---
title: "How to compare growth rates with nanofilm spectra"
description: "Recalculate Thiele’s Figure 1, check slender-thread dispersion and read a strong-slip thermal-fluctuation spectrum."
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last_modified: "2026-10-28"
---

[Dynamic interfaces](https://sandboxphysics.com/stories/dynamic-interfaces/)

L4 · Chapter 8

# How to compare growth rates with nanofilm spectra

Sandbox Physics · 2026-10-28

Start reading an interface paper by identifying its vertical axis. A curve against wavenumber may show a deterministic growth rate or a root-mean-square amplitude averaged over molecular simulations. This distinction determines what a simplified model can be compared with.

## Before you begin

Growth estimation from Chapter 3, thread dispersion from Chapter 6 and numerical error from Chapter 7. Open the original papers alongside this guide.

- Recalculate two analytic continuum benchmarks independently.
- Identify the extra physics and observations needed for strong slip and thermal noise.

## Recalculate Figure 1 of Thiele 2010

Open the author-hosted published article at section 2, equation (5) and Figure 1. Locate zero wavenumber, the crossing and the maximum. They mark mass conservation, the stability boundary and the fastest mode. The stable branch remains below zero.

Define $K=k/k_c$ and divide growth by $M_0\gamma k_c^4$. The unstable branch becomes a universal dimensionless curve for this linear model. At $K=0,1/2,1/\sqrt2,1,5/4$, its values are $0,3/16,1/4,0,-225/256$. These are newly calculated formula values, not digitized source points.

$$
\widehat\sigma=K^2(1-K^2)
$$

Our figure and benchmark use this analytic curve. Nonlinear continuation branches are outside the recalculation.

Analytic growth rate $\widehat\sigma=K^2(1-K^2)$

$K_*=1/\sqrt2$ $K_c=1$

- Unstable film
- Stable film (separate rate scale)

Newly calculated schematic of the analytic curves in Thiele (2010), Figure 1. The horizontal coordinate is wavenumber divided by the unstable cutoff. No experimental points are shown.

Sources: U. Thiele

## Why the thread relation has another time scale

Open the later repository copy of Eggers–Dupont’s 1994 paper, arXiv:physics/0110081v1. Section 2, equation (27), is quadratic in growth rate; equation (28) supplies its growing branch. Equations (30) and (31) check low- and high-viscosity limits. Retaining inertia requires an evolving velocity, unlike the overdamped film.

Use $q=kR_0$ and $s=\omega t_c$. At zero Ohnesorge number, the reduced model selects $q=1/\sqrt2$, or wavelength $8.89R_0$. The paper contrasts this with about $9.01R_0$ for the full inviscid cylinder. Even a successful reduced equation has approximation error.

Figure 1 instead compares nozzle-fed water-jet shapes at $\lambda/R_0=14.57$. Reproducing it requires inlet conditions, jet speed and alignment with photographs. The periodic experiment here tests equation (27); the nozzle figure is a separate reading task.

$$
s^2+3\mathrm{Oh}\,q^2s-\frac{q^2(1-q^2)}2=0
$$

The continuum linear relation of the axisymmetric one-dimensional model has its boundary at unit dimensionless wavenumber.

Sources: Eggers & Dupont

## Figure 4 of Zhang 2024 shows spectra

Open arXiv:2406.08179v1 and its published counterpart, JFM 996, A19. Section 2 derives a stochastic strong-slip equation; section 4 explains the molecular wall model. Figure 3 calibrates slip using pressure-driven velocity profiles before the dewetting comparison.

Figure 4 fixes thickness at $1.2\,\mathrm{nm}$ and compares slip lengths $400\,\mathrm{nm}$ and $0.2\,\mathrm{nm}$. Symbols are root-mean-square height spectra from 40 independent molecular simulations. Lines are stochastic continuum predictions. A time-dependent spectrum is a different observable from the deterministic growth rate in Chapter 3.

Panel (a) also compares predictions with and without inertia. Strong slip changes the leading flow balance, making the inertial terms relevant for these conditions. The question becomes which velocity and noise model predicts the observed spectrum.

Sources: Yixin Zhang

## Turn the next extension into a testable question

A first extension can remain deterministic: hold wetting energy, mean thickness and the continuous initial profile fixed, change the wall condition, and derive its appropriate flux. Compare early spectra and energy before adding thermal fluctuations. Strong slip also requires reconsidering inertia and long-wave scaling.

Continuous thermal forcing requires a stochastic flux, a fluctuation–dissipation relation and independent realizations. Random initial heights supply only one initial disturbance. Continuing through rupture additionally needs a treatment of changing topology; molecular calculations need independently checked interactions, system sizes and sampling errors.

For the next paper, write down geometry, conserved quantity, wall conditions, observable and averaging procedure before comparing curves. These choices turn an appealing figure into a calculation someone else can repeat.

Sources: Yixin Zhang · Eggers & Dupont

## Predict, change and explain

### [Turn stored heights into a comparable growth rate](https://sandboxphysics.com/labs/thin-film-instability/)

1. Export a small-amplitude record, calculate modal slopes and save the fitting interval.
2. Compare with normalized analytic growth, accounting for the discrete wavenumbers of a finite periodic domain.
3. Return to Zhang’s Figure 4 and identify the additional physics and statistics needed for that comparison.

## Exercises and explanations

1. Check the benchmark at $K=5/4$ and interpret its sign.

   Read the explanation

   The result is $(25/16)(1-25/16)=-225/256$. The negative value describes a short mode stabilized by curvature.

2. Can one deterministic record with a fixed random seed represent a thermal spectrum?

   Read the explanation

   No. It evolves a single initial perturbation. Thermal noise continues to drive the surface, and the paper averages squared amplitudes over independent realizations.

3. Which source results are recalculated here, and which are reading material?

   Read the explanation

   The analytic flat-film and periodic slender-thread relations are recalculated and tested against simulations. Nonlinear continuation branches, nozzle photographs and stochastic strong-slip molecular spectra remain explicitly located reading material.

[Previous: When should a shrinking-neck calculation stop?](https://sandboxphysics.com/stories/dynamic-interfaces/resolution-and-singularities/)

## Sources and calculation scope

Examples and diagrams are newly authored. The paper comparison recalculates analytic relations; the research reading distinguishes those curves from experimental photographs and molecular simulations.

1. [Yixin Zhang, Linear stability theory and molecular simulations of nanofilm dewetting with disjoining pressure, strong liquid–solid slip, and thermal fluctuations, J. Fluid Mech. 996, A19 (2024)](https://arxiv.org/html/2406.08179v1)

   Read alongside the published version, DOI 10.1017/jfm.2024.701. Figures 3–4 and sections 2, 4–5 define slip calibration and the ensemble spectrum. Reading only: molecular data and stochastic strong-slip curves are not reproduced.

2. [U. Thiele, Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth, J. Phys.: Condens. Matter 22, 084019 (2010)](https://www.uni-muenster.de/Physik.TP/~thiele/Paper/Thie2010jpm.pdf)

   Author-hosted published article, equations (1)–(3), section 2 potential and equation (5), Figure 1. Recalculation covers the analytic linear dispersion only; nonlinear continuation branches in Figures 2–3 are a reading topic.

3. [Eggers & Dupont, Drop formation in a one-dimensional approximation of the Navier–Stokes equation, J. Fluid Mech. 262, 205–221 (1994)](https://arxiv.org/pdf/physics/0110081v1)

   arXiv:physics/0110081v1 is a later repository copy of the 1994 paper. Section 2, equations (17)–(18), curvature (15) and dispersion (27)–(32); Figure 1 caption states the water-jet geometry. The course derives the periodic slender-thread benchmark, not the nozzle experiment or satellite drops.