On the Configurational Entropy of Nanoscale Solutions for More Accurate Surface and Bulk Nano-Thermodynamic Calculations

Abstract

Abstract:The configurational entropy of nanoscale solutions is discussed in this paper. As follows from the comparison of the exact equation of Boltzmann and its Stirling approximation (widely used for both macroscale and nanoscale solutions today), the latter significantly over-estimates the former for nano-phases and surface regions. On the other hand, the exact Boltzmann equation cannot be used for practical calculations, as it requires the calculation of the factorial of the number of atoms in a phase, and those factorials are such large numbers that they cannot be handled by commonly used computer codes. Herewith, a correction term is introduced in this paper to replace the Stirling approximation by the so-called "de Moivre approximation". This new approximation is a continuous function of the number of atoms/molecules and the composition of the nano-solution. This correction becomes negligible for phases larger than 15 nm in diameter. However, the correction term does not cause mathematical difficulties, even if it is used for macro-phases. Using this correction, future nano-thermodynamic calculations will become more precise. Equations are worked out for both integral and partial configurational entropies of multi-component nano-solutions. The equations are correct only for nano-solutions, which contain at least a single atom of each component (below this concentration, there is no sense to make any calculations).

Nonlinear analysis of the degree of order and chaos of morphology of porous silicon nanostructures

“…The informational entropy is a basic building block of complexity theory including theories of chaos and fractals [19,20]. To evaluate the informational entropy of nanostructured systems it is required more accurate analysis of surface and bulk nano-thermodynamics.…”

A new paradigm on the chemical potentials of components in multi-component nano-phases within multi-phase systems

“…Similarly to macro-systems, the equilibrium in nano-systems is also calculable through the chemical potentials of the components. [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] Therefore, the knowledge of accurate equations describing the chemical potential of components in nano-phases is of primary importance.…”

Thermodynamics in Material Science

“…This issue considers a number of highly relevant concerns which include: A study of the use of Boltzmann and Stirling approaches in theoretical nano-thermodynamic calculations [ 1 ] where the replacement of Stirling’s Approximation by the de Moivre approximation is shown to improve results. A study of Δ Η mixing of Al-Tb alloys in the liquid phase [ 2 ] A new procedure [ 3 ] is proposed for enhancing the performance of a microcooler in both homogeneous and hybrid systems.…”

“…A study of the use of Boltzmann and Stirling approaches in theoretical nano-thermodynamic calculations [ 1 ] where the replacement of Stirling’s Approximation by the de Moivre approximation is shown to improve results.…”