Limits of size scalability of diffusion and growth: Atoms versus molecules versus colloids

Understanding fundamental growth processes is key to the control of nonequilibrium structure formation for a wide range of materials on all length scales, from atomic to molecular and even colloidal systems. While atomic systems are relatively well studied, molecular and colloidal growth are currently moving more into the focus. This poses the question to what extent growth laws are size scalable between different material systems. We study this question by analyzing the potential energy landscape and performing kinetic Monte Carlo simulations for three representative systems. While submonolayer (island) growth is found to be essentially scalable, we find marked differences when moving into the third (vertical) dimension.

Figure 1

Part (a) depicts potential landscapes sampled along the lines across a closed substrate layer, as well as across a step edge, as illustrated in parts (b) and (c). Solid lines denote in-plane sampling, while dashed lines denote sampling across a step edge. The pair potentials used in part (a) are the Lennard-Jones potential (atomic) [27], the Girifalco potential (C60) [29, 30], and the Asakura-Oosawa potential [28] (see inset) with attraction range $\mathrm{\Delta }=0.05\sigma$ (colloidal). All pair potentials are scaled such that the well depth is $-1$ and the intersection with 0 is at 1 (defining the diameter $\sigma )$. The double arrow in part (a) illustrates the barrier $E_{\mathrm{s}}$ that an atom has to overcome to cross the step edge. The units of the $x$ axis are in-plane lattice constants.

Figure 2

Part (a) depicts the island density ${\rho }_{\mathrm{n}}\left(t\right)$ related to layer $n$ (in units of the area $A=0.866a^2$ of one unit cell with lattice constant $a)$ for atomic, C60, and colloidal growth, while part (b) depicts the layer coverage ${\mathrm{\Theta }}_{\mathrm{n}}\left(t\right)$. Time is measured relative to the time of the formation of one monolayer (ML). The gray block depicts the region in which ${\mathrm{\Theta }}_{\mathrm{n}}\left(t\right)$ evolves identically for all three systems, while the dashed line marks the time related to the snapshots (right). The length bars indicate 100 lattice constants.

Figure 3

The normalized gyration radius $R_{\mathrm{gyr}}/R$ as a function of time [38]. Each data point corresponds to the maximum of the data of different layers $n$. The gray vertical line indicates the time related to the inset, where $R_{\mathrm{gyr}}/R$ is plotted as a function of the attraction range $\mathrm{\Delta }$ (the gray dashed line is a guide to the eye). For the colloidal system, $\mathrm{\Delta }=0.05$. For C60 and the atomic system, $\mathrm{\Delta }$ has been chosen as the distance where the attractive part of $V\left(r\right)$ has dropped to 10% of the maximal attraction.