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48 Metallic nanowires: differencies between stretching and bending where C = 9ν2d20/(8L4), which has the same type of dependency on n and Z as Eq. (B.13) has. Then we obtain curves as in fig 2.1. For small bending Z we obtain ′ N √ 2 N 5 n 2 − 4 N n k(Z) = n=1 n N−n +CZ 2√N−n +... . (B.16) The spring constant k and hence Young’s modules E in our model is proportional to this sum yielding a size effect lowering E for small cantilevers. E is found to have the ability to increase as well as decrease with diameter depending on the temperature [54]. Setting the sum equal to zero using N = EF /E0 and E0 = 2π2/(md20) we find using D0 = d0(E = 0) the size effect 2π2 D0 = mE . (B.17) F For wires with diameter less than this the Young’s modulus is zero in the model. For gold we obtain from this formula D0 = 0.3 nm. Replacing a sum by an integral we have N√ 45/2 n N−n ≈ 15 N −N . (B.18) n=1 Using this approximation we then obtain the spring constant of the unbent wire including the size effect as √ 2 ν 2 d 20 m 3 E F5 d 0 3 D 03 k0=5π26L1−d3. (B.19) 0 We see that if ν = 0 this formula yields k0 = 0. This because of that the cross sectional area then does not change when the wire bends. A beam of length L with a perpendicular force F applied at the end bends the distance Z: Fbend L3 k0 Z L3 Z = 3EbendI = 3EbendI , (B.20) where Ebend is the Young’s modulus for the cantilever and I is the moment of inertia of the beams cross section. I = d40/12 for a beam of square cross section. F = k0Z, and k is in turn the transverse spring constant of the wire. Eqs. (B.19) and (B.20) yields an expression for Ebend neglecting the size effect: n=1 4 √ 2 ν 2 m 3 E F5 Ebend = 5π2 6 . (B.21) This formula yields a value of Young’s modulus for a cantilever wire which may differ from the value obtained by stretching the wire due to the different mechanisms that changes the cross sectional area. Because D0 is very small, the size effect may be neglected for most nanowires.PDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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