Stiffening of Nanowires by Quantum Mechanics

Sept. 5, 2012 - PRLog -- Background

Over the past decade, the observation of significant stiffening of nanowires has been
reported, although some findings suggest there is no stiffening. Because of this uncertainty, research on the mechanism for stiffening has been a subject of great interest. Numerous mechanisms [1] have been proposed including: high surface-to-volume ratio, surface stresses, bulk nonlinear elasticity, surface stiffness, surface tension, surface reconstruction, surface strain and stress, and skin depth energy pinning.


Generally, the stiffening of nanowires is not based on direct measurements of material properties, but rather inferred from indirect measurements of increased resistance to buckling, enhanced resonant frequencies, and the like. In contrast, the traditional uniaxial tensile test of a nanowire gives mechanical properties directly, but is difficult to perform because of the nanoscopic size of the tensile specimen. Nevertheless, Young’s modulus and yield stress of fivefold twinned silver nanowires was recently [2] measured in tensile tests and found to show stiffening consistent with indirect measurements, the stiffening mechanism thought to be the high surface-to-volume ratio in combination with the annihilation of dislocations at free surfaces and enhanced strain hardening from fivefold twinning.

Problem

The stiffening mechanisms proposed to date find basis in classical physics contrary to the fact stiffening is not observed at the macroscale, but rather a QM effect only observed in < 100 nm diameter nanowires. QM stands for quantum mechanics. Unlike classical physics, only QM having a natural size effect may explain the stiffening of nanowires.  

QED Induced Radiation

Stiffening of nanowires by QM is proposed to occur from the QED radiation induced in nanowires from EM energy that under the TIR confinement creates photons within the wire instead of increasing the wire temperature.  QED stands for quantum electrodynamics, EM for electromagnetic, and TIR for total internal reflection. QED radiation relies on the QM given by the Einstein-Hopf relation for the harmonic oscillator that requires the heat capacity of the atoms in nanostructures to vanish. See http://www.nanoqed.org, 2012.  

QM Stiffening Mechanism

In nanowires, the supports are the source of thermal heating. The dissipative energy in strain hardening is relatively negligible. Supports are typically macroscopic, and therefore instantly transfer their temperature continuously during contact to maintain the kT energy of atoms in the nanowire. T is absolute temperature and k Boltzmann’s constant. Cantilever supports transfer their temperature at one end while simple supports transfer temperature at both ends, the latter of importance in tensile tests as gripping is required to place the nanowire in tension.

Lacking heat capacity, the nanowire cannot conserve the kT energy acquired from supports by an increase in temperature. Instead, conservation proceeds by the creation of QED radiation. The TIR confinement creates QED photons having Planck energy E = hc /2nd, where h is Planck’s constant, c  the speed of light, n the refractive index of the material, and d the nanowire diameter. Since the nanowire diameter d < 100 nm ( E > 5 eV ), the nanowire charges by the photoelectric effect. The photoelectrons promptly escape leaving positive charges that by electrostatic repulsion pressurize the interior of the nanowire, the material placed in hydrostatic tension.

Nanowire tensile tests under the QM pressure P are not uniaxial, but rather biaxial. Classical elasticity theory therefore allows the biaxial Young’s modulus Y to exceed the uniaxial value Y_O as shown in the thumbnail. Uniaxial tests have Y/Y_O = 1 for all values of Poisson’s ratio, but the biaxial stress state gives Y/Y_O > 1 depending on Poisson’s ratio. The same enhancements are expected for the yield stress S_Y as the QM pressure must be overcome before yielding can begin, i.e., S_Y / S_YO = Y/Y_O
   
Discussion

The QM pressure P produced in the tensile specimen from thermal kT energy acquired from the grips is obtained by computing the electrostatic repulsion from positive charges produced by the photoelectric effect. Although the charge distribution is random, a uniform distribution may be assumed in molecular dynamics simulation including the QED radiation loss to the surroundings. But the pressure P may be upper bound by assuming all the atoms in the nanowire acquire kT energy at the support temperature T. Provided the QED photon energy E > 5 eV, the pressure P = (3/2) N kT/ V, where N is the number of atoms in the volume V of the tensile specimen. But V/N = D³ where D is the cubical atomic spacing, and therefore P < (3/2) kT / D³.

For the silver nanowire with D = 0.26 nm at T = 300 K, P < 3.53 x 10⁸ Pa ~ 51000 psi. Taking the yield stress S_YO  of silver as 6550 psi, (1/3)P/ S_YO = 2.6. The thumbnail shows QM enhances the Young’s modulus Y / Y_O and yield stress S_Y / S_YO < 25. However, experimental [2] data shows Y / Y_O ~ 3 and  S_Y / S_YO ~ 50.  

Conclusions

1. The QM stiffening mechanism for silver nanowires is reasonably consistent with the enhancement of yield stress, but not the Young’s modulus. However, the flexibility of the supports would reduce the QM enhancement of Young’s modulus, but not the yield stress. More study is required.

2. The QM enhancement of the mechanical properties of Young’s modulus and yield stress is consistent with classical elasticity theory for a uniaxial tensile specimen under internal hydrostatic pressure.

3. The hydrostatic pressure is a consequence of QM that requires the heat capacity of the atom to vanish in nanowires. The kT energy acquired from the temperature of the macroscopic grips in the tensile test cannot be conserved by an increase in nanowire temperature, and therefore conservation proceeds by QED creating photons in the nanowire.

4. By QM, the high surface-to-volume ratio of nanowires causes almost all of the absorbed thermal energy to reside in the surface of the nanowire, the surface corresponding to the circumferential TIR mode of the QED photons. In effect, the QED photons provide their own EM confinement during absorption of thermal energy, i.e., absent thermal absorption, there is no TIR confinement and QED radiation is not created.      

5. Charge created by the photoelectric effect produces pressure inside the nanowire from the electrostatic repulsion of positive charge. Indeed, excessive charge causes a Coulomb explosion. The pressure may be upper bound by considering the temperature of the grips to be transferred to the atoms over the volume of the nanowire.

6.  The QM stiffening shows enhancement of Young’s modulus and yield stress. Reports that stiffening is sometimes not observed are most likely caused by the nanowires not acquiring the temperature of the supports. QM requires EM energy of any kind to be absorbed by the nanowire to produce charge and stiffening, as otherwise the mechanical properties of the nanowire are not enhanced.

7. QM stiffening of nanowires may be verified if raising the temperature of cantilever or simple supports enhances mechanical properties. Alternatively, by applying a voltage across the tensile specimen, the Joule heat should significantly enhance mechanical properties.    

References

[1] X. J. Lu, et al., “Size-induced elastic stiffening of ZnO nanostructures: Skin-depth energy pinning,” App. Phys. Lett., 94, 131902, 2009.
[2] Y. Zhu, et al., "Size effects on elasticity, yielding, and fracture of silver nanowires: In situ experiments,” Phys. Rev. B, 85, 045443, 2012.