Controlling the Shape of Small Clusters with and without Macroscopic Fields

Despite major advances in the understanding of the formation and dynamics of nanoclusters in the past decades, theoretical bases for the control of their shape are still lacking. We investigate strategies for driving fluctuating few-particle clusters to an arbitrary target shape in minimum time with or without an external field. This question is recast into a first passage problem, solved numerically, and discussed within a high temperature expansion. Without field, large-enough low-energy target shapes exhibit an optimal temperature at which they are reached in minimum time. We then compute the optimal way to set an external field to minimize the time to reach the target, leading to a gain of time that grows when increasing cluster size or decreasing temperature. This gain can shift the optimal temperature or even create one. Our results could apply to clusters of atoms at equilibrium, and colloidal or nanoparticle clusters under thermo- or electrophoresis.

Figure 1

Graph of configurations for a tetramer cluster ($N=4$) at $T=0.24$. The node size is proportional to the expected residence time $t_{\mathit{\varphi }}(s)$. The thickness and shade of the edges are proportional to the transition probability $p_{\mathit{\varphi }}(s,s^{\prime })$. Arrows in the nodes represent an optimal policy ${\mathit{\varphi }}_{*}(s)$ to reach the orange target shape (crosses correspond to a zero force).

Figure 2

Expected return time to target as a function of $1/T$. Zero-force case ${\tau }_0^r(\overline{s})$ and under the optimal policy ${\tau }_{*}^r(\overline{s})$ with $F_0=0.4$. The $\times$ corresponds to ${\tau }_{\infty }^r(\overline{s})$ for $T\to \infty$.

Figure 3

Estimates of the high temperature slope. $M_0(\overline{s})$ and $M_{*}(\overline{s})$ correspond to the analytical expressions of Eqs. (5),(8). $M_0^{\mathrm{sim}}(\overline{s})$ and $M_{*}^{\mathrm{sim}}(\overline{s})$ are the slopes extracted from a high temperature fit using iterative numerical methods. $M_0^{\mathrm{appr}}(\overline{s})$ and $M_{*}^{\mathrm{appr}}(\overline{s})$ are approximate expressions from Eqs. (6), and (9).

Figure 4

Gain ${\tau }_0^r(\overline{s})/{\tau }_{*}^r(\overline{s})$ in the return time to target due to the optimization of the forces. (a) As a function of the force magnitude $F_0$, for a fixed target at different $T$. (b) For similar targets of increasing size, with $T=0.24$ and $F_0=0.4$.