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

TL;DR

This work addresses the challenge of achieving consistent drying rates for solution films on wafer-sized substrates using a linearly moving air knife. It develops a discretized, mass-transfer–based drying model that links local thickness evolution to a moving nozzle through a position-dependent mass-transfer coefficient derived from a Nusselt-number correlation, and it targets synchronization of the drying front with the nozzle at a critical thickness via a gradient-based optimization. The authors derive a set of equations for consistent drying rates and solve them with a Levenberg–Marquardt gradient descent, obtaining optimal air-knife trajectories that render drying nearly uniform across the substrate for several thickness scenarios; convex cases show limitations but can still benefit from optimized trajectories and trajectory smoothing. The findings offer a pathway to improved throughput and uniformity in batch drying of large-area thin films and motivate experimental validation, two-nozzle strategies, and extensions to include surface-tension effects and multi-parameter optimization.

Abstract

This work tackles the problem of achieving consistent drying rates of a solution film deposited on a $20\,\rm{cm}$-wide substrate ($\approx $ silicon-wafer size) that is driven under a narrow air flow ejected by a slot nozzle (or "air knife"). The main prerequisite of the work is that the drying rate of the solution film is highly decisive for a certain performance indicator of the deposited film at a particular, critical concentration $c_{\rm crit.}$. Empirically, this concentration can be associated with the visual observation of "the drying front" as, for the example of hybrid perovskite thin films, caused by the onset of a crystallization process. As a main result, a set of equations for achieving consistent drying rates, $\dot{d}_{\rm crit.}$, at critical concentration is presented that is solved by a simple two-staged least-squares gradient decent. From the resulting velocity vector, an optimal trajectory of the air knife, $\hat{x}(t)$, depending on the initial wet film thickness distribution over the substrate is derived. It is demonstrated that scenarios where the wet film thickness increases along the movement direction of the air knife have a consistent set of equations. Wet thin films that do not obey this constraint, as in the demonstrated scenarios with convex and concave shapes of wet film thickness over the substrate area, cannot always be dried in a fully consistent way by optimizing the air-knife trajectory alone. However, with the presented methods, optimal trajectories can still be derived that enable more homogeneous drying results.

Figures (6)

• Figure 1: Schematic depicting the one-dimensional, idealized problem addressed in this work and illustration of the one-dimensional spatial discretization employed. An air knife is moved with a certain speed $u_i$ from position $x_{i-1}$ to $x_i$ while exerting the gaseous mass transfer coefficient $\beta_{i}(t)$ on the film at this position. Evidently, the film will be dried fastest directly under the air knife, while the drying rate falls of toward the edges of the substrate.
• Figure 2: Mass Transport coefficient $\beta (x)$ for the chosen DMF-based solution as calculated from the empirically validated Nusselt correlation by M. Nirmalkumar el al. nirmalkumarLocalHeatTransfer2011. FWHM($\beta$) and $\beta_{\rm max}$ are indicated.
• Figure 3: Start scenarios for coated wet film thickness that will be dried in the simulated environment. All scenarios (I-IV) with a non-decreasing wet film thickness over the substrate width are consistently solvable, whereas the scenarios of a convex film thickness over the substrate (V-VIII) are potentially inconsistent. To make the scenarios comparable, the initial wet film thickness of the film is $d_0=4\,\rm{\mu m}$ is kept in every scenario and the volume of the thin films is kept constant throughout the variation.
• Figure 4: Ideal solutions of scenarios I-IV. The first column of the left depicts the starting wet film thickness distribution, the second column the calculated air knife velocities (solid lines, left black axis) and the associated residuals (dots, right gray axis), the third column the smoothed air knife trajectory, $\hat{x}(t)$, (solid lines, left black axis) the associated acceleration $a(t)$ (dots, right gray axis) and the last column the thicknesses $d_i(t)$ over time corresponding to the turquoise line in the other plots along with the calculated drying rates (violet triangles) over the critical thicknesses (crosses).
• Figure 5: Optimal solutions of scenarios V-VIII. The first column of the left depicts the starting wet film thickness distribution, the second column the calculated air knife velocities (solid lines, left black axis) and the associated residuals (dots, right gray axis), the third column the smoothed air knife trajectory, $\hat{x}(t)$, (solid lines, left black axis) the associated acceleration $a(t)$ (dots, right gray axis) and the last column the thicknesses $d_i(t)$ over time corresponding to the turquoise line in the other plots along with the calculated drying rates (violet triangles) over the critical thicknesses (crosses).