Effect of particle polydispersity on the irreversible adsorption of fine particles on patterned substrates

We performed extensive Monte Carlo simulations of the irreversible adsorption of polydispersed disks inside the cells of a patterned substrate. The model captures relevant features of the irreversible adsorption of spherical colloidal particles on patterned substrates. The pattern consists of (equal) square cells, where adsorption can take place, centered at the vertices of a square lattice. Two independent, dimensionless parameters are required to control the geometry of the pattern, namely, the cell size and cell-cell distance, measured in terms of the average particle diameter. However, to describe the phase diagram, two additional dimensionless parameters, the minimum and maximum particle radii are also required. We find that the transition between any two adjacent regions of the phase diagram solely depends on the largest and smallest particle sizes, but not on the shape of the distribution function of the radii. We consider size dispersions up-to 20% of the average radius using a physically motivated truncated Gaussian-size distribution, and focus on the regime where adsorbing particles do not interact with those previously adsorbed on neighboring cells to characterize the jammed state structure. The study generalizes previous exact relations on monodisperse particles to account for size dispersion. Due to the presence of the pattern, the coverage shows a non-monotonic dependence on the cell size. The pattern also affects the radius of adsorbed particles, where one observes preferential adsorption of smaller radii particularly at high polydispersity.

💡 Research Summary

The authors investigate irreversible adsorption of size‑dispersed circular particles onto a patterned substrate composed of equal square cells arranged on a square lattice. By scaling the cell‑to‑cell distance (α) and the cell side length (β) with the average particle diameter, the geometry of the pattern is captured with two dimensionless parameters. When particle radii are not monodisperse, two additional dimensionless quantities—the minimum and maximum radii (r_min, r_max)—must be introduced to fully describe the phase diagram. Using extensive Monte‑Carlo simulations, the study employs a truncated Gaussian distribution to generate size dispersions up to 20 % of the mean radius.

Key findings are: (1) The boundaries between adjacent regions of the phase diagram depend solely on r_min and r_max; the detailed shape of the size distribution does not affect the transition lines. This implies that the extreme particle sizes alone dictate whether a given (α, β) configuration falls into a “single‑particle per cell”, “multiple‑particle per cell”, or “blocked” regime. (2) Coverage as a function of β is non‑monotonic. For very small cells only the smallest particles can fit, leading to low jammed coverage. As β increases past a critical value, larger particles become admissible, causing a sharp rise in coverage. Further increase of β reduces the effective packing efficiency because the cells become too large relative to the particle ensemble, and the overall coverage declines. (3) At high polydispersity (σ/⟨r⟩≈0.2) a pronounced preferential adsorption of the smallest particles is observed. Large particles are frequently excluded by the cell boundaries, whereas small particles fill the residual space, lowering the average radius of adsorbed particles well below the ensemble mean. (4) The exact analytical relations previously derived for monodisperse systems (e.g., maximal jammed coverage πβ²/4) are successfully generalized by incorporating r_min and r_max, providing closed‑form expressions that remain valid across a wide range of dispersions.

The work therefore extends the theoretical description of irreversible adsorption from the ideal monodisperse case to realistic polydisperse colloids, offering practical guidelines for designing patterned substrates that maximize coverage or control the size selection of adsorbed particles. Potential applications include the fabrication of ordered colloidal arrays, photonic crystals, and biosensing platforms where precise spatial arrangement and size filtering are essential.