Line Tension of Twist-Free Carbon Nanotube Lyotropic Liquid Crystal

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Line tension of twist-free carbon nanotube lyotropic liquid crystal microdroplets on solid surfaces Vida Jamali, Evan Gregory Biggers, Paul van der Schoot, and Matteo Pasquali Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02109 • Publication Date (Web): 07 Aug 2017 Downloaded from http://pubs.acs.org on August 7, 2017

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Line tension of twist-free carbon nanotube lyotropic liquid crystal microdroplets on solid surfaces Vida Jamali,1 Evan G. Biggers,1 Paul van der Schoot,2,3 Matteo Pasquali1,4,5,* 1

Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas

77005, USA 2

Theory of Polymers and Soft Matter Group, Department of Applied Physics, Eindhoven

University of Technology, 5600 MB Eindhoven, The Netherlands 3

Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The

Netherlands 4

Department of Chemistry, Department of Materials Science and NanoEngineering, Rice

University, Houston, Texas 77005, USA 5

The Smalley-Curl Institute, Rice University, Houston, Texas 77005, USA

Abstract Line tension, i.e., the force on a three-phase contact line, has been a subject of extensive research due to its impact on technological applications including nanolithography and nanofluidics. However, there is no consensus on the sign and magnitude of the line tension, mainly because it only affects the shape of small droplets, below the length scale dictated by the ratio of line tension to surface tension /. This ratio is related to the size of constitutive molecules in the

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system, which translates to a nanometer for conventional fluids. Here, we show that this ratio is orders of magnitude larger in lyotropic liquid crystal systems comprising micron-long colloidal particles. Such systems are known to form spindle-shaped elongated liquid crystal droplets in coexistence with the isotropic phase, with the droplets flattening when in contact with flat solid surfaces. We propose a method to characterize the line tension by fitting measured droplet shape to a macroscopic theoretical model that incorporates interfacial forces and elastic deformation of the nematic phase. By applying this method to hundreds of droplets of carbon nanotubes dissolved in chlorosulfonic acid, we find that /~ − 0.84 ± 0.06 µm. This ratio is two orders of magnitude larger than what has been reported for conventional fluids, in agreement with theoretical scaling arguments.

INTRODUCTION Line tension , defined as the excess free energy per unit length of a three-phase contact line, is important in designing technological applications such as nanofluidics1,

2

and

nanolithography3. Despite its importance, there is no agreement on the sign and magnitude of the line tension in the literature.4-8 Direct measurements are difficult because line tension affects the droplet shape only when droplets fall below a characteristic length that is proportional to the ratio of line tension to surface tension , /.5 This length scale is on the order of the size of the fluid molecules, i.e., a nanometer for conventional fluids, which makes the accurate measurements of the line tension a challenge.5, 9, 10 Here we show that, in lyotropic liquid crystal systems, this length scale approaches a micrometer due to the presence of micron-sized mesogens. Lyotropic systems are solutions of individually dispersed, anisotropic colloidal


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particles (mesogens) that form liquid crystalline phases, such as uniaxial nematics, at sufficiently high concentrations.11 At its onset, the nematic phase appears as elongated, spindle-like droplets, called tactoids.12-17 Nematic tactoids in equilibrium with the isotropic phase and in contact with a flat solid surface (sessile tactoids) have a flattened elongated shape, characterized by noncircular contact lines; see the schematic in Figure 1. 18, 19

Characterizing the line tension in liquid crystal systems requires an appropriate model that can capture the elongated shape and non-constant contact angle of nematic tactoids. Line tension for conventional fluids is often estimated by measuring the contact angle θ of a spherical capshaped sessile droplet of radius and using the modified Young equation: cos = − − − / , with , , and , the liquid-vapor, solid-vapor, and solid-liquid surface tensions, respectively.5, 20-23 The elongated shape of a nematic tactoid, however, results from the interplay of elastic, interfacial, and anchoring free energies, some of which are not included in the modified Young equation. Therefore, the modified Young equation is insufficient for characterizing the line tension in liquid crystal systems.

We propose to characterize the line tension in liquid crystal systems by measuring the shape of tactoids via polarized optical microscopy and coupling shape measurements to a new macroscopic free energy model, incorporating nematic phase elasticity and all interfacial tensions in the system, including line tension and anchoring effects. By applying our method to hundreds of nematic tactoids formed in solutions of carbon nanotubes (CNTs) in chlorosulfonic acid (CSA), we find the ratio of line tension to nematic-isotropic surface tension.


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Moreover, we use our theoretical calculations to study the effect of sign and magnitude of the line tension on the shape of droplets. We show that theoretically for positive values of the line tension, the contact line is pulled inward leading to spheroidal droplets in the small size limit, while for negative values, the contact line is pulled outward leading to elongated droplets. Therefore, when the line tension is negative, liquid crystal droplets below a minimum size completely wet the solid surface. Isotropic

A)

Nematic

y x

R

z

y

y

B)

d r' h

boojum

C) d

R

Side view

x

h

R' ~ R

r

r'

z

Front View

Figure 1. (A) Schematic of a sessile nematic liquid crystal droplet (tactoid) of depth , minor axis , and major axis , in equilibrium with the isotropic phase and in contact with a solid surface. (B) Side view of an asymmetric sessile tactoid with the depth and the major axis that can be assumed as a portion of a symmetric tactoid of minor axis ’ and major axis ’ that is cut by a solid surface at distance from the axis of symmetry of the tactoid. The director field lines (dashed lines), showing the average orientation of the molecules, converge on two virtual point defects at vertical position and horizontal . (C) Front view of an asymmetric tactoid showing the depth , the minor axis position length , and the minor axis of the original tactoid cut by a solid surface at distance .

EXPERIMENTAL SYSTEM Our experimental system consists of 0.1% by mass single-walled CNTs dissolved in CSA. CSA acts as a true solvent for CNTs and spontaneously dissolves CNTs forming a nematic


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liquid crystal phase at high concentrations.24 CNTs with a diameter of ~2.18 nm and an average number of walls of 1.6 were purchased from Meijo Nano Carbon Co, batch Meijo EC 1.5-P. CNTs were dissolved in CSA (99% purity, Sigma Aldrich), speed mixed (DAC 150.1 FVK, Flack Tek Inc.) at room temperature for 1 hr and then ultra-sonicated (Bransonic M1800) to shorten the CNTs and speed up the kinetics of the system to form nematic tactoids. The aspect ratio of the CNTs, defined as length/diameter (!/), was measured with extensional rheometry25 as 509 ± 11. The prepared sample was then loaded into as-purchased 0.1 × 0.1 mm square and 0.2 × 2 mm rectangular glass capillaries (VitroCom, Inc.), flame-sealed, and characterized by polarized optical microscopy using a Zeiss AxioPlan 2 microscope. Images were recorded with a cooled charge-coupled device (CCD) camera. The aspect ratio and size of the tactoids were measured using ImageJ software package (available from the National Institute of Health website) by measuring the major and minor axes of an ellipse circumscribing a tactoid to estimate the major and minor axes of the tactoid. Due to the low nematic-solid surface tension, the glass walls of the square capillaries were populated by hundreds of tactoids within a few weeks (Figure 2). Tactoids were small (less than 150 µm long) with a small density difference from the isotropic phase. The effect of gravity is negligible and hence tactoids formed on all four walls of the capillary. In order to characterize the tactoids from two different views, we mounted the capillaries on square beads, which allowed us to rotate exactly 90° and image the top and side views of tactoids (Figure 2). Free tactoids formed in the rectangular capillaries were identified by changing the focal plane on the microscope.


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Figure 2. Polarized optical micrographs showing (A) top view (xz plane) and (B) side view (yz plane) of sessile tactoids formed in a 0.1% by mass solution of carbon nanotubes in chlorosulfonic acid. Crossed arrows show the orientation of crossed polarizers. (C-D) Polarized optical micrographs of two sessile tactoids of similar size from different viewpoints with a dark area at the center which remains dark when the polarizers are decrossed (% = &' and % = ((') indicating a symmetric twist-free director-field configuration. The schematic in (D) shows the average orientation of the rods inside the droplet. The dashed white line shows the nematic-solid interface. Scale bars are 20 µm. THEORY Tactoid shape is characterized by the depth ) (along * axis), minor axis + (along , axis), and major axis (along - axis) (Figure 1A). Tactoids are symmetric if ) = + and asymmetric if ) . +. A cylindrically asymmetric tactoid can be considered as a portion of an imaginary symmetric tactoid of major axis ’ and minor axis +’ that is cut by a solid surface at distance / = + − )

from

the

axis

of

symmetry


(Figure

1B-C),

where

6

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= 02+ − / 1 + + 1 /2+ − + − /1 and + = 3+ 1 − /1 . The spindle shape of a symmetric tactoid can be described in the bispherical coordinate system where the shape is the result of rotating a circle section about its chord.26 The average orientation of particles inside a tactoid is described by the director field 4. The director field lines converge on two point defects known as boojums separated by the distance 2 5 (see Figure 1B). Boojums can be virtual, when they are located outside the droplet along the

main axis of the bispherical coordinate system ( 5 > ′) (Figure 1B). Based on the theoretical

description of the director field for a symmetric tactoid (/ = 0, = ′), we expect a continuous crossover in the director-field configuration from a homogenous configuration ( 5 → ∞, to a

bipolar configuration ( 5 = ′).27 For a cylindrically asymmetric tactoid, the virtual boojums are located at distance / under the solid surface.

Free Energy Model. For a three-dimensional sessile nematic tactoid, the equilibrium shape corresponds to the minimum total free energy :: : = :; + :? + :@ .

(1)

Here, :; are nematic-isotropic, nematic-solid, and isotropic-solid interfacial free energies, respectively. :? is the three-phase contact line energy and :@ is the Frank-Oseen elastic energy of the nematic phase (including splay, bend, and saddle splay deformation). Rod-like molecules prefer to align parallel to the nematic-isotropic interface.28-31 Hence, the nematic-isotropic interfacial energy includes an anchoring term that penalizes the rods for any non-planar anchoring at the interface. Here, we use a nematic-isotropic surface free energy of the


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Rapini and Papoular type:32 :; 0 for the planar anchoring at the interface.28, 30 Following the Rapini’s formalism, the nematic-solid interfacial energy is:33 :; = AM dL[ F; + HF; I ∙ 41 K, where F; is the nematic-solid surface tension, I’ is the unit normal to the solid surface, and HF; is the nematic-solid anchoring strength that penalizes the rods for not being parallel to the solid surface. Theoretically, planar anchoring is preferable at the solid-nematic interface for rod-like molecules, and therefore HF; > 0.34, 35 The isotropic-solid interfacial free energy is identical to the one used in the original Young equation: :;=> = AM dL G; , where G; is the isotropic-solid surface tension and is incorporated to subtract the energy that the isotropic phase should have paid if it were in contact with a solid surface. Both :; and :;=> integrals are over the nematic-solid contact area L. The three-phase line tension energy is simply the integral of the line tension GF; , over the threephase contact line O: :? = ∮Q dO GF; . Finally, our description of sessile tactoids includes the Frank-Oseen elastic free energy :@ that the nematic phase pays for splay, bend, and saddle-splay elastic

deformations:36

T

T

:@ = AZ dR S1 UTT ∇ ∙ 41 + 1 UWW [4 × ∇ × 4K1 X − AZ U1Y ∇ ∙

[4∇ ∙ 4 + 4 × ∇ × 4K. The elastic integral is over the tactoid volume R, and UTT , UWW , and U1Y are the splay, bend, and saddle-splay elastic constants, respectively. We neglect any twist deformation, which is consistent with our observations of twist-free tactoids in CNT-CSA


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solutions (see Figure 2). The U1Y term renormalizes the UTT term for the bispherical type directorfield configuration considered in this case. Although, the U1Y term is essentially a surface term and is often presumed subdominant, the UTT and the U1Y terms together dictate the shape of the droplet in the large size limit. The free energy parameters are grouped into: ∆= G; − F; /FG , \FG = HFG /FG , \F; = HF; /FG , ] = UTT − U1Y /FG \FG , ^GF; = GF; /FG , and

_WW = UWW /UTT − U1Y .

The

solid surface affinity parameter ∆ shows the affinity strength of the solid surface for the nematic phase in comparison with the isotropic phase. Both \FG and \F; , determine the relative anchoring strength at the two-phase interfaces with respect to the nematic-isotropic surface tension FG . The extrapolation length ] is the ratio of elastic and surface forces and controls the crossover in shape and director-field configuration with increasing droplet size. The ratio of line tension to nematic-isotropic surface tension is described by the characteristic length scale ^GF; . The parameter _WW measures the relative magnitudes of the bend UWW , and the difference between splay and saddle-splay (UTT − U1Y ) elastic constants of the nematic phase. All the free energy integrals are carried out in the Cartesian coordinate system, except for the nematic-isotropic surface integrals, which are calculated in the bispherical coordinate system (`, a, b)26, for simplicity. To find the upper and lower bounds of the integrals we used the equation of the solid surface that cuts through the tactoid at * = / in the Cartesian coordinate system. The transformation from Cartesian

to

bispherical

is

given

by

, = sin ` sin a cos b /1 + cos a sin `,

*=

sin ` sin a sin b /1 + cos a sin `, and - = cos ` /1 + cos a sin `. The ranges of the bispherical coordinates describing the tactoid shape in general is 0 . ` . e, f . b . 2e, and


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0 . a . ag , where ag denotes the a at the surface of the tactoid in contact with the isotropic phase and is calculated as 2 tanjT + / ′, and f is the angle that the solid surface (* = /) makes with the central axis of a tactoid (f = cot jT + //) (see Fig. 1 C). The * = / equation is transformed to the bispherical coordinate system as sin ` sin a sin b /1 + cos a sin ` = /. It follows from the bispherical-Cartesian transformation that , = * cot b. The upper bound of the integrals on , and - are +′ cossinjT*/+′|lmn and cos o /1 + cos a sin o, respectively, where ` is calculated for any given value of , and * using ` = sinjT*/−/ cos ag + sin ag sincot jT ,/* and a = ag . For the nematic-solid integral we use the value * = / to find -pqr only as a function of ,. Both upper and lower bounds on o integrals are also calculated using the equation of the solid surface cutting through the tactoid for any given value of b and a. For a symmetric sessile tactoid o = 0 and os = e which are the foci of the bispherical coordinate system. The free energy integrals are functions of the tactoid’s shape and the director field 4. Director field lines are the lines of constant a in the bispherical coordinate system (eη) that are tangent to the circle sections rotated about their chords. The corresponding expression in the Cartesian coordinate

is:

4=

T

√F u

2*- vr + 2,- vl + - 1 − , 1 − * 1 − 5 1 vw ,

where

x =

y- 1 + , 1 + * 1 − 5 1 z + 4, 1 + * 1 5 1 . The integrands of the splay and bend elastic free energy integrals are ∇ ∙ 41 = 16- 1 /xT and [4 × ∇ × 4K1 = 4, 1 + * 1 /xT

,

where

xT = , 1 + * 1 + - 1 − / j1 1 + 4 /

j1 , 1 + * 1 . The integrand of the nematic-solid interfacial term is I′ ∙ 41 |lmn = 4* 1 - 1 /x1 |lmn with x1 = 4 { | 5 }

j1

j1

, 1 + * 1 + , 1 + * 1 + - 1 − { | 5 } 1, where the


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normal to the solid surface I is vl in the Cartesian coordinate system. The three-phase contact line and the nematic-isotropic surface integrals are calculated in the bispherical coordinate system,

where

the

integrand

of

the

nematic-isotropic

integral

is

I ∙ 41 |~m~ = cos 1 o sin1 a 1 − / 1 1 /4xT |~m~ . RESULTS AND DISCUSSION Figure 2 shows polarized optical micrographs of CNT tactoids on surfaces, imaged from the top (,- plane) and side (*- plane). The top view shows the non-circular contact line and a dark cross at the center when the tactoid is aligned with one of the crossed polarizers ( = 90° ) (Fig 2C). The side view shows a dark area at the center of the tactoid which remains dark even when the polarizers are de-crossed at = 70° and = 110° , confirming that these sessile tactoids have a mirror-symmetric, twist-free bipolar optical texture; there is no indication of chiral symmetry breaking (Figure 2 C-D). This is different from recent results on sessile tactoids of achiral chromonic liquid crystals that showed a twisted director-field configuration.18 Tortora et al.18 explained the symmetry broken twisted director field as a way to skip energetically expensive splay and bend elastic deformations for sessile nematic tactoids. However, elongated bipolar droplets have a large splay elastic deformation. We expect that the large anchoring energy at the nematic-solid interface enforces parallel alignment of the rods to the interface and prevents our tactoids from obtaining a twisted director-field configuration. Additionally, for nematic systems of hard rods with U11 /UTT ~1/3

37, 38

, we should not expect to see any twisted

nematic droplets39–see Supplementary Information for details. Figure 3 A-B displays the top-view and side-view measured aspect ratios ( /) and /+, respectively) vs. the tactoid major axis length for over three hundred sessile tactoids together


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with the aspect ratio of more than four hundred free tactoids in the bulk isotropic phase. The aspect ratio behavior of the droplets in the small size regime is significantly different for the two data sets (Figure 3 A) that are obtained from the same system of CNTs. Free tactoids in the bulk isotropic phase are known to have an aspect ratio that is independent of their size in the small size regime.16, 27 However, the presence of the solid interface leads to the formation of highly elongated partially wetting tactoids in the small size limit, where the aspect ratio behavior shows a strong size-dependence. To determine the theoretical equilibrium shape of these tactoids, we numerically minimize the total free energy : (eq.1) of tactoids with a prescribed shape and director-field configuration over a range of sizes. A global least squares fit of our theoretical predictions to the measurements on the aspect ratio and the size of the droplets allows us to determine quantitative characteristics of the nematic, isotropic, and solid surfaces, including the line tension (Figure 3). We note that there is an experimental uncertainty associated with the shape and size measurements and the quasi-equilibrium nature of the system that introduces scatter to the data–see Supplementary Information. We performed a bootstrap statistical analysis to find the standard deviation on each of the parameters.40 The original data set shown in Figure 3 was randomly sampled 100 times with replacement. The theory was then fitted to the resampled data to obtain the mean value and the standard deviation for the parameters–see boxplots in Figure S2 and S3. The first three parameters related to the nematic-isotropic interface for sessile tactoids are ] = 25.5 ± 0.83 µm, _WW = 0.34 ± 0.27, and \FG = 1.85 ± 0.08. These values are in agreement with what we found for nematic tactoids of CNT-CSA solution floating freely in the bulk isotropic phase (] = 18.5 ± 1.5 µm, _WW = 0.3 ± 0.22, and \FG = 3.16 ± 0.17). The solid surface affinity of ∆ = 0.0017 ± 0.0016 confirms the affinity of the solid surface for the ACS Paragon Plus Environment

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nematic phase and explains the formation of partially wetting tactoids. We also present our theoretical results on the effect of affinity parameter on the shape of tactoids in Figure S4 of the Supporting Information. The nematic-solid anchoring of \F; = 121 ± 11 shows that CNTs, similar to other rod-like particles, prefer to lie parallel to the nematic-solid interface (\F; > 0). Calculations based on the density functional theory show that for rod-like particles of an aspect ratio between 5 and 20, the ratio of \F; /\FG varies between 6 and 4.34, 41 The theoretical ratio of +/) shown in the inset of Figure 3B indicates that our tactoids are cylindrically symmetric in shape (+/) → 1) with boojums located on the solid surface and close to the droplet poles, forming a bipolar directorfield configuration (Figure S6). Therefore, there is no anchoring at the nematic-solid interface. Our theoretical results on the effect of \F; on the shape of the tactoids in Figure S5 of the Supporting Information shows that if \F; /\FG > 10 the aspect ratio of the droplets slightly changes with further increasing of the \F; . Therefore, the fitted curve is insensitive to the value of nematic-solid anchoring, which decreases the number of free energy parameters in the system. The line tension characteristic length scale, defined as the ratio of the three-phase contact line tension to nematic-isotropic surface tension, GF; /FG , is about −0.84 ± 0.06 µm for the sessile tactoids. To assess the method ability to distinguish sessile and free tactoids, we fit the model to free tactoids and extracted the line tension length scale in the bulk isotropic phase (which has no physical meaning); the result is 0.0013 ± 0.0006 µm, which is essentially zero, demonstrating that the method correctly distinguishes line tension in sessile from free tactoids (p-value. 10j). In order to calculate explicitly the three-phase contact line tension GF; , we use FG = 0.156 U /! for a monodisperse hard rod system with !/ ≫ 1.42 Here, is the absolute temperature and U is the Boltzmann constant. For our CNT system, with diameter ~2.18 nm


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and aspect ratio !/~509, this translates to FG ~0.265 µN/m. Therefore, the value of line

tension at the three-phase contact line is GF; ~ − 0.22 pN.

Figure 3. (A) Top-view aspect ratio / for free tactoids in the isotropic bulk phase (triangles) and top-view aspect ratio for sessile tactoids of the same system (circles) and (B) side-view aspect ratio / of the sessile tactoids (squares) vs. size represented by the major axis length of the tactoids formed in a 0.1 % by mass carbon nanotube-chlorosulfonic acid solution. The solid curves are the best theoretical fits to the data ( = µm, = '. ', = (. , = ((, ∆= '. '''(, and / = − '. µm for sessile tactoids and = (. µm, = (. ', = . , = . , ∆= '. ', and / = '. ''' µm for free tactoids). The inset in (B) shows the ratio of the depth to the minor axis of the sessile tactoids vs. major axis length .

The relative value of line tension and nematic-isotropic surface tension, |GF; /FG |~0.84 ± 0.06 µm, is a characteristic length scale of the system studied that is proportional to the droplet size below which the line tension effect becomes relevant.5, 10 This characteristic length scale is two orders of magnitude larger than what has been reported for the droplets of conventional fluids (0.05-1.4 nm).5, 10 Using the theoretical expressions for line tension and nematic-isotropic surface tension for a monodisperse system of rods, we can determine how this length scale depends on


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the dimension of the constitutive mesogens. The line tension scales with U /,43 while the nematic-isotropic surface tension for a monodisperse system of hard rods scales with U /!.28, 42

Therefore, the ratio GF; /FG scales with the size of mesogens in the system (GF; /FG ∝ !),

shifting this length scale to the micron range in colloidal systems and making the line tension characterization easier. We also did the same analysis on a different samples of CNTs with different lengths (see Figure S7). Our results show the strong length dependence of the line tension characteristic length GF; /FG .

The theory also enables us to analyze the impact of line tension on the shape of tactoids (Figure 4). We set the value of free energy parameters to those obtained for the CNT-CSA system reported earlier and varied the line tension characteristic length scale between −1.2 and 2.4 µm. For negative values of the line tension, the contact line is pulled outward, while for positive values of line tension the contact line is pulled inward. For sufficiently negative values of line tension, small tactoids spread to become highly elongated two-dimensional domains on the solid substrate, and the contact angle approaches zero. We can distinguish two regimes for the aspect ratio of the droplets depending on the sign of the line tension and droplet size (Figure 4). For positive values of the line tension GF; > 0 and increasing the droplet size, the shape of a sessile tactoid first transitions from spheroidal, where /) , /+ → 1, to elongated, where /) , /+ ≫ 1, then back towards a spherical shape. Small tactoids are line tension dominated, intermediate ones are elasticity-dominated, and large ones are surface tension dominated. Alternatively, for negative values of line tension, GF; . 0, the shape of a sessile tactoid transitions from highly elongated to spheroidal by increasing the tactoid size. The elongated shape is dominated by elasticity and line tension; the spheroidal shape is dominated by surface


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tension. Therefore, it is clear that for our data with a continuous transition from highly elongated to spheroidal shape, the line tension in our CNT system must be negative.

For both negative and positive values of line tension, the ratio of depth to minor axis, )/+ exhibits a minimum with increasing size of the tactoid (Figure 4B inset). The minimum value of the )/+ curve corresponds to the most asymmetric structure with less than 5% change in the level of symmetry. Figure 4B shows that varying the value of the line tension does not strongly affect the asymmetry of the sessile tactoids. If we further decrease the value of the line tension, that is, GF; /FG . −5 µm, small tactoids become two-dimensional domains with )/+ → 0 for the set of parameters used in our calculations, corresponding to a complete wetting regime.

Figure 4. Influence of the ratio of line tension to surface tension, / , and the tactoid size on (A) the top-view aspect ratio / and (B) the side-view aspect ratio / of sessile tactoids calculated for the sessile tactoids formed in the solutions of carbon nanotubes in chlorosulfonic acid. The inset shows the ratio of the depth to the minor axis / of the tactoids with a minimum point corresponding to the most asymmetric structure.

CONCLUSION


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In conclusion, we have shown that for lyotropic liquid crystal solutions comprised of micron-sized colloidal mesogens, the ratio of line tension to surface tension is shifted to larger length scales (micron range), which is orders of magnitude larger than that of conventional fluids. The value of the line tension characteristic length scale of ~ − 0.84 ± 0.06 µm agrees with theoretical predictions and explains the formation of partially wetting, highly elongated sessile droplets in the small size regime.

We also presented our theoretical model that captures the shape of three-dimensional liquid crystal droplets with anisotropic contact angles. Previous efforts on developing the equivalent of the Young equation for liquid crystal droplets considered only two-dimensional droplets44 or neglected the effect of line tension.45,

46

We showed that elasticity in competition with

anisotropic surface forces plays an important role in determining the spindle shape and internal structure of the sessile tactoids, as is the case for tactoids in bulk dispersions.16,

27

While our

proposed model is specific to sessile lyotropic liquid crystal droplets on flat non-deformable solid interfaces, it can be extended to curved interfaces formed in contact with other fluids.

ASSOCIATED CONTENT Supporting information Sources of inaccuracy in the experimental data, effect of solid surface affinity and solid-nematic anchoring on the shape of the droplets, the theoretical predicted value of the level of bipolarness for sessile tactoids formed in CNT-CSA solutions, boxplots for the bootstrapped parameters of sessile and free tactoids in CNT-CSA system, additional data on the length dependence of the line tension characteristic length, and theoretical calculations for transition size from an un-


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twisted to a twisted configuration. The supporting Information is available free of charge on the ACS Publication website at DOI: XXX.

AUTHOR INFORMATION Corresponding author *E-mail [email protected]. Tel.: +1 713 348 5830. Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS We thank Dan Marincel, Francesca Mirri, Robert Pinnick, and Lauren Taylor for helpful discussions. This research was supported by Air Force Office of Scientific Research (AFOSR) grants FA9550-12-1-0035, FA9550-15-1-0370, the Robert A. Welch Foundation (C-1668), National Science Foundation (NSF) grant number CMMI-1025020.

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Line Tension of Twist-Free Carbon Nanotube Lyotropic Liquid Crystal

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