Interplay between nematic and cholesteric interactions in self-consistent field theory

Chirality is a design feature of a number of biomolecules (e.g., collagen). In these molecules, cholesteric (chiral-nematic) behavior emerges from a combination of the tendency for the biopolymers to align (nematic interactions) and for the alignment direction to change with position, rotating around an axis normal to the alignment direction. This paper presents self-consistent field theory (SCFT) of chiral-nematic polymers, which takes into account polymer flexibility and the orientational degrees of freedom of polymer segments. Using the resulting SCFT, we construct a phase diagram showing regions of stability for isotropic, nematic, and cholesteric phases. Furthermore, we find that nematic interactions can stabilize the cholesteric phase, pushing the isotropic-cholesteric phase transition to lower cholesteric interaction strength, until the isotropic-nematic-cholesteric triple point is reached.

Figure 1

Molecules that form cholesteric phases tend to be chiral, such as the pentuple helix (resembling type 1 collagen [10, 11, 12]) shown at the top. These may align parallel to one another (left) or at an angle (right). If the molecules are thought of as simple solids with the geometry shown, adjacent helices could pack all in parallel (nematic, left) or if successive layers are rotated by an angle $2\mathrm{\Theta }$ (with $\mathrm{\Theta }$ being the twist angle, as defined at the top) then the subunits of one helix would fit into the grooves between subunits of the adjacent helix. Interactions between real molecules are undoubtedly more complicated, involving weak interactions, such as hydrogen bonding and van der Waals interactions, as well as other complications. This simple picture is only meant to provide an intuitive caricature of the formation of cholesteric phases.

Figure 2

Collagenlike chains (in a nematic phase) (left) and a field representation (right). The polymers are modeled as wormlike chains with persistent length ${\ell }_{\mathrm{p}}$ and contour length $L$, interacting with each other through chiral-nematic interactions. The field representation focuses on one chain and replaces the influence of others by a field $w(\mathbf{r},\mathbf{u})$ it is subject to. In self-consistent field theory (SCFT), $w(\mathbf{r},\mathbf{u})$ and the spatial distribution of “particles” (e.g., chain segments), denoted as $\psi (\mathbf{r},\mathbf{u})$, are determined self-consistently. We adjust $w(\mathbf{r},\mathbf{u})$ and $\psi (\mathbf{r},\mathbf{u})$ iteratively until they are “correctly” related. Reprinted with modifications from Ref. [28], with the permission of AIP Publishing.

Figure 3

Distributions, $\psi (\mathbf{r},\mathbf{u})$, are shown for the cholesteric phase (single peak) and the nematic phase (double peak), calculated at $\gamma =4.5133$ and $\nu =132.528$. The horizontal plot on the left shows the distribution in $\varphi \text{−}\theta$ space whereas the plot on the right shows a slice through $\theta =\pi /2$. The illustration in the middle is to define the angles $\varphi$ and $\theta$. The rod on top has $\varphi =\pi /2$ and $\theta =\pi /2$. This orientation represents the peak location in the $\varphi \text{−}\theta$ space in the distribution on the left. The distribution is more strongly peaked for the cholesteric phase.

Figure 4

Free energy, per chain, of the cholesteric phase relative to the nematic phase, as it varies with $k$, the magnitude of the helical wave vector. The curve was calculated at $\gamma =4.5133$ and $\nu =132.528$.

Figure 5

Left: Optimal twist wave vector, $k^{*}$, and rescaled $k^{*}{A}^2$ (along with the rescaling factor $A$), obtained with $\nu =132.528$, as a function of cholesteric interaction strength, $\gamma$. The rescaled wave vector $k^{*}{A}^2$ tends to the energy minimizing value of $k^{*}{A}^2=1/L$ as $\gamma \to \infty$. We did not extend to higher $\gamma$ as calculations become unstable. Right: Helical pitch, $p=2\pi /k^{*}{A}^{2}L$, and $A^2$ at $\nu =132.528$ as a function of $\gamma$. The helical pitch approaches the expected (energy-minimizing) value $2\pi$, as $\gamma$ increases.

Figure 6

Left: Optimal twist wave vector $k^{*}$, rescaled wave vector $k^{*}{A}^2$, and $A$, obtained with $\gamma =4.5133$, as a function of nematic interaction strength, $\nu$. The rescaled $k^{*}{A}^2$ increases and tends to $1/L$ as $\nu$ increases. We did not extend to higher $\nu$ as calculations become unstable. Right: Helical pitch, $p=2\pi /k^{*}{A}^{2}L$, and $A^2$ at $\gamma =4.5133$ as a function of $\nu$. The pitch $p$ approaches $2\pi$, as $\nu \to \infty$.

Figure 7

Left: Free energy, per chain, of the cholesteric phase relative to the nematic phase, as it varies with cholesteric interaction strength, $\gamma$. Free energies are calculated at the optimal twist angle, with the same $\nu =132.528$ as in Fig. 5. Right: Free energy, per chain, of the cholesteric phase relative to the nematic phase, as it varies with nematic interaction strength, $\nu$. Free energies are calculated at the optimal twist angle, with the same $\gamma =4.5133$ as in Fig. 6.

Figure 8

Phase diagram, depicting the boundaries between the cholesteric, isotropic, and nematic phases in the $\nu \text{−}\gamma$ space. The diagrams are shown for (a) $L/{\ell }_{\mathrm{p}}=0.5$ (rodlike), (b) 1, (c) 10, and (d) 20 (collagenlike). Here ISO labels the region where the isotropic phase is stable; similarly, CHOL and NEM stand for cholesteric and nematic, respectively. As $L/{\ell }_{\mathrm{p}}$ increases, the polymers become more flexible. As a result, the isotropic phase is stable in a wider space. The dot in each diagram is the triple point, where the three phases coexist.