Polarized epifluorescence microscopy and the imaging of nematic liquid crystals in highly curved geometries

We develop polarized epifluorescence microscopy (PFM), a technique to qualitatively determine a director field, even when refraction effects are too strong to use optical polarized microscopy. We present the basic theory behind the technique and cover in detail the experimental setup. We validate PFM on the well-studied cases of a planar nematic cell, spherical nematic droplets, and a cylindrical capillary filled with nematic liquid crystal. Last, we use nematic capillary bridges to demonstrate that PFM can indeed provide measurements of the director field, even when refraction effects are large.

Figure 1

(a) The director, $\mathbf{n}$, describes the average alignment direction of the NLC molecules (gray ellipses). Due to their shape anisotropy, fluorophores (red rounded rectangles) will also be aligned along $\mathbf{n}$. (b) Coordinate system defining $\delta$ and ${\mathrm{\Phi }}_A$, the orientation of a fluorophore and the analyzer in the $xy$ plane, respectively. (c–f) Schematics of (c, d) radial and (e, f) hyperbolic (c, e) hedgehog and (d, f) ring defect structures.

Figure 2

(a) Schematic of a microscope setup for polarized epifluorescence microscopy (PFM) [4]. (b, c) PFM images of a planar cell filled with Nile red-doped 5CB oriented with rubbing direction along $\widehat{x}$ and the analyzer at $0^{\circ }$ and ${90}^{\circ }$ relative to $\widehat{y}$, respectively.

Figure 3

(a–d) Grayscale transmitted intensity vs. analyzer orientation for PFM images of a planar cell oriented with the rubbing direction along (a, d) $\widehat{x}$, (b) ${25}^{\circ }$ counterclockwise from $\widehat{y}$, and (c) ${45}^{\circ }$ counterclockwise from $\widehat{y}$. The cells in panels (a–c) were measured in the nematic phase while the cell in panel (d) was melted to the isotropic phase before measuring. The solid blue curves in panels (a–c) are fits of the data to Eq. (1), returning ${\delta }^{\prime }={89}^{\circ }$, ${\delta }^{\prime }=2^{\circ }$, and ${\delta }^{\prime }={67}^{\circ }$, respectively. The black dashed lines in panels (b, c) correspond to the theoretical curves for ${\delta }^{\prime }={25}^{\circ }$ and ${\delta }^{\prime }={45}^{\circ }$, respectively.

Figure 4

(a, b) Optical polarized microscopy (OPM) images with no sample, the polarizer (P) and analyzer (A) crossed, and the dichroic mirror in the light path. ${\mathrm{\Phi }}_A$ is indicated by the white arrow in the bottom-right of each image. (c) Transmitted intensity as a function of ${\mathrm{\Phi }}_A$ for the same setup as in panels (a, b).

Figure 5

(a) Bright-field images of radial droplets made from an emulsion of Nile red-doped 5CB dispersed in water with 8 mM sodium dodecyl sulfate added to enforce homeotropic anchoring. (b, c) Corresponding OPM images of the droplets in (a) with the P and A analyzer indicated in each image. (d–i) PFM images of the droplets in panel (a) with the A orientation indicated in each image. (d, e) Superpositions of two PFM images, where the images have the same ${\mathrm{\Phi }}_A$ but opposite analyzer wedge-angle orientations. (d) is composed of the raw data and is blurry, indicating a net displacement due to the wedge character of the analyzer. The images forming (e) have been shifted to correct for the displacement due to the analyzer wedge angle, resulting in a superposition with less blurriness. (f) PFM image after converting to grayscale and blurring with a square Gaussian filter with side length 7 px. (g) is the result of downsampling the image in panel (f) with panels (h, i) similarly processed images with different ${\mathrm{\Phi }}_A$. (j) PFM intensity as a function of ${\mathrm{\Phi }}_A$ from the highlighted pixel in panels (g–i). The blue curve is a fit to Eq. (1), returning ${\delta }^{\prime }={88}^{\circ }$. (k) The ${\delta }^{\prime }$ from a PFM analysis of every pixel in panels (g–i) plotted on top of an epifluorescence image of the droplets. Scale bar in panel (a) is 25 $\mu \mathrm{m}$.

Figure 6

(a) Schematic from the side showing how the wedge angle of the analyzer causes light to be translated by $\mathrm{\Delta }\rho$ in the direction of the wedge, ${\mathrm{\Phi }}_W$. (b) Schematic from the top showing the angular displacement $\psi$, between the analyzer pass axis, ${\mathrm{\Phi }}_A$, and the wedge angle, ${\mathrm{\Phi }}_W$.

Figure 7

(a) OPM image taken with a $5\times$ objective of a 600 $\mu \mathrm{m}$ inner diameter cylindrical capillary filled with Nile red-doped 5CB under homeotropic anchoring. (b) Associated bright field image with ${\delta }^{\prime }$ from the PFM analysis plotted on top of the bright field image. (c) PFM image of the capillary in panel (a) taken with a $20\times$ objective and centered on the defect. The ${\delta }^{\prime }$ from the associated PFM analysis are plotted on top of the image. (d) ${\delta }^{\prime }$ plotted vs. the position across the capillary for the cross-section within the white box in (b). The blue curve is $2arctan(x/R)$, the theoretical expectation for an escaped radial configuration in the one-constant approximation. Scale bar is 250 $\mu \mathrm{m}$ for (a, b) and 125 $\mu \mathrm{m}$ for (c). Panels (b) and (c) reproduced from Ref. [4].

Figure 8

(a) Schematic of the experimental setup to make a capillary bridge. (b–d) Bright field, OPM, and PFM images, respectively, of an example capillary bridge with $\mathrm{\Gamma }\approx 1.2$ and made from Nile red-doped 5CB. Scale bar is 150 $\mu \mathrm{m}$.

Figure 9

Bright-field images of a (a) waist-shaped and a (c) barrel-shaped bridge, with the radius $R$ and height $H$ defined in the image. The waist in panel (a) has $R=180\mu \mathrm{m}$ and $H=170\mu \mathrm{m}$, and the barrel in panel (c) has $R=280\mu \mathrm{m}$ and $H=300\mu \mathrm{m}$. (b, d) Contours of the bridge shown in panels (a, c), respectively, at different $\mathrm{\Gamma }=2R/H$, where the positions have been scaled by $H$ and the (open symbols) lower contours have been reflected and shifted to line up with the (filled symbols) upper contours. The contours in panel (b) have (squares) $\mathrm{\Gamma }=7.3$, (circles) $\mathrm{\Gamma }=4.6$, (upward triangles) $\mathrm{\Gamma }=2.0$, (downward triangles) $\mathrm{\Gamma }=1.0$, (diamonds) $\mathrm{\Gamma }=0.6$, and (stars) $\mathrm{\Gamma }=0.4$. The contours in panel (d) have (squares) $\mathrm{\Gamma }=4.5$, (circles) $\mathrm{\Gamma }=3.9$, (upward triangles) $\mathrm{\Gamma }=1.9$, (downward triangles) $\mathrm{\Gamma }=1.5$, and (diamonds) $\mathrm{\Gamma }=1.3$.

Figure 10

(a) Elevation angle $\theta$ and arclength parameter $s$ defined for a surface of revolution in cylindrical coordinates $\{r,\phi ,z\}$, with a contour of the surface drawn as a solid line. $\theta$ is the angle between the tangent of the solid curve at $s$ and $\widehat{r}$. The contact angle ${\theta }_0=\theta (s=0)$ is shown in the magnified section in the circle. (b, c) Numerically calculated contours for a constant mean curvature surface of revolution plotted on experimentally measured contours for (b) waist- and (c) barrel-shaped bridges. The numerical solutions have contact angles (b) (upper contour) ${\theta }_0={37}^{\circ }$, (lower contour) ${\theta }_0={35}^{\circ }$, and (c) (upper contour) ${\theta }_0^{\mathrm{barrel}}={149}^{\circ }$, (lower contour) ${\theta }_0^{\mathrm{barrel}}={120}^{\circ }$. The measured contours in panels (b, c) are reproduced from and form the envelope of those in Figs. 9 and 9, respectively.

Figure 11

(a–d) PFM analysis on waist-shaped bridges with (a, b) $\mathrm{\Gamma }\approx 0.8$ and (c, d) $\mathrm{\Gamma }\approx 4.0$. The ${\delta }^{\prime }$ are plotted on the epifluorescence images in panels (b, d) with the associated bright field images in panels (a, c). The scale bars are $250\mu \mathrm{m}$, with those in panels (a, c) and panels (b, d) the same.

Figure 12

(a–d) PFM analysis on barrel-shaped bridges with (a, b) $\mathrm{\Gamma }\approx 1.5$ and (c, d) $\mathrm{\Gamma }\approx 6.6$. The ${\delta }^{\prime }$ are plotted on the epifluorescence images in panels (b, d) with the associated bright field images in panels (a, c). The scale bars are $250\mu \mathrm{m}$, with those in panels (a, c) and panels (b, d) the same. Panels (a, b) are reproduced from Ref. [4].