Critical Size Total Energy Storage

Assuming a hollow sphere of K exterior radius with a shell of C l + C k thick, or three contracted atomic layers, the total energy stored in this hollow sphere at 0 K in comparison with that stored in an ideal solid sphere of the same size without surface effect being considered. 

Calculations based on the given C and the curvature-dependent z, values derive the trends in Fig. 29.6. One can find the critical size below which the energy stored in the shell of the hollow sphere is greater than that stored in the ideal bulk of the same volume without considering the surface and temperature effects. 

The estimation indicates that the critical size is bond nature dependent. The critical size is = 6, 8 and 10 for m = 1, 3, and 5, respectively. Similarly, for hollow tubes, the corresponding critical K values are estimated to be 7, 10, and 14. For the single-walled hollow structure, the integration crosses only the diameter of the wall atom. 

The elasticity of the skin shell of the pore is always higher than the bulk interior because the elasticity is proportional to the energy density though the total energy stored in the shell may be lower than the entire sphere beyond the critical size. For plastic deformation, the hollow sphere could be tougher than the ideal bulk because of the long-distance effect in the indentation measurement. On the other hand, the thermal stability of the hollow sphere is always lower than the solid sphere [71]. 

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