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50 Metallic nanowires: differencies between stretching and bending Table B.1: Poisson’s ratio ν, Fermi energy EF and calculated Young’s modulus for different metals. Using ν and EF in the table, we calculate using the free electron model Young’s modulus when stretching the wire Estretch and Ebend when bending the cantilever wire, from Eqs. (B.29) and (B.21) respectively. Free electron bulk values Ebulk from Eq. (B.31). ν Ag 0.37 [56] Al 0.35 [56] Au 0.44 [56] Be 0.032 [56] Bi 0.33 [56] Ca 0.31 [56] Cd 0.30 [56] Cu 0.34 [56] Fe 0.29 [56] Mg 0.29 [56] Nb 0.40 [56] Pb 0.44 [56] Sn 0.36 [56] Sr 0.28 [56] Tl 0.45 [56] Zn 0.25 [56] Estretch (GPa) Ebend (GPa) 3.94 8.41 28.0 49.1 2.20 12.1 EF (eV) 5.48 [57] 11.6 [57] 5.51[57] 14.3 [53] 1.58 [53] 0.205 4.68 [57] 3.25 7.46 [57] 10.7 7.00 [57] 8.22 11.1 [53] 29.2 7.13 [57] 9.64 5.32 [53] 1.83 9.37 [57] 8.27 10.0 [57] 18.6 3.95 [57] 2.23 8.15 [53] 4.98 9.39 [57] 19.7 Ebulk (GPa) 26.6 200 12.5 30.2 251 9.98 83.1 9.13 19.0 45.5 47.0 35.8 129 2.13 19.9 33.6 27.7 14.8 197 13.5 0.692 1054 0.299 1.56 3.98 26.2 12.0 88.9 13.1 60.4 Comparing(B.21)with(B.29)weseethatthedependencyonEF arethesameinthe two formulas but the dependency on Poisson’s number ν are different. In the can- tilever case Young’s module increases as ν2, but in the elongation due to stretching case Young’s modulus is proportional to ν (1 − 2ν). For metal in bulk we have the well known free electron bulk modulus B, see ref. [53, 57]. We then obtain 4 √ 2 m 3 E F5 B = 9π2 6 , 4 √ 2 m 3 E F5 Ebulk = 2 (1−2ν) 6 , (B.30) (B.31) using Ebulk = 3B (1 − 2ν), see ref. [58]. For macroscopical materials we have 3π Estretch = Ebend = Ebulk. For our nanowires we have found Estretch = 18Bν(1−2ν), (B.32) 15 Ebend = 95 B ν 2 . (B.33) When ν = 72 ≈ 0.286 the two formulas yields the same result. Note the large dif- ferences in E for beryllium (Be) in Table B.1 due to the metal’s low Poisson’s ratio ν = 0.032.PDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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