Consistent control of drying rates of solution thin films on wafer-sized substrates by dynamic air-knife drying with optimal trajectories

This paper presents a mathematical framework and a two-staged gradient descent algorithm to derive optimal air-knife trajectories that ensure consistent drying rates at a critical concentration for solution thin films on wafer-sized substrates, while also addressing the limitations of achieving uniform drying when initial wet film thickness distributions are non-monotonic.

The Gist

Imagine you are trying to dry a very large, wet painting on a canvas the size of a standard silicon computer chip (about 8 inches wide). You have a powerful hairdryer (called an "air knife" in the industry) that blows a narrow, focused stream of air.

The problem is that this hairdryer doesn't blow air evenly. The air is strongest right in the center of the stream and gets weaker as you move toward the edges. If you just hold the hairdryer still or move it at a constant speed, some parts of the painting will dry too fast, and others too slow.

For certain types of "paint" (specifically, special chemical solutions used to make solar cells and electronics), the speed at which they dry at a specific moment is critical. If they dry too fast or too slow at that exact moment, the final product will be flawed. The goal of this research is to figure out exactly how to move the hairdryer so that every single spot on the canvas hits that "perfect drying speed" at the exact same time.

Here is how the author, Simon Ternes, solved this puzzle:

1. The "Drying Front" Race

Think of the wet paint as a runner. As the air hits it, the paint dries and shrinks. There is a specific moment in the race—let's call it the "finish line"—where the paint reaches a critical thickness. The author wants the hairdryer to be right next to every runner exactly when they cross that finish line.

If the paint is thin in one spot and thick in another, the thin spot will reach the finish line faster. To keep the race fair, the hairdryer needs to move faster over the thin spots and slower over the thick spots. It's like a conductor leading an orchestra: if the violins play fast, the conductor speeds up; if the drums play slow, the conductor slows down, so everyone stays in sync.

2. The "Smart Hairdryer" Strategy

The paper proposes a method to calculate the perfect path for this hairdryer. Instead of moving in a straight line at a constant speed, the hairdryer needs to:

• Speed up and slow down dynamically.
• Accelerate (change speed) in a very specific, smooth way.

The author created a set of math equations to act as a GPS for the hairdryer. This GPS tells the machine exactly how fast to go at every single millimeter of the canvas to ensure the drying rate is perfect everywhere.

3. Different Shapes of Wet Paint

The author tested this idea with different "landscapes" of wet paint:

• The Slope (Easy Mode): Imagine the paint is a ramp, getting thicker from left to right. The math shows the hairdryer should start slow and gradually speed up. This works perfectly, like a car smoothly accelerating up a hill.
• The Jump (Academic Mode): Imagine the paint suddenly gets thicker in the middle, like a step. The hairdryer would need to instantly slow down to catch up with the thicker paint. In the real world, you can't stop instantly, so the machine would have to smooth out that jump, making the drying slightly less perfect at that exact spot.
• The Hill and Valley (Hard Mode):
  • The Hill (Convex): Imagine the paint is thick in the middle and thin on the edges. The hairdryer has to speed up, then slow down to handle the thick middle, then speed up again for the thin edges. This is tricky. The math shows that for the very end of the canvas, the hairdryer might not be able to move fast enough to keep up perfectly. It's like trying to run a race where the finish line keeps moving away from you; you do your best, but you might not be perfectly synchronized at the very end.
  • The Valley (Concave): Imagine the paint is thin in the middle and thick on the edges. This is actually easier to control! The hairdryer speeds up to handle the thin middle, then slows down for the thick edges. This works very well.

4. The Result

The paper concludes that by using these calculated, changing speeds (trajectories), you can get a much more uniform result than just moving the hairdryer at a constant speed.

• For simple slopes: You can get a perfect, consistent dry.
• For tricky shapes (hills): You might not get perfection, but you will get a result that is far better than the old "constant speed" method.

The Takeaway

If you are making high-tech films on a large, rigid board (like a silicon wafer), don't just move your drying tool at a steady pace. Instead, use a robot arm that knows the shape of your wet film and moves with a "smart" rhythm—speeding up and slowing down precisely—to ensure the whole film dries evenly at the most critical moment. This could lead to better solar cells and electronics with fewer defects.

Technical Summary

Technical Summary: Consistent Control of Drying Rates via Dynamic Air-Knife Trajectories

Problem Statement
This work addresses the challenge of achieving consistent drying rates for solution thin films deposited on wafer-sized substrates (specifically 20 cm wide, approximating an 8-inch silicon wafer). In industrial batch processing, particularly for organic and hybrid semiconductor films like perovskites, the film morphology is critically dependent on the drying rate at a specific "critical concentration" ($c_{crit}$). This concentration often corresponds to the onset of crystallization, visually observable as a "drying front."

Standard drying methods using static air knives or constant-speed substrate movement introduce spatial inhomogeneities in drying rates due to the non-uniform mass transfer profile of the air flow (highest at the nozzle center, falling off toward the edges). While periodic movement can average drying speeds, it fails when the morphology formation is time-sensitive, requiring a specific drying rate at the precise moment of crystallization onset. The core problem is determining the optimal time-dependent trajectory of a moving air knife to synchronize the drying front across the entire substrate width, ensuring a uniform drying rate at the critical thickness ($d_{i,crit}$).

Methodology
The study employs a one-dimensional, idealized mathematical model of the drying process, discretizing the substrate into $N$ segments. The drying dynamics are based on an empirically validated equation for methylammonium lead iodide perovskite films in N,N-Dimethylformamide (DMF), where the rate-limiting step is the diffusion of gaseous species into the surrounding volume.

• Mass Transfer Modeling: The spatial dependence of the mass transfer coefficient, $\beta(x)$, is modeled using a Nusselt number correlation for a slot nozzle, accounting for stagnation, turbulent transition, and wall jet regions.
• Optimization Goal: The objective is to maximize the mean absolute drying rate ($\dot{d}_{i,crit}$) at the critical thickness across all substrate positions. The ideal scenario involves the air knife center synchronizing with the drying front as it propagates.
• Trajectory Calculation: The problem is solved by determining the velocity vector $u_i$ for the air knife moving from position $x_{i-1}$ to $x_i$. This involves transforming the integral drying equation into the spatial domain to create a system of $N$ equations.
• Numerical Solution: A two-staged least-squares gradient descent approach is utilized:
  1. Stage 1: A logarithmic transformation of the governing equations is used to solve for an initial velocity vector ($u_{opt,1}$) using the Levenberg–Marquardt algorithm. This stage assumes a consistent solution exists where the air knife arrives exactly when the film reaches $d_{i,crit}$.
  2. Stage 2: To handle scenarios where a perfectly consistent solution is mathematically impossible (e.g., convex film profiles), a second residual function is minimized. This stage uses the result from Stage 1 as an initial guess to find an optimal trajectory that maximizes the mean drying rate even when residuals cannot reach machine precision.
• Trajectory Smoothing: The discrete velocity steps are converted into a continuous, physically realizable trajectory $\hat{x}(t)$ using a windowed third-degree polynomial fit to ensure regularized acceleration (finite jerk), suitable for robotic actuation.

Key Results
The study evaluates eight distinct scenarios of initial wet film thickness distributions:

1. Non-Decreasing Thickness (Scenarios I–IV): For films with constant, linearly increasing, step-jump, or exponentially increasing thickness, the optimization yields a consistently solvable set of equations.

   • The resulting air knife velocities increase monotonically as the film thickness decreases (thinner films dry faster).
   • In scenarios with linearly increasing thickness (Scenario II), the required velocity profile can exhibit a shallow maximum or linear increase, balancing the drying time against the travel time.
   • The residuals in these cases are negligible ($<10^{-14}$), confirming that a constant drying rate at $d_{i,crit}$ is achievable across the entire substrate.
   • The required acceleration profiles generally show a linear increase (constant jerk), which is feasible for state-of-the-art robot axes.

2. Convex Thickness Profiles (Scenarios V–VIII): For films that are thicker in the center and thinner at the edges, a fully consistent solution is not always possible.

   • As the air knife moves past the center toward the thinner edges, the drying front moves faster than the nozzle can adjust without exceeding physical velocity limits (capped at 1 m/s in the simulation).
   • Residuals indicate deviations: the air knife may be too fast for the final thin sections or too slow to catch the drying front, depending on the specific slope.
   • Despite the lack of perfect consistency, the optimized trajectory still provides a significant improvement in homogeneity compared to constant-speed drying.

3. Concave Thickness Profiles (Scenarios V'–VIII'): Films that are thinner in the center and thicker at the edges are easier to control.

   • The drying front reaches the thin center early. The subsequent increase in thickness at the edges allows the air knife to simply reduce its velocity to maintain synchronization, resulting in more consistent drying than convex profiles.

Significance and Claims
The paper claims that dynamic air-knife drying with optimized, non-constant trajectories offers a viable strategy to mitigate spatial inhomogeneities in drying rates, which are critical for morphology formation in materials like perovskite solar cells.

• General Applicability: While demonstrated on perovskite films, the methodology is general and applicable to any thin-film drying process where a critical concentration dictates morphology, provided the film thickness and solvent parameters are adjusted.
• Feasibility: The required trajectories involve accelerations and jerks that are achievable with current robotic positioning systems, suggesting that the transition from constant-speed to dynamic drying is an accessible experimental upgrade.
• Limitations: The work acknowledges that fully consistent drying is mathematically constrained by the initial film thickness distribution. Specifically, convex thickness profiles (thick center, thin edges) present a fundamental challenge where the drying front can outpace the nozzle's ability to decelerate or accelerate within physical limits.
• Future Directions: The authors suggest that for highly irregular films, future strategies could involve dual air knives approaching from both edges or optimizing gas speed in parallel with trajectory. They also note that future work should incorporate surface tension effects and film deformation, which are currently neglected in the idealized model.

In summary, the work provides a mathematical framework and numerical evidence that optimizing the trajectory of a moving air knife can significantly improve the homogeneity of drying rates on wafer-sized substrates, particularly for films with non-decreasing thickness gradients, and offers a "best-effort" optimization for more complex geometries.