# Self-organization of a mesoscale bristle into ordered hierarchical helical assemblies

A well known phenomenon is the clumping of wet hair.

This can be explained by studying the elasticity of each hair strand and the adhesion among the neighboring hair strands. When the hair strands are immersed in water, and the water is allowed to evaporate, capillary forces between the strands cause them to deform, adhere to each other, and form clumps.

This paper demonstrates that this elastocapillary force can be used to self-assemble stiff and upright nano-pillars to helical clusters and even coiled large-area structures. The fundamental theory involves the following parameters:

$r$: circular radius of the nano-pillars

$L$: length of nano-pillars

$d$: distance between adjacent nano-pillars

$E$: Yong's modulus

$B\simeq Er^4$: bending stiffness of the nano-pillars

$J$: adhesive energy per unit area

$\gamma$: interfacial tension of the evaporating liquid

There is a competition between the capillary force due to the menisci that wants to pull the free ends of adjacent nano-pillars together, and the force due to the bending stiffness of the nano-pillars that wants to pull them apart and return to the straight position. These forces are related to the above parameters through the following:

Bending force: $F_B\simeq Bd/L^3$

Capillary force: $F_C\simeq \gamma r$

when the two forces are equal to each other, we arrive at the critical length of the nano-pillars, given by

critical nano-pillar length: $L_C \simeq (Bd/\gamma r)^{1/3} \simeq (Ed/\gamma)^{1/3}r$

At this critical length, when the adhesive energy per unit area is equal to the interfacial tension $J=\gamma$, we arrive at another important length

$L_a\simeq(Ed/J)^{1/3}r$

These critical lengths can be used to guide the design of nano-pillars where such bending is possible and maintained even after the liquid evaporates.

$\tau=\pi$