🔬 applied physics

Coupling Evaporation, Chemical Reactions, and Crystallization: A Population-Balance Framework Applied to Two-Step Perovskite Crystallization

This paper introduces a zero-dimensional population-balance model that couples evaporation, chemical reactions, and crystallization to analyze two-step perovskite formation, offering a quantitative framework and a new vocabulary of "supply" and "rate limitations" to interpret crystallization kinetics and optimize solar-cell performance.

Original authors: K. Segadlo, O. J. J. Ronsin, J. Harting

Published 2026-10-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: K. Segadlo, O. J. J. Ronsin, J. Harting

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

Solar cells are becoming a cornerstone of the global energy transition, offering a way to harvest power from the sun that is increasingly affordable and efficient. Among the most promising materials for next-generation solar panels are metal halide perovskites, a class of crystals that can be manufactured using liquid solutions rather than expensive, high-vacuum machinery. This liquid-based approach allows for flexible, roll-to-roll production, much like printing newspapers. However, turning a liquid mixture into a high-performance solar film is a delicate balancing act. The quality of the final device depends entirely on how the crystals form as the liquid dries. If the crystals grow too small, too large, or with too many defects, the solar cell will fail to perform or degrade quickly. The challenge lies in the fact that the drying process is dynamic and complex: as the solvent evaporates, the concentration of the remaining chemicals changes, triggering a chain reaction of crystallization that is difficult to predict or control.

To solve this puzzle, researchers Kai Segadlo and Olivier Ronsin, working alongside Jens Harting, developed a new way to model the formation of these films. Instead of trying to track every single molecule in a three-dimensional space, they created a simplified, zero-dimensional framework that treats the entire film as a single, well-mixed system. This approach allows them to simulate how three distinct processes—chemical reactions, the evaporation of liquid, and the growth of crystals—interact over time. By focusing on the rates at which these events happen, they could explore the specific pathways a film takes from a wet liquid to a solid crystal, a process that is often too fast and complex to observe directly in a lab.

The researchers applied their model to a common scenario in perovskite manufacturing: a two-step crystallization process. In this pathway, the dissolved precursor chemicals first react with the solvent to form a temporary, intermediate solid phase. Only later does this intermediate transform into the final, desired perovskite crystal. This is a crucial distinction because many existing studies assume the final crystal forms directly from the liquid. The team's simulations revealed that the speed of the solvent's evaporation is the master switch that dictates which path the material takes. If the solvent evaporates slowly, the system has time to form the intermediate phase, which then slowly converts into the final crystal. If the solvent evaporates too quickly, the intermediate phase may never fully form, or it may be trapped in a state that prevents the final crystal from growing correctly.

One of the most significant findings from these simulations is that the absence of a detectable intermediate phase does not prove that it never existed. The model showed that if the second step of the process—the conversion of the intermediate to the final crystal—is fast enough, the intermediate phase can appear and disappear so quickly that it leaves no trace in the final film. This means that scientists looking at a finished solar film might see no evidence of an intermediate step and incorrectly conclude that the crystal formed directly from the liquid. The researchers argue that without measuring the evaporation rate alongside the crystallization, it is impossible to know which mechanism is actually at work.

The study also introduced a new way to think about what limits the speed of crystal growth. The team distinguished between "supply limitations," where the process waits for more material to become available, and "rate limitations," where the process is waiting for the chemical reaction or crystal growth to happen fast enough. In the context of drying films, the evaporation of the solvent often acts as the supply limit. If the solvent leaves the film faster than the crystals can grow, the process stalls, leaving behind unconverted liquid residues. Conversely, if the crystals grow faster than the solvent can leave, the process becomes limited by how fast the evaporation can create the necessary conditions for growth. This interplay means that simply changing the drying conditions can completely alter the final structure of the film, even if the chemical ingredients remain the same.

By mapping out these different scenarios, the researchers provided a vocabulary and a set of tools to interpret experimental data more accurately. Their work suggests that to truly control the formation of perovskite solar cells, scientists must stop looking at crystallization in isolation. Instead, they must view it as a coupled system where the removal of the solvent is just as important as the chemical reactions happening within the liquid. This understanding could help engineers design better drying protocols, ensuring that the crystals form with the perfect size and shape needed for high-efficiency, durable solar panels. While the model is currently based on simulations rather than direct physical measurements of every variable, it offers a robust framework for testing hypotheses and guiding future experiments, moving the field closer to mastering the art of solution-processed solar energy.

Technical Summary: Coupling Evaporation, Chemical Reactions, and Crystallization in Perovskite Films

Problem Statement
The formation of solution-processed perovskite photoactive layers is fundamentally challenged by a limited understanding of how solvent evaporation couples to crystallization pathways. Precise control over film morphology (grain size, density, roughness) is essential for solar cell performance and stability, yet current deposition protocols often rely on dynamic solvent evaporation to trigger nucleation and growth without a quantitative grasp of the underlying mechanisms. While in situ characterization techniques (e.g., GIWAXS, reflectometry) have advanced mechanistic understanding, many investigations remain qualitative. Crucially, evaporation kinetics are rarely measured consistently, and proposed film-formation pathways are seldom quantitatively tested against data. Furthermore, existing modeling frameworks in photovoltaics typically decouple chemical reactions, evaporation, and crystallization or rely on oversimplified descriptions, lacking a holistic approach to describe these jointly affecting processes.

Methodology
To address these gaps, the authors develop a general, zero-dimensional population-balance model (PBM) that couples evaporation, chemical reactions, and crystallization.

Framework: The model treats reagents and products as defined by their chemical nature and phase state (liquid, solid, gas). It utilizes a population balance equation where the evolution of species populations is determined by the sum of advancement rates ( $\xi_p$ ) of active processes.
Process Rates:
- Chemical Reactions: Modeled using a generalized mass-action rate equation based on chemical potentials (Flory-Huggins-Gibbs free energy density) rather than simple concentrations.
- Evaporation: Described via a semi-empirical Hertz–Knudsen expression, accounting for the reduction in surface area as the film solidifies and the pressure difference between the film surface and the environment.
- Crystallization: Derived from Classical Nucleation Theory (CNT) and the Kolmogorov-Johnson-Mehl-Avrami (KJMA) framework. The model distinguishes between nucleation and growth, incorporating an "impingement function" to account for geometric confinement when crystals collide. It introduces specific "mother phase" volumes and "impingement volumes" to handle complex scenarios where crystallization occurs within distinct liquid or solid domains (e.g., solid-state transformation within a precursor crystal).
Application to Perovskites: The authors propose a reference process system for Methylammonium lead halide (MAPI) formation involving solvent evaporation, precursor-solvent complexation, intermediate phase formation (solid-state precursor), and final perovskite crystallization. To manage the high dimensionality of the full system, the study focuses on a "two-step" successive crystallization subsystem: solvent evaporation inducing a crystalline precursor, which subsequently recrystallizes into the final perovskite phase, releasing solvent.

Key Contributions

Integrated Modeling Framework: The paper introduces a unified PBM capable of simultaneously handling multiple chemical reactions, multi-solvent evaporation, and successive or competitive crystallization processes. This fills a gap in photovoltaic research where such holistic approaches were previously lacking.
Vocabulary of Limitations: The authors introduce a conceptual vocabulary of "supply limitations" (e.g., solvent depletion) and "rate limitations" (e.g., evaporation vs. crystallization speed) to interpret crystallization kinetics based on governing process conditions.
Quantitative LaMer Curves: The model generates quantitative LaMer curves (precursor concentration vs. time) that explicitly link drying conditions to nucleation and growth regimes.
Mechanistic Decoupling: The study systematically dissects the interplay between evaporation, intermediate formation, and final crystallization, demonstrating how relative process speeds dictate the observable kinetics and final material state.

Results
Through parameter sweeps and simulation studies, the authors identify several critical behaviors:

Nucleation vs. Growth Regimes: Allowing for finite-size nuclei reveals a nucleation-dominated regime alongside the classical growth-dominated regime. In the nucleation-dominated regime, the Avrami exponent decreases (ranging from 1 to 3) without changing the growth dimension, rendering the Avrami exponent an ambiguous indicator of growth dimensionality.
Invisibility of Intermediates: In successive crystallization pathways, if the secondary step (precursor to perovskite) is kinetically limited by the primary step (precursor formation), the intermediate phase may be practically invisible (low peak concentration). Consequently, the absence of an intermediate signal in in situ data is insufficient evidence to conclude that formation occurs via direct crystallization.
Evaporation-Induced Coupling:
- Slow Evaporation: When evaporation is slow relative to crystallization, the system enters an evaporation-limited regime. The precursor is fully converted, and the crystallization rate is dictated by the rate of supersaturation buildup.
- Fast Evaporation: When evaporation is fast, the system becomes solvent-limited. The solvent may deplete before the precursor is fully converted, leaving liquid residues and resulting in incomplete conversion.
Solvent Recycling Feedback: In solvent-coupled successive crystallization, the secondary step (releasing solvent) can feedback to the primary step. If the secondary step is fast enough, it can resupply solvent, lifting solvent limitations on the primary step and enabling complete conversion even when initial solvent amounts are low.
Kinetic Signatures: The shape of the crystallization curves (e.g., kinks, plateaus, or extended tails) depends heavily on the relative speeds of evaporation, precursor formation, and final crystallization. Neither the intermediate nor the final phase curve alone can uniquely identify the mechanism; evaporation kinetics must be measured concurrently.

Significance
The paper claims that these findings provide a route to distinguishing crystallization pathways from measurable drying and phase-resolved composition curves. By establishing that the observed kinetics depend primarily on relative timescales rather than absolute parameter values, the authors suggest their framework is robust across different material systems and processing conditions. The work emphasizes that to control perovskite film formation, one must not only tune crystallization kinetics but also explicitly measure and model evaporation, as the two are inextricably coupled. The proposed framework serves as a mechanistic prior for analyzing complex solution-processed systems, moving beyond qualitative descriptions to quantitative, process-condition-dependent interpretations.

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