On the statistical thermodynamics of a free-standing nanocrystal: Silicon
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MATERIALS AND CERAMICS
On the Statistical Thermodynamics of a Free-Standing Nanocrystal: Silicon M. N. Magomedov Institute for Geothermal Problems, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, 367030 Russia e-mail: [email protected] Received June 10, 2015
Abstract—The dependence of the thermodynamic parameters of a free-standing nanocrystal of simple matter on its size, density, temperature, and surface shape has been studied. The following parameters have been analyzed: Debye temperature Θ, Gruneisen parameter γ, melting temperature Tm, surface energy σ, surface pressure Psf, elastic modulus BT, Poisson ratio μ, thermal expansion coefficient αp, and specific heats cv and cp. Calculations performed for silicon have shown that the functions Θ, Tm, σ, and BT decrease, whereas the functions γ, |Psf|, μ, αp, cv , and cp increase with an isomorphic decrease in the number of atoms N. The stronger the nanocrystal shape deviates from the most energetically stable shape, the more pronounced the change in the aforementioned functions with an isothermal decrease in N is. DOI: 10.1134/S1063774517030142
INTRODUCTION Despite the fact that nanocrystals have experimentally been studied for a long time, their thermodynamics is still debated [1–3]. The main question stated even in the works by J.W. Gibbs and E.A. Guggenheim is as follows: should the matter of the surface layer be considered as a phase different from the “bulk” one? The questions about the surface layer thickness and density, which stem from this issue, have not been clearly answered even for an infinite flat surface of a simple single-component matter. At the same time, if the nanocrystal surface is considered as a geometric surface of zero volume (i.e., Gibbs surface [4, 5]), the dependence of the thermodynamic properties of nanocrystal on its size at different temperatures is either not quite clear. We will analyze a condensed nanosystem of N identical atoms limited by a surface. The change in the free Helmholtz energy (F) of this system with variation in temperature (T), volume (V), number of atoms, and surface area (Σ) is generally presented in the form [4, 5]
( ) ( )
( ) ( )
dF = ∂ F dT + ∂ F dV ∂ T N ,V ,Σ ∂ V N ,T ,Σ dN + ∂ F dΣ + ∂F ∂ N T ,V ,Σ ∂Σ N ,V ,T = −SdT − PdV + μ g dN + σ d Σ,
(1)
from which the values of entropy S, pressure P, chemical potential μg, and specific (per unit area) free energy σ are determined.
For a finite system limited by a surface of certain shape, one cannot change isomorphically (i.e., at a given surface shape) the surface area at constant T, N, and V values. For a finite system, one cannot either physically change the volume at constant T, N, and Σ values. Moreover, N cannot be changed at constant T, V, and Σ values for a nanosystem. Specifically these limitations restrict the application of thermodynamics to the description of nanosystems. Even when the thermodynamic description is applied to relatively large nanosystems with a constant N value, the question about correct definition of specific sur
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