Effect of electric fields on the director field and shape of nematic tactoids

Tactoids are spindle-shaped droplets of a uniaxial nematic phase suspended in the coexisting isotropic phase. They are found in dispersions of a wide variety of elongated colloidal particles, including actin, fd virus, carbon nanotubes, vanadium peroxide, and chitin nanocrystals. Recent experiments on tactoids of chitin nanocrystals in water show that electric fields can very strongly elongate tactoids even though the dielectric properties of the coexisting isotropic and nematic phases differ only subtly. We develop a model for partially bipolar tactoids, where the degree of bipolarness of the director field is free to adjust to optimize the sum of the elastic, surface, and Coulomb energies of the system. By means of a combination of a scaling analysis and a numerical study, we investigate the elongation and director field's behavior of the tactoids as a function of their size, the strength of the electric field, the surface tension, anchoring strength, the elastic constants, and the electric susceptibility anisotropy. We find that tactoids cannot elongate significantly due to an external electric field, unless the director field is bipolar or quasibipolar and somehow frozen in the field-free configuration. Presuming that this is the case, we find reasonable agreement with experimental data.

Figure 1

(a) Polarized optical micrograph illustrating a number of nematic tactoids of different size and extinction pattern associated with the director field conformation, in a solution of carbon nanotubes in chlorosulfonic acid at 1000 ppm. Arrows show the orientation of crossed polarizers. Schematics of (b) a bipolar tactoid with boojum surface defects at the poles, (c) a homogeneous tactoid, and (d) an intermediate tactoid described by virtual boojums outside of the droplet. The parameters $R$ and $r$ represent the major and minor axes of the tactoid, respectively. Adopted from Ref. [5].

Figure 2

Cross section (solid) and director field (dashed) of a tactoid presumed in our calculations. The droplet is cylindrically symmetric about its main axis. $2R$ denotes the length of a tactoid and $2r$ its width. $2\tilde{R}$ is the distance between the virtual boojums, which are the focal points of the (extrapolated) director field. For $R=\tilde{R}$, the virtual boojums become actual boojums, i.e., surface point defects. Also indicated is $\alpha$, the opening angle of the spindle-shaped droplet (see also the main text).

Figure 3

Dependence of (a) the degree of bipolarness $y$ and (b) the aspect ratio $x$ of tactoids on the scaled volume $v$ in the presence of an electric field, according to the scaling theory. Blue solid line: the field strength $\mathrm{\Gamma }$ is below the critical value ${\mathrm{\Gamma }}_c$, and the anchoring strength $\omega$ is somewhat larger than unity. The volumes $v_{-}$ and $v_{+}$ demarcate crossovers from quasibipolar director fields to bipolar ones, and $v_{<}$ and $v_{>}$ those from elongated to spherical droplet shapes. Dashed-dotted line: same curve, but now for $\mathrm{\Gamma }={\mathrm{\Gamma }}_c$. Dashed lines: $\mathrm{\Gamma }>{\mathrm{\Gamma }}_c$. For fields above the critical field strength ${\mathrm{\Gamma }}_{*}$, the tactoids are always elongated provided the anchoring strength $\omega$ is sufficiently large. See the main text. The slopes of the various curves are listed in Tables 1, 2, and 3, and the values of the various crossover volumes and field strengths in Table 4.

Figure 4

Schematic “phase” diagram of director fields of tactoids in an external field. Plotted is the scaled volume $v$ versus the scaled field strength $\mathrm{\Gamma }$. Indicated are the crossovers between the five regimes, demarcated by the crossover volumes $v_{-}, v_{+}, v_{<}, v_{>}, v_{*}$, and $v_c$. Expressions for the crossover volumes $v_{-}, v_{+}, v_{<}$, and $v_{>}$, and the critical field strengths ${\mathrm{\Gamma }}_c$ and ${\mathrm{\Gamma }}_{*}$, are listed in Table 4. See also the main text. In the region bounded by $v_{-}$ and $v_c$, tactoids with sufficiently large anchoring strength $\omega \gg 1$ are elongated with a director field that is quasibipolar with a bipolarness $y$ that decreases with increasing volume. In the region bounded by $v_c$ and $v_{+}$ in the upper right-hand corner it is quasibipolar with a bipolarness that increases with droplet volume. In region bounded by $v_{-}$ and $v_{+}$ in the upper left-hand corner. the director field is for all intents and purposes bipolar. The tactoids are more or less spherical in the region bounded by $v_{<}$ and $v_{>}$ in the uppermost left-hand corner, and elongated outside of that region, at least if $\omega \gg 1$. The volume $v_c$ demarcates the crossover from decreasing to increasing bipolarness for quasibipolar director fields, the volume $v_{*}$ that between decreasing aspect ratio to increasing aspect ratio.

Figure 5

The bipolarness $y$ of the director field of a tactoid in the presence of an electric field as a function of its dimensionless volume $v$. Indicated by the different symbols are results for different values of the dimensionless electric-field strength $\mathrm{\Gamma }$. The dimensionless anchoring strength is fixed at $\omega =14$ and the dimensionless bend constant at $\kappa =10$.

Figure 6

Bipolarness $y$ of a tactiod as a function of the dimensionless electric field $\mathrm{\Gamma }$ for a dimensionless volume $v={10}^7$. Anchoring strength $\omega =14$ and dimensionless bend constant $\kappa =10$. The solid line shows the scaling of $y$ as with ${\mathrm{\Gamma }}^{0.5}$.

Figure 7

Bipolarness $y$ of the droplet as a function of the dimensionless volume $v$ of tactoids in the presence of an electric field for different values of the dimensionless bend constants $\kappa$ indicated by the symbols. Anchoring strength $\omega =14$ and dimensionless field strength $\mathrm{\Gamma }=100$.

Figure 8

Bipolarness of a tactoid as a function of the dimensionless bend constant $\kappa$ in the presence of an electric field. Anchoring strength $\omega =1.4$, dimensionless field strength $\mathrm{\Gamma }=100$, and dimensionless volume $v={10}^{-4}$.

Figure 9

Bipolarness $y$ of a tactoid as a function of the dimensionless volume $v$ in the presence of an electric field for different values of the anchoring strength, indicated by the symbols. Dimensionless field strength $\mathrm{\Gamma }=100$ and bend constant $\kappa =10$.

Figure 10

Bipolarness $y$ of a tactoid as a function of an anchoring strength $\omega >1$ for small and large droplets, with dimensionless volumes $v={10}^{-2}$ and ${10}^6$. The dimensionless electric field strength is fixed at $\mathrm{\Gamma }=10$ and the dimensionless bend constant at $\kappa =0$. Indicated are also the scaling relations $y\sim {\omega }^{-0.43}$ for the small volume and $y\sim {\omega }^{-0.62}$ for the large volume. See also the main text.

Figure 11

Bipolarness $y$ of a tactoid as a function of anchoring strength $\left(\omega <1\right)$ for small and large droplets, with dimensionless volumes $v={10}^{-2}$ and ${10}^6$. The dimensionless electric field strength is fixed at $\mathrm{\Gamma }=10$ and the dimensionless bend constant at $\kappa =0$. Indicated are also the scaling relations $y\sim {\omega }^{-0.51}$ and $y\sim {\omega }^{-0.52}$. See also the main text.

Figure 12

Bipolarness $y$ as a function of anchoring strength $\omega$ for two different dimensionless bend constants of $\kappa =0.1$ and ${10}^3$. The dimensionless volume is set at a value $v={10}^{-4}$. Also indicated are the scaling exponent of $-0.43$ for the small value of the bend constant and of $-0.90$ for the large value of the constant. See also the main text.

Figure 13

The aspect ratio $x$ of a tactoid as a function of the dimensionless volume $v$ for different (dimensionless) electric field strengths $\mathrm{\Gamma }$ indicated by the symbols. The anchoring strength is set at $\omega =14$ and the dimensionless bend constant at $\kappa =10$.

Figure 14

Aspect ratio $x$ and bipolarness $y$ of the director field as a function of volume of a droplet in the absence of the electric field with $\mathrm{\Gamma }=0$. Left vertical axis shows us the aspect ratio (we use blue triangles for the aspect ratio) and the right vertical axis shows us bipolarness (cross signs for bipolarness) and the red circles represent the experimental data of [39]. The best fits we obtain by eye are for the parameter values $\kappa =20, \omega =1.3$, and $\left(K_{11}-K_{24}\right)/\sigma =4\mu \mathrm{m}$. Notice that the largest tactoids are bipolar because $y\to 1$ and the smallest ones quasibipolar with $y\approx 3$.

Figure 15

Aspect ratio $x$ of tactoids as a function of volume $V$ in the presence of an electric field. We compare our numerical results with the experimental data of [39]. The best fit by eye we obtain taking as parameter values $\mathrm{\Gamma }=500, \omega =1.3, \kappa =20$, and $\left(K_{11}-K_{24}\right)/\sigma =4\mu \mathrm{m}$.

Figure 16

Aspect ratio $x$ of a tactoid as a function of the dimensionless volume $v$ for different electric-field strengths $\mathrm{\Gamma }$ according to the restricted equilibrium model. See the main text. Anchoring strength $\omega =14$ and bend elastic constant $\kappa =10$.

Figure 17

Aspect ratio $x$ of a tactoid as a function of the electric field $\mathrm{\Gamma }$ according to the restricted-equilibrium model. Anchoring strength $\omega =14$ and dimensionless bend constant $\kappa =10$. Different symbols show different volumes: triangles $v={10}^7$, squares $v={10}^8$, and circles $v={10}^9$. Indicated are also the scaling exponents, which are close to 0.4 for the three tactoid volumes.