Unveiling the physics of hair shapesJournal Club

Krissia de Zawadzki

Instituto de Fısica de Sao Carlos - Universidade de Sao Paulo

April 3, 2014

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Outline

1 Introduction

2 Theoretical Background

3 Results

4 Theoretical Analysis

5 Conclusions

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Shapes of suspended curly hair

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Shape of a Ponytail and the Statistical Physics ofHair Fiber Bundles

Raymond E. Goldstein, Patrick B. Warren, and Robin C. BallPRL 108, 078101 (2012)

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Motivation

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Shapes of a suspended curly hair

Is it possible to predict the shape of a naturally curved rod?

Model for an elastic rodsuspended under its own weight

Metaphor for a curly hair!

precisionexperiments

+computational

simulations+

theoreticalanalysis

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Mechanical properties of rods

Curvature, Young’s modulus, Poisson Number

𝜅(𝑠) = ||T′(𝑠)|| = ||𝛾′′(𝑠)||

𝜈 = − 𝑑𝜀𝑡𝑟𝑑𝜀𝑎𝑥

𝜀𝑡𝑟 transverse strain𝜀𝑎𝑥 axial strain

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Mechanical properties and coordinate system

𝑟(𝑠) = position of the centerline

orthonormal director basis(d1(𝑠),d3(𝑠),d3(𝑠))

r′3 = d3 tangent vector

cartesian basis e𝑖

𝑟(𝐿) = 0

(d1,d2,d3)𝑠=𝐿 = (e𝑦,−e𝑥, e𝑧)

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Mechanical equilibrium: minimization problem

𝐸 = Young’s modulus

𝐼 = moments of inertia

𝐸3 = 𝐺 = 𝐸/2(1+𝜈) =shear modulus

𝐼3 = 𝐽 = the momentof twist

𝜈 = Poisson’s number

ℰ𝑒 =3∑

k=1

𝐸𝑘𝐽𝐾2

∫ 𝐿

0

(𝜅𝑘(𝑠)− ��𝑘(𝑠))2𝑑𝑠

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Reescalings

Rationalizing → reescalings:

Arc length: 𝑠 = 𝑠𝜅𝑛

Length: �� = 𝜅𝑛𝐿

Energies: �� = 𝐵𝜅𝑛,

with 𝐵 = 𝐸𝐼 is the bending stiffness and𝐼 = 𝜋𝑟4/4 moment of inertia

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Total energy of the rod

ℰ =

∫ ��

0

(1

2

[(��1 − 1)2 + ��2

2 + 𝐶��3

]− ��𝑠 cos𝛽

)𝑑𝑠

strainenergy

gravitationalpotential

Controlparameters

𝐿, 𝜌, 𝑟, 𝐸, 𝜈 and𝜅𝑛 (𝜅1, 𝜅2)

𝐶 = the radio between the twisting and bendingmoduli�� = 𝑤/𝐵𝜅3

𝑛 = dimensionless weight𝑤 = 𝜌𝜋𝑟2𝑔 = weight per unit of length𝐶(𝜈) = (1 + 𝜈)−1 = 2/3

3 Stationary points of ℰ

1 < 𝐿[𝑐𝑚] < 20

𝜌 = 1200 kg/m3

𝑟 = 1.55 mm

𝐸 = 1290± 12 kPa

𝜈 ≈ 0.5

0 < 𝜅𝑛[𝑚−1] < 62

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Experiments

Physical experiments:

3 custom fabricationof rods (hair)

3 fine control of 𝜅𝑛

PVC flexible tubesVinylpolyxiloxane

Comparison with simulations:

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Simulations

Finite-difference method

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Equilibrium shapes of suspended rods

𝐿, 𝜌, 𝑟, 𝐸, 𝜈 ctes

𝜅𝑛 varying

3 𝜅𝑛 = 0 → straight

3 𝜅𝑛 small → planar

3 increasing 𝜅𝑛 → nonplanar 3D helical shape

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Vertical elevation versus arc length

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Phase diagram in the parameter space (��, ��)

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Comparison between experiments and simulations

3 11 110 equilibriumshapes

0 < 𝜅𝑛[m−1] < 100

Curvature 𝜅𝑛 influence in shapes

Vertical elevation versus length

Phase diagram in the parameter space(��, ��)

3 quantitative agreement betweenexperiments and simulations

3 small 𝜅𝑛 → planar shapes for alllengths

high value of 𝜅𝑛 → planar for𝐿 . 0.1m, non-planar 𝐿 & 0.1m

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Shape classification

(i) planar

2D curvesrecov. 3D model��1 = 𝛽′ and��2 = ��3 = 0

(ii) nonplanar localizedhelical

localized helices nearfree end

helical portion < 95 %of the total length

(iii) nonplanar globalhelical

typical 3D helix helicalportion > 95 % of the

total length

3 2D - 3D transition: it’s predictable!

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Local helical configuration quantified by 𝛽(𝑠)

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Classification of helical configurations

�� ≪ 1

d2.e𝑧 ≈ 0

Parametrization

Euler angles 𝛽(𝑠) and 𝛾(𝑠)d1 = cos𝛽(− sin 𝛾e𝑥 + cos 𝛾ey) + sin𝛽e𝑧

d2 = − cos𝛽e𝑥(− sin 𝛾e𝑦d3 = − sin𝛽(− sin 𝛾e𝑥 + cos 𝛾ey)

��1 = 𝛾′

��2 = −𝛽′

��3 = 𝛾′ cos𝛽

𝛾′ = 𝜅𝑛 sin𝛽/ sin2 𝛽+𝐶 cos2 𝛽

ℰ3𝐷 =

∫ ��

0

(𝑓(��𝑠, 𝛽(𝑠)) +

1

2𝛽′(𝑠)2

)𝑑𝑠

𝑓(𝑢, 𝛽) = 12(1 + tan2 𝛽/𝐶)−1 − 𝑢 cos𝛽

Equilibrium configurations: stationary points of ℰ3𝐷 with respect to 𝛽(𝑠)

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Local helix approximation failure

7 Instability at 𝑠*𝐿𝐻 = (��𝐶)−1: 𝛽 varies quickly

7 𝑠 ≤ 𝑠*𝐿𝐻 → upper part of the rod remains vertical

7 0 ≤ 𝑠 ≤ 𝑠*𝐿𝐻 → 𝛽 = 0 → helical configuration at the end

Inner layer approximation

restoring 𝛽′(𝑠)

𝑓(𝑠 ≈ 𝑠𝐿𝐻) ≈ 𝑓0 +1

2

𝜕2𝑓

𝜕𝛽2𝛽2 +

1

24

𝜕4𝑓

𝜕𝛽4𝛽4

𝜕2𝑓𝜕𝛽2 = ��(𝑠− 𝑠*)𝜕4𝑓𝜕𝛽4 = 3(4− 3𝐶/𝐶2)

We have to minimize the functional∫((��(𝑠− 𝑠*)/2)𝛽2 + 𝑓4/24𝛽

4 + 1/2𝛽′2)𝑑𝑠

Changing variables:

𝑆 = 𝑠− 𝑠*/��−1/3

𝐵(𝑆) =√𝑓4/12��

−1/3𝛽(𝑠)

New functional∫(𝑆𝐵2 +𝐵4 +𝐵′2)𝑑𝑆

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Introduction Theoretical Background Results Theoretical Analysis Conclusions

Inner layer approximation

3 New functional

∫(𝑆𝐵2 +𝐵4 +𝐵′2)𝑑𝑆

3 Euler-Lagrange condition → second Painleve equation

3 𝐵′′(𝑆) = 𝑆𝐵(𝑆) + 2𝐵3

3 It has a unique solution connecting 𝐵 → 0 for 𝑆 → ∞ (symmetric)to the 𝐵

√−𝑆/2 for 𝑆 → ∞ (bifurcated)

Hastings-McLeon solution

𝛽𝐼𝐿(𝑠) =2𝐶��1/3

√4− 3𝐶

𝐵𝐻𝑀𝐿

(𝑠− 𝑠*𝐿𝐻

��−1/3

)3 IL solution successfully

describes the smoothtransition between thehelical and straight portionsof rod near 𝑠*𝐿𝐻

3 We now are able to predict the transition local → global !

3 𝛽𝐼𝐿(0.95��) = 1.5𝑜

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Local helical configuration quantified by 𝛽(𝑠)

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Conclusions

3 Agreement between experiments and simulations

3 Model rcovers both 2D and 3D shapes and allow to explain thetransition planar → helical configurations

3 Model applicable of a variety of engineering, naturally curvedsystems, wires, cables, pipes

3 Helpful to the inverse problem of manufacture rodlike structures

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