Theory and simulation of the confined Lebwohl-Lasher model

We discuss the Lebwohl-Lasher model of nematic liquid crystals in a confined geometry, using Monte Carlo simulation and mean-field theory. A film of material is sandwiched between two planar, parallel plates that couple to the adjacent spins via a surface strength ${\epsilon }_s$. We consider the cases where the favored alignments at the two walls are the same (symmetric cell) or different (asymmetric cell). In the latter case, we demonstrate the existence of a single phase transition in the slab for all values of the cell thickness. This transition has been observed before in the regime of narrow cells, where the two structures involved correspond to different arrangements of the nematic director. By studying wider cells, we show that the transition is in fact the usual isotropic-to-nematic (capillary) transition under confinement in the case of antagonistic surface forces. We show results for a wide range of values of film thickness and discuss the phenomenology using a mean-field model.

Figure 1

Schematic representation of the confined Lebwohl-Lasher model. $h$ is the number of spin layers sandwiched between the two external plates (shaded). $h+1$ is the film thickness in units of the cubic lattice parameter. The unit vectors along the Cartesian coordinates are indicated.

Figure 2

Excess heat capacity per spin in reduced units, $c_v^{*}$, as a function of reduced temperature $T^{*}$, for the symmetric case and for various plate separations (indicated as labeled). Several lateral sizes, given in the inset, are considered in each case.

Figure 3

Nematic uniaxial order parameter $P_i$ (left panels) and tilt angle of the nematic director $\varphi$ along the $z$ direction for the slit pore with $h=8$. (a) and (b) $T^{*}=0.850$; (c) and (d) $T^{*}=1.076$; (e) and (f) $T^{*}=1.200$. The lateral size $L$ used in the simulations is indicated in the insets.

Figure 4

(a) Excess heat capacity per spin in reduced units, $c_v^{*}$, for the system with $h=8$, as a function of reduced temperature $T^{*}$ and for different lateral system sizes $L$ (indicated in the inset). (b) Maximum of the heat capacity per spin in reduced units, $c_v^{*\max }$, for the system with $h=8$, as a function of lateral system sizes $L$. The straight line is a linear fit.

Figure 5

Dependence of the normalized fourth-order cumulant $G_4=\langle Q_{\mathit{xy}}^4\rangle /\langle Q_{\mathit{xy}}^2\rangle {}^2$ with reduced temperature $T^{*}=\mathit{kT}/\epsilon$ and lateral size $L$ for the system with $h=8$. The lateral size of the systems is quoted in the legend.

Figure 6

Order parameters (a) $P_{\mathit{xy}}$ and (b) $P$ as a function of temperature $T$, both for pore width $h=8$. (a) Different curves give $P_{\mathit{xy}}$ for different values of lateral size $L$ (see key). (b) Order parameter $P_i$ for $i=1$ (triangles), $i=2$ (open circles), $i=3$ (open squares), and $i=4$ (filled squares) for lateral size $L=32$. The global order parameter $P$ is represented by filled circles. In both (a) and (b) the vertical arrow indicates the location of the phase transition as estimated from the heat capacity.

Figure 7

Dependence of the order parameter $P_{\mathit{xy}}$ on the lateral size $L$ of the system for two different values of scaled temperature which are close to the true critical temperature, for the case $h=8$. Filled circles: $T^{*}=1.076$. Open circles: $T^{*}=1.075$. Error bars are included in each case. The horizontal line indicates an approximate value of $L^{1/8}{P}_{\mathit{xy}}$ for the latter temperature in the thermodynamic limit $L\to \infty$.

Figure 8

Variation of the excess heat capacity per particle $c_v^{*}$ with reduced temperature $T^{*}$ for $h=1$ and different values of $L$ (indicated in the inset).

Figure 9

Behavior of the nematic order parameter $P$ as a function of reduced temperature $T^{*}$ for $h=1$ and various lateral sizes $L$ (indicated in the inset).

Figure 10

Phase diagram for the hybrid cell in the $T^{*}$-$h^{-1}$ plane for the case ${\epsilon }_s^{(1)}={\epsilon }_s^{(2)}=\epsilon$, showing temperatures at which the LS transition occurs for each value of plate separation. L (S) phase is stable below (above) the corresponding symbol. Circles: Present MC simulation results. Triangles: MC simulation results by Chiccoli et al. [4]. Squares: Present MF results. Filled symbols represent first-order phase transitions, while open ones refer to continuous phase transitions. Horizontal dotted lines: Bulk temperatures as obtained from the MF and MC calculations (upper and lower lines, respectively). Continuous line: Modified Kelvin equation. Dashed line is a guide to the eye.

Figure 11

Local order parameters $P_i$, $(P_{\mathit{xy}}){}_i$ and director tilt angle ${\varphi }_i$ obtained from the MC simulations for two different pore widths, $h=8$ ($L=64$) and $h=32$ ($L=48$), at various temperatures in the neighborhood of the corresponding critical temperature $T_c(h)$. (a) $h=8$ and $T^{*}=1$; (b) $h=8$ and $T^{*}=1.076$; (c) $h=8$ and $T^{*}=1.15$; (d) $h=32$ and $T^{*}=1.101$; (e) $h=32$ and $T^{*}=1.118$; (f) $h=32$ and $T^{*}=1.135$. Notice that for $T>T_c$ the jump in $\varphi$ is system-size ($L$) dependent, and becomes steeper as $L$ approaches the thermodynamic limit.

Figure 12

Order parameters $P_i$ and $B_i$ and director tilt angle ${\varphi }_i$ for the two phases coexisting at the LS phase transition, in the case ${\epsilon }_s^{(1)}={\epsilon }_s^{(2)}=\epsilon$ and as obtained from MF calculations. (a) L phase for $h=8$; (b) S phase for $h=8$; (c) L phase for $h=9$; (d) S phase for $h=9$.

Figure 13

Order parameters $P_i$ and $B_i$ and director tilt angle ${\varphi }_i$ for the two phases coexisting for a pore width $h=32$, as obtained from the MF calculations in the case ${\epsilon }_s^{(1)}={\epsilon }_s^{(2)}=\epsilon$. (a) L phase. (b) S phase.

Figure 14

Order parameters $P$ and $P_{\mathit{xy}}$ as a function of reduced temperature $T^{*}$ from mean-field theory. (a) $h=60$; (b) $h=5$. Continuous curves: $P$. Dashed curves: $P_{\mathit{xy}}$. In (a), vertical dotted lines indicate location of first-order LS phase transition, while arrow points to bulk temperature. In (b), arrow indicates temperature of continuous LS transition.

Figure 15

Phase diagram for the hybrid cell in the $t$-$h^{-1}$ plane, with $t=(T-T_{\mathrm{IN}})/T_{\mathrm{IN}}$, for different values of the surface couplings ${\epsilon }_s^{(1)}$ and ${\epsilon }_s^{(2)}$ (indicated in the inset), as obtained from mean-field theory. Dashed lines correspond to the lowest-order Kelvin equation in each case (see text).

Figure 16

Order parameter $P_i$ and director tilt angle ${\varphi }_i$ for the two phases coexisting at the LS transition for a pore of width $h=15$ and different values of the surface coupling constants, as obtained from mean-field theory. Upper panels: Linear-like phase. Lower panels: Step-like phase. The surface parameters $({\epsilon }_s^{(1)},{\epsilon }_s^{(2)})$ are as follows: (a) and (b), $(\epsilon ,\epsilon )$; (c) and (d), $(\epsilon ,0.4\epsilon )$; (e) and (f), $(\epsilon ,0.219\epsilon )$; and (g) and (h), $(0.219\epsilon ,0.219\epsilon )$.

Figure 17

Order-parameter profile of the isotropic-nematic interface in the MF theory for the Lebwohl-Lasher model.

Figure 18

Wetting regime of the Lebwohl-Lasher model in mean-field theory. For ${\epsilon }_s⩽0$ the substrate is wet by the isotropic phase. For ${\epsilon }_s\gtrsim 0.430\epsilon$ the substrate is wet by the nematic phase. In between a partial-wetting regime occurs. The arrow indicates the case where the surface-nematic and surface-isotropic interfaces are equal, which occurs at ${\epsilon }_s=0.219\epsilon$.

Figure 19

Scaled elastic constant $K^{*}=\mathit{Ka}/{\epsilon }^{*}$ for Lebwohl-Lasher model as a function of relative temperature $t$ (as defined in caption of Fig. 10). Circles: Simulation results of Cleaver and Allen [20]. Continuous curve: Present MF results.