattern is complete
when no candidate seed locations remain. For the sake of
time, the scribing area is broken into bins that correspond in
size to the maximum distance between lines. All points in
each line are associated with one of these bins. This allows the
script to check for adjacent lines by examining only points in
nearest neighbor bins, rather than every point in the pattern. If
a point in a line is too close to any point in another line, the
two lines are considered to be too close at that location. This
simplification dictates that the distance between lines must be
greater than the step size that determines the distance between
any two points in the same line. To produce sufficiently strong
anchoring, the line density must be very high (a small distance
between lines), and consequently the step size is constrained
to be very small. This eliminates the possibility of using an
adaptive step size algorithm. As a practical consideration, each
additional point in the pattern increases the amount of time
it takes for the AFM to complete the lithography. A dense
pattern, if left unaltered, would take an unreasonable amount
of time to scribe; therefore, each completed pattern is filtered to
remove unnecessary points in areas of relatively low curvature,
yielding a pattern that can be scribed in 2–4 h.
III. EXPERIMENTAL
Turning to the experiment, a glass microscope slide was
cleaned, spin coated with the polyamic acid RN-1175 (Nissan
Chemical Industries), and baked according to the manufacturer’s specifications. The resulting polyimide surface was then
gently and uniformly rubbed with a polyester rubbing cloth
(Yoshikawa YA-20-R). The purpose of the light cloth rubbing
was to provide a uniform orientation outside of the AFM
scribed region. The cloth-rubbed polyimide then was partially
overwritten by the stylus of an atomic force microscope
according to the algorithm described above. Lines defined by
Eq. (1) were scribed with an Agilent 5500 AFM in contact
mode over a region 65 × 65 μm using an AFM probe (Bruker
MPP-11100, approximately 300 kHz resonance) with a rigid
cantilever (40 N/m spring constant); this probe ordinarily
is used for tapping mode applications, but here it remained
in contact with the polyimide to effect scribing. For the
work described herein, we scribed two different 2D arrays
of +1 and −1 defects, such that the scribed substrates served
as the “master” surface. Because of the complexity of the
pattern, the spacing between scribe lines was not constant,
being denser in regions of high curvature (near the defect
cores) and less dense in regions of smaller curvature (the
regions between cores). Moreover, we scribed the +1/−1
patterns using both a low density of lines with spacings
varying between 250 (in high curvature regions) and 500
nm (in low curvature regions), as well as +1/−1 patterns
using a higher density of lines, between 120 and 300 nm.
We tested various set-point voltages for the AFM stylus’
contact force, ranging from as low as 0.5 up to 1.75 V,
corresponding to approximately 1–3.5 μN. This parameter
range yielded characteristic polyimide groove depths from
4–5 up to 15–20 nm. Empirically we found that the deeper
grooves tended to provide more reliable formation of the defect
cores. The scribing speed was set to be as slow as possible for
the instrument using a scan size of 65 μm with a resolution
of 4096 × 4096 pixels, about 18 μm/s. Following scribing
we examined the substrate using AFM in noncontact mode.
Figure 1 shows a noncontact mode AFM image of a section of
the polyimide surface that was patterned using a low-density
line spacing of 250–500 nm. On examining regions very close
to the defect cores, where the curvature of the scribed lines is
higher, we often found a buildup of polyimide debris that is
a by-product of the plowing process; this was not a problem
away from the cores where the line curvature is lower, as
seen in Fig. 1. Cells constructed with debris buildup tended to
exhibit less-than-ideal microscope textures in the core region,
as the diverging director gradients were overwhelmed by poor
liquid crystal alignment at the substrate due to the presence
of polyimide debris. We found that sonication (1–3 min) in
ethanol of the patterned substrates greatly reduced the debris
problem, although on occasion sonication did not entirely
eliminate it.
A second glass substrate was cleaned and
spin
coated
with
an
ethanol/“GLYMO”
[(3Glycidyloxypropyl)trimethoxysilane] mixture, with the
GLYMO obtained from Sigma-Aldrich. The coating, which
was baked at 170 °C for 1 h, provides a good planar-degenerate
alignment surface for the liquid crystal director [27], and
serves as a “slave” surface to the patterned RN-1175 “master”
surface. The two substrates were placed together, separated by
mylar spacers, where the cell gap was estimated to be 3–5 μm
based on interferometric measurements of a sampling of
cells. It is important to have the gap spacing somewhat
smaller than the spacing between the defect cores so that
the resulting disclination lines begin and terminate at defect
cores on the opposing substrates [28]; if the cell is too thick,
the disclination lines may commence and terminate on the
same substrate. The cell was then filled with the liquid crystal
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CREATING ARBITRARY ARRAYS OF TWO-DIMENSIONAL . . .
PHYSICAL REVIEW E 90, 052501 (2014)
FIG. 1. (Color online) AFM image of a section of the polyimide surface scribed at low line density.
pentylcyanobiphyenyl (“5CB”) in the isotropic phase, cooled
to room temperature, and placed on the stage of a polarizing
microscope.
IV. RESULTS AND DISCUSSION
In Figs. 2 and 3, Figs. 2(a) and 3(a) represent the computergenerated scribing patterns, Figs. 2(b) and 3(b) show the
theoretical microscope view for the liquid crystal between
crossed polarizers, and Figs. 2(c) and 3(c) show the actual
polarized microscope view of the pattern scribed with a
higher density of lines (120–300 nm separation). We make
two remarks. First, for purposes of clarity, the line density
displayed in Figs. 2(a) and 3(a) is greatly reduced from the
actual scribing density. But it should be emphasized that
the actual scribing density is proportional to the computergenerated line density, i.e., the scribing density is highest in the
same regions in which the computer-generated line density in
Figs. 2(a) and 3(a) is highest. Second, the theoretical images in
Figs. 2(b) and 3(b) assume only phase retardation effects for a
plane wave of light traversing the cell. In other words, we made
the simple approximation of a plane-wave traversing each
pixel, such that the intensity after the analyzer is proportional
to sin2 (2α), where α is the angle between the polarizer and
the director n̂ for the pixel. The images in Figs. 2(b) and 3(b)
do not account for diffraction effects, finite anchoring effects
at the substrate, possible out-of-plane director orientations, or
the tendency for elastic effects to cause the director to become
spatially uniform as one traverses the cell away from the master
surface [29].
FIG. 2. (Color online) (a) Computer-generated scribing pattern for a checkerboard of s = +1 (red solid circles) and −1 (blue open circles)
defects, where the constant ci is set equal to zero; (b) theoretical polarized micrograph for the liquid crystal between crossed polarizers;
(c) actual experimental polarized micrograph of the liquid crystal cell scribed with a high density of lines (120–300 nm line spacing). The
initial soft cloth rubbing was horizontal with respect to the image in (c). The lines then were scribed at a stylus speed of 18 μm/s and with a
force of 3 μN. The bar represents 10 μm.
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MURRAY, PELCOVITS, AND ROSENBLATT
PHYSICAL REVIEW E 90, 052501 (2014)
FIG. 3. (Color online) Same as Fig. 2 except that the s = +1 and −1 defects have been offset from each other.
Specifically, Fig. 2 shows the results for a square 2D array
of +1/−1 defects that is approximately “charge neutral”—
actually, the limited size of the pattern results in a net charge
of +1—and Fig. 3 shows the results for a 2D “offset array” of
+1/−1 defects. But when a low density of scribe lines (spaced
250–500 nm apart) is used, permanent domain walls can occur
and result in a very different appearance than the theoretical
textures; this can be seen in Fig. 4. The primary effect of
low line density is to reduce the azimuthal anchoring strength
ϕ
coefficient W2 , which has been shown to vary monotonically
with line density [30], increasing very sharply for line spaces
below 400 nm. Additionally, the lower line density may have
been insufficient to overcome the initial weak polyester cloth
rubbing of the surface, as it is known that there is a yield
stress required to establish a new easy axis after an initial easy
axis had been created [31]. On the other hand, a well-defined
easy axis and anchoring are stronger in the cells with the
higher line density having 120–300 nm line spacing, as seen
in Figs. 2(c) and 3(c). These microscope images clearly show
the expected features of the theoretical patterns in Figs. 2(b)
FIG. 4. (Color online) Photomicrograph of a cell with low density of scribe lines, corresponding to the pattern in Fig. 3. The bar
represents 10 μm.
and 3(b) with good fidelity. Nevertheless, the renditions are
not perfect. Although a few domain walls appear in these
images, they tend to occur near the edges of the patterns
where the director has to transition from the AFM scribed
region to the uniformly rubbed background region. This can
be especially problematic in the small patterns, where the net
charge is nonzero. We remark, however, that an unrubbed
background would have been more problematic, resulting in
an uncontrolled random planar alignment outside the square.
Additionally, the boundaries in the actual scribed images tend
to be a bit ragged compared to the computer-generated images,
and the regions immediately around some of the actual defect
cores are less well defined than their computer-generated
counterparts. This undoubtedly comes from optical diffraction
limitations in the actual cell and the increasing difficulty in
scribing lines of smaller and smaller radius of curvature.
Figure 3(c) also shows a small brighter region around the +1
defect cores. We believe that this behavior is due to an escape
of the director into the z direction [32], which is supported
by measurements of the light intensity—and therefore of
the optical retardation—near the defect core compared with
the intercore region. For example, for a cell thickness 3
μm, the intercore region would have a phase retardation in
the neighborhood of 2π in the visible part of the spectrum,
whereas the retardation for a nonzero z component of the
director would be reduced and result in a larger transmission
through crossed polarizers. We have observed that the tendency
toward nonplanar alignment can be reduced or eliminated by
using thinner cells and by sonicating the substrate before
cell assembly. Additionally, in principle the tendency for a
nonplanar director configuration can be reduced with the use
of a negative dielectric anisotropy liquid crystal in conjunction
with an electric field applied across the cell thickness.
Some defect cores, such as several of those in Fig. 2(c),
appear to be closely spaced pairs of two-brush defects (see
Fig. 5 for a theoretical polarized micrograph) rather than single
four-brush defects that would be expected for an integer value
of s. This issue, which can be explained on the basis of defect
bifurcation, seems to be separate from the escaped director
issue discussed above. It is known that integer defects can
break into multiple half-integer defects [28], as the elastic
energy of the 2D director field (excluding the defect’s core,
which has a diameter of several nanometers) is proportional
to s 2 [24]. Therefore, a pair of half-integer defects would have
an energy one-half that of an s = ±1 defect. Because of this
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CREATING ARBITRARY ARRAYS OF TWO-DIMENSIONAL . . .
FIG. 5. Theoretical polarized micrograph for a pair of s = ±1/2
defects offset diagonally from each other. A similar intensity profile
can be seen in some individual defect cores in Fig. 2(c).
reduced energy configuration, integer defects will tend to yield
a pair of half-integer cores. What varies from defect to defect
is the length scale over which this bifurcation occurs, and
whether this separation can be resolved with visible light. For
the case of scribed defects, the separation distance between the
half-integer defects is governed most likely by the anchoring
ϕ
strength W2 in the core region and the liquid crystal’s elasticity.
PHYSICAL REVIEW E 90, 052501 (2014)
charge. In principle, the spacing between defects can be
reduced to a few micrometers when sufficiently small spacings
between AFM scribed lines are used—we have achieved line
spacings of 20 nm using reduced AFM stylus force to limit the
displacement of the polyimide material during plowing [33]—
but the distance between defect cores probably cannot be
made smaller than a few micrometers due to the inherent
extrapolation length of the liquid crystal [21]. We should point
out that other techniques can provide similar results, with some
advantages and some disadvantages. Wei’s group at Kent State
University, for example, is using a photolithographic technique that involves a mask of ultrathin slits in a silver film to
create tightly patterned alignment layers [34]. This approach
can be used to create millimeter or even centimeter scale
patterns, much larger than patterns created by a single AFM
scan. (Of course, we can build up a larger pattern by scribing
multiple patterns side by side, although the duration of the
process could become impractical.) The photolithography
approach also requires mask preparation, and the lower limit
of the pattern’s resolution is about 100 nm. Nevertheless,
the resulting defect arrays, whether created by AFM or
photolithography, present a variety of opportunities, as they
can then be used as gratings, as periodically spaced traps for
nanoparticles, or for the study of disclination lines emanating
from the defects, such as a +1 splitting into a pair of + 12 lines.
ACKNOWLEDGMENTS
The patterns demonstrated herein represent only two of an
infinite variety of possible complex defect arrays, including
arrays of topological defects having arbitrary location and
The authors thank Professor Emmanuelle Lacaze, Professor
Qi-Huo Wei, and Professor Ivan Smalyukh for useful discussions and suggestions. B.S.M. also thanks Professor Lacaze
for her hospitality at Université Pierre et Marie Curie and
the Partner University Fund for providing funds for his travel
and residency in Paris. This work was funded by the National
Science Foundation’s Condensed Matter Physics and Solid
State and Materials Chemistry Programs under Grant No.
DMR1065491.
[1] J. Nehring and A. Saupe, J. Chem. Soc., Faraday Trans. 2 68, 1
(1972).
[2] See, e.g., P. J. Collings and J. S. Patel, Handbook of Liquid
Crystal Physics (Oxford University Press, Oxford, UK, 1997).
[3] M. Kléman and O. D. Lavrentovich, Soft Matter Physics
(Springer, New York, 2003).
[4] F. C. Frank, Discuss. Faraday Soc. 25, 19 (1958).
[5] See, e.g., G. P. Crawford and S. Zumer, Liquid Crystals in
Complex Geometries (Taylor and Francis, London, 1996).
[6] J. F. Li, V. Percec, C. Rosenblatt, and O. D. Lavrentovich,
Europhys. Lett. 25, 199 (1994).
[7] P. E. Cladis, W. van Saarloos, P. L. Finn, and A. R. Kortan, Phys.
Rev. Lett. 58, 222 (1987).
[8] C. D. Muzny and N. A. Clark, Phys. Rev. Lett. 68, 804 (1992).
[9] D. Coursault, J. Grand, B. Zappone, H. Ayeb, G. Levi, N. Félidj,
and E. Lacaze, Adv. Mater. 24, 1461 (2012).
[10] D. K. Yoon, M. C. Choi, Y. H. Kim, M. W. Kim, O. D.
Lavrentovich, and H.-T. Jung, Nat. Mater. 6, 866 (2007).
[11] J.-B. Fleury, D. Pires, and Y. Galerne, Phys. Rev. Lett. 103,
267801 (2009).
[12] P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Science
275, 1770 (1997).
[13] A. Jakli, B. Senyuk, G. Liao, and O. D. Lavrentovich, Soft
Matter 4, 2471 (2008).
[14] M. Škarabot and I. Muševič, Soft Matter 6, 5476 (2010).
[15] B. Senyuk, J. S. Evans, P. J. Ackerman, T. Lee, P. Manna,
L. Vigderman, E. R. Zubarev, J. van de Lagemaat, and I. I.
Smalyukh, Nano Lett. 12, 955 (2012).
[16] I. I. Smalyukh, S. Chernyshuk, B. I. Lev, A. B. Nych, U. Ognysta,
V. G. Nazarenko, and O. D. Lavrentovich, Phys. Rev. Lett. 93,
117801 (2004).
[17] J. N. Eakin, Y. Xie, R. A. Pelcovits, M. D. Radcliffe, and G. P.
Crawford, Appl. Phys. Lett. 85, 1671 (2004).
[18] G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. Callan-Jones,
and R. A. Pelcovits, J. Appl. Phys. 98, 123102 (2005).
[19] S. P. Gorkhali, Ph.D. thesis, Brown University, 2007.
[20] P. J. Ackerman, Z. Qi, and I. I. Smalyukh, Phys. Rev. E 86,
021703 (2012).
[21] Y. Choi and C. Rosenblatt, Phys. Rev. E 81, 051708
(2010).
V. CONCLUSIONS
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MURRAY, PELCOVITS, AND ROSENBLATT
PHYSICAL REVIEW E 90, 052501 (2014)
[22] J.-H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, Appl.
Phys. Lett. 78, 3055 (2001).
[23] J. S. Evans, P. J. Ackerman, D. J. Broer, J. van de Lagemaat,
and I. I. Smalyukh, Phys. Rev. E 87, 032503 (2013).
[24] P. G. DeGennes and J. Prost, The Physics of Liquid Crystals
(Clarendon, Oxford, UK, 1994).
[25] I. E. Dzyaloshinskii, Sov. Phys. JETP 31, 773 (1970).
[26] B. Jobard and W. Lefer, in Visualization in Scientific Computing
‘97, edited by W. Lefer and M. Grave (Springer, Berlin, 1997).
[27] D. N. Stoenescu, H. T. Nguyen, P. Barois, L. Navailles,
M. Nobili, Ph. Martinot-Lagarde, and I. Dozov, Mol. Cryst.
Liq. Cryst. 358, 275 (2001).
[28] A. S. Backer, A. C. Callan-Jones, and R. A. Pelcovits, Phys.
Rev. E 77, 021701 (2008).
[29] T. J. Atherton, R. Wang, and C. Rosenblatt, Phys. Rev. E 77,
061702 (2008).
[30] B. Wen and C. Rosenblatt, J. Appl. Phys. 89, 4747 (2001).
[31] M. P. Mahajan and C. Rosenblatt, Appl. Phys. Lett. 75, 3623
(1999).
[32] R. J. Ondris-Crawford, G. P. Crawford, S. Zumer, and J. W.
Doane, Phys. Rev. Lett. 70, 194 (1993).
[33] J. S. Pendery, I. R. Nemitz, T. J. Atherton, M. Nobili, E. Lacaze,
R. G. Petschek, and C. Rosenblatt (unpublished).
[34] Q. Wei (private communication).
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