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arxiv: 2510.00085 · v1 · submitted 2025-09-30 · ⚛️ physics.flu-dyn · physics.app-ph

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

Simon Ternes This is my paper

Pith reviewed 2026-05-18 12:44 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.app-ph
keywords air knife dryingthin film dryingsolution processingoptimal trajectorydrying rate controlperovskite filmswafer substratesfluid dynamics
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0 comments X

The pith

Optimal air-knife trajectories achieve consistent drying rates on wafer-sized substrates by compensating for initial thickness variations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops equations that relate air-knife speed to the drying rate at a critical concentration in solution films on 20 cm substrates. The equations are solved with a two-stage least-squares gradient descent to produce a velocity vector, from which the knife's position as a function of time is calculated based on the starting wet-film thickness map. A reader would care because the drying rate at that critical point controls crystallization onset and final film quality in processes such as perovskite deposition, where inconsistent rates create defects across large areas. The method works exactly when film thickness increases along the knife's travel direction; for convex or concave profiles it yields the best achievable approximation.

Core claim

A set of equations for achieving consistent drying rates, dot d_crit., at critical concentration is presented that is solved by a simple two-staged least-squares gradient descent. 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. Scenarios where the wet film thickness increases along the movement direction of the air knife have a consistent set of equations, while convex and concave shapes cannot always be dried in a fully consistent way by optimizing the air-knife trajectory alone.

What carries the argument

The optimal trajectory hat x(t) of the air knife, obtained from the velocity vector that satisfies the equations for constant drying rate dot d_crit. at the critical concentration.

If this is right

  • When initial wet-film thickness increases along the knife movement direction, the derived trajectory produces identical drying rates at the critical concentration everywhere on the substrate.
  • For convex or concave thickness profiles the same procedure still supplies a trajectory that minimizes spatial variation in drying rate.
  • The approach applies directly to any solution-coating process in which drying rate at a single critical concentration governs final film quality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same trajectory-calculation steps could be adapted to other drying hardware such as infrared lamps or heated air streams to enforce uniform rates on large substrates.
  • In roll-to-roll or sheet-fed manufacturing lines the method offers a software-only route to reduce thickness or crystallization nonuniformity without hardware redesign.
  • Direct measurement of local solvent evaporation rate during a trial run with the computed path would test whether the predicted consistency is realized in practice.

Load-bearing premise

The drying rate at the critical concentration determines the performance indicator of the final deposited film.

What would settle it

Compare local drying rates or final film properties measured across the substrate when the air knife follows the computed optimal trajectory versus a constant-speed pass; the prediction is that uniformity improves exactly where the thickness profile obeys the monotonic-increase condition.

Figures

Figures reproduced from arXiv: 2510.00085 by Simon Ternes.

Figure 1
Figure 1. Figure 1: Schematic depicting the one-dimensional, idealized problem addressed in this work and [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mass Transport coefficient β(x) for the chosen DMF-based solution as calculated from the empirically validated Nusselt correlation by M. Nirmalkumar el al. [25]. FWHM(β) and βmax are indicated. 2.4 Goal of optimization of air-knife trajectory To optimize the drying process in the above described system, we need to define the goal of op￾timization. Due to the asymmetry of the setup explained above, it is no… view at source ↗
Figure 3
Figure 3. Figure 3: Start scenarios for coated wet film thickness that will be dried in the simulated environ [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ideal solutions of scenarios I-IV. The first column of the left depicts the starting wet film [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: As compared to the convex film shapes, the drying of concave films is easier to control. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Optimal solutions of scenarios V-VIII. The first column of the left depicts the starting [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimal solutions of scenarios V’-VIII’ for concave film shapes. The first column of the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a mathematical framework for controlling drying rates of solution thin films on ~20 cm substrates via dynamic air-knife motion. A set of equations is derived to enforce constant drying rate dot d_crit. at critical concentration c_crit.; these are solved by two-staged least-squares gradient descent to obtain a velocity vector, which is integrated to yield an optimal knife trajectory hat x(t) that depends on the initial thickness distribution. The equations are shown to be consistent when thickness increases along the knife direction, while convex and concave profiles cannot always be dried fully consistently by trajectory optimization alone, although improved homogeneity is still claimed to be achievable.

Significance. If the optimization reliably produces trajectories that reduce spatial variation in drying rate at c_crit., the method could be useful for solution-processed thin films (e.g., hybrid perovskites) where drying dynamics at the crystallization onset affect final film quality. The explicit equation set and simple descent procedure provide a concrete, implementable approach for wafer-scale dynamic drying control.

major comments (2)
  1. [Optimization procedure and trajectory derivation] The two-staged least-squares gradient descent is stated to solve the equations for constant dot d_crit., yet the manuscript acknowledges that convex and concave thickness profiles cannot always be dried fully consistently. No demonstration is given that the resulting velocity vector, once integrated, produces provably small residual inhomogeneity in drying rate at the moment each location reaches c_crit., nor that the descent reaches a global minimizer of rate variation (see abstract description of the procedure and the non-monotonic profile scenarios).
  2. [Physical model and prerequisites] The central prerequisite—that drying rate at c_crit. is decisive for film performance—is introduced without explicit model equations relating local drying rate to air-knife velocity, position, or film thickness. This makes it difficult to judge how generally the derived trajectories apply beyond the specific cases examined.
minor comments (3)
  1. Typographical error in the abstract: 'gradient decent' should read 'gradient descent'.
  2. Notation such as dot d_crit., hat x(t), and c_crit. would benefit from a dedicated nomenclature table or clearer inline definitions.
  3. The manuscript would be strengthened by inclusion of quantitative metrics (e.g., standard deviation of achieved drying rates or comparison to constant-velocity drying) for the convex/concave cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the scope and limitations of our work. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [Optimization procedure and trajectory derivation] The two-staged least-squares gradient descent is stated to solve the equations for constant dot d_crit., yet the manuscript acknowledges that convex and concave thickness profiles cannot always be dried fully consistently. No demonstration is given that the resulting velocity vector, once integrated, produces provably small residual inhomogeneity in drying rate at the moment each location reaches c_crit., nor that the descent reaches a global minimizer of rate variation (see abstract description of the procedure and the non-monotonic profile scenarios).

    Authors: We agree that the two-staged least-squares gradient descent yields a local solution and does not guarantee a global minimizer of the rate variation. The manuscript explicitly notes that fully consistent drying is not always possible for convex and concave profiles because the underlying system of equations becomes over- or under-determined. For the monotonic cases the solution satisfies the constant-rate condition exactly; for the non-monotonic examples we report numerical results showing reduced spatial variation relative to constant-velocity drying. We do not claim a formal proof of residual bounds or global optimality. In the revised manuscript we have added a short discussion of the optimization character, the observed residual inhomogeneity in the presented examples, and the fact that the procedure provides a practical, implementable trajectory rather than a provably optimal one. revision: partial

  2. Referee: [Physical model and prerequisites] The central prerequisite—that drying rate at c_crit. is decisive for film performance—is introduced without explicit model equations relating local drying rate to air-knife velocity, position, or film thickness. This makes it difficult to judge how generally the derived trajectories apply beyond the specific cases examined.

    Authors: The prerequisite is grounded in experimental literature on solution-processed films (e.g., hybrid perovskites), where morphology and performance are sensitive to the drying rate at the onset of crystallization. The present work develops the optimization framework under the assumption that a functional dependence between local drying rate, air-knife position, velocity, and film thickness exists and can be controlled. We acknowledge that an explicit statement of this dependence would improve clarity on generality. In the revised manuscript we have inserted a concise subsection that outlines the physical basis (convective mass transfer under the air knife) together with references to supporting studies, while keeping the focus on the trajectory-optimization procedure. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation derives trajectories from independent physical model

full rationale

The paper sets up equations enforcing constant drying rate dot d_crit at c_crit based on a physical model relating drying rate to air-knife velocity and position, then applies two-staged least-squares gradient descent to solve for the velocity vector and integrate to hat x(t). No step reduces a prediction to a fitted input by construction, no self-citation is load-bearing for the central claim, and the optimization is presented as a numerical method to achieve the stated goal rather than assuming the result. Limitations for non-monotonic profiles are explicitly noted, confirming the derivation does not smuggle in its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that consistent drying rate at c_crit determines film performance and on an implicit physical model relating local drying rate to air-knife velocity and initial thickness; no free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption 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_crit.
    Explicitly stated as the main prerequisite of the work.

pith-pipeline@v0.9.0 · 5808 in / 1443 out tokens · 70542 ms · 2026-05-18T12:44:08.084338+00:00 · methodology

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    a set of equations for achieving consistent drying rates, ˙d_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, ˆx(t), ...

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Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Drying of aqueous films, an application of heat and mass transfer

    Rodolphe Heyd, Julie Fichot, Driss Lahboub, Abderrahim Bakak, Christophe Josserand, and Marie-Louise Saboungi. Drying of aqueous films, an application of heat and mass transfer. In Diana Enescu, editor,Heat and Mass Transfer - from Fundamentals to Advanced Applications, chapter 8. IntechOpen, London, 2025

    work page 2025

  2. [2]

    Muhammad A. Butt. Thin-Film Coating Methods: A Successful Marriage of High-Quality and Cost-Effectiveness—A Brief Exploration.Coatings, 12(8):1115, August 2022

    work page 2022

  3. [3]

    Monitoring drying process of varnish by immersion solid matching method.Progress in Organic Coatings, 136:105299, November 2019

    Ilpo Niskanen, Janne Lauri, Jukka R¨ aty, Rauno Heikkil¨ a, Henrikki Liimatainen, Taro Hashimoto, Tapio Fabritius, Kaitao Zhang, and Masayuki Yokota. Monitoring drying process of varnish by immersion solid matching method.Progress in Organic Coatings, 136:105299, November 2019. 14

    work page 2019

  4. [4]

    Recent Advances in Printed Thin-Film Batteries

    Benoit Clement, Miaoqiang Lyu, Eeshan Sandeep Kulkarni, Tongen Lin, Yuxiang Hu, Vera Lockett, Chris Greig, and Lianzhou Wang. Recent Advances in Printed Thin-Film Batteries. Engineering, 13:238–261, June 2022

    work page 2022

  5. [5]

    Recent Advances and Challenges in Thin-Film Fabrication Techniques for Low-Temperature Solid Oxide Fuel Cells.Crystals, 13(7):1008, July 2023

    Mohammadmehdi Choolaei, Mohsen Fallah Vostakola, and Bahman Amini Horri. Recent Advances and Challenges in Thin-Film Fabrication Techniques for Low-Temperature Solid Oxide Fuel Cells.Crystals, 13(7):1008, July 2023

    work page 2023

  6. [6]

    Verboven and W

    I. Verboven and W. Deferme. Printing of flexible light emitting devices: A review on different technologies and devices, printing technologies and state-of-the-art applications and future prospects.Progress in Materials Science, 118:100760, May 2021

    work page 2021

  7. [7]

    Recent progress of organic and hybrid thermoelectric materials.Synthetic Metals, 225:3–21, March 2017

    Naoki Toshima. Recent progress of organic and hybrid thermoelectric materials.Synthetic Metals, 225:3–21, March 2017

    work page 2017

  8. [8]

    Recent Progress in Emerging Organic Semiconductors.Advanced Materials, 34(22):2108701, 2022

    Qichun Zhang, Wenping Hu, Henning Sirringhaus, and Klaus M¨ ullen. Recent Progress in Emerging Organic Semiconductors.Advanced Materials, 34(22):2108701, 2022

    work page 2022

  9. [9]

    Howard, Tobias Abzieher, Ihteaz M

    Ian A. Howard, Tobias Abzieher, Ihteaz M. Hossain, Helge Eggers, Fabian Schackmar, Si- mon Ternes, Bryce S. Richards, Uli Lemmer, and Ulrich W. Paetzold. Coated and Printed Perovskites for Photovoltaic Applications.Advanced Materials, 31(26):1806702, 2019

    work page 2019

  10. [10]

    An efficient and precise solution-vacuum hybrid batch fabrication of 2D/3D perovskite submodules.Nature Communications, 16(1):7019, July 2025

    Yingping Fan, Zhixiao Qin, Lei Lu, Ni Zhang, Yugang Liang, Shaowei Wang, Wenji Zhan, Jiahao Guo, Haifei Wang, Yuetian Chen, Yanfeng Miao, and Yixin Zhao. An efficient and precise solution-vacuum hybrid batch fabrication of 2D/3D perovskite submodules.Nature Communications, 16(1):7019, July 2025

    work page 2025

  11. [11]

    Søndergaard, Markus H¨ osel, and Frederik C

    Roar R. Søndergaard, Markus H¨ osel, and Frederik C. Krebs. Roll-to-Roll fabrication of large area functional organic materials.Journal of Polymer Science Part B: Polymer Physics, 51(1):16–34, 2013

    work page 2013

  12. [12]

    The roll-to-roll revolution to tackle the industrial leap for perovskite solar cells.Nature Communications, 15(1):3983, May 2024

    Ershad Parvazian and Trystan Watson. The roll-to-roll revolution to tackle the industrial leap for perovskite solar cells.Nature Communications, 15(1):3983, May 2024

    work page 2024

  13. [13]

    Xiaoyu Ding, Jianhua Liu, and Tequila A. L. Harris. A review of the operating limits in slot die coating processes.AIChE Journal, 62(7):2508–2524, 2016

    work page 2016

  14. [14]

    Practical operations for intermittent dual-layer slot coating processes.Korea-Australia Rheology Journal, 34(3):181– 186, August 2022

    Jin Seok Park, Sanghun Jee, Byoungjin Chun, and Hyun Wook Jung. Practical operations for intermittent dual-layer slot coating processes.Korea-Australia Rheology Journal, 34(3):181– 186, August 2022

    work page 2022

  15. [15]

    Edge Formation in High-Speed Intermittent Slot-Die Coating of Disruptively Stacked Thick Battery Electrodes.Energy Technology, 8(2):1900137, 2020

    Ralf Diehm, Hannes Weinmann, Jana Kumberg, Marcel Schmitt, J¨ urgen Fleischer, Philip Scharfer, and Wilhelm Schabel. Edge Formation in High-Speed Intermittent Slot-Die Coating of Disruptively Stacked Thick Battery Electrodes.Energy Technology, 8(2):1900137, 2020

    work page 2020

  16. [16]

    Richards, Uli Lemmer, Gerardo Hernandez-Sosa, and Ulrich W

    Fabian Schackmar, Helge Eggers, Markus Frericks, Bryce S. Richards, Uli Lemmer, Gerardo Hernandez-Sosa, and Ulrich W. Paetzold. Perovskite Solar Cells with All-Inkjet-Printed Ab- sorber and Charge Transport Layers.Advanced Materials Technologies, 6(2):2000271, 2021

    work page 2021

  17. [17]

    Ritzer, and Ulrich W

    Kristina Geistert, Simon Ternes, David B. Ritzer, and Ulrich W. Paetzold. Controlling Thin Film Morphology Formation during Gas Quenching of Slot-Die Coated Perovskite Solar Mod- ules.ACS Applied Materials & Interfaces, 15(45):52519–52529, November 2023. 15

    work page 2023

  18. [18]

    Minh Pham, Meri¸ c Arslan, Philip Scharfer, Wilhelm Schabel, Bryce S

    Simon Ternes, Jonas Mohacsi, Nico L¨ udtke, H. Minh Pham, Meri¸ c Arslan, Philip Scharfer, Wilhelm Schabel, Bryce S. Richards, and Ulrich W. Paetzold. Drying and Coating of Perovskite Thin Films: How to Control the Thin Film Morphology in Scalable Dynamic Coating Systems. ACS Applied Materials & Interfaces, 14(9):11300–11312, March 2022

    work page 2022

  19. [19]

    Schwenzer, Ihteaz M

    Simon Ternes, Tobias B¨ ornhorst, Jonas A. Schwenzer, Ihteaz M. Hossain, Tobias Abzieher, Waldemar Mehlmann, Uli Lemmer, Philip Scharfer, Wilhelm Schabel, Bryce S. Richards, and Ulrich W. Paetzold. Drying Dynamics of Solution-Processed Perovskite Thin-Film Photo- voltaics: In Situ Characterization, Modeling, and Process Control.Advanced Energy Materials, ...

    work page 2019

  20. [20]

    Majewski, S

    M. Majewski, S. Qiu, O. Ronsin, L. L¨ uer, V. M. Le Corre, T. Du, C. J. Brabec, H.-J. Egelhaaf, and J. Harting. Simulation of perovskite thin layer crystallization with varying evaporation rates.Materials Horizons, 12(2):555–564, 2025

    work page 2025

  21. [21]

    Starger, Amy E

    Jesse L. Starger, Amy E. Louks, Kelly Schutt, E. Ashley Gaulding, Robert W. Epps, Rose- mary C. Bramante, Ross A. Kerner, Kai Zhu, Joseph J. Berry, Nicolas J. Alvarez, Richard A. Cairncross, and Axel F. Palmstrom. Formation trajectories of solution-processed perovskite thin films from mixed solvents.Cell Reports Physical Science, 6(7):102655, July 2025

    work page 2025

  22. [22]

    Paetzold

    Simon Ternes, Felix Laufer, and Ulrich W. Paetzold. Modeling and Fundamental Dynamics of Vacuum, Gas, and Antisolvent Quenching for Scalable Perovskite Processes.Advanced Science, 11(14):2308901, 2024

    work page 2024

  23. [23]

    Drying kinetics of polymer films.AIChE Journal, 44(4):791–798, 1998

    B´ eatrice Guerrier, Charles Bouchard, Catherine Allain, and Christine B´ enard. Drying kinetics of polymer films.AIChE Journal, 44(4):791–798, 1998

    work page 1998

  24. [24]

    In situ characterization and modelling of drying dynamics for scalable printing of hybrid perovskite photovoltaics.KIT Scientific Publishing, March 2023

    Simon Ternes. In situ characterization and modelling of drying dynamics for scalable printing of hybrid perovskite photovoltaics.KIT Scientific Publishing, March 2023

    work page 2023

  25. [25]

    Nirmalkumar, Vadiraj Katti, and S

    M. Nirmalkumar, Vadiraj Katti, and S. V. Prabhu. Local heat transfer distribution on a smooth flat plate impinged by a slot jet.International Journal of Heat and Mass Transfer, 54(1):727–738, January 2011

    work page 2011

  26. [26]

    Progress in blade-coating method for perovskite solar cells toward commercialization

    Runsheng Wu, Chunhua Wang, Minhua Jiang, Chengyu Liu, Dongyang Liu, Shuigen Li, Qin- grong Kong, Wei He, Changjun Zhan, Fayun Zhang, Xiaohong Liu, Bingchu Yang, and Wei Hu. Progress in blade-coating method for perovskite solar cells toward commercialization. Journal of Renewable and Sustainable Energy, 13(1):012701, February 2021

    work page 2021

  27. [27]

    Subbiah, Luis V

    Anand S. Subbiah, Luis V. Torres Merino, Anil R. Pininti, Vladyslav Hnapovskyi, Subhashri Mannar, Erkan Aydin, Arsalan Razzaq, Thomas G. Allen, and Stefaan De Wolf. Enhancing the Performance of Blade-Coated Perovskite/Silicon Tandems via Molecular Doping and Interfacial Energy Alignment.ACS Energy Letters, 9(2):727–731, February 2024

    work page 2024

  28. [28]

    Yu, Salman Manzoor, Shen Wang, William Weigand, Zhenhua Yu, Guang Yang, Zhenyi Ni, Xuezeng Dai, Zachary C

    Bo Chen, Zhengshan J. Yu, Salman Manzoor, Shen Wang, William Weigand, Zhenhua Yu, Guang Yang, Zhenyi Ni, Xuezeng Dai, Zachary C. Holman, and Jinsong Huang. Blade-Coated Perovskites on Textured Silicon for 26%-Efficient Monolithic Perovskite/Silicon Tandem Solar Cells.Joule, 4(4):850–864, April 2020. 16

    work page 2020