```SUPPLEMENTARY MATERIAL
Morphology Engineering of Hollow Carbon Nanotube Pillars by Oxygen Plasma
Treatment
1
USA
‡
These authors contributed equally to this work
Methodology for Geometric Model
The geometric model presented here aims to elucidate the transformation of the CNT pillar
morphology that occurs on the macroscopic scale during Region II. The ultimate goal of the
model is to develop a tool for predicting the shape of tapered CNT pillars fabricated through the
oxygen plasma treatment.
CNT pillars are approximated as an array of vertically-aligned strings with constant properties
throughout the length of the pillar. As a string, the CNTs can be deformed into any arbitrary
curved geometry and maintain a constant arc length throughout the etching process. From this
approximation, the model aims to predict the profile of the pillar geometry through the
progression of Phase II and examine secondary effects of etching that cannot be readily resolved
through experimental observation.
*
Corresponding author: Morteza Gharib. Tel: (626) 395-4450 email: [email protected]
The profile of the pillar is modeled as a hyperbolic cosine (Equation 1). A hyperbolic cosine is
typically used in models to represent the curvature of strings [1].

=  cosh ()
(1)
The origin of the coordinate system is placed at the center of the pillar tip (Figure S1). The
direction along the length of the pillar is given by x. The radius of the pillar at a given height is
given by y. Equation 1 describes the profile for the right side of the pillar. The pillar is
axisymmetric such that the profile for the left side of the pillar is given by reflecting the profile
generated by Equation 1 across the x-axis. The radius at the tip of the pillar is given by a. From
the experimental results, it was observed that the base is constant throughout Region II. This
constraint is present in the model such that the base diameter for all generated microneedles is
constant, d0.
2a
y
x
s
d0
Figure S1. Coordinate and variable definitions for geometric model.
The parameter b is set to keep the arc length, s, of the CNT string constant throughout the
etching process. The consistency of the base diameter in the experimental results indicates that
the primary method of morphology change during Phase II is not the large-scale destruction of
individual CNTs but rather the densification of the CNT network causing the CNT pillar to
compress. The arc length of the CNT string is set as the initial height of the pillar before etching.
The parameter b is related implicitly to arc length by Equation 2. The parameter b is solved for
numerically to within an error of 0.01% in arc length.
cosh−1 (0 /2)
= ∫0
2
√( sinh ( ) ) + 1

(2)
Figure S2 depicts the model results for the evolution through Region II of a cylindrical pillar of
initial height 400 µm with a base diameter of 150 µm. As the O/C ratio of the pillar increases,
the tip diameter begins to contract. The side profile of the pillar is initially linear and then
becomes more concave as etching progresses. Interior CNTs are represented as a set of aligned,
similar curves based on the computed profile of the pillar. The termination of Region II is
depicted in the final pillar profile with a tip diameter of 10 µm.
Figure S2. Model results for the progression of a CNT pillar with an initial diameter and height
of 150 µm and 400 µm, respectively, through Region II. Starting with the initial pillar
morphology (upper left) progressing to the final morphology (bottom right) the increased
concavity of the profile is observed concurrent with a reduction in the total height of the pillar.
References
[1]
Ng HN, Grimsdale RL. Computer graphics techniques for modeling cloth. Computer
Graphics and Applications, IEEE. 1996;16(5):28-41.
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