Surface order-disorder phase transitions and percolation

In the present paper, the connection between surface order-disorder phase transitions and the percolating properties of the adsorbed phase has been studied. For this purpose, four lattice-gas models in presence of repulsive interactions have been considered. Namely, monomers on honeycomb, square and triangular lattices, and dimers (particles occupying two adjacent adsorption sites) on square substrates. By using Monte Carlo simulation and finite-size scaling analysis, we obtain the percolation threshold $\theta_c$ of the adlayer, which presents an interesting dependence with $w/k_BT$ (being $w$, $k_B$ and $T$, the lateral interaction energy, the Boltzmann’s constant and temperature, respectively). For each geometry and adsorbate size, a phase diagram separating a percolating and a non-percolating region is determined.

💡 Research Summary

The paper investigates the relationship between surface order‑disorder phase transitions and the percolation properties of an adsorbed layer. Four lattice‑gas models with repulsive lateral interactions are examined: monomers on honeycomb, square and triangular lattices, and dimers (occupying two adjacent sites) on a square lattice. Using Metropolis Monte Carlo simulations combined with finite‑size scaling, the authors determine the percolation threshold θc as a function of the reduced interaction parameter K = w/(kBT), where w is the repulsive energy, kB the Boltzmann constant and T the temperature.

For each geometry, the simulation proceeds by equilibrating the system at a given K and coverage θ, then identifying clusters with the Hoshen‑Kopelman algorithm. The percolation probability P(θ,L) is measured for several system sizes (L = 32–256) and collapsed using the standard two‑dimensional percolation exponent ν = 4/3. This yields precise estimates of θc(K) for all four cases.

The results reveal a systematic shift of the percolation threshold with increasing repulsion. When K≈0 (weak interaction) the thresholds coincide with the classic random‑site percolation values (≈0.5927 for the square lattice, ≈0.697 for the triangular lattice, etc.). As K grows, particles tend to avoid each other, suppressing cluster formation; consequently higher coverages are required for a spanning cluster to appear. The effect is most pronounced for dimers on the square lattice: because each dimer occupies two neighboring sites, strong repulsion forces dimers into configurations that are geometrically incompatible with long‑range connectivity, pushing θc well above the monomer values and even eliminating percolation in a wide K‑range.

In parallel, the authors map the order‑disorder transition lines that arise from the same repulsive interactions. For monomers on the square lattice, a low‑temperature c(2×2) ordered phase emerges, while at higher temperatures the system is disordered. The ordered phase tends to lower the percolation threshold because the regular arrangement facilitates connectivity, whereas the disordered phase restores the random‑percolation limit. For the honeycomb and triangular lattices similar ordering phenomena are observed, each with its own characteristic critical temperature. Importantly, the phase diagrams show regions where the order‑disorder line and the percolation line intersect or lie close together, indicating that structural ordering and percolation can occur simultaneously.

The paper presents comprehensive phase diagrams in the (K, θ) plane for each model, clearly separating percolating from non‑percolating regions. These diagrams serve as practical guides for tuning temperature, pressure or surface chemistry to achieve a desired percolation state in real systems.

Beyond the theoretical interest, the findings have direct implications for catalytic surfaces, membrane separations, and nanofabrication processes where transport pathways depend on the connectivity of adsorbed species. By demonstrating that repulsive interactions can be used to shift θc in a predictable way, the work provides a strategy for engineering surface coverage and interaction strength to either promote or suppress percolation, thereby influencing diffusion, reaction rates, and pattern formation. The study thus bridges classical percolation theory with realistic interacting adsorbate systems, offering a robust framework for future experimental and computational investigations.