1 I. SUPPLEMENTARY MATERIAL 1: entations. This …10.1038...1 I. SUPPLEMENTARY MATERIAL 1:...

1 I. SUPPLEMENTARY MATERIAL 1: COMPARISON OF SIMULATIONS WITH REAL SNOWFLAKES (a) Simple star from [1] (b) Simple star (c) Stellar dendrite from [1] (d) Stellar dendrite (e) Sectored plate from [1] (f) Sectored plate FIG. 1. Comparison between real snowflakes photographs (left) taken from [1], and our phase field simulations (right). Visual rendering for our simulations uses the software Blender. SUPPLEMENTARY MATERIAL 2: 3D FACETING ALGORITHM Faceting requires the anisotropy constants xy and z values in A(n), to exceed 1/35 for 6-fold horizontal symmetry, and 1/3 for 2-fold vertical symmetry [2]. Above such values, metastable and unstable crystal orientations n where the stiffness becomes negative, emerge. A(n) must hence be modified, so that the stiffness remains positive. A preliminary step is to determine missing orientations. This can be done by detecting sign change in the stiffness. This causes a local convexity inversion in the 2D polar plot of 1/A(θ,φ 0 ), for any fixed azimuthal angle φ 0 , or in the polar plot of 1/A(θ 0 ,φ) for any fixed polar angle θ 0 . This reads A+∂ 2 θθ A = 0 and A+∂ 2 φφ A =0 respectively, providing us with the maximum anisotropy constants m xy and m z before missing orientation: m xy = 1+ z cos(2φ) 35 m z = 1+ xy cos(6θ) 3 . (1) Restricting the study to θ ∈ [-π/6,π/6] and φ ∈ [0,π] due to rotation periodicity of A(θ, φ), missing θ orientations lie within a φ-dependant angular sector [-θ m (φ)+ kπ/3,θ m (φ)+ kπ/3] (k ∈ Z) for xy > m xy . Same goes for φ orientations lying within [-φ m (θ)+ kπ, φ m (θ)+ kπ] for z > m z . Following the procedure developed by Eggleston in [3], θ m and φ m respectively satisfy ∂ θ [cos(θ)/A(θ,φ)] = 0 and ∂ φ [cos(φ)/A(θ,φ)] = 0, lead- ing to: 6 xy sin(6θ m ) cos(θ m ) sin(θ m ) = (1 + xy cos(6θ m )+ z cos(2φ)) 2 z sin(2φ m ) cos(φ m ) sin(φ m ) = (1 + xy cos(6θ)+ z cos(2φ m )) . (2) The regularization step finally consists in replacing A(θ,φ) by a simple trigonometric function for missing orientation, as suggested by Debierre et al. [4]: A(n)= A θ (θ,φ)= A 1 (φ)+ B 1 cos(θ), |θ| <θ m , |φ|≥ φ m A φ (θ,φ)= A 2 (θ)+ B 2 cos(φ), |θ|≥ θ m , |φ| <φ m α(θ,φ)A θ + (1 - α(θ,φ))A φ , |θ| <θ m , |φ| <φ m 1+ xy cos(6θ)+ z cos(2ψ) otherwise. (3) Coefficients A 1 (φ), B 1 , A 2 (θ), B 2 and α(θ,φ) are set to ensure continuity of A(n) and its derivatives: A 1 (φ)=1+ xy cos(6θ m )+ z cos(2φ) - 6 xy sin(6θ m ) sin(θ m ) cos(θ m ) B 1 = 6 xy sin(6θ m ) sin(θ m ) A 2 (θ)=1+ xy cos(6θ)+ z cos(2φ m ) - 2 z sin(2φ m ) sin(φ m ) cos(φ m ) B 2 = 2 z sin(2φ m ) sin(φ m ) α(θ, φ)= |θ - θ m | p (θ - θ m ) 2 +(φ - φ m ) 2 . (4)

Upload
dangdiep
Category

Documents
view
214
download
0

Embed Size (px):

Transcript of 1 I. SUPPLEMENTARY MATERIAL 1: entations. This …10.1038...1 I. SUPPLEMENTARY MATERIAL 1:...

I. SUPPLEMENTARY MATERIAL 1:COMPARISON OF SIMULATIONS WITH REAL

SNOWFLAKES

(a) Simple star from [1] (b) Simple star

(e) Sectored plate from [1] (f) Sectored plate

FIG. 1. Comparison between real snowflakes photographs(left) taken from [1], and our phase field simulations (right).Visual rendering for our simulations uses the softwareBlender.

SUPPLEMENTARY MATERIAL 2: 3DFACETING ALGORITHM

Faceting requires the anisotropy constants εxy and εzvalues in A(n), to exceed 1/35 for 6-fold horizontal sym-metry, and 1/3 for 2-fold vertical symmetry [2]. Abovesuch values, metastable and unstable crystal orientationsn where the stiffness becomes negative, emerge. A(n)must hence be modified, so that the stiffness remainspositive. A preliminary step is to determine missing ori-

entations. This can be done by detecting sign change inthe stiffness. This causes a local convexity inversion inthe 2D polar plot of 1/A(θ, φ0), for any fixed azimuthalangle φ0, or in the polar plot of 1/A(θ0, φ) for any fixedpolar angle θ0. This reads A+∂2θθA = 0 and A+∂2φφA = 0respectively, providing us with the maximum anisotropyconstants εmxy and εmz before missing orientation:

εmxy =

1 + εz cos(2φ)

εmz =1 + εxy cos(6θ)

(1)

Restricting the study to θ ∈ [−π/6, π/6] and φ ∈ [0, π]due to rotation periodicity of A(θ, φ), missing θ orienta-tions lie within a φ-dependant angular sector [−θm(φ) +kπ/3, θm(φ) + kπ/3] (k ∈ Z) for εxy > εmxy. Same goesfor φ orientations lying within [−φm(θ) + kπ, φm(θ) +kπ] for εz > εmz . Following the procedure developedby Eggleston in [3], θm and φm respectively satisfy∂θ[cos(θ)/A(θ, φ)] = 0 and ∂φ[cos(φ)/A(θ, φ)] = 0, lead-ing to:

6εxy sin(6θm) cos(θm)

sin(θm)= (1 + εxy cos(6θm) + εzcos(2φ))

2εz sin(2φm) cos(φm)

sin(φm)= (1 + εxy cos(6θ) + εzcos(2φ

m)) .

(2)The regularization step finally consists in replacingA(θ, φ) by a simple trigonometric function for missingorientation, as suggested by Debierre et al. [4]:

A(n) =

Aθ(θ, φ) = A1(φ) +B1 cos(θ), |θ| < θm, |φ| ≥ φm

Aφ(θ, φ) = A2(θ) +B2 cos(φ), |θ| ≥ θm, |φ| < φm

α(θ, φ)Aθ + (1− α(θ, φ))Aφ, |θ| < θm, |φ| < φm

1 + εxy cos(6θ) + εz cos(2ψ) otherwise.(3)

Coefficients A1(φ), B1, A2(θ), B2 and α(θ, φ) are set toensure continuity of A(n) and its derivatives:

A1(φ) = 1 + εxy cos(6θm) + εz cos(2φ)− 6εxy sin(6θm)

sin(θm)cos(θm)

B1 =6εxy sin(6θm)

sin(θm)

A2(θ) = 1 + εxy cos(6θ) + εz cos(2φm)− 2εz sin(2φm)

sin(φm)cos(φm)