Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles

A general continuum theory for the distribution of hairs in a bundle is developed, treating individual fibers as elastic filaments with random intrinsic curvatures. Applying this formalism to the iconic problem of the ponytail, the combined effects of bending elasticity, gravity, and orientational disorder are recast as a differential equation for the envelope of the bundle, in which the compressibility enters through an “equation of state.” From this, we identify the balance of forces in various regions of the ponytail, extract a remarkably simple equation of state from laboratory measurements of human ponytails, and relate the pressure to the measured random curvatures of individual hairs.

Figure 1

A ponytail. (a) Rotationally averaged image of a switch of $N\approx 9500$ fibers, approximately 25 cm long. Coordinate system for envelope shape $R(s)$ in terms of arc length $s(z)$. (b) Meanderings of hairs near ponytail edge.

Figure 2

Geometry of hairs. (a) Representative projections of hair contours, thickened for clarity. (b) Mean squared radial excursion and curvature as functions of arc length, from processing stereoscopic image pairs [9]. The reconstructed arc lengths cluster tightly around $24.50\pm 0.05\mathrm{cm}$ (inset histogram), demonstrating the accuracy of the image processing and analysis methodology. Error bars are standard errors from ensemble averaging ($N=115$ fibers in total).

Figure 3

Analysis of ponytail shapes. (a) Measured hair switch profiles (thin black lines), compared to the prediction of Eq. (3) with $\Pi (R)$ as given (solid blue line), with $\Pi (R)=0$ (lower dashed blue line), and with the van Wyk EOS (upper dashed red line) [5, 13]. (b) Dimensionless swelling pressure $\Pi (R)$ from cubic and quartic fits to the measured profiles, using the procedure outlined in the text. (c) The magnitude of the four terms in Eq. (3) for the calculated profile (solid blue line) in (a).

Figure 4

Trimming a ponytail. (a) Superimposed rotationally averaged outlines of four hair switches, cut down from 25 cm in steps of 5 cm. (b) Predicted profiles from Eq. (3) with $\Pi (R)={\Pi }_0(1-R/R^{*})$. (c) Predicted profiles with $\Pi (R,s)={\Pi }_0(1-R/R^{*})\times 2s/L^{*}$, where $L^{*}=25\mathrm{cm}$.