Figure 1

(Color online) A comparison of the terms $\frac{\left(2+\kappa \right)}{\left(3+2\kappa \right)}{\partial }_z^{2}T$ (dark line) and $\frac{\kappa }{\left(3+2\kappa \right)}{\partial }_z^{2}S$ (light line) obtained within the PSW solution to the GLdG equations, for a $\kappa$ value of 18.0. The PSW approximation consists of ignoring the $\frac{\left(2+\kappa \right)}{\left(3+2\kappa \right)}{\partial }_z^{2}T$ term in comparison to the $\frac{\kappa }{\left(3+2\kappa \right)}{\partial }_z^{2}S$ term. Both terms, however, are of comparable magnitude.Reuse & Permissions

Figure 2

(Color online) Biaxial and uniaxial profiles for $\kappa =0\left(a\right)$, $0.4\left(b\right)$, $4\left(c\right)$, and $18.0\left(d\right)$, comparing results from our numerical computations $(\times )$, with our analytic formula (dashed line) and the formula of PSW (solid line). The main figure shows the biaxial profile whereas the inset shows the uniaxial profile. In (a), for $\kappa =0$, the solution has $T=0$, with the $S$ profile exactly given by the tanh form. In (b), for $\kappa =0.4$, the computed biaxial profile $\left(t\right)$ (main panel) is fit remarkably well by our analytic form, whereas the PSW approximation tends to overestimate the peak value. The uniaxial $\left(s\right)$ profile is shown in the inset of (a); here the results obtained by us and by PSW are identical and the fit to a tanh profile is accurate over the entire region. In (c) (main panel), for $\kappa =4.0$, the numerical data are fit well by the analytic forms, particularly away from the main peak, yielding essentially exact agreement deep into the isotropic and nematic sides. The PSW approximation is still an overestimate to the peak value, and also differs sharply in relation to the numerical data deep into the isotropic side. The inset shows the uniaxial $\left(s\right)$ profile for this case. In (d) (main panel), for $\kappa =18.0$, the PSW form appears to fit the peak better for larger $\kappa$, but again fails to capture the decay toward the isotropic side.Reuse & Permissions

Figure 3

(Color online) A comparison of the results of our analytic calculation to profiles of $T$ obtained from a density-functional calculation for the isotropic-nematic interface. Profiles obtained for two values of $\kappa$, $\kappa =5.8$ (for $z<0$) and $\kappa =0.69$ (for $z>0$) are shown. The larger $\kappa$ value essentially fits the $T$ profile exactly on the isotropic side, whereas the smaller $\kappa$ value provides an accurate fit on the nematic side. The inset shows the $S$ profile obtained from the density functional calculation, together with an optimum fit varying the value of $l_c$.Reuse & Permissions