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, i.e., 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 nonmonotonic 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.

Figure 1

Generalized version of the phase diagram for cell linear dimension $\alpha$ vs intercell distance $\beta$. Cell-cell separation $\beta <{\beta }_c$ yields the interacting cell-cell adsorption (ICCA) regime, while $\beta ⩾{\beta }_c⩾1$ gives the noninteracting cell-cell adsorption (NICCA) regime. For cell sizes $\alpha <{\alpha }_c⩽\sqrt{2}/2$, we find the single-particle-per-cell adsorption (SPCA) regime, while for $\alpha ⩾{\alpha }_c$, we find the multiparticle-per-cell adsorption (MPCA) regime.

Figure 2

Particles attempting adsorption have a truncated Gaussian-size distribution, as described in the text. The two snapshots represent $20\times 20$ unit cell regions from larger simulations showing a typical configuration at the jammed state with $\alpha =0.96$. Cell-cell separation is large enough for the simulations to take place in the NICCA regime: (a) At a size dispersion of $1\%$, $\beta =1.5>{\beta }_c=1.02$, while (b) for a size dispersion of $20\%$, $\beta =1.5>{\beta }_c=1.4$. In (a), only monomers, dimers, and trimers within cells are possible (in fact, trimers are not observed in this snapshot), while in (b) tetramers and pentamers are also observed.

Figure 3

Adsorbed radii distribution for patterned surface with $\alpha$ equal to $0.4$ (solid, red line) and $1.7$ (dashed, green line), and the bulk case ($\alpha \to \infty$) (dotted, blue line). For size dispersion: (a) $1\%$, (b) $5\%$, (c) $10\%$, (d) $15\%$, and (e) $20\%$. The vertical lines represent the mean normalized radius $\langle \eta \rangle$ of adsorbed particles. Notice that for $\alpha =0.4$, the corresponding value is $1/2$, regardless of the size dispersion.

Figure 4

For the noninteracting cell-cell adsorption regime, at the jammed state, as a function of the cell size: (a) coverage (for $\beta =1.5$), (b) mean number of particles per cell, and (c) mean adsorbed radius. Each line type corresponds to different values of the size dispersion, namely, solid (red) to $1\%$, dashed (green) to $5\%$, dotted (blue) to $10\%$, fine dotted (purple) to $15\%$, and dash-dotted (light blue) to $20\%$. In (a), the solid (black) line reproduces the monodisperse case in Ref. [17] (vertically displaced for clarity). In the inset of (a), the region $0.8⩽\alpha ⩽1.2$ for $5\%$ (lower line) and $10\%$ (upper line) of the size dispersion is magnified (details in the text). The error bars are smaller than the line thickness.

Figure 5

Density of monomers (red, solid line), dimers (green, dashed line), trimers (blue, dotted line), tetramers (magenta, fine dotted line), and pentamers (light blue, dash-dotted line) for different values of the size dispersion: (a) $1\%$, (b) $5\%$, (c) $10\%$, (d) $15\%$, and (e) $20\%$.