Adsorption of Self-Assembled Rigid Rods on Two-Dimensional Lattices

Monte Carlo (MC) simulations have been carried out to study the adsorption on square and triangular lattices of particles with two bonding sites that, by decreasing temperature or increasing density, polymerize reversibly into chains with a discrete number of allowed directions and, at the same time, undergo a continuous isotropic-nematic (IN) transition. The process has been monitored by following the behavior of the adsorption isotherms for different values of lateral interaction energy/temperature. The numerical data were compared with mean-field analytical predictions and exact functions for noninteracting and 1D systems. The obtained results revealed the existence of three adsorption regimes in temperature. (1) At high temperatures, above the critical one characterizing the IN transition at full coverage Tc(\theta=1), the particles are distributed at random on the surface and the adlayer behaves as a noninteracting 2D system. (2) At very low temperatures, the asymmetric monomers adsorb forming chains over almost the entire range of coverage, and the adsorption process behaves as a 1D problem. (3) In the intermediate regime, the system exhibits a mixed regime and the filling of the lattice proceeds according to two different processes. In the first stage, the monomers adsorb isotropically on the lattice until the IN transition occurs in the system and, from this point, particles adsorb forming chains so that the adlayer behaves as a 1D fluid. The two adsorption processes are present in the adsorption isotherms, and a marked singularity can be observed that separates both regimes. Thus, the adsorption isotherms appear as sensitive quantities with respect to the IN phase transition, allowing us (i) to reproduce the phase diagram of the system for square lattices and (ii) to obtain an accurate determination of the phase diagram for triangular lattices.

💡 Research Summary

The authors investigate the adsorption of asymmetric monomers possessing two bonding sites on two‑dimensional lattices (square and triangular) using extensive Monte Carlo simulations. By varying temperature (or equivalently the ratio of lateral interaction energy ε to thermal energy kBT) and surface coverage θ, the particles can reversibly polymerize into linear chains while simultaneously undergoing a continuous isotropic‑nematic (IN) transition. The central observable is the adsorption isotherm μ(θ), i.e., the relationship between chemical potential (or pressure) and coverage, which is recorded for a wide range of ε/kBT values. The simulated isotherms are benchmarked against three theoretical references: (i) the exact non‑interacting 2D lattice‑gas solution, (ii) the exact one‑dimensional (1D) solution for a chain of hard rods, and (iii) a mean‑field (MF) approximation that treats lateral interactions as an average field.

Three distinct temperature regimes emerge from the analysis. In the high‑temperature regime (T > Tc(θ = 1), where Tc is the critical temperature for the IN transition at full coverage), the lateral attraction is negligible; particles distribute randomly, the isotherm follows the ideal 2D lattice‑gas law, and the nematic order parameter S remains near zero. In the low‑temperature limit (ε ≫ kBT), the bonding energy dominates, causing monomers to bind into chains over essentially the whole coverage range. The system behaves as a 1D fluid: the isotherm collapses onto the exact 1D solution, the average chain length diverges with coverage, and S approaches unity, indicating global nematic order. The intermediate temperature regime is the most intriguing. At low coverage the monomers adsorb isotropically, reproducing the 2D non‑interacting isotherm, but once the coverage reaches a critical value θc(T) the IN transition is triggered. Beyond this point the adsorption proceeds by adding monomers to existing chains, and the isotherm switches to the 1D functional form. This crossover appears as a pronounced kink or singularity in μ(θ), providing a highly sensitive signature of the IN transition. By locating this singularity for many temperatures, the authors reconstruct the full phase diagram for square lattices (in agreement with previous studies) and, for the first time, obtain a precise phase diagram for triangular lattices.

Comparison with the MF theory shows that MF captures the limiting high‑ and low‑temperature behaviors but fails to reproduce the mixed regime because it cannot account for the simultaneous emergence of one‑dimensional chain ordering and two‑dimensional isotropic fluctuations. The study therefore demonstrates that adsorption isotherms are not merely thermodynamic curves but can serve as direct probes of orientational ordering phenomena in adsorbed layers.

The implications are twofold. Practically, measuring adsorption isotherms (e.g., via gravimetric or quartz‑crystal microbalance techniques) offers a straightforward experimental route to detect the IN transition without requiring direct imaging of nematic domains. Theoretically, the work clarifies how reversible polymerization couples to orientational order and how dimensional crossover (2D → 1D) manifests in surface thermodynamics. This insight is relevant for designing functional nanostructured coatings, directional catalysts, and anisotropic conductive films where controlled chain formation on a substrate is desired. In summary, the paper provides a comprehensive quantitative description of temperature‑driven dimensional crossover and isotropic‑nematic ordering in self‑assembled rigid rods on 2D lattices, establishing adsorption isotherms as a powerful diagnostic tool for surface phase transitions.