Orientational phase transition in monolayers of multipolar straight rigid rods: The case of 2-thiophene molecule adsorption on the Au(111) surface

Monte Carlo simulations and finite-size scaling theory have been carried out to study the critical behavior and universality for the isotropic-nematic (IN) phase transition in a system of straight rigid pentamers adsorbed on a triangular lattice with polarized nonhomogeneous intermolecular interactions. The model was inspired by the deposition of 2-thiophene molecules over the Au(111) surface, which was previously characterized by experimental techniques and density functional theory. A nematic phase, observed experimentally by the formation of a self-assembled monolayer of parallel molecules, is separated from the isotropic state by a continuous transition occurring at a finite density. The precise determination of the critical exponents indicates that the transition belongs to the three-state Potts universality class. The finite-size scaling analysis includes the study of mutability and diversity. These two quantities are derived from information theory and they have not previously been considered as part of the conventional treatment of critical phenomena for the determination of critical exponents. The results obtained here contribute to the understanding of formation processes of self-assembled monolayers, phase transitions, and critical phenomena from novel compression algorithms for studying mutual information in sequences of data.

Figure 1

(a) Simplified schematic diagram showing two curcuminoid molecules adsorbed on the gold surface. (b) Lattice-gas representation of the situation in part (a). The FCC(111) surface is represented by a triangular lattice and the 2-thiophene molecules are modeled as pentamers. Inset: For each 2-thiophene molecule (pentamer), three interaction centers are considered: two at the ends and one at the center.

Figure 2

4 snapshots of the simulation in order of increasing coverage: (a) $\theta \ll {\theta }_c$, (b) $\theta$ just under ${\theta }_c$, (c) $\theta$ just over ${\theta }_c$, and (d) $\theta \gg {\theta }_c$. The polar pentamers are sketched as rods of three segments, representing the two heads and the center element and are colored according to their orientation.

Figure 3

(a) Log-log plot of the maximum value of the derivative of the Binder cumulant $dU/d\theta$, the logarithmic derivatives of $\langle \delta \rangle$ and $\langle {\delta }^2\rangle$, and the diversity of the surface coverage $D\left(\theta \right)$. Dashed lines represent linear fits proportional to $L^{1/\nu }$. (b) Log-log plot of the maximum value of the derivative of the order parameter $\langle \delta \rangle$ and the maximum of susceptibility $\chi$. Dashed lines represent linear fits proportional to $L^{(1-\beta )/\nu }$ for the maximum value of $\frac{d\langle \delta \rangle }{d\theta }$ and proportional to $L^{-\gamma /\nu }$ for the maximum value of $\chi$.

Figure 4

Fourth-order Binder cumulant $U_L\left(\theta \right)$ curves as a function of the lattice coverage $\theta$ for different sizes: $L=50$, solid squares; $L=75$, open triangles; $L=100$, open circles; and $L=125$, spheres. The critical lattice coverage ${\theta }_c$ is obtained from the intersection point of the curves.

Figure 5

Full data collapsing of the cumulants, $U_L$ vs $\varepsilon L^{1/\nu }$ for different sizes: $L=50$, solid squares; $L=75$, open triangles; $L=100$, open circles; and $L=125$, spheres. The plots were made using ${\theta }_c=0.792$ and $\nu =5/6$.

Figure 6

Data collapsing of the nematic order parameter $\delta$ in logarithmic scale for different sizes: $L=50$, solid squares; $L=75$, open triangles; $L=100$, open circles; and $L=125$, spheres. The collapse is produced for ${\theta }_c=0.792, \nu =5/6$, and $\beta =1/9$.

Figure 7

Data collapsing of the susceptibility $\chi$ for different sizes: $L=50$, solid squares; $L=75$, open triangles; $L=100$, open circles; and $L=125$, spheres. The collapse is produced for ${\theta }_c=0.792, \nu =5/6, \beta =1/9$, and $\chi =13/9$.

Figure 8

Location of the maximum value ${\theta }_c$ vs $L^{1/\nu }$ from several quantities, susceptibility $\chi$, the logarithmic derivatives of $\delta$ and ${\delta }^2$, the mutability ${\zeta }_{\delta }$, and the diversity $D_{\delta }$ of the order parameter. Dashed lines correspond to linear fits of the data. The data corresponding to $L=50$ were excluded from the fits of the mutability and diversity. From extrapolation one obtains the critical coverage at the thermodynamic limit ${\theta }_c(L=\infty )$.