Deformation free energy and elastic description of a self-assembled system

A continuum model is proposed to describe orientational states of a self-assembled system formed by rodlike molecules, in contact with an isotropic solid substrate. The total free energy is determined by taking into account the interactions between the molecules forming the film and between the molecules and the substrate. A phase diagram is presented, demonstrating that a critical surface molecular density exists, depending on the the ratio between the surface and the bulk free energy, separating homeotropic from tilted phases. The behavior of the elastic constants is investigated as a function of the surface molecular density. The elastic description leads to the presence of a linear term in the free energy, which accounts for the existence of possible spontaneous elastic distortions induced in the system.

Figure 1

Orientations of two rodlike molecules, of length $L$, whose long axes are $\mathbf{m}$ and ${\mathbf{m}}^{\prime }$. The Cartesian axes $x$ and $y$ are parallel to the flat solid substrate. $\theta$ is the angle by which a molecule of the membrane may bend under the action of the interactions.

Figure 2

Definition of the interparticle distance $\mathbf{r}$ and of the integration elements $d\ell$ and $d{\ell }^{\prime }$ relative to two identical molecules of length $L$, whose long axes are $\mathbf{m}$ and ${\mathbf{m}}^{\prime }$.

Figure 3

Molecular orientation near the solid substrate. $\zeta$ is in the molecular scale of length.

Figure 4

Behavior of the direct interaction of the molecule vs $\theta$ for $L∕R_0=10$ and $\zeta ∕R_0=1.0$.

Figure 5

Phase diagram exhibiting the value ${\theta }_{\mathrm{c}}$ for which $F_{\mathrm{u}}\left(\theta \right)$ is minimum vs the inverse of the dimensionless surface molecular density $\underset{̱}{\sigma }=\sigma R_0^2$ (see the text).

Figure 6

Behavior of the elastic coefficients vs ${\theta }_0$, for $\underset{̱}{\sigma }=0.1$. (a) ${10}^{-3}{b}_{yy}^{\varphi \varphi }$, (b) ${10}^{-3}{b}_{yy}^{\theta \theta }$, (c) ${10}^{-6}{b}_{xx}^{\varphi \varphi }$, (d) ${10}^{-6}{b}_{xx}^{\theta \theta }$, and (e) ${10}^{-3}{b}_{xy}^{\theta \theta }$. The scale is arbitrary just to show all the coefficients in the same plot. However, all the elastic coefficients are positive in the range $0⩽{\theta }_0⩽\pi ∕2$. The curves were depicted for $L∕R_0=9.0$ and $\zeta ∕R_0=1.0$.

Figure 7

Behavior of the elastic coefficients vs $\underset{̱}{\sigma }$ for ${\theta }_0=1.56\mathrm{rad}$. Solid line refers to ${10}^{-6}{b}_{yy}^{\theta \theta }$, dotted line to ${10}^{-8}{b}_{xx}^{\theta \theta }$, dashed line to ${10}^{-4}{b}_{yy}^{\varphi \varphi }$, dash-dotted line to ${10}^{-7}{b}_{xx}^{\varphi \varphi }$, and short-dotted line to ${10}^{-5}\times b_{xy}^{\theta \theta }$. The scale is arbitrary and the set of parameters is the same as in Fig. 6.

Figure 8

Behavior of the elastic constant $K_1$ vs $\underset{̱}{\sigma }$. (a) $L∕R_0=10$, (b) $L∕R_0=20$, and (c) $L∕R_0=100$.

Figure 9

Behavior of the elastic constant $K_{2x}$ and $K_{2y}$ vs $\underset{̱}{\sigma }$ for the case in which (a) $L∕R_0=20$ and (b) $L∕R_0=100$.