Angular optical response of cellulose nanocrystal films explained by the distortion of the arrested suspension upon drying

Cellulose nanocrystals (CNCs) are biosourced chiral nanorods that can form stable colloidal suspensions able to spontaneously assemble above a critical concentration into a cholesteric liquid crystal, with a cholesteric pitch usually in the micron range. When these suspensions are dried on a substrate, solid films with a pitch of the order of few hundreds of nanometers can be produced, leading to intense reflection in the visible range. However, the resulting cholesteric nanostructure is usually not homogeneous within a sample and comports important variations of the cholesteric domain orientation and pitch, which affect the photonic properties. In this work, we first propose a model accounting for the formation of the photonic structure from the vertical compression of the cholesteric suspension upon solvent evaporation, starting at the onset of the kinetic arrest of the drying suspension and ending when solvent evaporation is complete. From that assumption, various structural features of the films can be derived, such as the variation of the cholesteric pitch with the domain tilt, the orientation distribution density of the final cholesteric domains and the distortion of the helix from the unperturbed cholesteric case. The angular-resolved optical response of such films is then derived, including the iridescence and the generation of higher-order reflection bands, and a simulation of the angular optical response is provided, including its tailoring under external magnetic fields. Second, we conducted an experimental investigation of CNC films covering a structural and optical analysis of the films. The macroscopic appearance of the films is discussed and complemented with angular-resolved optical spectroscopy, optical and electron microscopy, and our quantitative analysis shows an excellent agreement with the proposed model. This allows us to access the precise composition and the pitch of the suspension when it transited into a kinetically arrested phase directly from the optical analysis of the film. This work highlights the key role that the anisotropic compression of the kinetically arrested state plays in the formation of CNC films and is relevant to the broader case of structure formation in cast dispersions and colloidal self-assembly upon solvent evaporation.

Figure 1

Schematic of a CNC suspension (in blue) drying in a petri dish. Black arrows represent the vertical compression experienced upon drying after the kinetic arrest has occurred.

Figure 2

Schematic diagram of a tilted and left-handed cholesteric domain (a) before deformation, (b) after a unidirectional compression along $\widehat{\mathbf{z}}$ scaling with a factor $\alpha$, and (c) after redefining the helical axis as ${\widehat{\mathbf{m}}}^{\prime }$. The directors $\widehat{\mathbf{n}}$ and ${\widehat{\mathbf{n}}}^{\prime }$ are depicted by double-headed arrows (in blue) to account for their symmetry by inversion.

Figure 3

Pitch compression ratio $p^{\prime }/p$ for two different values of $\alpha$. The ratio $p^{\prime }/p$ is reported in terms of the final tilt ${\beta }^{\prime }$ of the domain. Dashed lines illustrate the trajectories of the combined pitch decrease and domain reorientation as $\alpha$ decreases upon compression, exemplified for a set of different initial angles (open symbols).

Figure 4

Distortion of the helical order illustrated by the variation of the helical angle ${\varphi }^{\prime }$ after unidirectional deformation. (Top) Different compression ratios $\alpha$ at fixed initial tilt $\beta ={45}^{\circ }$. (Bottom) Constant $\alpha =0.1$ for different initial tilts $\beta$.

Figure 5

Multiple cross-section views of a distorted cholesteric domain with final orientation ${\beta }^{\prime }\approx 9.8^{\circ },{\phi }^{\prime }=0^{\circ }$, after a compression of $\alpha =0.1$ from an original orientation $\beta ={60}^{\circ },\phi =0^{\circ }$. The local orientation of ${\widehat{\mathbf{n}}}^{\prime }$ is depicted by small arrows, while the continuity of ${\widehat{\mathbf{n}}}^{\prime }$ across each face leads to the Bouligand arches, depicted by full lines. (a) Corner view in perspective; (b) top view, generating periodic variations $\mathrm{\Delta }=p^{\prime }\left({\beta }^{\prime }\right)/(2sin{\beta }^{\prime })$ of ${\widehat{\mathbf{n}}}^{\prime }$, usually in the micron range, (c) cross-section in the plane containing ${\widehat{\mathbf{m}}}^{\prime }$ (i.e., at an azimuthal angle $\mathrm{\Delta }{\phi }^{\prime }=0^{\circ }$), (d) cross-section at an azimuthal angle $\mathrm{\Delta }{\phi }^{\prime }={30}^{\circ }$ leading to asymmetric Bouligand arches, arising from the distortion of the cholesteric order (the dotted line highlights the asymmetry), and (e) cross-section in the (${\widehat{\mathbf{m}}}^{\prime },\widehat{\mathbf{z}}$) plane (i.e., at an azimuthal angle $\mathrm{\Delta }{\phi }^{\prime }={90}^{\circ }$), leading to symmetric Bouligand arches, in spite of the distortion of the cholesteric order.

Figure 6

Random sample of $N=1200$ configurations illustrating on the left an initially isotropic orientation distribution function (ODF) [i.e., $f_0\left(\beta \right)=1/2\pi$] and on the right its transformation $f_d\left({\beta }^{\prime }\right)$ after compression (assuming no anchoring at the interfaces).

Figure 7

(a) Orientation distribution function (ODF) of the helices in the initial isotropic state [$f_0\left(\beta \right)=1/2\pi$, in red] and its transformation after compression [$f_d\left({\beta }^{\prime }\right)$, in blue], assuming no anchoring at the interfaces. (b) Same ODFs multiplied by the spherical Jacobian (full lines) and their corresponding cumulative distribution functions (CDF, in dashed lines). The histograms correspond to the sampling of the configurations shown in Fig. 6.

Figure 8

Average pitch $\langle p^{\prime }\rangle$ compared with the minimal pitch $p^{\prime }\left(0\right)=\alpha p$ in the film and computed for different $\alpha$ ratios using Eqs. (C5) and (C6), assuming different initial distributions of the domains: purely isotropic ($\gamma =0$) and under a vertical magnetic alignment with $\gamma =1$ and 10 and no anchoring. Note that a range of values for $\langle p^{\prime }\rangle /\left(\alpha p\right)$ closer to unity than the isotropic case are also possible from the sole effect of anchoring (cf. Sec. 2g3).

Figure 9

Random sample of $N=1200$ configurations with on the left an initially anisotropic ODF [i.e., $f_H\left(\beta \right)\sim e^{\gamma {(\widehat{\mathbf{m}}·\widehat{\mathbf{H}})}^2}$] resulting from the magnetic alignment of the helices along an external magnetic field $\mathbf{H}$ tilted by an angle ${\beta }_H$ and on the right its transformation after compression ($\alpha =0.1$), assuming no anchoring at the interfaces.

Figure 10

Schematic of the light propagation inside the film in the plane ($\widehat{\mathbf{x}},\widehat{\mathbf{z}}$), with the angle definitions used to apply Fergason's law (i.e., Bragg's law corrected by Snell's law).

Figure 11

Intensity factors $C_l$ of the different orders $l$ of the reflected light, as given by Eq. (E7), in the case of incident and outgoing light propagation parallel to ${\widehat{\mathbf{m}}}^{\prime }$, implying ${\theta }_{\mathrm{o}}=-{\theta }_{\mathrm{i}}$. The reported intensity is not corrected by the Fresnel transmission factors or the solid angle transformations.

Figure 12

Schematic of the optical path inside the film, connecting the local cholesteric orientation of a given domain after deformation (${\beta }^{\prime },{\phi }^{\prime }$) and the external angles of incident and outgoing beams (${\theta }_{\mathrm{i}},{\theta }_{\mathrm{o}},{\psi }_{\mathrm{o}}$).

Figure 13

Simulation of the angular optical response of the films obtained assuming $\alpha =0.1$ and no anchoring at the interfaces with an incident light at [(a), (c), (e), and (g)] ${\theta }_{\mathrm{i}}=0$ and [(b), (d), (f), and (h)] ${\theta }_{\mathrm{i}}={30}^{\circ }$, and a magnetic alignment obtained for [(a) and (b)] $\gamma =0$ (no $H$ field), [(c) and (d)] $\gamma =10,{\beta }_H=0^{\circ },{\psi }_H=0^{\circ }$, [(e) and (f)] $\gamma =10,{\beta }_H=-{30}^{\circ },{\psi }_H=0^{\circ }$, and [(g) and (h)] $\gamma =10,{\beta }_H=-{30}^{\circ },{\psi }_H=-{90}^{\circ }$. The colored regions of the half-spheres visible at specific solid angles indicate the color observed in the direction pointing away from the half-sphere center. The incident and specularly reflected beam directions are represented with gray arrows.

Figure 14

Macroscopic photographs of CNC films prepared from a suspension of [(a)–(c)] higher salt concentration or [(d)–(f)] lower salt concentration, and in absence of magnetic field [(a) and (d)], in a vertical field [(b) and (e)], and in a field locally exploring different tilts [(c) and (f)]. Each film was observed in specular (left half) and off-specular (right half) conditions. Pictures (a)–(e) are adapted with authorization under the terms of the CC-BY 4.0 license [23].

Figure 15

Macroscopic photographs of a single flake of a polydomain CNC film [R = 100 $\mu \mathrm{mol}/\mathrm{g}$, $H=0$, cf. Fig. 14]. Reflection in (a)–(c) specular and (d)–(f) off-specular conditions using either [(a) and (d)] no polarizing filter, [(b) and (e)] a left-circularly polarization (LCP) filter, and [(c) and (f)] a right-circularly polarization (RCP) filter (scale bar 1 cm). The polarization of the reflected light by the cholesteric structure is LCP in specular conditions, whereas in off-specular both polarization states are present.

Figure 16

Optical microscopy images of CNC films observed in reflection through a filter selecting left-circularly polarized light (the brightness was enhanced for gray scale printing purposes). (a) Polydomain film displaying disordered stripy patterns with various periodicities $\mathrm{\Delta }$ ranging from 0.85–2 $\mu \mathrm{m}$ [$H=0$, cf. Fig. 14]. (b) Monodomain film made under a tilted magnetic field, displaying uniform stripy patterns with regular periodicities $\mathrm{\Delta }\approx 2$ $\mu \mathrm{m}$ $[{\mu }_{0}H\approx 0.5$ T, ${\beta }_H\approx {50}^{\circ }$, cf. Fig. 14]. (c) Monodomain film made under vertical field, displaying uniform color with a diverging $\mathrm{\Delta }$ [${\mu }_{0}H\approx 0.5$ T, ${\beta }_H\approx 0^{\circ }$, cf. Fig. 14].

Figure 17

Angular-resolved optical spectroscopy performed on samples made in absence of magnetic field [(top) sample made with $R=100$ $\mu \mathrm{mol}$ ${\mathrm{g}}^{-1}$ and (bottom) $R=25$ $\mu \mathrm{mol}$ ${\mathrm{g}}^{-1}$], illuminating at an angle ${\theta }_{\mathrm{i}}={15}^{\circ }$ (intensity reported using a red-yellow-blue heat map in ${\log }_{{}_{10}}$ scale). The fits for the first order (plain curve) and the much weaker second order (dashed) correspond to the model from (39). The specular reflection is observed at ${\theta }_{\mathrm{o}}={\theta }_{\mathrm{i}}={15}^{\circ }$ for all wavelengths. The horizontal dashed lines represent the angles ${\beta }^{\prime }$ of the probed domains, while their initial angles $\beta$ are indicated by the curved dashed lines on the right.

Figure 18

Angular-resolved optical spectroscopy performed on samples prepared under a vertical magnetic field (sample made from $R=25$ $\mu \mathrm{mol}$ ${\mathrm{g}}^{-1}$, intensity reported using a heat map in ${\log }_{{}_{10}}$ scale).

Figure 19

Angular-resolved optical spectroscopy performed on samples prepared under a tilted magnetic field [(top) position $x=10$ mm, $\langle {\beta }_H\rangle ={33}^{\circ }$ and (bottom) $x=0$ mm, $\langle {\beta }_H\rangle ={90}^{\circ }$]. The fits of the first, second, and third orders are represented with full, dashed, and dotted lines, respectively. The dash-dot line corresponds to Eq. (D1). Sample made from $R=25$ $\mu \mathrm{mol}$ ${\mathrm{g}}^{-1}$, intensity reported using a heat map in ${\log }_{{}_{10}}$ scale. The arrow indicates a second order reflection observed for an angle ${\theta }_{\mathrm{loc}}\approx 0^{\circ }$, i.e., parallel to ${\widehat{\mathbf{m}}}^{\prime }$.

Figure 20

SEM image of a cross-section of the sample from Fig. 14. It can be seen that the pitch varies significantly depending on the tilt of the helical axis ${\widehat{\mathbf{m}}}^{\prime }$ (scale bar 2 $\mu \mathrm{m}$).

Figure 21

Asymmetric Bouligand arches resulting from the distortion of the cholesteric order. (Left) SEM observation in cross-section of the sample from Fig. 14. (Right) Asymmetric Bouligand arches reproducing similar asymmetric arches (the asymmetry is illustrated by the position of the dashed line, drawn at the apex of the arch highlighted in red; see details in Appendix pp2).