Lattice Monte Carlo study of orientational order in a confined system of biaxial particles: Effect of an external electric field

In this work we have used lattice Monte Carlo to determine the orientational order of a system of biaxial particles confined between two walls inducing perfect order and subjected to an electric field perpendicular to the walls. The particles are set to interact with their nearest neighbors through a biaxial version of the Lebwohl-Lasher potential. A particular set of values for the molecular reduced polarizabilities defining the potential used was considered; the Metropolis sampling algorithm was used in the Monte Carlo simulations. The relevant order parameters were determined in the middle plane of the sample and for some cases across the whole thickness of the sample. We have determined the temperature–electric field phase diagram for this system and found, as expected, five different system configurations corresponding to three different mesophases. At low temperatures and low fields the system finds itself in an undistorted biaxial phase. On increasing the field at low temperatures, a Freedericksz transition takes place and the secondary directors reorient towards the field while the primary director stays undistorted and parallel to the walls. On increasing the field further, a second Freedericksz transition occurs and the primary director orients also towards the field direction. The orientational order measured at the field strengths tested is not affected by the field. On increasing the temperature, a transition to a uniaxial phase occurs and within the range of this phase a field increase leads also to a Freedericksz transition where the main director reorients towards the field. At higher temperature a transition to a disordered phase is found. We have performed finite size scaling analysis for the Freedericksz critical fields and found that they scale with the distance $L$ between the walls as $L^{–1}$ as expected from continuum theory. From these fields we have also determined the temperature dependence of two elastic constant ratios. Critical exponents and critical temperatures for the order parameter and the correlation length for the biaxial-uniaxial phase transition and the uniaxial to disordered phase transition were also determined by finite size scaling and are discussed.

Figure 1

Biaxial molecules with orthorhombic symmetry are considered; $x,y,z$: molecular principal axis.

Figure 2

Geometry of the sample.

Figure 3

Comparison between the temperature dependencies of the order parameters $S$ and $C$ for our $y$-bounded samples (filled circles and triangles) with $\lambda =0.306, L=n_y=30, n_x=n_z=10$, determined halfway between the $y$ bounds and the $S$ and $C$ values (marked with ${}^{*}$) for unbounded samples, with $\lambda =0.3$ and $n_x=n_y=n_z=10$ [17].

Figure 4

Comparison between the temperature dependencies of the order parameters $S$ and $C$ for our $y$-bounded samples (filled symbols) with $\lambda =0.4, L=n_y=40, n_x=n_z=10$, determined halfway between the walls and the $S$ and $C$ values (marked with ${}^{*}$) for unbounded samples with $\lambda =0.408$ and $n_x=n_y=n_z=10$ [17].

Figure 5

$Y$ dependence values for the order parameter $S^{\prime }$ for $t=0.1$ and different fields with strengths indicated, going above the Freedericksz transition field for the main director. $n_x=n_z=10, L=n_y=30$.

Figure 6

$Y$ dependence of the apparent values for the order parameter $C^{\prime }/6$ for $t=0.1$ and different fields strengths as indicated. $n_x=n_z=10$ and $L=n_y=30$.

Figure 7

Field dependence of the apparent order parameter $S^{\prime }$ determined halfway between the sample walls for $t=0.1, n_y=30$ and different values of $l=n_x=n_z$ ranging from 6 to 18.

Figure 8

Field dependence of the apparent order parameter $C^{\prime }/6$ determined halfway between the sample walls for $t=0.1, n_y=30$ and different values of $l=n_x=n_z$ ranging from 6 to 18.

Figure 9

Case (a) reports the initial director configuration, as in Fig. 2. [$S^{\prime }=1, C^{\prime }/6=1/2$ (athermal)]. Case (b) shows the directors configuration after a ${90}^{\circ }$ rotation around ${\mathbf{n}}_0$ ($S^{\prime }=1, C^{\prime }/6=–(1/2)$ (athermal)]. Case (c) reports the director configuration after a second ${90}^{\circ }$ rotation around ${\mathbf{m}}_1$ [$S^{\prime }=–(1/2), C^{\prime }/6=–\left(1/4\right)$ (athermal).

Figure 10

Order parameters' susceptibilities as a function of applied field obtained from the fluctuation-dissipation relation, for the apparent order parameters $S^{\prime }$ and $C^{\prime }/6$ reported in Figs. 7 and 8.

Figure 11

Order parameters’ susceptibilities as a function of applied field obtained from $–d{S}^{\prime }/dh, –d\left(C^{\prime }/6\right)/dh$, for the apparent order parameters $S^{\prime }$ and $C^{\prime }/6$ reported in Figs. 7 and 8.

Figure 12

Reduced temperature dependence of the apparent order parameter $S^{\prime }$ determined halfway between the sample walls for $n_y=30$ and different values of $l=n_x=n_z$ ranging from 6 to 16 and $h=0$. The error bars indicate the standard deviation of the reported values.

Figure 13

Reduced temperature dependence of the order parameter $S$ determined halfway between the sample walls for $n_y=30$ and different values of $l=n_x=n_z$ ranging from 6 to 16 and $h=0$. The error bars indicate the standard deviation of the reported values.

Figure 14

Reduced temperature dependence of the apparent order parameter $C^{\prime }$ determined halfway between the sample walls for $n_y=30$ and different values of $l=n_x=n_z$ ranging from 6 to 16 and $h=0$. The error bars indicate the standard deviation of the reported values.

Figure 15

Reduced temperature dependence of the order parameter $C$ determined halfway between the sample walls for $n_y=30$ and different values of $l=n_x=n_z$ ranging from 6 to 16 and $h=0$. The error bars indicate the standard deviation of the reported values.

Figure 16

System's phase diagram in the $t\text{−}h$ plane. $N_{B1}$: undistorted biaxial nematic phase; $N_{B2}$: biaxial nematic with secondary director aligned with the field; $N_{B3}$: biaxial nematic with main director aligned with the field. $N_{U1}$: undistorted uniaxial nematic; $N_{U2}$: uniaxial nematic with director aligned with the field. $I$: isotropic phase.

Figure 17

Temperature dependence of the elastic constant ratios $K_m/K_{n1}$ and $\left(K_{n1}+K_m\right)/K_a$, showing a sharp decrease of $K_m$ as the $N_B\text{−}{N}_U$ phase transition is approached.

Figure 18

$1/h_{cS}$ dependence on $1/n_x$ for the samples with different layers ($n_y=10$ to 40) used in the scaling analysis of the Freedericksz critical fields for the main director reorientation.

Figure 19

$1/h_{cC}$ dependence on $1/n_x$ for the samples with different layers ($n_y=10$ to 40) used in the scaling analysis of the Freedericksz critical fields for the secondary director reorientation.

Figure 20

Fit of $\ln \left[h_{c\infty }\left(n_y\right)\right]$ vs $\ln \left(n_y\right), L=n_y=10$, 20, 30, 40—circles: $h_{cS}$; squares: $h_{cC}$. Linear fit equations: $S$: 1.80–$1.0036x$; $C$: 1.3747–$1.0147x$. Thus $h_{c\infty }\left(n_y\right)\propto {n_y}^{-1}$, as expected from the continuum theory [25].

Figure 21

Reduced temperature dependence of the order parameter $S$ over the transition $N_U\text{−}I$ for eight different values of the system size in the $x$ and $z$ directions, $l=n_x=n_z$ ranging from 6 to 20 and $n_y=40$.

Figure 22

$\tilde{S}$ scaling function curves as a function of $\left(l^{1/\nu }{t}_r\right)$ for eight different values of $l=n_x=n_z$ ranging from 6 to 20 and $n_y=40$; the curves collapse can be observed.

Figure 23

Sample thickness $L=n_y$ dependence of the critical temperature $T_c$ and the critical exponents $\nu$ and $\beta$ for the order parameter $S$ at the $N_U\text{−}I$ phase transition. The dotted lines are guides to the eye.

Figure 24

Reduced temperature dependence of the order parameter $C$ over the transition $N_B\text{−}{N}_U$ for eight different values of the system size in the $x$ and $z$ directions; $l=n_x=n_z$ ranging from 6 to 20 and $n_y=40$.

Figure 25

$\tilde{C}$ scaling function curves as a function of $\left(l^{(1/v)}{t}_r\right)$ for seven different values of $l=n_x=n_z$ ranging from 6 to 20 and $n_y=40$; the curve collapse can be observed.

Figure 26

Sample thickness $L=n_y$ dependence of the critical temperature $T_c$ and the critical exponents $\nu$ and $\beta$ for the order parameter $C$ at the $N_B\text{−}{N}_U$ phase transition. The dotted lines are guides to the eye.