🌀 Spin Coating Approximation Tool

⚠️ Status: Under Development

Predicts the final dry film thickness of a spin-coated layer and how it changes with spin speed. Three models of increasing realism, live plot, statistical deviation and saveable parameter presets. It tries to turn the theoretical thin-film PDE into a tool that is usable in the lab.

Available as a desktop ui and as a single-file web version. Both use the same models and give the same numbers.

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✨ Features

• Thickness-rpm plot: $h(\omega)$ over a chosen range, current operating point marked.
• Two ways to enter the solids fraction: directly as volume fraction, or from weight fraction plus the two pure-component densities.
• Statistical deviation:
  • Gauss (analytic): linearised error propagation
  • Monte Carlo: samples every uncertain parameter (normal, uniform or triangular) and propagates it through the full, non-linear model; reports P05 / P50 / P95. See the caveat in src/model/deviation.py about mean/std under large relative uncertainties, the percentiles stay reliable regardless.
  • The Flack model solves an ODE per sample, so its Monte Carlo runs on demand (button Run Monte Carlo, about 4 s for 3000 samples).
• Parameter presets: save/load a complete parameter set (model, values, uncertainties, note), e.g. one per resin.
• Themes: multiple color schemes: dark_purple, dark_blue, black_white, choosable style.py.

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🧭 At a glance

Model	Computes	Evaporation	Viscosity	Typical use
Emslie-Bonner-Peck (1958)	wet film thickness $h(t)$ after a spin time	none	constant	understanding the flow-driven thinning
Meyerhofer (1978)	final dry film thickness $h_\mathrm{f}$ (closed form)	constant rate $E$	constant	quick estimate, first guess
Flack et al. (1984), depth-averaged	final dry film thickness $h_\mathrm{f}$ (numerical ODE)	slows down as the film concentrates	rises with solids content	resins whose viscosity changes a lot while drying

Rule of thumb: start with Meyerhofer. Switch to Flack if the result is clearly off and you can calibrate $k_\eta$ and $n$ against your own measurements.

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🚀 Quick start

Web

Enter via the link next to About or click here for the web version. Note: There are some issues with preset loaders in the web version!

Desktop

Since Python is already a perfect habitat for scientific calculations, only matplotlib is needed:

pip install -r requirements.txt  # or just matplotlib

and run main.py:

python main.py

• Python 3.9+ (tested with 3.12). tkinter ships with Python on Windows/macOS; on Linux install it first (e.g. sudo apt install python3-tk).
• Pick a colour theme without editing code: SPIN_COATING_THEME=dark_blue python main.py (dark_purple, dark_blue, black_white).
• Presets are read from / written to data/*.json.

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🗂️ Project structure

spin-coating/
├── index.html              # web version (single self-contained file; mirrors src/model line by line)
├── main.py                 # desktop entry point
├── requirements.txt
├── data/                   # parameter presets (JSON, one file each)
├── src/
│   ├── settings.py         # app defaults: Monte Carlo, plot range, preset folder
│   ├── store.py            # read/write presets
│   └── model/              # pure physics, no UI
│       ├── parameters.py   # units, parameter specs + defaults, data shapes
│       ├── emslie.py       # model 1
│       ├── meyerhofer.py   # model 2
│       ├── flack.py        # model 3 (numerical ODE)
│       ├── deviation.py    # Gauss propagation + Monte Carlo for every model
│       └── __init__.py     # dispatch by model id, spin curves
├── ui/                     # tkinter desktop app
│   ├── app.py              # main window, live recompute
│   ├── widgets.py  plot.py  # fields, dropdowns, result display, thickness-rpm plot
│   ├── forms.py  presets.py  info.py  dialogs.py   # input forms, preset window, parameter help, popups
│   └── style.py  tex.py  texts.py   # themes, formula rendering, UI texts
├── tests/                  # see below
└── *.pdf                   # reference papers

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🔬 Physics

1. Emslie, Bonner and Peck (1958): Newtonian, non-volatile

Outward flow of a Newtonian liquid film on a spinning disk [1]:

$$ \frac{\partial h}{\partial t} + \frac{\rho\omega^2 r h^2}{\eta}\frac{\partial h}{\partial r} = -\frac{2\rho\omega^2}{3\eta}h^3 $$

$t$ time, $\omega$ angular velocity, $r$ radius, $\rho$ density, $\eta$ viscosity, $h$ thickness of the liquid film. For a film that is uniform in $r$:

$$ h(t) = h_0\left(1 + \frac{4\rho\omega^2}{3\eta}h_0^2,t\right)^{-1/2} $$

Without evaporation the film keeps thinning (about $t^{-1/2}$) and never reaches a dry value, so this model gives the wet thickness after a given spin time only.

2. Meyerhofer (1978): constant evaporation

Adds a solvent evaporation rate $E$ (solvent volume removed per substrate area and time) [1]:

$$ 0 = \frac{\mathrm{d}h}{\mathrm{d}t} + \frac{2\rho\omega^2}{3\eta}h^3 + E $$

Early on, flow dominates; late, evaporation does. Where both solvent losses are equal, the film freezes:

$$ E = \frac{(1-C_0),2\rho\omega^2}{3\eta_0},h_s^3 \quad\Rightarrow\quad h_s = \left(\frac{3\eta_0 E}{2(1-C_0)\rho\omega^2}\right)^{1/3} $$

$h_s$ is the wet thickness at that point ($C_0$ = initial solids volume fraction). The solute volume stays constant afterwards, so the dry film is

$$ h_\mathrm{f} = C_0,h_s $$

With $E\propto\sqrt{\omega}$ this gives the widely observed $h_\mathrm{f}\propto\omega^{-1/2}$; with constant $E$ it is $\omega^{-2/3}$. The closed form assumes an abrupt switch; integrating the full ODE gives a dry film about 3 % thinner at $C_0=10$ %.

3. Flack et al. (1984): concentration-dependent viscosity (depth-averaged)

Flack, Soong, Bell and Hess [2] model a resist whose viscosity and solvent diffusivity change strongly with polymer concentration, resolved through the film depth and with non-Newtonian behaviour.

What this tool implements is the depth-averaged ("well-mixed") version of that mechanism, not the full paper model. It keeps "viscosity rises and evaporation slows as the film concentrates", but has no depth profile, no solid skin, no shear thinning and no Fujita-Doolittle diffusivity, so the paper's five parameters ($D_0$, $A$, $B$, $\eta_0$, $\kappa_0$) are not used.

State: solute and solvent volume per area $q$, $s$ (so $h=q+s$, $\varphi=q/h$):

$$ Q = \frac{2\rho\omega^2h^3}{3\eta(\varphi)},\qquad \frac{\mathrm{d}q}{\mathrm{d}t} = -\varphi,Q,\qquad \frac{\mathrm{d}s}{\mathrm{d}t} = -(1-\varphi),Q - E(\varphi) $$

solved with an adaptive RK4 until the solvent is gone; $q$ is then the dry film thickness.

Law	Formula	Parameters
Evaporation	$E(\varphi) = E_0,(1-\varphi)^n$	$E_0$ = rate of the pure solvent, $n$ = slowdown ($n=0$: constant)
Viscosity, exponential (default, generic)	$\eta(\varphi) = \eta_0,e^{k_\eta(\varphi - C_0)}$	$k_\eta$ (0 = constant; about 18 reproduces the paper's PMMA/chlorobenzene curve between 10 and 50 wt %)
Viscosity, Flack Table I (measured PMMA, zero shear)	$\eta_{p0}(w)=\eta_\mathrm{ref},e^{-c/(0.043+0.040c)},w^{2.33}$, $c=1-w$	only the shape is used, $\eta_0$ still anchors the solution as dispensed; $k_\eta$ ignored

With $k_\eta = n = 0$ the model collapses to Meyerhofer's equation before the closed-form approximation. Both viscosity laws stay finite for $\varphi\to1$, unlike real resins (divergence near vitrification). The defaults come from one literature system, so absolute numbers for other resins need calibration ($k_\eta$, $n$ against your own ellipsometry / profilometry data).

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🧪 Tests

python tests/test_model.py      # models, deviation, unit conversion (no framework needed)
python tests/test_store.py      # presets
python tests/test_themes.py     # theme palettes (needs tkinter)
python tests/test_ui_smoke.py   # app runs through all models/methods (tkinter stub, no display needed)
pytest tests/test_web_matches_python.py     # index.html == Python models (needs node + pytest)

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📚 References

1. Ossila, "Spin Coating: A Complete Guide". https://www.ossila.com/pages/spin-coating (original: A. G. Emslie, F. T. Bonner, L. G. Peck, J. Appl. Phys. 29, 858 (1958); D. Meyerhofer, J. Appl. Phys. 49, 3993 (1978))
2. W. W. Flack, D. S. Soong, A. T. Bell and D. W. Hess, "A mathematical model for spin coating of polymer resists", J. Appl. Phys. 56, 1199-1206 (1984). https://www.researchgate.net/publication/224533748_A_mathematical_model_for_spin_coating_of_polymer_resists

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📄 License

Distributed under the MIT License.