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![Page 1: Raymond E. Goldstein University of Cambridge · Raymond E. Goldstein University of Cambridge . Computer Animation of the Dynamics of Hair Super-Helices for Predicting the Dynamics](https://reader031.fdocuments.in/reader031/viewer/2022021706/5b8a5bdb7f8b9ac1328bfd93/html5/thumbnails/1.jpg)
Leonardo, Rapunzel,
and the Mathematics
of Hair
Raymond E. Goldstein
University of Cambridge
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Computer Animation of the Dynamics of Hair Super-Helices for Predicting the Dynamics of Natural Hair
Bertails, Audoly, Cani, Querleux, Leroy, Lévêque (SIGGRAPH 2006)
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Opportunity Knocks…
Dear Prof. Goldstein,
I work at Unilever's R&D labs in Port Sunlight in the UK in the Hair Research
Division. My personal background being in the Soft Matter Physics area.
Some of the challenging technical problems in the Hair Care area depends upon
us better understanding hair array statistical mechanics under various
conditions. From your publications and your current research interests I see that
your research interests lies in quite varied and challenging areas. I was wondering
if the area of hair array statistical mechanics may be something you might
possibly be interested in? …
…
Samiul Amin
Unilever R&D Port Sunlight, Quarry Road East, Bebington, Wirral CH63 3JW
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The Team
Patrick Warren, Unilever Robin Ball, Warwick Joe Keller REG
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Drawings of Hair, From Hooke’s Micrographia (1665)
Observ. XXXII. Of the Figure of several
sorts of Hair, and of the texture of the skin.
Viewing some of the Hairs of my Head with a very
good Microscope, I took notice of these particulars:
1. That they were, for the most part, Cylindrical,
some of them were somewhat Prismatical, but
generally they were very neer round, such as are
represented in the second Figure of the 5.
Scheme, by the Cylinders EEE. nor could I find
any that had sharp angules.
.
.
5. That the top when split (which is common in
long Hair) appear'd like the end of a stick,
beaten till it be all flitter'd, there being not
onely two splinters, but sometimes half a
score and more.
6. That they were all, as farr as I was able to
find, solid Cylindrical bodies, not pervious,
like a Cane or Bulrush; nor could I find that
they had any Pith, or distinction of Rind, or
the like, such as I had observ'd in Horse-
hairs, the Bristles of a Cat, the Indian
Deer's Hair, &c.
Wellcome Library, London
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Hair Has a Complex Structure! Cuticle
Cortex Cortical Cell
Macrofibril
Microfibril
Keratin
coiled-coil
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Interesting Facts About Hair* • Adults have 50,000-100,000 head hairs
• Growth of 1 cm/month ≈ 4 nm/sec per hair
• Hair density is 1.3 g/cm3, is elliptical in x-section,
with an average major axis diameter d ≈ 80 µm
and a linear mass density λ ≈ 65 µg/cm ≈ 6.5 g/km
• Bulk modulus is like nylon, E ≈ 4 GPa, so its
bending modulus is A ≈ 10-8 Nm2.
*Co
urte
sy o
f Su
san
Welc
h, U
nile
ver R
&D
R
.E.G
.
Energy density of an approx. horizontal filament,
implies a characteristic length
Hence, we introduce the “Rapunzel number”
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Hair Has Random Intrinsic Curvatures
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“Annealed” vs. “Quenched” Curvatures
Thermal fluctuations Intrinsic curvature
Equipartition:
Euler-Lagrange equation:
Projected length deficit:
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Statistics of Random Curvatures
PRL 108, 078101 (2012)
Measurements on 115 hairs from a commercial* switch,
using high-resolution stereographic imaging.
Filament reconstruction based in part on an algorithm
due to W.S. Ryu for C. elegans tracking.
*International Hair Importers & Products, Inc. (Glendale, NY)
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Density Functional Theory of Fiber Bundles
Fiber length density
(#/unit area crossing a plane ⊥ to fibers)
Local mean orientation of hairs
Absence of free ends →
Hypothesis: a local energy functional,
filament elasticity
external potential
disorder
curvature: pressure:
θ
t
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Leonardo’s Observation “Observe the motion of the surface of the water which resembles
that of hair, and has two motions, of which one goes on with the flow
of the surface, the other forms the lines of the eddies…”
The N
ote
books o
f Leonard
o d
a V
inci –
“On t
he p
roport
ions a
nd o
n the
movem
ents
of
the h
um
an
fig
ure
”
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Application to an Axisymmetric Ponytail
Ansatz of a self-similar density profile:
A well-studied problem
(L&L, Audoly & Pomeau)
Yields an equivalent single-fibre energy for envelope:
Minimization → The Ponytail Shape Equation
Let n(r,z) be # of fibres within radius r at depth z.
elasticity tension weight pressure
PRL 108, 078101 (2012)
Van Wyk (1946) – wool
Beckrich et al. (2003) - 2D
Average over 5
72o rotations
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Balance of Forces Along Length of a Ponytail
PRL 108, 078101 (2012)
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Testing Equations of State
Swelling due to pressure
PRL 108, 078101 (2012)
Fixed “launch angle”
Away from the clamp, ignore elasticity
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Empirical Equation of State of Hair
PRL 108, 078101 (2012)
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Experimentum crucis: Trimming Ponytails P
RL
10
8, 0
78
10
1 (
20
12
)
Experiment Constant Π0 Graded Π0
Ra ≈ 5
Ra ≈ 1
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Interpreting the Equation of State
Consistent with the essential features of “tube models”
Elastic energy density with spontaneous curvature:
d
parabola
Similar result holds for a helical filament confined to a cylinder:
Integrated EOS:
Hence, effective tube is some fraction of the ponytail radius
x
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Ponytail Motion
© Aila Images
Shutterstock.com
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The Faraday Instability
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(1923-2016)
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The Maths of a “Parametric Excitation”
George William Hill
1838-1914
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A Controlled Experiment
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A Controlled Experiment
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The Next Frontier: Tangling
comb
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The Amontons-Coulomb Percolation Transition: How
a Staple Yarn Transmits Tension and Why Our
Clothes Don't Fall Apart
Bayman, Am. J. Phys. 45, 185 (1977).
Maddocks and Keller, SIAM J. Appl. Math. 47, 1185 (1987).