Nonequilibrium growth of patchy-colloid networks on substrates

Patchy colloids with highly directional interactions are ideal building blocks to control the local arrangements resulting from their spontaneous self-organization. Here we propose their use, combined with substrates and nonequilibrium conditions, to obtain structures, different from those of equilibrium thermodynamic phases. Specifically, we investigate numerically the irreversible adhesion of three-patch colloids near attractive substrates, and analyze the fractal network of connected particles that is formed. The network density profile exhibits three distinct regimes, with different structural and scaling properties, which we characterize in detail. The adsorption of a mixture of three- and two-patch colloids is also considered. An optimal fraction of two-patch colloids is found where the total density of the film is maximized, in contrast to the equilibrium gel structures where a monotonic decrease of the density has been reported.

Figure 1

Schematic representation of a patch (red circle) on the surface of a colloid [blue (dark gray)] and its interaction range [green (light gray)]. The limits of the interaction range are defined by an angle $\theta$ with the center of the patch.

Figure 2

Density profile of the colloidal network on a substrate showing three different regimes: surface layer (green, left); liquid film (red, middle); and interfacial region (blue, right). Top: Snapshot of a typical configuration, where each stick corresponds to a colloid-colloid connection. Bottom: Density ${\rho }^{*}\left(z\right)$ as a function of the height $z$ after the adsorption of 40 particle layers, averaged over 500 independent realizations. ${\rho }_l$ is the density of the liquid film and $z_{1/2}$ is the film thickness, defined as the height at which the density is ${\rho }_l/2$.

Figure 3

Left: Dependence of the density of the first adsorbed layer, ${\rho }^{*}\left(0\right)$, on $D$, for values ranging from ${10}^{-3}$ to 1, and substrates with lateral size $L$, ranging from 16 to 128. Right: Snapshots of the first adsorbed layer (directly in contact with the substrate) for different diffusion coefficients $D$, namely, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$, and 1, on a substrate with lateral size 32 in units of the particle diameter.

Figure 4

(a) Liquid-film density for substrates with lateral size ranging from 16 to 128, and diffusion coefficient $D$ between ${10}^{-3}$ and 1. (b) Dependence of the interfacial decay length ${\xi }_{\mathrm{inter}}$ on $L$ and $D$.

Figure 5

Data collapse for the liquid film and interfacial region, including the dependence on the substrate lateral size $L$ and the diffusion coefficient $D$. Results obtained after adsorbing 60 particle layers and averaging over 500 samples for $L=\{16,32\}$, 200 samples for $L=64$, and 100 samples for $L=128$.

Figure 6

Top: Density profile of the colloidal network on a substrate with anisotropic particle-substrate interaction. Results after the adsorption of 40 particle layers, averaged over 500 independent realizations. ${\rho }_l$ is the density of the liquid film. Bottom: Snapshot of the particles in the surface layer regime.

Figure 7

Data collapse for the liquid film and interfacial region, where bonds are considered optimal. Results are obtained after adsorbing 60 particle layers and averaging over 500 samples for $L=32$ and 200 for $L=64$.

Figure 8

Snapshots of a region in the liquid film for different fractions of two-patch colloids, $r_D$. From left to right, top to bottom: 0, $0.3$, $0.6$, and $0.8$. Snapshots obtained from the adsorption on a substrate with a lateral size of 32 particle diameters and 40 particle layers. Three-patch colloids are in blue (dark gray), two-patch colloids are in green (light gray), the (red) spheres on the surface of the colloids represent the patches, and the (red) sticks are the connections between colloids.

Figure 9

Mass scaling as a function of the inverse box size in the box counting algorithm, for networks, with different concentrations of two-patch colloids, $r_D$. Results obtained with 32 layers of the 60 layers of adsorbed particles in the liquid film on a substrate with $L=32$, averaged over 20 samples. A fractal dimension $d_f=2.57\pm 0.04$ was found.

Figure 10

Analysis of the dependence on the fraction of two-patch colloids, $r_D$. (a) Liquid-film density (black squares) and the contribution from two- (red triangles) and three-patch (green spheres) colloids. (b) Total number of chains of two-patch colloids between three-patch ones with sizes 0, 1, and 2. The peak of the distribution of chains of unit size coincides with the peak of the film density. Results for 60 particle layers adsorbed on a substrate with $L=32$ averaged over 500 samples.

Figure 11

(a) Fraction of free (unconnected) patches on two-patch (red triangles) and three-patch (green circles) colloids in the liquid film. (b) Exponential increase of the average size of chains of two-patch colloids.