Size-Dependent Materials Properties Toward a Universal Equation
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DOI: 10.1007/s11671-010-9614-1
Abstract
Due to the lack of experimental values concerning some material properties at the nanoscale, it is interesting to evaluate this theoretically. Through a “top–down” approach, a universal equation is developed here which is particularly helpful when experiments are difficult to lead on a specific material property. It only requires the knowledge of the surface area to volume ratio of the nanomaterial, its size as well as the statistic (Fermi–Dirac or Bose–Einstein) followed by the particles involved in the considered material property. Comparison between different existing theoretical models and the proposed equation is done.
Keywords
Nanomaterials Size effect Shape effect Theory Top–downIntroduction
Understanding how materials behave at tiny length scales is crucial for developing future nanotechnologies. The advances in nanomaterials modeling coupled with new characterization tools are the key to study new properties and capabilities and then to design devices with improved performance [1]. This study of size and shape effects on material properties has attracted enormous attention due to their scientific and industrial importance [2–4]. Nanomaterials have different properties from the bulk due to their high surface area over volume ratio and possible appearance of quantum effects at the nanoscale [5–7]. The determination of nanomaterials properties is still in its infancy and many materials properties are unknown or ill-characterized at the nanoscale [8, 9]. Therefore, modeling different phenomena by only one general equation could be particularly helpful at the nanoscale when experimental data is lacking.
Theory
When modeling nanomaterials, there exist two main approaches. In the “top–down” approach, one looks at the variation of the properties of systems that change when going from the macro to the nano dimensions. At the opposite, in the “bottom-up” approach, one starts from atoms and one adds more and more atoms, in order to see how the properties are modified. The first makes use of classical thermodynamics, whereas the second relies on computational methods like molecular dynamics. Molecular dynamics generally considers less than one million atoms [10] in order to keep calculation time within reasonable values. This factor limits the nanostructure size modeled until values around 100 nm [11]. By using classical thermodynamics, the “top–down” approach ceases to be valid when thermal energy kT becomes smaller than the energetic gap between two successive levels, δ. Generally for metals, according to Halperin [12], when δ/k ~ 1 K, the band energy splitting appears for diameter values between ~4–20 nm depending on the material considered. When δ/k ~ 100 K, this diameter is between ~1 and 4 nm in agreement with the value announced by Wautelet et al. [13]. The size limit considered in this manuscript will be 4 nm. Therefore, the “top–down” approach emerges as a simple complementary method which can give useful insights into nanosciences and nanotechnology.
where X represents melting, Debye, Curie or superconducting. α_{shape} is the parameter quantifying the size effect on the material property and depending on the nanostructure’s shape. α_{shape} is defined as α _{shape} = [D(γ _{ s } − γ _{ l })/ΔH _{ m,∞}](A/V) where A/V is the surface area over volume ratio, ΔH _{ m,∞} is the bulk melting enthalpy and γ _{ s(l)} the surface energy in the solid (liquid) phase. D is the size of the nanostructure. S equals to one half or one if the particles involved in the considered phenomena follow a statistic of Fermi–Dirac or Bose–Einstein. For melting and ferromagnetism (Curie), S equals to one-half, whereas for superconducting and vibration (Debye) S equals to one.
Distinction between “fermionic” and “bosonic” material properties
S = 1/2 (“fermionic properties”) |
S = 1 (“bosonic properties”) | |
---|---|---|
Material property |
Melting |
Superconductivity |
Ferromagnetism |
Vibration | |
Cohesion | ||
Diffusion | ||
Vacancies |
where ξ represents the size/shape-dependent material property and ξ_{∞} represents the bulk material property. The material properties considered here are the melting temperature, Curie temperature, Debye temperature, superconductive temperature, cohesive energy, activation energy of diffusion, vacancy formation energy.
Results and Discussion
where p is the ratio between the interface surface energy per unit area at 0K over the surface energy per unit energy at 0K. d _{ hkl } is the interplanar distance of hkl. β equals to 3κ/D, 2/w or 1/t for a nanoparticle, nanowire or nanofilm, respectively. D w and t are the size of the nanoparticle, width of nanowire and thickness of the nanofilm, respectively. κ is the shape factor of the nanoparticle defined as the surface area ratio between non-spherical and spherical nanoparticles in an identical volume.
where d is the atomic diameter, R is the ideal gas constant. S _{ b } is the bulk evaporation entropy.
where Z _{ SB } is the ratio of the surface coordination number over the bulk coordination number. D _{0} is the size of the nanoparticle for which all the atoms are located on the surface. D _{0} = (2/3)(3 − λ)(P _{ S }/P _{ L })d. λ is a parameter representing the dimension of the nanostructure: λ = 0 for nanoparticles, λ = 1 for nanowires and λ = 2 for nanofilms. P _{ S } is the packing fraction of the surface crystalline plane. P _{ L } is the lattice packing fraction. d is the atomic diameter.
where i is counted up to 3 from the outermost atomic layer to the center of the solid because no coordination imperfection is expected for i > 3. γ _{ i } = τc _{ i } d/D is the portion of the atoms in the i th layer from the surface compared to the total number of atoms in the entire solid. τ is a parameter representing the dimension of the nanostructure (τ = 1 for a film, τ = 2 for a wire and τ = 3 for a particle). d is the bond length or the atomic diameter (without coordination number imperfection). Z _{ iB } is the ratio of the coordination number of the i th layer (Z _{ i }) over the bulk coordination number (Z _{ B }). is the bond contraction coefficient. m is a parameter representing the nature of the bond.
where E _{ s } = πd ^{2} γ is the cohesive energy of an atom at the surface and γ is the surface energy of the material. d is the atomic diameter.
Conclusion
In summary, it is shown that there exists a universal relation between many materials properties, the inverse of the particle size and the spin of the particles involved in the considered material property. Whatever the nature of the material, Figs. 1 and 2 are general maps summarizing the size and shape effects on the mentioned materials properties from the bulk to the nanoscale. The prediction from the universal relation (Eq. 3) has been validated by comparison with available experimental results and existing theoretical models. Describing different phenomena with only one equation is the “Holy Grail” for all physicists and maybe a more sophisticated equation may exist by considering other material properties. Nevertheless, the great advantage of the present equation is that it is free of any adjustable parameters!
Acknowledgments
The author thanks the Belgian Federal Science Policy Office (BELSPO) for financial support through the “Mandats de retour” action. Dr. Steve Arscott is greatly acknowledged for proof reading this manuscript.
Open Access
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