Monte Carlo simulation of selective adsorption in a binary hard-disk mixture on patterned adhesive surfaces

Problem Statement
The paper addresses the fundamental problem of selective adsorption in multicomponent systems on chemically patterned surfaces. While surface heterogeneity is known to influence adsorption isotherms and ordering, real-world systems often involve complex interplays between particle size, shape, charge, and specific binding affinities. The authors aim to isolate the specific contribution of "affinity-driven selectivity" from other geometric and thermodynamic factors. Specifically, they investigate how the geometry of adhesive domains (size, surface coverage, and spatial arrangement) influences the preferential adsorption of one species over another in a binary mixture, independent of particle size asymmetry or differences in bulk chemical potential.

Methodology
The study employs Grand Canonical Monte Carlo (GCMC) simulations to model a two-dimensional binary mixture of hard disks adsorbing onto a surface patterned with circular adhesive domains.

• System Model: The system consists of two species of hard disks ('1' and '2') with identical diameters ($D_1 = D_2 = D_p$) and identical bulk chemical potentials ($\mu_1 = \mu_2 = \mu$). The only distinguishing factor is the strength of their attractive interaction with the adhesive domains. Species '1' interacts twice as strongly as species '2' ($A_{1d} = 2A_{2d}$).
• Interaction Potential: The attraction is modeled as being proportional to the overlap area between a particle and an adhesive domain. The interaction energy is the sum of pair interactions with all overlapping domains.
• Surface Geometry: The adhesive domains are arranged in either ordered (square lattice) or disordered configurations. The study varies the domain size ($D_d$) relative to the particle size ($D_p$) with ratios of 0.5, 1.0, and 2.0. Two domain surface coverages ($\sigma_d$) are examined: 0.349 and 0.503.
• Simulation Parameters: Simulations are performed in a square box with periodic boundaries. The reduced chemical potential ($\mu^*$) is varied from -12.0 to 5.0. The system is equilibrated using displacement, insertion, and deletion moves until average particle numbers stabilize.
• Metrics: The primary outputs are the adsorption isotherms (surface coverage $\sigma_\alpha$ vs. $\mu^*$) and the selectivity coefficient $S_{1/2} = \sigma_1 / \sigma_2$.

Key Results

1. Affinity-Driven Selectivity: Even with identical sizes and bulk chemical potentials, the species with stronger domain affinity (Species '1') consistently enriches the adsorbed layer ($S_{1/2} > 1$).
2. Dependence on Domain Size:
   • Comparable Sizes ($D_d/D_p = 1.0$): This geometry yields the most pronounced selectivity at low and intermediate chemical potentials. The formation of adsorption sites with large particle-domain overlap allows the strongly attracted species to preferentially occupy these favorable locations.
   • Large Domains ($D_d/D_p = 2.0$): High selectivity is observed at low chemical potentials (dilute regime), as the few available particles are predominantly the strongly attracted species. However, as the surface becomes crowded, selectivity decreases.
   • Small Domains ($D_d/D_p = 0.5$): Selectivity is generally weaker compared to the $D_d/D_p = 1.0$ case. The adhesive regions are numerous and uniformly distributed, creating an averaged attractive field that reduces the discrimination between species. However, the paper notes that further reducing the domain size (approaching the limit $D_d \to 0$) can slightly increase selectivity compared to the $D_d/D_p = 0.5$ case, as the surface approaches a uniform attractive substrate with effective affinity parameters.
3. Surface Coverage and Order: Increasing the domain surface coverage ($\sigma_d$) enhances the overall adsorption and the magnitude of selectivity, particularly for $D_d/D_p = 1.0$. The distinction between ordered and disordered domain arrangements is significant only at low and intermediate chemical potentials; at high coverages, excluded volume effects dominate, masking the influence of the specific domain arrangement.
4. Deviation from Uniform Surfaces: The adsorption behavior on patterned surfaces deviates significantly from that of a uniform attractive surface with equivalent effective attraction strength, particularly for domain sizes comparable to or larger than the particle size.

Significance and Claims
The paper claims to provide a minimal statistical mechanical model that successfully separates energetic contributions (affinity) from geometric factors in selective adsorption. The primary significance lies in demonstrating that surface geometry is a tunable parameter for controlling selectivity, independent of chemical affinity differences.

• The authors conclude that optimizing the geometry of the adhesive pattern, specifically the domain size, can enhance selectivity.
• They assert that domains comparable to the particle size are particularly effective for discrimination because they create distinct, high-overlap adsorption sites.
• The study suggests that for particle-sorting applications, selectivity can be improved not only by increasing the difference in particle-domain affinity but also by tailoring the adhesive domain size.
• The work establishes a baseline for future extensions, such as mixtures with different particle sizes or asymmetric bulk conditions, to examine how size effects and bulk composition jointly determine selective adsorption.