Emergent Rate Laws for Collective Lying-Standing Transitions

Abstract

Lying-standing transitions in the first molecular monolayer at organic-inorganic interfaces strongly influence interface dipoles, energy-level alignment, and growth modes, yet their collective kinetics remain difficult to predict. Here, we establish a quantitative adsorbate-to-kinetics relationship using first-principles-based kinetic Monte Carlo simulations combined with a mean-field coarse-graining strategy. Focusing on tetracyanoethylene on Cu(111), we show that the collective transition rate cannot be inferred from any single elementary step but emerges from coupled microscopic processes, including reorientation, adsorption, and diffusion. A local two-step reorientation mechanism captures the diffusion-limited regime, while diffusion of lying molecules accelerates the transition in diffusion-enhanced regimes by suppressing back-reorientation via vacancy-molecule decoupling. This effect is described by a regime-dependent geometric factor accounting for deviations between single-molecule and collective rate constants. By varying molecular size and footprint ratio, we demonstrate that geometry is an intrinsic control parameter. While the collective rate scales approximately with molecular area, increasing the footprint ratio between lying and standing configurations yields order-of-magnitude accelerations due to enhanced vacancy creation and diffusion-assisted stabilization. Finally, we derive an analytical expression for the collective reorientation rate constant linking temperature- and pressure-dependent microscopic rate constants to geometric parameters. The formulation reproduces the simulations across kinetic regimes and provides transferable design principles for engineering lying-standing transition timescales at organic-inorganic interfaces.

Figures (15)

• Figure 1: Reference model system illustrating (a) geometry, (b) single-molecule energetics, and (c) thermodynamic stability. (a) Molecular geometry: the planar adsorbate occupies a square footprint of side length $l=6.8\angstrom$ and area $a_\mathrm{L}=l_L^2$ in the lying orientation (blue), while the standing orientation occupies a smaller area $a_\mathrm{S}$, such that two standing molecules fit into $a_\mathrm{L}$. (b) Schematic potential energy landscape showing adsorption energies and activation barriers for diffusion, reorientation, rotation, and desorption of individual molecules. All on-surface processes are assigned a common attempt frequency of $1e12\per s$ (see \ref{['sec:p3_methods']}). (c) Thermodynamic phase diagram indicating the orientation with lowest Gibbs free energy of adsorption per area. Orange regions indicate thermodynamically stable standing molecules, blue regions lying molecules, and grey regions conditions under which the empty surface is thermodynamically favored.
• Figure 1: kMC representation of model systems with footprint ratios $f = 2$, 3, and 4. This figure extends Figure \ref{['fig:pub3_fig6']} of the main text, where footprint areas of molecules in lying ($a_{\mathrm{L}}$) and standing ($a_{\mathrm{S}}$) orientations are denoted. For a footprint ratio $f = 2$, the lying molecule is represented by a $2 \times 2$ square of the kMC lattice (square lattice with lattice constant $l_{\mathrm{uc}}$), and by $3 \times 3$ and $4 \times 4$ squares for $f = 3$ and 4, respectively. Standing molecules occupy either $1 \times f$ or $f \times 1$ areas of the kMC lattice, corresponding to the two adsorption orientations of standing molecules (horizontal and vertical).
• Figure 2: (a) Coverage fraction of standing adsorbates $\theta_\mathrm{S}$ as a function of time at $T=300K$ and $p~=~1m\bar{$. Snapshots of five independent kMC simulations (faded markers) are averaged to obtain a statistically robust coverage profile (orange solid line) for fitting via the IPL2SA (dashed red line). (b) Temperature-dependence of the collective rate constant for the lying-standing transition (dashed) relative to the single-molecule rate constants (solid, with annotated microscopic processes) for a pressure of $p=1m\bar{$. The intersections of single-molecule rate constants indicate changes of kinetic regimes between diffusion-limited (DL), reorientation-limited (RL) and adsorption-limited (AL). The red marker highlights the collective rate constant obtained from the IPL2SA fit at $T=300K$ and $p~=~1m\bar{$ (panel a).
• Figure 2: Definition of on-surface processes for the reference model system (footprint ratio $f = 2$) showing the three possible adsorption geometries lying, standing horizontal, and standing vertical. Shown are diffusion in lying (a) and standing (b-c) orientations, rotation in standing orientation (d-e), and reorientation from lying to standing (forward transition; f-i), referred to as stand-up, with the reverse transition termed lay-down. Initial adsorption sites are visualized as filled blue squares or orange rectangles, respectively, while final adsorption sites are shown with hatched shapes. Arrows indicate the transition directions. Only forward transitions are shown explicitly, whereas reverse transitions correspond to the inverted pathways. This schematic representation is adapted from the Supporting Information of ref werkovits_kinetic_2024.
• Figure 3: Dominant reaction channel illustrating the local two-step reorientation mechanism in the absence of diffusion. Following reorientation (State 1 $\rightarrow$ State 2), the system either returns to the lying configuration via back-reorientation or proceeds by adsorption to complete the effective reorientation (State 3). The local reorientation event is treated in isolation from the surrounding adsorbates (black area), whose orientations are assumed arbitrary and whose coupling to the transition site is neglected. Annotated arrows indicate directions and total rate constants ($n_{\mathrm{i} \rightarrow \mathrm{j}} \cdot k_{\mathrm{i} \rightarrow \mathrm{j}}$) of single-molecule transitions between microstates $i$ and $j$. Blue: lying; orange: standing; grey: empty.