Calculation of the absolute free energy of a smectic-$A$ phase

In this paper, we provide a scheme to compute the absolute free energy of a smectic-$A$ phase via the “indirect method.” The state of interest is connected through a three-step reversible path to a reference state. This state consists of a low-density layer of rods coupled to two external fields maintaining these rods close to the layer's plane and oriented preferably normal to the layer. The low-density free energy of the reference state can be computed on the basis of the relevant second virial coefficients between two rods coupled to the two external fields. We apply this technique to the Gay-Berne potential for calamitics with a parameter set leading to stable isotropic (I), nematic (N), smectic-$A$ (SmA), and crystal (Cr) phases. We locate the I-SmA phase transition at low pressure and the sequence of phase transitions I-N-SmA along higher-pressure isobars and we establish the location of the I-N-SmA triple point. Close to this triple point, we show that the N-SmA transition is clearly first order. Our results are compared to the coexistence lines of the approximate phase diagram elucidated by de Miguel et al. [J. Chem. Phys. 121, 11183 (2004)] established through the direct observation of the sequence of phase transitions occurring along isobars under heating or cooling sequences of runs. Finally, we discuss the potential of our technique in studying similar transitions observed on layered phases under confinement.

Figure 1

Plot of $-\beta {\Sigma }^{-2}{\gamma }^{\mathrm{int}}$ for a smectic layer of rods subjected to the two external fields as a function of surface density $\Sigma$ for different reference temperatures and/or field intensities (green open triangles for $T_{\mathrm{ref}}=1.40$, ${\mu }_0=50$, ${\lambda }_0=10$; red open circles for $T_{\mathrm{ref}}=1.12$, ${\mu }_0=15$, ${\lambda }_0=10$; and blue open squares for $T_{\mathrm{ref}}=1.04$, ${\mu }_0=15$, ${\lambda }_0=10$). This function does start from $B_2^{\mathrm{Sm}}(T_{\mathrm{ref}},{\mu }_0,{\lambda }_0)$ at $\Sigma =0$ as shown in the main figure. Second virial coefficient values obtained by numerical integration of Eq. (18) and gathered in Table 1 are indicated on the vertical axis by a larger filled symbol of same color and are seen to be perfectly coherent with simulation data at finite $\Sigma$. The dashed line, dashed-dotted line, and continuous line are polynomial fits of order $n=5$. The integral of this function (see inset where color code and line style follow main figure data) directly yields the reduced free energy $\beta {\tilde{f}}_{1,1}^{\mathrm{int}}$ [Eq. (23)] which starts like $B_2^{\mathrm{Sm}}\Sigma$ at low densities [see Eq. (17)].

Figure 2

Temperature dependence of the phase-dependent part of the chemical potential ${\mu }^{*}$(see text) along the isobar at $p=0.75$. Both the isotropic (filled blue circles) and smectic A (filled red squares) branches are shown over their (meta)stability range. Continuous lines correspond to polynomial fits [degree 3(SmA) or 4(I)]. The reference state point per branch ($p,T_{\mathrm{ref}}$) at which the absolute free energy has been determined is indicated by a black empty symbol.

Figure 3

Temperature dependence of the phase-dependent part of the chemical potential ${\mu }^{*}$ (see text) along the isobar at $p=1.00$. The isotropic (filled blue circles), nematic (filled green lozenges), and smectic A (filled red squares) branches are shown over their (meta)stability range. Continuous lines correspond to polynomial fits [degree 3 or 2(N)]. The reference state point per branch ($p,T_{\mathrm{ref}}$) at which the absolute free energy has been determined is indicated by a black empty symbol.

Figure 4

Temperature dependence of the phase-dependent free energy ${\mu }^{*}(p,T)$ at $p=1.50$, centered on the SmA- (red squares) to-N (green lozenges) transition. Data point obtained by an absolute free-energy calculation are indicated as open black symbols. The reference state point per branch ($p,T_{\mathrm{ref}}$) at which the absolute free energy has been determined is indicated by a black empty symbol. The phase transition temperature is found to be $T_{\mathrm{N}-\mathrm{Sm}A}=1.325\left(3\right)$. The entropy difference at transition is $\Delta s=0.58\left(1\right)$ and expected enthalpy difference is thus $\Delta h=T_{\mathrm{N}-\mathrm{Sm}A}\Delta s=0.77\left(1\right)$. Polynomials fits are second order.

Figure 5

Temperature dependence of the phase-dependent enthalpy $h(p,T)$ (a), volume per rod $v(p,T)$ (b), and order parameters $S(p,T)$ and $\tau (p,T)$ (c) along isobars at $p=1.50$. The N-SmA transition located at $T_{\mathrm{N}-\mathrm{Sm}A}=1.325$ is indicated by a vertical dashed line. Polynomial fits are second order. SmA data are denoted by red squares and N data by green lozenges. For the order parameters, the open symbols denote the orientational order parameter $S$ and the filled symbols the translational order parameter $\tau$. Polynomial fits are second order except for the N branch for $h$ and $v$, which are first order. Finite differences at transition nonreported in Table 2 are $\Delta S=0.07\left(2\right)$ and $\Delta \tau =0.38\left(5\right)$.

Figure 6

Temperature dependence of the phase-dependent part of the chemical potential ${\mu }^{*}$(see text) along the isobar at $p=2.00$. The I (filled green lozenges), N (filled blue circles), and SmA (filled red squares) branches are shown over their (meta)stability range. The reference state point per branch ($p,T_{\mathrm{ref}}$) at which the absolute free energy has been determined is indicated by a black empty symbol. Continuous lines correspond to polynomial fits [degree 3 or 4(I)].

Figure 7

GB phase diagram close to the I-N-SmA triple point. Results based on free-energy calculations ($N=432$ molecules) derived in this work are all noted by filled symbols. The interpolated triple point (see text) is denoted by a diamond while $p\text{−}T$ coexistence points are denoted differently for N-SmA (circles), SmA-I (squares), and N-I (triangle) transitions. Reference [22] data on $N=1620$ molecules are indicated by pairs of open symbols using the same symbol shape convention, the one on the left (right) being the observed transition temperature on cooling (heating). Note that the Cr-SmA coexistence line which should appear on the left of the I-N-SmA triple point [22] is not indicated on this figure as no attempt was done to locate it by absolute free-energy calculations (note, however, that Cr-I-SmA and Cr-SmA-N triple points are expected to lie outside the shown $p\text{−}T$ area).