A Quantum Mechanical Model for the Calculation of the Surface Free Energy and Surface Stress of Nanoparticles
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As a model for a nanoparticle we use a nanocube. We assume that the potential of the cube is separable. Therefore, we can split Schrödinger’s equation into three one-dimensional ones. The potential in each of the one-dimensional sub-systems has a Meander-like run. For the calculation of the surface free energy of the nanoparticles we split a nanocube into smaller ones. We calculate the difference of the electron energies in the original nanocube and in the disintegrated cubes. If we divide the energy increment by the newly generated surface we have a very easy and correct procedure for the calculation of the surface free energy resulting from the cleavage of a nanocube into nanocubes. For an exact calculation of the surface free energy emerging in the disintegration of a macroscopic body into nanocubes we have to calculate the energy levels in an infinitely extended body with the same type of potential. Also a formula for the calculation of the surface stress for a nanocube has been derived. Approximating methods are given for the calculation of the surface free energy and the surface stress of nanocubes. In the case that the mean electron energy for a material with completely occupied energy bands is known it is possible to calculate, at least approximately, with this easy method the surface free energy and the surface stress. In this way, the surface free energy as a function of the chemical potential and temperature has been calculated. For a nanocube with 10 × 10 × 10 atoms with a completely occupied energy band the surface free energy amounts to 0.4 J/m2 and the surface stress is – 0.19 N/m.