Chapter 6 Nanowires and Nanotubes
6.2. Designing metal nanowires with negative Poisson’s ratio
6.2.5. Hollow nanowires
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We have seen auxeticity in rectangular nanowires. Such behavior is strongly dependent on geometry and surface stress. With higher surface stress, the induced compressive stresses inside the nanowires are larger and thus the structures become more auxetic. Therefore, changing a material with higher surface stress is a possible way to enhance auxeticity in rectangular nanowires. This issue will be discussed later. One can easily choose other ways e.g.
increasing the aspect ratio r=a/b (Figure 6.10) or reducing nanowires size (Figure 6.11). In this section, we introduce another way to heighten surface effect on the mechanical property. In particular, a hole inside nanowire is introduce by deleting a volume at the center of nanowire L x c x d (Figure 6.16). The nanowire becomes a rectangular metal hollow nanowire or rectangular metal nanotube (RMNT). With the hole inside, surface area is larger and it can provide large asymmetric induced stresses along the lateral directions, and therefore it can enhance the auxeticity of the structure.
Metal nanotubes have been become an attractive research recently210–213. Solid part of the RMNT can be divided into 3 kinds of section which can be seen in Figure 6.16. For RMNT with infinite length, there are 8 free surfaces instead of 4 like the case of a “solid” rectangular nanowire. Due to the tensile stress at free surfaces, there are also compressive stress in atoms in the interior part of the structure. Here, interior parts means the parts not at the free surfaces.
Approximately, the average stresses in the RMNT along y-direction
y and z-direction zcan be calculated as:) 3 3 ( ) 2 2 ( ) 1
1 ( 4 2
2
y y
y
y V
V V
V V
V
(6.3)
) 3 3 ( ) 2 2 ( ) 1
1 ( 4 2
2
z z
z
z V
V V
V V
V
(6.4)
where V1, V2, and V3 are volume of the section (1), (2), and (3) (presented in Figure 6.16) respectively;
b d
f
y(1) 4
;
b f
y ) 2
2
(
;
b f
y ) 2
3
(
(6.5)
a f
z ) 2
1
(
;
a f
z ) 2
2
(
;
a c
f
z(3) 4
(6.6)
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Substituting Eqs. (6.5) to (6.3) and (6.6) to (6.4) we obtain:
cd ab
c f a
y
2
(6.7)
cd ab
d f b
z
2
(6.8)
Clearly, the average stresses
yand zare functions of a, b, c, and d. Note that if c=0 then d should be vanished and vice versa. For a given a and b e.g. a=19.6 nm and b=9.8 nm, we plot changes of
yand zwith respect to c and d in Figure 6.17a and 6.17b respectively.Note that
yand zare plotted in term of their magnitudes. While
yis more sensitive to c and it increases with c, zshows strongly dependence on d and increases with d. As can be seen in Figure 6.17, slopes of the contour lines of
yare quite large. Especially the contour lines are nearly parallel to the vertical axis with small values of c and it increase with c. In contrast, slopes of the contour lines of zare quite small and the contour lines seem to be normal to the vertical axis with small value of d and it increases with d.Again, the key idea here to enhance auxeticity is changing the geometry of the nanoscale material such that the induced stresses are more asymmetric and larger as much as possible. For the RMNTs, with given a and b, the design parameters now are only c and d.
With the characteristics of
yand zas mentioned above, a way to increasing the compressive
yand lowering the compressive zis selecting larger c and smaller d. We illustrate the design by considering the change of the Poisson’s ratio of nanotubes with a=19.6 nm, b=9.8 nm and various c and d. RMNTs with different c and d are marked as P1 to P7 in Figure 6.17a and Figure 6.17b. In Figure 6.18, we present the dependence of the Poisson’s ratio on c and d. For all considered cases, the nanotubes are more auxetic than the nanowire. The rectangular nanowire has auxetic strain of 0.044. On the other hand, the RMNT with c=9.8 nm and d=1.6 nm (P7) can show auxetic behavior at a strain of 0.025. Poisson’s ratio of the nanotube gradually changes with changing of size of the hole. Clearly,
yof the RMNTs can be much larger than that of the rectangular nanowire especially at large c, while zof RMNT is just slightly larger103
than that of the rectangular nanowire. In comparison to induced stresses of rectangular nanowires,
yof RMNTs can be much larger than that of rectangular nanowires of while zslightly changes. Consequently, the induced stresses in the RMNTs can be more asymmetric.
The dependence of Poisson’s ratio on c and d in Figure 6.18 is excellently consistent with the changes of induced stresses of the RMNTs described in Figure 6.17a and Figure 6.17b.
In Figure 6.19, deformation behavior of a RMNT is plotted. We show the unstrained configuration and configuration at a strain of 0.065. Unlike a square nanowire, asymmetric induced stresses inside the RMNT dilute the sudden branching of the material in the case it at macroscale. As the result, the RMNT shows a negative Poisson’s ratio. Effects of elliptical voids inside structure on Poisson’s ratio of materials were reported in a theoretical work by Grima and Gatt214 or by simulation and experiment by Taylor et al.125. The structure with the voids in those studies can show a negative Poisson’s ratio due to rotation of a part of the structure. Here, the auxetic mechanism is based on the asymmetry of stresses inside rectangular nanowires or RMNTs induced by surface stress and their geometry, and the branching phenomenon of cubic materials.
In Figure 6.20, we show size effects on auxeticity of RMNTs. We calculated auxetic strains of different RMNTs which have different values of a, the same aspect ratio r=a/b=2, c=0.6a and d=1nm. Smaller nanowires have smaller critical auxetic strain. This tendency is the same as that of the rectangular nanowires (showed in Figure 6.11).
We have seen that rectangular nanowire and nanotube can exhibit negative Poisson’s ratio and metal nanotube can be more auxetic than metal nanowire. We believe that negative Poisson’s ratio can be found not only in rectangular nanowire and rectangular nanotube but also in “any polygon with asymmetric cross-section”. The asymmetry of cross-section provides asymmetrically induced stresses. This is an intrinsic property of nanoscale materials. Figure 6.21 shows Poisson’s ratios of the elliptical nanowire with major radius of a=7.4 nm, minor radius of b=3.7 nm, and the corresponding elliptical metal nanotube (the same a, b) with the thickness of the tube t=2.5 nm. Auxeticity is observed in both nanowire and nanotube. Similar
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to the case of rectangular cross-section structures, the elliptical nanotube is more auxetic than the elliptical nanowire.