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Outline

The long and winding road (of research)

Acetone Droplets in a Leidenfrost state
Soap film singularities

Suspended rings

Classical Catenoid
Calculus of variations formulation
Experimental verification

Gravity-mediated platonic solid assemblies

Surface Evolver simulations

Displacement-driven prisms and pyramids

Transient tile angle simulations and modeling
Effective radii of structures

Additional geometries

Starting Point: Leidenfrost Effect on a Liquid Substrate

Starting Point: Leidenfrost Effect on a Liquid Substrate

Analogous Soap Film Experiments

First Gravity Mediated Assembly

Gravity mediated assembly process

Davies, October 2018
Davies and Raufaste, June 2021

Catenoid

\[ y = a \cosh \left( \frac{x}{a} \right) \]

Goldstein et al. September 2021

Raufaste et al. June 2022

Goldschmidt solutions

The red region represents combinations of R^* and h^* yielding Goldschmidt solutions, and in blue are pairs for which a catenoid solution exists. The inset images are simulation results for R^*=0.5 and h^*=0.91, 0.915 corresponding to catenoidal and Goldschmidt solutions.

Calculus of variations formulation

\[ \begin{align} \mathcal{E} (R,h) &= \int_{0}^{l}2 \sigma( 2\pi R) ds - m g h \nonumber\\ &= \int_{0}^{h}\left[ 4\pi\sigma R\sqrt{1 + (R')^2}\right] \,dz - m g h \nonumber \end{align} \]

Variations \(R+\epsilon\eta\) and \(h+\epsilon k\) are considered for some parameter \(\epsilon\ll1\) with two constraints on the variations.

\[ \eta (0) = 0 \qquad \qquad \eta (h) = -k R'(h) \]

First Variation Condition and BVP

The first-variation condition for the energy functional can be written as \[ \begin{equation} \scriptsize{ 0=\bbox[5px, border: 2px solid red]{\int_{0}^{h} 4\pi\sigma \eta \left[ \sqrt{1 + (R')^2} - \frac{\mathrm{d}}{\mathrm{d} z} \left(\frac{R R'}{\sqrt{1 + (R')^2}}\right)\right] \,dz} + \bbox[5px, border: 2px solid blue]{ k\left[ 4\pi\sigma R_i\sqrt{1 + (R'(h))^2} - mg - \frac{4 \pi \sigma R_i (R'(h))^2}{\sqrt{1 + (R'(h))^2}} \right]} } \end{equation} \]

with the following corresponding terms of the boundary value problem \[ \begin{equation} \bbox[5px, border: 2px solid red]{0 = 1 + R'(z)^2 - R(z)R''(z)} \end{equation} \]

\[ \begin{equation} \bbox[5px, border: 2px solid blue]{(R'(h))^2 = \frac{1}{B^{2}} - 1} \end{equation} \]

Where the Bond number of the system is given by \(B=mg/ 4\pi\sigma R_i\) and \(R(0)=R_i\) is the boundary condition at the supporting ring.

Closed form solution

\[ \begin{equation} R(z) = \frac{mg}{4\pi\sigma} \cosh\left( \frac{4\pi\sigma z}{mg}+\cosh^{-1}\left(\frac{4\pi\sigma R_o}{mg}\right) \right) \end{equation} \]

Evaluating the above equation at \((R_i,h)\) and solving for \(h\) yields \[ \begin{equation} h = \frac{mg}{4\pi\sigma}\left[ \cosh^{-1}\left(\frac{4\pi\sigma R_i}{mg}\right)-\cosh^{-1}\left(\frac{4\pi\sigma R_o}{mg}\right)\right] \end{equation} \]

Letting \( R^*=R_i /R_o \) and \( h^*=h /R_o \) in addition to \(B=mg/ 4\pi\sigma R_i\) \[ \begin{equation} \bbox[5px, border: 2px solid red] { h^* = B R^* \left(\cosh^{-1}\frac{1}{B}-\cosh^{-1}\frac{1}{B R^*}\right) } \end{equation} \]

Contour Plot

\[ \scriptsize{ \begin{equation} h^* = B R^* \left(\cosh^{-1}\frac{1}{B}-\cosh^{-1}\frac{1}{B R^*}\right) \end{equation} } \]

Experimental results of suspended rings

Platonic solid geometries

Platonic solid geometries

Surface Evolver simulations

Gravity-mediated assemblies phase space

Displacement-driven assembly setup

Displacement-driven cube assembly

Legrain et al. June 2014

Rectangular prisms

\[ \begin{equation} \mathcal{T}_g=\frac{Lmg\cos\theta}{2}, \qquad \mathcal{T}_\sigma=2Lw\sigma\cos(\theta+\phi), \qquad B_t = \frac{mg}{4w\sigma} \end{equation} \]

Surface Evolver simulations of transient tile angles

Surface Evolver simulations compared to experimental results

Addition of mathematical model

Effective radius for prismatic pinch-off

Effective radius for pyramidal pinch-off

Soap film mediated assembly

Additional geometries

Thank you!

ありがとうございます

Surface Evolver simulations

Examples of refinement (r) and iteration (g) operations within a Surface Evolver simulation of a catenoid where R^*=0.5 and h^*=0.9.

Calculus of variations formulation continued

Let \(\mathcal{F}\) be defined by \(\mathcal{F} ( \epsilon) =\mathcal{E}(R+\epsilon\eta,h+\epsilon k)\) \[ \begin{align} \mathcal{F} &= \int_{0}^{h + \epsilon k}\left[ 4\pi\sigma (R+\epsilon \eta)\sqrt{1 + (R'+\epsilon \eta')^2} \right] \,dz - mg(h+\epsilon k) \\ \end{align} \]

Granted that R and h correspond to a minimum of the potential energy functional, the Gateaux derivative of \(\mathcal{F} \) must vanish at \(\epsilon=0\): \[ \begin{equation} \frac{\mathrm{d}\mathcal{F}(\epsilon)}{\mathrm{d}\epsilon}\bigg|_{\epsilon=0}=0 \end{equation} \]

Two constraints on the variations arise from the boundary conditions \[ \eta (0) = 0 \qquad \qquad \eta (h) = -k R'(h) \]

Verification of Surface Evolver simulations

Model of transient tile angle

The equilibrium condition \(\mathcal{T}_g=\mathcal{T}_\sigma \) can be expressed as \[ \begin{equation} B_t = \frac{mg}{4 w \sigma}=\frac{\cos(\phi+\theta)}{\cos\theta} \end{equation} \]

Under the assumption that a cross-section of the soap film can be represented as a catenary curve \[ \scriptsize{ \begin{equation} d^* = (R^*+L^*\cos\theta) g(B_t,\theta)\Big[ \cosh^{-1} \left(\frac{1}{g(B_t,\theta)}\right) - \cosh^{-1} \left( \frac{1}{ (R^*+L^*\cos\theta) g(B_t,\theta) }\right) \Big] + L^*\sin\theta \end{equation} } \]

where the function \( g(B_t,\theta) \) is given by \[ \begin{equation} g(B_t,\theta)=\cos[\cos^{-1} ( B_t\cos\theta)-\theta], \qquad 0\le B_t\le1.1547, \qquad -\frac{\pi}{6}\le\theta\le\frac{\pi}{2} \end{equation} \]

Transient tile angle experiments