Chapter 6 Nanowires and Nanotubes
6.2. Designing metal nanowires with negative Poisson’s ratio
6.2.3. Rectangular nanowires
We firstly present Poisson’s ratio components νxy and νxz of an Au (001) nanoplate and Au (001) nanowires with cross-sectional area of axb where a is the width along y- direction and b is the thickness along z- direction in Figure 6.10. b is kept as 10a0 where a0 is the lattice parameter while b is various with the ratio a/b= 1, 1.5, 2, ∞. When r=1, the cross-sectional shape is square, νxy= νxz= ν=0.49 at unstrained state. And it seems that the Poisson’s ratios do not change with applied strain. Poisson’s ratio components of a square nanowire are larger than that of the bulk counterpart (0.46) due to surface effect. Detail of surface effect on Poisson’s ratio of square nanowire can be seen in Ref. 57. As r=∞ (nanoplate), we can see that the nanoplate has two distinguished Poisson’s ratio and they show strong dependence on applied strain. The component νxy starts with 0.64 at unstrained state and then it increases. On the other hand, νxz is 0.31 at zero strain, it decreases with increasing of strain, reaches a negative value at a strain of 0.034. The strain at which materials show negative Poisson’s ratio is called as auxetic strain. The thickness effect on the mechanical property is shown in Figure 6.11.When r is larger than 1 (but still a nanowire), it is interesting that a negative Poisson’s ratio (νxz) is still observed. Poisson’s ratio behavior is dependent on the ratio a/b and it approaches to that of nanoplates as a/b increases. It is noteworthy to mention that as the aspect ratio is 2.0, difference of Poisson’s ratios between the nanoplate and the nanowire is quite small. For
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example, at unstrained state, νxz is about 0.30, and the critical strain of the nanowire is also the same as that of the nanoplate. This is the first time metal nanowires are found to show auxeticity.
We explain the auxetic behavior in the rectangular nanowires by firstly considering auxeticity of the bulk counterpart under a proper loading condition. It is obvious that the cubic materials do not exhibit a negative Poisson’s ratio along principle directions at unstrained state.
However, they can show auxeticity by controlling multi-axial loading in the principal directions i.e. the stress tensor components are
x0,
y 0,
z0,
y
z. In this case, stresses are assigned with different amounts in two transverse principal directions. Under uniaxial stress condition, cubic materials can fails with a suddenly large contraction along a transverse principal direction and a suddenly large expansion along the other transverse principal direction (tetragonal to orthorhombic phase transformation), as a result of an elastic instability (Chapter 3). However, under multi-axial loading, with the asymmetrically applied stresses along transverse principal directions, there is no longer the sudden contraction/expansion of crystal lattices. Instead, the lattices along lateral directions deform differently and at a certain strain, they can deform with different directions. Consequently, negative Poisson’s ratio is observed in a principal direction. Larger asymmetry of the applied stresses induces smoother branching if the lattice parameter and therefore smaller auxetic strain is obtained. Details of this mechanism was presented in Section 3.3 in Chapter 3.We conducted simulations for Au bulk under multi-axial loading in which tensile strain is applied along x-direction, stress is kept -800 MPa in y-direction; while along z-direction we consider several cases with different amounts of applied stress ranging from -800 MPa to 0 MPa. In Figure 6.12, we plot changes of Poisson’s ratios of the bulk material under loading. In the case the applied stresses along two lateral directions are equivalent
y
z, two Poisson’s ratio components νxy and νxz are the same which is about 0.45 and they seemly do not change with loading. One there is asymmetry between applied stresses along y- and z-directions, we can observe negative Poisson’s ratio. In the case the asymmetry is at most (stress free along z- direction), the material show the most auxetic at certain strain while it shows less auxetic in the case there is compressive stress along z-direction. We repeated the bulk simulation with98
different amount of applied stresses and plot critical auxetic strain as a function of the applied stresses in Figure 6.13. The material shows auxeticity along the direction which the applied stress is less compressive. And, when the applied stresses are the same, no negative Poisson’s ratio is observed. The Poisson’s ratio does not change much with applied strain until the material loses its elastic instability with a phase transformation (see Section 3.2 in Chapter 3).
We now turn back to the nanoscale materials. For a material under uniaxial stress condition, only one stress component along x-direction σxx is zero, the others are zeros. For bulk material, stress at any point in its domain also follow the condition. However, in nanoscale materials, stress at any point is not necessary to have single none-zero stress component.
Instead, effective stress tensor has one and only one none zero stress component along x- direction. Due to large tensile stress at free surfaces, stress inside nanoscale materials are usually compressive due to tensile surface stress. The compressive stress along in-plane lateral direction inside a (001) nanoplate induced by tensile surface stress is found to be inversely proportional to its thickness (Chapter 5). Approximately, stresses induced by surface stress inside a (001) rectangular nanowire axb are
y 2f b and z 2f aalong y- and z- directions respectively. Note that as a approaches to infinite, there is no induced stress along z-direction. Approximately, a nanoplate under uniaxial stress condition can be regarded as the corresponding bulk material under multiaxial stress condition in which tensile strain is applied along x-direction, an applied stress
y 2f balong y-direction and stress is free along z- direction. The compressive stresses induced by surface stress dilute the sudden branching of crystal structure as described above. That is why negative Poisson’s ratio can be observed in metal nanoplate. In the case of square nanowire with the width of b, the compressive stresses induced by the surface stress are
y
z2f b. Because the induced stresses are equivalent, we never see the auxetic behavior of the square nanowire. With increasing of applied strain, the nanowire deforms gradually with a negative Poisson’s ratio and it may fail with the phase transformation or by other failure mechanism such as slip, twining etc. On the other hand, for a rectangular nanowire with cross-section axb (supposing a>b), the induced stresses are not the same. Now the rectangular nanowire can be regarded as the bulk material under multiaxial99
loading which is combination of a strain along x-direction, an applied stress along y-direction and an applied stress along z-direction, and the applied stresses are different from each other.
As the result, it can show auxeticity, as plotted in Figure 6.10. Therefore, the rectangular nanowire can show a negative Poisson’s ratio. Auxeticity in metal (001) nanoplates is just a special case of rectangular nanowires.