In the boundary layer, Laplace's equation (1) is conveniently solved around the hyperboloidal tip with prolate ... where ξs represents the hyperboloid surface.
Supplementary Informations for “Singular sublimation of ice and snow crystals” Etienne Jambon-Puillet, Noushine Shahidzadeh, Daniel Bonn Institute of Physics, Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
SUPPLEMENTARY NOTE 1: SHORT TIME TIP SMOOTHING MODEL
We model the tip of our drops as half of a two sheet hyperboloid of revolution whose equation is given in cylindrical coordinates by z 2 /a2 − r2 /b2 = 1. The initial cone of semi angle α is drawn by the asymptotes of the projected hyperbola such that b/a = tan α (see Supplementary Figure 1A) and the sharpness of the tip is given by its curvature κ = a/b2 (the two principal curvatures are equal at the tip κ1 = κ2 = κ [1]). The rest of the drop sits below the tip and the full vapor field ρ(r) is a combination of the hyperboloidal tip field and the drop field. Since the gradient of the field is very strong close to the tip and decays very fast, we use a boundary layer approach similar to the one developped for Taylor cones [2] to compute the field (Supplementary Figure 1B). In the boundary layer, Laplace’s equation (1) is conveniently solved around the hyperboloidal tip with prolate spheroidal coordinates (ξ, η, φ)[3] since surfaces of constant ξ are hyperboloids. The general solution is ρ = C1 + C2 ln [(1 + ξ)/(1 − ξ)] with C1 and C2 integration constants [1, 2, 4]. The first boundary condition is ρ(ξs ) = ρsat where ξs represents the hyperboloid surface. The remaining constant is equal to the value of the field on the horizontal plane were the asymptotes meet (ξ = 0) that we consider as the edge of the boundary layer (Supplementary Figure 1B). The field around the hyperboloid is finally given by 1+ξ ln 1−ξ ρ = ρξ=0 + (ρsat − ρξ=0 ) 1+ξs ln 1−ξ s and the flux at the tip surface by s D ∂ρ 1 − ξ 2 ∂ρ D jn = = =− ρice ∂n ξs ρice c η 2 − ξ 2 ∂ξ ξs ρ
ρ − ρξ=0 2D p sat 1+ξs c (η 2 − ξ 2 )(1 − ξ 2 ) s s ice ln 1−ξs
(1)
√ ∂ with n denoting the normal direction, ∂n = n · ∇ the normal derivative and c = a2 + b2 the linear eccentricity of the projected hyperbola. To close the problem we need to match this solution with the outer field to know ρξ=0 .
A
B
C
Supplementary Figure 1. Hyperboloidal tip smoothing model. A Schematic of the tip of the hyperboloid with its conical asymptotes. The cone semi-angle α and the parameters defining the hyperboloid a and b are drawn (tan α = b/a). B Schematic of the model: an hyperboloidal tip sits on a hemispherical drop. Far from the tip, the outer field is given by the hemisphere and must match the inner tip field at the edge of the boundary layer. The tip corresponds to the surface ξ = ξs , the dotted lines are equipotentials (ξ = cst) and the solid line is the equipotential at the edge of the boundary layer: the horizontal plane ξ = 0 corresponding z = H0 + a0 . C Full field for this geometry solved numerically. The lines represent equipotentials.
2 Matching with an hemisphere: For simplicity we first assume that the drop below the tip is a hemisphere. The √ outer field is thus ρ − ρ∞ = (ρsat − ρ∞ )R/ r2 + z 2 and neglecting its curvature we have on the ξ = 0 or z = H0 + a0 plane ρξ=0 − ρ∞ = (ρsat − ρ∞ )R/(H0 + a0 ) (the subscript “0” denotes the initial value of the parameter). Because the tip is small compared to the drop we have H0 + a0 ≈ R and ρsat ≈ ρξ=0 . As a result, jn is extremely sensitive to the exact value of ρξ=0 and tiny errors, for instance introduced by our simple matching procedure, results in appreciable differences for the evaporation rate. We thus introduce a parameter q to finely tune the value of the matching field around the predicted value: ρξ=0 − ρ∞ = q(ρsat − ρ∞ )R/(H0 + a0 ). This free parameter should be close to 1. It was proven by Ham [5] that hyperboloidal surfaces are shape preserving solutions of the time dependent diffusion equation both during growth and shrinkage provided ρξ=0 = cst (though p the solution provided has no closed form). R02 − 2βt where β = D (ρsat − ρ∞ ) /ρice is This is not strictly the case here as the drop shrinks with R(t) = an evaporation parameter including all the thermodynamic quantities. Though the evaporation being faster at the tip, this is a reasonable approximation at very short time close to the singularity, explaining the self-similar shapes observed in the experiments. The tip is thus characterized by only one parameter whose evolution is given by evaluating Supplementary Equation (1) at its apex (η = 1). Coming back to the usual hyperboloids parameters (ξs = a/c) we get 2βa0 c0 R(t) da 1−q = . (2) +a0 dt H0 + a0 a(t)b20 ln cc00 −a 0 Because our tips are initially very sharp (a0 ∼ 1 µm while H0 ∼ 1 mm) H0 + a0 ≈ H0 and integrating Supplementary Equation (2) between 0 and t we obtain ! p Z t 2 − 2βt∗ R 4βa c 0 0 0 1−q dt∗ a(t)2 − a20 = +a0 H0 0 b20 ln cc00 −a 0 v u R(t)3 − R03 4βa0 c0 u t+q a(t) = ta20 + . (3) 3βH0 b2 ln c0 +a0 0
c0 −a0
From Supplementary Equation (3) we finally calculate the tip curvature κ(t) = 1/(a(t) tan2 α): −1/2 2 3 3 R(t) − R 4β tan α 0 t+q κ(t) = κ−2 . 0 + 1+cos α 3βH0 ln 1−cos cos α α
(4)
If we look close to the singularity, the variation of the outer field can be neglected for t
