Mechanogeometry of nanowrinkling in cholesteric liquid crystal surfaces

Biological plywoods are multifunctional fibrous composites materials, ubiquitous in nature. The chiral fibrous organization is found in chitin (insects), cellulosics (plants), and collagen I (cornea and bone of mammals) and is a solid analog of that of cholesteric liquid crystals. The surface and interfaces of plywoods are distinguished by hierarchical topographies and nanowrinkling. In this paper, we present a theory to model the emergence of these surfaces and interfaces using liquid crystal-based shape equations that directly connect material properties with geometric wrinkling. The model applies to liquid crystal precursors of the plywood solid analoges. We focus on wrinkling geometry, wrinkling mechanics, and the mechanogeometry relationships that underlie multifunctionality ubiquitous in biological surfaces. Scaling wrinkling laws that connect mechanical pressures and stresses to folding and bending are formulated and quantified. A synthesis of the connections between mechanics and geometry is achieved using the topology of stress curves and curvature of the wrinkles. Taken together the results show that anchoring is a versatile surface morphing mechanism with a rich surface bending stress field, two ingredients behind many potential multifunctionalities.

Figure 1

Schematic of surface wrinkling of a cholesteric liquid crystal; surface relief $\left[h\right(x\left)\right]$, geometric frame $(\mathbf{t},\mathbf{k})$, surface curvature ($\kappa$), surface mechanical fields $(\mathit{\Xi },{\mathbf{T}}_v)$; $\mathbf{t}$ is the unit surface tangent vector, $\mathbf{k}$ is the outward unit normal, $\mathit{\Xi }$ is the capillary vector, and ${\mathbf{T}}_v$ is the surface stress vector. Far from the interface, the cholesteric helix unit vector $\mathbf{H}={\widehat{\mathit{\delta }}}_x$ is horizontal and the director field is a pure twist: $\mathbf{n}\left(x\right)=(0,n_y,n_z)$; $\mathbf{n}·\mathbf{H}=0$. In the interfacial region the helical director field uncoils $\mathbf{n}\left(x\right)=(n_x,n_y,0)$ into a planar splay-bend director field and $\mathbf{n}·{\widehat{\mathit{\delta }}}_z=0$. The surface and bulk region are separated by an array of low-energy nonsingular ${\lambda }^{+}$ disclination lines [11, 12]. The chirality of the bulk director is transferred to the surface director in terms of spatial gradient information. The surface curvature is periodic and the period is equal to half the cholesteric pitch $P_0/2$. Capillary vector $\mathit{\Xi }$ generalizes the surface tension under anisotropy. A counterclockwise rotation of $\mathit{\Xi }$ gives the stress vector ${\mathbf{T}}_v$. The orthonormal geometrical frame $(\mathbf{t},\mathbf{k})$ is related to the orthogonal mechanical frame $(\mathit{\Xi },{\mathbf{T}}_v)$ by a dilation-rotation transformation, as shown in this paper.

Figure 2

Organization and flow of information in this paper. App. denotes Appendix. Geo. denotes geometry, and Mech. denotes mechanics.

Figure 3

Wrinkling phase diagram as a function of anchoring coefficients. The top semicircle shows the surface profile $h\left(x\right)$ and the bottom semicircle shows the surface tension profiles $\gamma \left(x\right)$. The resonance line $\mathfrak{R}$ is ${\mu }_{2*}+{\mu }_{4*}=0$ and the critical line $\mathfrak{C}$ is ${\mu }_{2*}+2{\mu }_{4*}=0$ (see text). At the resonance line $\mathfrak{R}$ the quartic and quadratic harmonics is a maximum. The sectors within the $\mathfrak{C}$ and ${\mu }_{4*}$ lines display two waves, and the rest of the sectors only one wave. According to Eq. (36) the surface tension $\gamma$ is proportional to the surface relief $h\left(x\right)$ and hence essentially the same; thus we show $h\left(x\right)$ on the top semicircle and $\gamma$ on the bottom semi-circle. The symmetry of the $h\left(x\right)$ profiles (or the $\gamma$ profiles) follow the graphical equations in Sec. 3a.

Figure 4

Mean value $m$ (a) and arithmetic mean Ra (b) as a function of two dimensionless anchoring coefficients. See Table 1 and Appendix pp3 for definitions and details. The following discussion refers to Fig. 5, where we fix the magnitude of $\mathit{\mu }$ as a constant to discuss the contributions of ${\mu }_2$ and ${\mu }_4$ to the statistical properties of the surface. The mean vanishes close to the resonance line and the maximum is along a line normal to this line. The arithmetic mean is a minimum along the resonance line and increases normal to this line. The labels $N1\text{–}N4$ denote characteristic roughness values used to characterize rough surfaces, with $N1$ having low and $N4$ high roughness. The location of $N1\text{–}N4$ in the anchoring diagram shows the practical applicability of anchoring.

Figure 5

Extremum values distribution for $m$ and $\sigma$ restricted by ${\mu }_{2*}^2+{\mu }_{4*}^2=C^2$ within ${\mathrm{\Lambda }}_4$ space, with number next to each line representing its slope. The mini-max lines crisscrossing the anchoring phase diagram show the versatility of anchoring as a wrinkling force. The radial lines also show that essential wrinkling qualities depend on the ratio of the anchoring coefficients (azimuthal angle in this diagram) and not on $C$ (sum of anchoring amplitudes square or radial direction in this diagram).

Figure 6

Computed skewness Rsk (a) and kurtosis Rku (b) surfaces as a function of the two anchoring coefficients. Representative surface profiles with different skewness (c) and kurtosis (d). Skewness as a function of kurtosis (e), as the anchoring coefficients change, showing mirror symmetry.

Figure 7

Capillary pressures visualized in polar plot where rotation angle $\theta =\varphi -qx$. Blue, red, and magenta paths represent dilation (a), director (b), and rotation (c) pressures, respectively. (${\mu }_{2*}=-0.002$ and ${\mu }_{4*}=+0.0015$). (d) Shows the three pressures (here we compute the real value instead of absolute value) in a single polar coordinate system where 0 is the ring close to the magenta curve. It also shows how the total capillary pressure vanishes [Eq. (19)].

Figure 8

Maximum winding number $W_p$ as a function of the highest anchoring model order $n$ and Limaçon curves. $W_p=0$ represents a single point in $\beta$ plot and further a flat surface. All the possible patterns are limited between a flat surface and red line. For example, for the quartic model, the possible winding numbers of the stress loops are 1 and 2, which corresponds to single and double wrinkling in Fig. 3.

Figure 9

High winding number stress loops in the anchoring space for the sextic anchoring model $(n=3)$. Front view and back view of a subspace $(+-+)$ within ${\mathrm{\Lambda }}_6$ where $W_p=3$ is possible; the signs corresponds to the three anchoring coefficients of the $n=3$ model. $W_p=3$ can only be found in the region sandwiched in the middle of $\mathfrak{S}$ and $\mathfrak{X}$. The Limaçon stress curves for $W_p=1,2,3$ are shown on the right back view figure. The figure was computed using the intersections of $\mathfrak{S}$, $\mathfrak{I}$, and $\mathfrak{X}$ (see text and Appendix pp5).

Figure 10

The first and second quadrant of ${\mathrm{\Lambda }}_4$. $\mathfrak{R}$ is the resonance line while $\mathfrak{C}$ is the critical line. (1) to (8) are eight points chosen for further discussion in Figs. 11 and 12 to relate geometry $\left[h\right(x\left)\right]$, mechanics (imaginary part of $\beta$ as a function of its real $\mathbb{R}$ part), and mechanogeometry [curvature $\kappa$ as a function of stress ratio $u=\text{Re}\left(\beta \right)/\text{Im}\left(\beta \right)$].

Figure 11

The comparison of $h\text{−}x$ (geometry), $\beta$ (mechanics), and $\kappa \text{−}u$ (mechanogeometry relationship), for eight (1–8) conditions identified in Fig. 10. In (1–4) the left column is $h$, the middle is $\beta ,$ and on the right we find $\kappa \text{−}u$; the same applies to (5–8). In (4) we see that double wrinkling corresponds to inner loops (crossings) in stress and curvatures-stress loops.

Figure 12

Magnitudes of surface profile $h\left(x\right)$ (left), Lamé stress curve $\beta$ (middle) and curvature-stress ratio loop $\kappa \text{−}u$ (right) and how they change along the resonance line $\mathfrak{R}$ while maintaining their topology ($3\times {\mu }_{\text{Red}}=2\times {\mu }_{\text{Green}}={\mu }_{\text{Blue}}$). Moving radially in the anchoring diagram (Fig. 10) only changes amplitudes. This figure only shows half the period $P_0/2$.

Figure 13

A sketch explaining how $h\left(x\right)$ and $\beta$ can be seen as a direct superposition of their components in quartic model. Here, red point represents the center introduced by each anchoring coefficient. Black point represents the starting point, which is also the point where ${\beta }_2$ and ${\beta }_4$ are glued together.

Figure 14

Stress loop superposition and resulting Limaçon curves. A summary of how five conditions of ${\beta }_2$ and ${\beta }_4$ in the second quadrant of ${\mathrm{\Lambda }}_4$. The number on the right represents its corresponding position in Fig. 10. And the black point on the loop represents the starting point, which is the first invariant point in Table 2, and the point where ${\beta }_2$ and ${\beta }_4$ are glued together. The subfigure on the bottom left represents how the $\pi$ rotation symmetric point (fourth quadrant) behave.

Figure 15

Surface profile when ${\mu }_2<0$.

Figure 16

Surface vector sketch when ${\mu }_2>0$.

Figure 17

Surface profile when ${\mu }_2>0$.

Figure 18

The upper space of parametric space ${\mathrm{\Lambda }}_6$ is composed by four small cubes showing on the right.

Figure 19

Font view and back view of winding number $W_p$ distribution of $\beta$ under $(+-+)$ and $(--+)$ conditions.