Page 148 - Carbon Nanotube Fibres and Yarns

Carbon nanotube yarn structures and properties 141

7.1.3 Nanotube packing density, bulk density, and porosity
If all nanotubes are perfectly straight and aligned in one direction (a bun-
dle of parallel cylinders), they can lie next to each other and the nano-
tube assembly achieves the maximum packing density. Fig. 7.3 shows a
cross-sectional view of parallel cylinders closely packed in a hexagonal
array. Using the parallelogram “unit cell” [14], we can find that the pro-
portion of unfilled area (including spaces inside the nanotubes), or the
minimum porosity of the closely packed yarn, is

−
π 1 (/ ) 2  (7.1)
dD


ϕ min =−
1
23
where D (nm) is the outer diameter of the CNT and d (nm) is the inner di-
ameter of the CNT. If the nanotubes are treated as solid cylinders, i.e., d = 0,
Eq. (7.1) gives the minimum porosity of 9.3%, or the maximum packing
density of 90.7%. If the space inside the nanotube is counted as voids in the
yarn, the minimum porosity will be higher. For example, when d/D = 0.4,
the minimum porosity is 23.8% [14].
2
The maximum number of nanotubes that can be packed in a 1 μm cross
2
section (n max , tubes/μm ) can be calculated from
23 (7.2)
n = ×10 6
max
3 D 2

Unit
cell

Fig. 7.3 Close hexagonal packing of parallel cylinders [14]. (Reprinted with permission
from M. Miao, J. McDonnell, L. Vuckovic, S.C. Hawkins, Poisson’s ratio and porosity of carbon
nanotube dry-spun yarns, Carbon 48 (10) (2010) 2802–2811.)