Fabrication of slender elastic shells by the coating of curved surfaces

Various manufacturing techniques exist to produce double-curvature shells, including injection, rotational and blow molding, as well as dip coating. However, these industrial processes are typically geared for mass production and are not directly applicable to laboratory research settings, where adaptable, inexpensive and predictable prototyping tools are desirable. Here, we study the rapid fabrication of hemispherical elastic shells by coating a curved surface with a polymer solution that yields a nearly uniform shell, upon polymerization of the resulting thin film. We experimentally characterize how the curing of the polymer affects its drainage dynamics and eventually selects the shell thickness. The coating process is then rationalized through a theoretical analysis that predicts the final thickness, in quantitative agreement with experiments and numerical simulations of the lubrication flow field. This robust fabrication framework should be invaluable for future studies on the mechanics of thin elastic shells and their intrinsic geometric nonlinearities.


Supplementary Note 3: Supporting analytical results for the lubrication model
The derivation of the underlying equation of our model presented in the Methods of the manuscript is briefly outlined in this section. We assume a thin liquid film on a sphere of radius, R, invariant in the azimuthal direction. Its initial characteristic thickness of the film is hi; the resulting film aspect ratio is ε = hi/R. As mentioned in the Discussion section of the main text, the time evolution of a thin-film on the outside and underside of the mold produces identical results.
Here, we focus on the derivation of the first case. Considering a small aspect ratio ε of the film, mass conservation indicates that the velocity normal to the interface is significantly smaller than the tangential component. Furthermore, the low Reynolds number conditions for this flow allows for the Stokes equations to be used. The equation for momentum balance in the radial direction is and the boundary condition for the pressure is p(R+h) = p0 + γκ (p0 is the external pressure, γ is the surface tension of the fluid, and κ is the curvature of the interface). Integrating Supplementary Eq. (2) along the radial direction and using the above boundary condition yields the pressure distribution, p(r, ϕ) = p0 + γκ + ρg cos ϕ (R +h −r). By integrating twice, the ϕ component of the momentum equation, and considering the no-slip boundary condition at the sphere surface, u(R, ϕ) = 0, as well as the zero-shear stress interface, u(R + h, ϕ)/r = 0, we obtain the tangential velocity component: The depth-integrated velocity is given by ( ) = ∫ ( , ) +ℎ . Using the local mass conservation in spherical coordinates, h/t +(R sin ϕ) -1 (sin ϕ Q)/ϕ = 0, we eventually obtain the lubrication equation: where the leading order curvature derivative is κϕ = −R -2 (h ϕϕϕ + 2hϕ + hϕϕ cot ϕ − hϕ csc 2 ϕ). The term I in the spatial variation of the flux corresponds to the surface tension effects, term II represents the variation of the hydrostatic pressure distribution and term III accounts for the drainage. In the case of a liquid film on the underside of a sphere, the hydrostatic pressure variation term would have an opposite sign, but the rationale would otherwise be identical.
The film thickness and time can be non-dimensionalized by hi and the initial drainage time, τd = μR/(ρghi 2 ), respectively, such that the lubrication equation expressed with non-dimensional quantities is written as where B = ρghiR/γ is the modified Bond number.
For a time-varying viscosity, the initial drainage time is built upon μ0 and the factor μ0/μ(t) appears ahead of the flux variation terms.
From the mean velocity, the velocity at the interface can be computed as: where the parameter c depends on the initial condition.
Note that the homogeneous solution only influences the transient regime; for large times the solution decreases as (4̃3 ⁄ ) −1/2 and is independent of the initial condition. In dimensional form, the asymptotic solution for the film thickness is given by: where the parameter d depends on the initial condition. At late times, the spatial variation of the film thickness is therefore of the form 1 + (1/10) 2 + (41/4800) 4 + ( 6 ).

Supplementary Note 5: Considering curing effects for the predictions of the final film thickness
In order to obtain the prediction for the final film thickness given by equation (6) in the main text, Supplementary Eq. (12) needs to be modified to take into account the rheology of the polymer fluid. For this purpose, we make use of the following empirical description for the evolution of the viscosity, which we found to fit our experimental data well: with μ1 = μ0.exp(βτc)τc -α , where τc is the curing time and α and β are parameters which have to be fitted depending on the specific details of the fluid (see Supplementary Table 1 If the curing time τc is large enough so that the term I is much larger than unity and if the final time t is larger than τc so that term II becomes negligible, then the asymptotic solution for the film thickness is given by where we highlight the fast temporal decrease of the velocity u ~ t -α and its independence on the viscosity, density and gravity.