Adsorbed films of three-patch colloids: Continuous and discontinuous transitions between thick and thin films

We investigate numerically the role of spatial arrangement of the patches on the irreversible adsorption of patchy colloids on a substrate. We consider spherical three-patch colloids and study the dependence of the kinetics on the opening angle between patches. We show that growth is suppressed below and above minimum and maximum opening angles, revealing two absorbing phase transitions between thick and thin film regimes. While the transition at the minimum angle is continuous, in the directed percolation class, that at the maximum angle is clearly discontinuous. For intermediate values of the opening angle, a rough colloidal network in the Kardar-Parisi-Zhang universality class grows indefinitely. The nature of the transitions was analyzed in detail by considering bond flexibility, defined as the dispersion of the angle between the bond and the center of the patch. For the range of flexibilities considered we always observe two phase transitions. However, the range of opening angles where growth is sustained increases with flexibility. At a tricritical flexibility, the discontinuous transition becomes continuous. The practical implications of our findings and the relation to other nonequilibrium transitions are discussed.

Figure 1

Schematic representation of patches (red) on the surface of a three-patch colloid (blue) and their interaction range $\theta$ (green). The distribution of patches is described by an opening angle $\delta$, in units of $\pi \text{rad}$, from the center of the two patches and the center of the reference one. The (red) patch is the bonding site and its interaction range (green) represents the extent of the attractive interaction between patches.

Figure 2

Adsorption of three-patch colloids for different values of the opening angle $\left(\delta \right)$. Two absorbing phases are found for $\delta <{\delta }_{\min }$ and $\delta >{\delta }_{\max }$, where growth is suppressed at a finite thickness. For ${\delta }_{\min }<\delta <{\delta }_{\max }$, a sustained growth is observed (active phase). The data points are for the data collapse of the roughness in the active phase using $w_{\mathrm{sat}}=L^{{\alpha }_{\mathrm{KPZ}}}\mathcal{F}\left[\delta \right]$, where $\mathcal{F}$ is a scaling function and ${\alpha }_{\mathrm{KPZ}}$ is the roughness exponent for the Kardar-Parisi-Zhang universality class. We considered three different lengths of the substrate $L=\{256,512,1024\}$ and results are averages over $\{320000,80000,40000\}$ samples.

Figure 3

Snapshots for different regimes. From left to right $\delta =\{0.4,0.468,0.666,0.833,0.85\}$. For a system size $L=128$ and $10L$ deposited colloids.

Figure 4

(a) Lower and upper bounds for ${\delta }_{\min }$ as a function of $1/L$, for $L=\{128,256,512,1024,2048\}$ and $\{1600,800,400,200,100\}$ samples. (b) Dependence of the growth rate $\left(r\right)$ on the opening angle $\left(\delta \right)$ for $L=2048$ averaged over ${10}^2$ samples. (c) Finite-size scaling for the growth rate, where $\beta =0.58,{\nu }_{\perp }=0.73$, and ${\delta }_c=0.468$, consistent with the directed percolation universality class. The system sizes and number of samples are the same as in (a).

Figure 5

Main plot: Histogram of the growth rate for four values of opening angle close to ${\delta }_{\max }$, namely, $\delta =\{0.83,0.82,0.81,0.8\}$, and $L=512$, averaged over ${10}^5$ samples. Inset: Histogram of the growth rate at ${\delta }_{\max }=\frac{5}{6}$ for $L=\{128,256,512,1024,2048\}$, averaged over $\{16,8,4,2,1\}\times {10}^5$ samples.

Figure 6

Schematic representation of the flexibility mechanism between bonds: The patch attempting binding will be at a position defined by the angle $\gamma$ generated by a Gaussian distribution centered at zero, with dispersion $F\theta$, truncated at $F\theta$. For both colloids, the orientation of the bond is shifted from the ideal one by a randomly generated angle $\gamma$.

Figure 7

Finite-size scaling for the growth rate rescaled by the exponents of the absorbing transition for (a) $F=0.1$ with linear rescaling and (b) $F=0.5$ with directed percolation rescaling. Results are for $L=\{128,256,512,1024,2048\}$ and averaged over $\{1600,800,400,200,100\}$ samples. (c) Phase diagram in the two-parameter space: flexibility $\left(F\right)$ and opening angle $\left(\delta \right)$. The (blue-)solid curve in the left-hand side corresponds to the lower threshold $\left({\delta }_{\min }\right)$ and the (red-)solid curve in the right-hand side to the higher one $\left({\delta }_{\max }\right)$. The data points are extrapolations for the thermodynamic limit from the behavior of the size dependence of the thresholds. The (black) dashed curve is the theoretical prediction for ${\delta }_{\max }$.