Elasticity of nematic phases with fundamental measure theory

In a previous publication [R. Wittmann, M. Marechal, and K. Mecke, Europhys. Lett. 109, 26003 (2015)], we introduced fundamental mixed measure theory (FMMT) for mixtures of anisotropic hard bodies, which shows that earlier results with an empirical parameter are inaccurate. Now we provide a deeper insight into the background of this theory in integral geometry. We study the Frank elastic coefficients in the nematic phase of the hard spherocylinder fluid. The framework of FMMT provides us with the required direct correlation function without additional input of an equation of state. A series representation of the mixed measure gives rise to closed analytical formulas for the elastic constants that only depend on the density, order parameters, and the particle geometry, pointing out a significant advantage of our geometry-based approach compared to other density functionals. Our elastic coefficients are in good agreement with computer simulations and increase with the density and the nematic order parameter. We confirm earlier mean-field predictions in the limits of low orientational order and infinitely long rods.

Figure 1

Oriented bodies ${\mathcal{B}}_i\subset {\mathbb{R}}^3$ in a coordinate system. We denote the centers of the bodies by a dot and the origin of the coordinates by a solid square. (a) Illustration of ${\mathcal{B}}_i({\mathbf{r}}_1,{\mathbf{O}}_1)$ centered at ${\mathbf{r}}_1$ and parametrized by ${\mathbf{R}}_i\left(\widehat{\mathbf{r}-{\mathbf{r}}_1}\right)$, as used in the weight functions found in Eq. (8). If $\mathbf{r}\in {\mathbb{R}}^3$ lies on $\partial {\mathcal{B}}_i({\mathbf{r}}_1,{\mathbf{O}}_1)$, then ${\mathbf{R}}_i\left(\widehat{\mathbf{r}-{\mathbf{r}}_1}\right)=\mathbf{r}-{\mathbf{r}}_1$. (b) Comparison of a body ${\mathcal{B}}_i(-\mathbf{r},{\mathbf{O}}_1)$ centered at $-\mathbf{r}$ with its inversion ${\overline{\mathcal{B}}}_i(\mathbf{r},{\mathbf{O}}_1)$ at $\mathbf{r}$. It can be seen that the elements $d{\mathbf{r}}_1\cap {\overline{\mathcal{B}}}_i(\mathbf{r},{\mathbf{O}}_1)$ and $d(-{\mathbf{r}}_1)\cap {\mathcal{B}}_i(-\mathbf{r},{\mathbf{O}}_1)$ are each other's image under point reflection.

Figure 2

Isotropic (solid line) and nematic equations of state of hard spherocylinders with aspect ratio $l=5$. We use the generalized $\zeta$ correction from Eq. (52) with different truncation orders $n_{\text{t}}$ and the expression ${\varphi }_3^{\left(\text{TR}\right)}$ from Eq. (55) by Tarazona and Rosenfeld. The convergence of this expansion towards the FMMT result with the full mixed weighted density, Eq. (50), becomes slower with increasing packing fraction $\eta$ and nematic order parameter. The horizontal dotted lines indicate the isotropic-nematic coexistence pressure and densities. The vertical dotted line denotes the limit of stability with respect to the smectic-$A$ phase obtained with FMMT [59]. The circles and triangles denote the simulation results [70] for the isotropic and nematic branch, respectively.

Figure 3

(a) Splay elastic constant $K_1$ and (b) bend elastic constant $K_3$ as a function of the packing fraction $\eta$. We compare the analytic formulas from the expansion in Eq. (95) at different truncation orders $n_{\text{t}}$ and $m_{\text{t}}$ to the numeric FMMT values according to Eq. (93), which involves the full mixed weight function. Unless mentioned otherwise, $m_{\text{t}}=0$. As in Fig. 2, we use ${\varphi }_3^{\left(\text{TR}\right)}$ and verify a quick convergence of the even-order terms in Eq. (95), setting $m_{\text{t}}=0$. As our best analytic approximation for $K_1$, we add odd-order terms up to $m_{\text{t}}=12$ to the result for $n_{\text{t}}=7$. The result for $K_3$ for $m_{\text{t}}=0$ and $n_{\text{t}}=7$ is indistinguishable from the full FMMT result. The diamonds denote the result, which we obtain using only the formulas we provide in Appendix pp2. The nematic phase is stable to the left of the vertical dashed line.

Figure 4

Elastic coefficients $K_{\epsilon }^{\left(\text{MM}\right)}$ of hard spherocylinders using ${\varphi }_3^{\left(\text{TR}\right)}$ with an aspect ratio of $l=5$. The solid, dotted, and dashed lines or symbols correspond to splay, twist, and bend, respectively. As a function of (a) the packing fraction $\eta$ and (b) the nematic order parameter $S$, we compare our FMMT results to computer simulations for $\eta =0.4418$ [34] and other DFT values [22, 24] corresponding to $\eta =0.4418$ and $\eta =0.5181$. Note that these functionals, as well as FMMT, predict different equilibrium order parameters.

Figure 5

(a) Splay, (b) twist, and (c) bend elastic coefficients of hard spherocylinders as a function of the packing fraction $\eta$. We show the FMMT results $K_{\epsilon }^{\left(\text{MM}\right)}$ using ${\varphi }_3^{\left(\text{TR}\right)}$ for different aspect ratios $l=L/D$. The gray extension is beyond the stability limit of the nematic phase [59].

Figure 6

Illustration of the original body $K+\mathbf{x}$ (solid loop), its point inversion $\mathbf{r}-K$ (dashed loop), and the two Borel sets $A$ and $B$ (shaded regions). The arrow denotes $\mathbf{r}-\mathbf{x}=\mathbf{p}(K+\mathbf{x},\mathbf{y})-\mathbf{x}=\mathbf{r}-\mathbf{p}(\mathbf{r}-K,\mathbf{z})$.