PHYSICAL REVIEW E 97, 032702 (2018)

N-Sm A-SmC phase transitions probed by a pair of elastically bound colloids
Muhammed Rasi M,1 K. P. Zuhail,1 Arun Roy,2 and Surajit Dhara1,*
1
2

School of Physics, University of Hyderabad, Hyderabad 500046, India
Soft Matter Group, Raman Research Institute, Bangalore 500080, India(Received 2 January 2018; published 13 March 2018)

The competing effect of surface anchoring of dispersed microparticles and elasticity of nematic and cholestericliquid crystals has been shown to stabilize a variety of topological defects. Here we study a pair of colloidal
microparticles with homeotropic and planar surface anchoring across N -SmA-SmC phase transitions. We showthat below the SmA-SmC phase transition the temperature dependence of interparticle separation (D) of colloids
with homeotropic anchoring shows a power-law behavior; D ∼ (1 − T /TAC )α , with an exponent α ≈ 0.5. For
colloids with planar surface anchoring the angle between the joining line of the centers of the two colloids andthe far field director shows characteristic variation elucidating the phase transitions.
DOI: 10.1103/PhysRevE.97.032702
I. INTRODUCTION

When foreign particles are dispersed in an uniformlyaligned nematic liquid crystal the average orientation of themolecules (i.e., the director) is locally deformed around theparticles due to the strong surface anchoring. Each particle
stabilizes topological points or loop defects creating an elastic
deformation in the surrounding medium [1–11]. The deformation increases the elastic energy of the liquid crystals. When the
distortion regions of two physically separated particles tend tooverlap they exhibit long-range elastic interaction [3,4,12,13].
The interaction energy is anisotropic and typically of the orderof a few thousand kB T , where kB is the Boltzmann constant andT is the absolute temperature [2,14]. The system minimizes thetotal elastic energy by sharing the topological defects associated with the particles, and based on this principle various
interesting two- and three-dimensional colloidal assemblies
have been reported [8,13,15–19].
When the colloidal particles are dispersed in a SmA liquid
crystal, the situation is very different due to the translational order in addition to the orientational order of themolecules [20–28]. The surface anchoring of the SmA layers
at the colloid’s surface is not well defined, and understandingof the induced defects and equilibrium separation among theparticles is incomplete. Very recently we have shown that thesurface anchoring and defects in the SmA phase can be studied
by dispersing the colloids in the nematic phase of a liquid
crystal that exhibits N -SmA phase transition [29–33]. In thenematic phase the well-defined anchoring of the molecules on
the particle’s surface guides the layer orientation and elastic
deformation in the SmA phase around the colloids when cooled
across the N -SmA phase transition. We showed that the point
defects such as hyperbolic hedgehog and boojum defects aretransformed to focal conic line defects, and the elasticity of therespective media controls the equilibrium separation between
the particles.

*

sdsp@uohyd.ernet.in

2470-0045/2018/97(3)/032702(6)In the SmC phase the molecules are titled with respect tothe layer normal, and consequently it becomes weakly biaxial
due to the anisotropic fluctuations of the orientation of thelong axes. There are some theoretical and experimental studiesof the spherical inclusions in the SmC/SmC ∗ liquid crystals
but mostly in free-standing films [28,34–38]. In this paper wereport experimental studies on a pair of colloids with planar andhomeotropic surface anchoring across the N -SmA-SmC phasetransitions. Our study shows that the temperature dependenceof the equilibrium separation and the angle of the colloid pairwith respect to the far field director depend on the type of
surface anchoring and are highly sensitive to the elasticity and
SmC order parameter.
II. EXPERIMENTAL

Silica microspheres of diameter 5.2 μm obtained from
Bangs Chemicals (USA) were coated with octadecyldimethyl3-trimethoxysilylpropyl-ammonium chloride (DMOAP),
which provides homeotropic surface anchoring of liquid
crystal molecules. The coated particles were dispersed in aliquid crystal that exhibits the following phase transitions:
Cr 33.2 ◦ C, SmC 57.7 ◦ C, SmA 65.5 ◦ C, N 68 ◦ C I. The
microspheres were coated with N-methyl-3 aminopropyl
trimethoxy-silane (MAP) to induce planar anchoring of themolecules. Liquid crystal cells were made of two parallel
glass plates coated with polyimide AL-1254 and rubbed in the
antiparallel way for getting planar alignment of the director.
Colloidal mixture was introduced into the cell of thickness
in the range 11–14 μm by capillary action. A laser tweezer
was built around an inverted optical polarizing microscope(Nikon Eclipse Ti-U) using a cw solid-state laser operating at
1064 nm (Aresis, Tweez 250Si). An acousto-optic deflector
interfaced with a computer was used to control the trap
movement. The positions of the colloids were tracked byusing the video-microscopy technique with a resolution of±10 nm [39,40].
The temperature of the sample was controlled by aproportional-integral-derivative controller (Instec Inc.) with

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PHYSICAL REVIEW E 97, 032702 (2018)

FIG. 1. (a) Molecular structure and phase transition temperaturesof the compound. (b) Temperature profile of the heat capacity (Cp )on fast cooling (0.7 K/h). (Inset) Slow cooling (0.12 K/h) across the
SmA-SmC phase transition.

an accuracy of ±0.1 ◦ C. The trajectories of the particles asa function of temperature were video recorded with a CCD
camera at 25 frames per second while the sample was cooledat the rate of 0.2 ◦ C/min. The x-ray diffraction studies werecarried out on unoriented samples by using Cu-Kα radiationfrom a PANalytical instrument (DY 1042-Empyrean) and alinear detector (PIXcel 3D). The sample temperature wascontrolled with a precision of 0.1 ◦ C using a Linkam heaterand a temperature controller. The molecular tilt angle (θ )was calculated from the temperature-dependent layer spacing
across the SmA-SmC phase transition. The chemical structureof the compound used in the experiment is shown in Fig. 1(a).
The sample was obtained from Prof. R. Dabrowski, Instituteof Chemistry, Military University of Technology, Warsaw,
Poland. The heat capacity (Cp ) measurements were made bya fully automated adiabatic scanning calorimeter. To achieve
maximum insulation from the environment the air between
them was removed by a vacuum pump [41]. The almost real
thermal equilibrium state of the sample was achieved due tothe high-temperature stability (50 mK) and the slow scanning
rates. The heat capacity of the empty cell was measured in a
separate experiment and subtracted from the measured data to
get actual Cp of the sample.
III. RESULTS AND DISCUSSION

The variation of specific heat (Cp ) across the phase transitions obtained from high-resolution ac calorimetry is shownin Fig. 1(b). It shows a first order N-SmA and second order
SmA-SmC phase transitions. First we looked at the DMOAPcoated colloids in the nematic phase. It was observed that the

FIG. 2. Optical photomicrographs of a pair of collinear colloids
with homeotropic anchoring at different temperatures in a planar cell.
Images are taken (a–c) with crossed polarizers, (d–f) with crossed
polarizers and a λ-plate, and (g–i) without polarizers and the λ-plate.
Cell thickness and diameters of the colloids are 5.2 μm and 14 μm,
respectively.

colloids induce a Saturn ring defect (i.e., quadrupolar defectstructure) irrespective of cell thickness. This is in contrast
with the observation in conventional liquid crystals such as
5CB or 8CB [2,4,8,9]. Using optical tweezer we converted a
few colloids from quadrupolar to dipolar structure and formeda collinear pair of elastic dipoles. The dipoles are bound byelastic forces of the liquid crystal medium with a equilibrium
separation due to the elasticity of the medium and the hyperbolic hedgehog defects. An isolated dipolar pair was studied
across the N -SmA-SmC phase transitions while cooling at therate of 0.2 ◦ C/min. Figure 2 shows some representative imagesof a pair of dipoles in different phases. When the temperature
is decreased from the nematic to SmA phase the hyperbolic
hedgehogs defects are transformed into smectic focal line
defects. We reported similar results in 8CB liquid crystal in ourprevious study [30]. It was shown that the transformation took
place through the formation of a splay soliton at the expenseof energy cost associated with elastic bend deformation. The
length of the defect line is extended up to to the several times
the diameter of the colloid [Fig. 2(b)]. The homeotropic surface
anchoring of the molecules in the SmA phase is retained only
on the left-half part of the left colloid, and it is broken in theremaining part creating a strong layer distortion around thecolloids. The λ-plate (530 nm) images at the corresponding
temperatures are shown in Figs. 2(d)–2(f). The bluish andyellowish colors in narrow regions on the upper and lower sidesof the line defect indicate that the molecules in the layers aretilted in the opposite direction creating an angle discontinuity
across the defect line.
In the SmC phase the layer distortion becomes nonuniform
surrounding the colloidal pair [Fig. 2(f)]. This could arise dueto the random variation of tilting direction of the molecules
and the weak biaxiality of SmC phase [42]. The length ofthe defect line in the SmC phase has decreased compared to032702-2

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PHYSICAL REVIEW E 97, 032702 (2018)

FIG. 3. (a) Temperature dependence of center-to-center separation (D) of a pair of dipolar colloids across the N -SmA-SmC phasetransitions in a planar cell. Temperature variation of D below (b) the
N -SmA and (c) the SmA-SmC phase transition in logarithmic scale.
The red lines are the best fits to the power laws: (D − D0 ) ∼ t α , withan exponent α = 0.36, where t = (1 − T /TNA ) and (D − D0 ) ∼ τ β ,with an exponent β = 0.53, where τ = (1 − T /TAC ). Cell thickness
and the diameter of the colloids are 14 μm and 5.2 μm, respectively.
The sample was cooled at a rate of 0.2 ◦ C/min.

that of the SmA, and the center-to-center separation is not thesame in all the phases [Figs. 2(h)–2(i)]. Here we define an
extrapolation length l = W/B, where W and B are the surface
anchoring energy and elastic layer compressional modulus of
the smectics, respectively. In the SmA phase [43,44], W ∼10−2 –10−3 J/m3 and B ∼ 105 –106 J/m3 and estimated l ∼0.1 μm, which is much smaller than the size of the colloids.
Thus strong surface anchoring guides the transformation of thedefect across the phase transition.
We measured the interparticle separation (D) across the
N-SmA-SmC phase transitions as a function of temperature
while cooling the sample at the rate of 0.2 ◦ C/min as shown in
Fig. 3(a). In the nematic phase D increases linearly followed byalmost a discontinuous change at the N-SmA phase transition.
We recently explained the pretransitional behavior of D basedon the increase in the ratio of the bend to splay elastic
constant (i.e., K33 /K11 ) [30]. Since the equilibrium separationis governed by the elasticity of the medium, the discontinuity
in D marks the discontinuity of the elastic constants of therespective phases. The distance of closest approach between
the particles at the N -SmA transition (TNA ) is D0 = 6.8 μm,
and this is about 1.6 μm larger than the center-to-center distance between the two colloids (5.2 μm). This indicates that at
the transition there are about 800 layers present between themassuming the length of a molecule is about 2 nm. Below the
N-SmA transition D increases continuously. The temperaturedependence of D − D0 , just below the N-SmA transition isFIG. 4. (a) Schematic diagrams of two collinear colloids in thenematic phase with homeotropic surface anchoring. The continuouslines represent the director field. (b) Schematic diagram showing thelayer distortion and focal conic line around the colloid pair in the
SmA phase. The continuous lines represent SmA layers. Orientationof a few molecules within the layers is shown inside the circle in theupper right-hand corner. The blue regions indicate the breaking of
homeotropic anchoring. The red line represents the focal conic line.
The gray regions represent the deviation (or distortion) of the layers
from the uniform orientation. dd and d0 are the thickness of the dilated
and undilated layers, respectively.

shown in Fig. 3(b) using a logarithmic scale. It shows apower-law behavior, (D − D0 ) ∼ t α , with an exponent α =0.36, where t = (1 − T /TNA ). A similar exponent was also obtained in the case of 8CB liquid crystal [30]. In the SmA phase
there is an angle discontinuity across the focal line due to the
opposite tilting of the molecules. A schematic representationof this is presented in Fig. 4(b). For the purpose of comparison
a schematic diagram in the nematic phase is also shown in
Fig. 4(a). The blue region around the colloids in Fig. 4(b)indicates the region where the homeotropic anchoring of thedirector is broken to reconcile with the far field director. The
angle discontinuity causes a layer dilation given by dd − d0 ,where dd and d0 are the thickness of the dilated and undilated
layers respectively [21,45]. A schematic representation ofsuch layer dilation is shown in Fig. 4(b). Consequently theelastic energy B(1 − dd /d0 )2 increases where B is the SmA
elastic compression or dilation modulus. Below the N -SmA
transition the elastic modulus varies as B ∼ t 0.38 [43]. If the
discontinuity angle exceeds a critical value, it may further
create edge dislocations in the confined region [46]. In thepresent system the length of the defect line increases and thecolloidal particles are pushed apart, thereby increasing thecenter-to-center separation.
As the temperature is decreased in the SmA phase D
increases very slowly till the SmA-SmC transition temperatureis reached [Fig. 3(a)]. For example, a small negative slope is032702-3

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FIG. 5. (a) Temperature variation of tilt angle (θ ) below the
SmA-SmC phase transition. (b) Variation of θ with reduced temperature below the SmA-SmC transition (logarithmic scale) and thebest fit to the equation θ = θ0 τ η with an exponent η = 0.48.

observed between 60 ◦ C and 57.7 ◦ C. At the transition point(TAC = 57.7 ◦ C), D changes slope and increases continuously
as the temperature is decreased. For example, at the SmA-SmC
transition temperature D = 7.2 μm, and it is increased to
7.6 μm when the temperature is decreased to 56 ◦ C. The
N-SmA and SmA-SmC phase transitions in this compound
are weakly first order and second order, respectively. It appears
that the temperature dependence of D is a pointer to the discontinuous and continuous phase transitions at the respective
temperatures. In Fig. 3(c) we show the temperature variationof D − D0 in the SmC phase using a logarithmic scale. It canalso be fitted to a power law, (D − D0 ) ∼ τ β , with β = 0.53,
where τ = (1 − T /TAC ). The model based on the de Gennes
theory suggests that the SmA-SmC phase transition belong
to the 3D XY universality class, whereas many experiments
showed a classic mean-field behavior with the tilt angle θ is anorder parameter which is given by θ = θ0 |τ |η , where η = 0.5.
To verify it we measured the temperature dependence of θ from
x-ray diffraction studies, presented in Fig. 5(a). The exponentof the order parameter obtained from the fitting [Fig. 5(b)]is η = 0.48, and it is very close to the exponent β = 0.53 of
the temperature dependence of D [see Fig. 3(c)]. Thus we
conjecture that the interparticle separation D is coupled tothe order parameter (θ ) of the SmA-SmC phase transition.
It may be mentioned that several physical properties such as
birefringence and elastic modulus of the SmA phase shows theeffect of strong pretransitional fluctuations across SmA to thetilted phase (SmC or SmC*) transition [47–50]. In the present
system we do not observe any significant pretransitional effectexcept a small and linear variation of D just above the SmC to
SmA phase transition.
When colloidal particles with planar anchoring (coated with
MAP) are introduced in a planar cell, each particle creates apair of antipodal surface defects known as boojums. Figure 6
shows some representative optical images of a pair of colloids
with boojums and the transformation of the boojums acrossthe N -SmA-SmC phase transitions. The transformation ofboojums across the N -SmA phase has been studied by us recently [29]. In the SmA phase the boojums are transformed into
two line defects which are extended on the opposite sides ofthe pair along the rubbing direction up to several micrometers.
Subsequent λ-plate images [Figs. 6(d)–6(f)] clearly indicatethat the SmA layers are tilted in the opposite direction of the

FIG. 6. Optical photomicrographs of a pair of colloids withboojums defects in a planar cell at different temperatures in the
N , SmA, and SmC phases in a planar cell. Images are taken (a–c)
with crossed polarizers, (d–f) with crossed polarizers and a λ-plate,and (g–i) without polarizers and the λ-plate. The diameters of thecolloids and the cell thickness are 5.2 μm and 11.5 μm respectively.
(j) Schematic diagram showing the director field around two colloids
with planar anchoring in the nematic phase. (k) Schematic diagram
showing subsequent layer orientation in the SmA phase.

defect lines. Schematic diagrams showing the director field and
SmA layer orientation are shown in Figs. 6(j) and 6(k). It may
be noted that in the case of homeotropic surface anchoring,in both SmA and SmC phases, the focal lines are collinear,whereas in the case of planar anchoring they are parallel to eachother and are separated by a distance nearly equal to the radiusof the colloidal particle. In the SmC phase, the nonuniformly
colored regions around the colloids indicate that the layers
are distorted randomly [Fig. 6(c)]. This could again be dueto the weak biaxiality of the SmC phase as we discussed in
the case of homeotropic anchoring previously [Fig. 2(c)]. In
addition to this the defect lines have become shorter and lessdistinct than the SmA phase [Fig. 6(i)]. It suggests that the angle
discontinuity in the SmC phase has decreased with respect to
the SmA phase.
To see the effect of phase transitions on the interparticle
separation we measured D across the N -SmA-SmC phasetransition as a function of temperature, presented in Fig. 7(a).
The interparticle separation D decreases very sharply followed
by a slope change at the N -SmA phase transition (TNA =65.5 ◦ C) and becomes independent of temperature in the SmA
phase and across the SmA-SmC transition. Hence there is no
noticeable change of D across the SmA-SmC phase transition,and this is in contrast to the homeotropic colloids discussed
in the previous section. In the case of planar anchoring since
the line defects in the SmA and SmC phases are parallel to032702-4

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PHYSICAL REVIEW E 97, 032702 (2018)

φ decreases continuously with a change of slope at thephase transition temperature (57.7 ◦ C). Hence in the case of
homeotropic surface anchoring D is sensitive to both the
N -SmA and SmA-SmC phase transitions. In the case of planar
surface anchoring D is sensitive only to N -SmA transition,
whereas φ is sensitive to both the N -SmA and SmA-SmC phasetransitions. The structural change that directly affects the colloid pair across the SmA-SmC transition is the molecular tilt
angle (θ ), which is the order parameter of the SmA-SmC phasetransition. Therefore, it is expected that the relative variation of
D and φ across the phase transitions is connected to the orderparameters of the respective phase transitions. Theoretical andcomputer simulation studies could be useful for understanding
the direct linking of these physical quantities.
IV. CONCLUSION

FIG. 7. (a) Temperature dependence of center-to-center separation (D) of a pair of colloids with planar anchoring in a planar cell.
Temperature variation of φ across the (b) SmA-SmC and (c) N -SmA
phase transitions. φ is the angle between the joining the far field
director and the joining line between the two centers of the colloids(shown in inset). The sample was cooled at a rate of 0.2 ◦ C/min.
The diameters of the colloids and the cell thickness are 5.2 μm and11.5 μm, respectively.

We have studied a pair of spherical colloids with homeotropic
and planar surface anchoring across N -SmA-SmC phasetransitions. The hyperbolic point defect is transformed to asmectic focal line below the N -SmA transition. The line defectis slightly shortened without any significant change in the
structure below the SmA-SmC transition. The temperaturedependence of the interparticle separation of a collinear colloids shows a similar power-law as that of SmA-SmC orderparameter. For colloids with planar surface anchoring there
is no observable change in the interparticle separation acrossthe SmA-SmC transition, although the angle made by thejoining line with respect to the far field director decreases
continuously with a characteristic slope change. This suggeststhat the interparticle separation of a pair of colloids and theirorientation angle are coupled to the order parameters of thephase transitions.

each other the colloids are not pushed apart unlike in theprevious case [see Figs. 2(b) and 2(c)], hence D remains
unchanged. This suggests that the temperature dependence ofD depends on how the two colloids are bound by defects.
The line joining the center-to-center of the colloids makes
an angle φ with respect to the far field director. To see theeffect of phase transitions on φ, we measured it as a functionof temperature, presented in Figs. 7(b) and 7(c). The angle (φ)
decreases discontinuously and continuously across the N -SmA
and SmA-SmC phase transitions, respectively. For example, at
the N -SmA transition [Fig. 7(c)], φ discontinuously decreases
from 19◦ to 17.5◦ . Across the SmA-SmC transition [Fig. 7(b)]We gratefully acknowledge support from the Departmentof Science and Technology, Govt. of India (DST/SJF/PSA02/2014-2015), and DST-PURSE. M.R.M acknowledges DST
for support through a fellowship grant. We thank Prof.
Dabrowaski for providing the sample. We also thank George
Cordoyiannis for helping us by doing the ac calorimetry
measurements.

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