🔬 materials science

Computational Determination of Optimal Growth Protocols for Metastable Polymorphs

This paper demonstrates that combining kinetic Monte Carlo simulations with optimal control theory enables the design of time-dependent temperature and pressure protocols that significantly increase the yield of metastable organic-inorganic interface polymorphs by guiding the system through phase space to avoid thermodynamically stable structures.

Original authors: Simon B. Hollweger, Anna Werkovits, Tadeas Lesovsky, Oliver T. Hofmann

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Simon B. Hollweger, Anna Werkovits, Tadeas Lesovsky, Oliver T. Hofmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a specific type of Lego castle. You have a box of Lego bricks, and you want to build a "Metastable Castle" (a structure that is tricky to build but has amazing features, like glowing windows). However, if you just leave the bricks alone or build them too slowly, they naturally want to snap together into a "Stable Castle" (a boring, dull structure that is very hard to break apart but lacks the cool features).

The problem is that the "Stable Castle" is the path of least resistance. If you just follow the natural rules of physics, you'll almost always end up with the boring one. The scientists in this paper wanted to find a way to force the bricks to build the "Metastable Castle" instead, and they wanted to do it quickly and efficiently.

Here is how they did it, explained simply:

The Challenge: The Rush vs. The Trap

Think of building these structures like a race.

Thermodynamics (The Slow, Safe Path): This is like taking a long, flat road. It's easy, but it leads to the boring "Stable Castle."
Kinetics (The Fast, Risky Path): This is like taking a shortcut over a bumpy hill. It's harder, but if you time it right, you can reach the "Metastable Castle" before the race is over.

Usually, scientists try to build the Metastable Castle by just slowing everything down (lowering the temperature). But this takes forever, and sometimes you still end up with the wrong castle.

The Solution: A Smart GPS for Temperature and Pressure

The authors realized that instead of just picking a starting point and an ending point, they needed a smart GPS route that changes speed and direction constantly. They used a computer method called "Optimal Control Theory" to calculate the perfect recipe of Temperature (how hot the bricks are) and Pressure (how many bricks are being thrown at the building site) over time.

They tested this on a model system that acts like a metal surface covered in organic molecules. These molecules can lie flat or stand up, and they can arrange themselves in different patterns (like a brick wall or a herringbone pattern).

The Goal: Get the molecules to stand up in a specific "brick wall" pattern (the Metastable target).
The Enemy: The molecules lying flat in a different pattern (the Stable, boring outcome).

The "Manual" Attempt vs. The "Smart" Route

First, the scientists tried a "manual" approach. They guessed a three-step plan:

Heat it up fast.
Hold it there for a while.
Cool it down fast.
This worked okay, getting them 73% of the way to the perfect Metastable Castle. The rest was the wrong, boring structure.

Then, they let the computer find the perfect route. The computer's plan was surprisingly clever and counter-intuitive:

The Detour: Instead of rushing straight to the "cool and high pressure" zone where the Metastable Castle grows, the computer said, "Wait, let's drop the pressure first!"
- Why? This keeps the molecules in a temporary "waiting room" (a flat-lying state) where they don't accidentally snap into the boring Stable Castle. It's like waiting in a safe zone before crossing a dangerous bridge.
The Sprint: Once the temperature dropped low enough to make the Metastable Castle grow faster than the Stable one, the computer slammed the pressure up to the maximum and held it there. This forced the molecules to snap into the desired shape quickly.
The Cool Down: Finally, it slowly adjusted the conditions back to normal to lock everything in place.

The Result

By following this computer-generated "smart route," the yield of the desired Metastable Castle jumped from 73% to 97%.

The Catch (The Reality Check)

The scientists also tested this "smart route" in a more detailed, realistic simulation (like a full-scale video game of the building process).

The Good News: The computer's prediction was almost perfect for most of the process.
The Bad News: At the very end, the prediction was slightly off. The computer thought it would get 97%, but the detailed simulation showed a tiny bit less.
Why? The computer model was trained on data that didn't include the very specific conditions of the final step. It was like a map that was perfect for the highway but slightly inaccurate for the final driveway. However, the scientists confirmed that the route still successfully guides the system to a very high yield, proving the method works.

The Bottom Line

This paper shows that by using math to design a dynamic, changing recipe of heat and pressure, we can trick molecules into building the "cool" structures we want, rather than the "boring" ones they naturally prefer. It's like giving the molecules a strict, perfectly timed dance instructor so they don't trip and fall into the wrong formation.

Technical Summary: Computational Determination of Optimal Growth Protocols for Metastable Polymorphs

Problem Statement
The reliable growth of specific target structures at organic–inorganic interfaces remains a central challenge, particularly when the desired structure is thermodynamically metastable. While thermodynamics drives systems toward the lowest Gibbs free energy state, kinetic effects often trap systems in metastable configurations. Although such metastable structures frequently exhibit superior properties (e.g., higher bio-availability or charge carrier mobility), identifying experimental protocols to selectively grow them is difficult. Standard strategies involving reduced temperatures often lead to unacceptably long experimental timescales or low yields due to the simultaneous formation of competing stable phases. Furthermore, multiple competing metastable structures can exist, further reducing the yield of the target material. Experimentally identifying optimal time-varying temperature and pressure protocols to navigate these kinetic regimes is time-consuming and costly.

Methodology
The authors propose a computational framework that combines kinetic Monte Carlo (kMC) simulations with a parameterized nucleation-and-growth model, optimized via Optimal Control Theory (OCT).

Model System: The study utilizes a prototypical model of $\pi$ $π$ -conjugated organic molecules on a metal surface, abstracted into four potential polymorphs:
- LHB: Lying molecules in a herringbone motif (negligible role).
- LBW: Lying molecules in a brickwall motif (thermodynamically stable at high $T$ , low $p$ ).
- SHB: Standing molecules in a herringbone motif (stable at low $T$ , high $p$ ).
- SBW: Standing molecules in a brickwall motif (metastable; the target structure).
Kinetic Modeling: A parameterized effective nucleation-and-growth model is formulated to describe phase transition rates. The rate of change in surface fraction $c_i$ is governed by nucleation rates ( $n_{ji}$ ) and growth rates ( $k_{ji}$ ) dependent on temperature ( $T$ ) and partial pressure ( $p$ ). These rates follow Arrhenius-like expressions involving free energy barriers and attempt frequencies.
Parameterization: Elementary rates (diffusion, reorientation, adsorption) are derived from kMC simulations performed on a fine grid of $(T, p)$ conditions. The parameters for the effective model are obtained via least-squares fitting to the kMC data.
Optimal Control Formulation: The growth protocol is treated as an optimal control problem. The objective is to maximize the final yield of the metastable SBW phase ( $c_{SBW}(t_f)$ $c_{S B W} (t_{f})$ ) within a fixed duration ( $t_f = 1800$ $t_{f} = 1800$ s). The control variables are the time-dependent temperature $T(t)$ $T (t)$ and the logarithmic pressure $q(t) = \log_{10} p(t)$ $q (t) = lo g_{10} p (t)$ .
- Constraints: Realistic experimental bounds are imposed, limiting temperature changes to $1 \text{ K s}^{-1}$ and pressure changes to one order of magnitude per 100 s. The temperature range is restricted to $300\text{--}525$ K, and pressure to $10^{-8}\text{--}10^{-6}$ bar. Smooth transitions are enforced by requiring vanishing time derivatives at the start and end points.
- Solver: The problem is solved using a direct pseudospectral approach implemented in the Python framework yapss, utilizing the Ipopt solver.

Key Results

Baseline Performance: A manually optimized three-step protocol (rapid transition to a "growth point," constant conditions, rapid return) was established as a baseline. This approach yielded a maximum of 73% SBW phase, with the remaining 27% consisting of the undesired stable SHB phase.
Optimized Protocol: The OCT-derived protocol significantly improved the yield to 97% for the same total duration. The optimized trajectory differs fundamentally from the manual approach:
- Initial Stage: Instead of immediately moving to the low-temperature, high-pressure regime favorable for SBW, the protocol initially lowers pressure. This keeps the system in the LBW-stable region while minimizing the accumulation of the stable SHB phase, which is kinetically suppressed at low pressures.
- Growth Stage: Once the temperature drops sufficiently to enter the regime where the LBW-to-SBW transition rate exceeds the LBW-to-SHB rate ( $k_{LBW\to SBW} > k_{LBW\to SHB}$ ), the protocol rapidly increases pressure to the maximum allowed ( $10^{-6}$ bar) to drive the conversion.
- Final Stage: Towards the end, temperature is gradually increased to ensure complete conversion of the initial phase before the protocol concludes, followed by a rapid return to the final experimental conditions.
Validation: The optimized protocol was validated against kMC simulations. While the effective model predicted a 97% yield, the kMC simulation showed a slight deviation in the final stage, resulting in a slightly lower yield. The authors attribute this to the effective model being extrapolated into a regime (high $T$ , near-full SBW coverage) not fully represented in the training data, leading to an underestimation of SHB nucleation in the model. Nevertheless, the kMC results confirmed that the optimized trajectory successfully identifies conditions for high-yield growth.

Significance and Claims
The paper demonstrates that time-dependent temperature and pressure protocols can be computationally optimized to selectively promote the formation of metastable surface polymorphs with high yield. By formulating the growth process as an optimal control problem, the authors show that high-yield growth is not achieved by simply moving as quickly as possible to a single favorable growth point. Instead, the optimal trajectory exploits specific kinetic regimes, guiding the system through phase space to avoid regions where the thermodynamically stable structure is favored.

The authors claim that this approach provides a rational method for controlling metastable structure formation in organic monolayers. They highlight its relevance to organic electronics, where the microscopic structure of the interface dictates device performance. The ability to selectively grow desired interface polymorphs offers a pathway toward more controlled interface design, potentially overcoming current limitations in fabricating high-performance organic electronic devices. The study emphasizes that this method relies on combining physically interpretable kinetic models with optimal control theory, offering a more efficient alternative to trial-and-error experimental optimization.