- Nano Express
- Open Access
Size-Dependent Materials Properties Toward a Universal Equation
- G Guisbiers^{1}Email author
- Received: 11 March 2010
- Accepted: 15 April 2010
- Published: 4 May 2010
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–down
Introduction
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!
Declarations
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
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Authors’ Affiliations
References
- Pitkethly MJ: Mate. Today. 2004,7(Supplement 1):20. 10.1016/S1369-7021(04)00627-3View ArticleGoogle Scholar
- Gleiter H: Acta Mater.. 2000, 48: 1. COI number [1:CAS:528:DC%2BD3cXmtVCltw%3D%3D] COI number [1:CAS:528:DC%2BD3cXmtVCltw%3D%3D] 10.1016/S1359-6454(99)00285-2View ArticleGoogle Scholar
- Lieber CM, Wang ZL: Bulletin. 2007, 32: 99. COI number [1:CAS:528:DC%2BD2sXjsVOlsrk%3D] COI number [1:CAS:528:DC%2BD2sXjsVOlsrk%3D] 10.1557/mrs2007.41Google Scholar
- Sutter E, Sutter P: Nano Lett.. 2008, 8: 411. COI number [1:CAS:528:DC%2BD1cXltlCmtQ%3D%3D]; Bibcode number [2008NanoL...8..411S] COI number [1:CAS:528:DC%2BD1cXltlCmtQ%3D%3D]; Bibcode number [2008NanoL...8..411S] 10.1021/nl0719630View ArticleGoogle Scholar
- Yacaman MJ, Ascencio JA, Liu HB, Gardea-Torresdey J: J. Vac. Sci. Technol. B. 2001, 19: 1091. COI number [1:CAS:528:DC%2BD3MXlvVehur8%3D] COI number [1:CAS:528:DC%2BD3MXlvVehur8%3D] 10.1116/1.1387089View ArticleGoogle Scholar
- Sun CQ: Prog. Solid State Chem.. 2007, 35: 1. 10.1016/j.progsolidstchem.2006.03.001View ArticleGoogle Scholar
- Roduner E: Chem. Soc. Rev.. 2006, 35: 583. COI number [1:CAS:528:DC%2BD28Xmt1Gmu7Y%3D] COI number [1:CAS:528:DC%2BD28Xmt1Gmu7Y%3D] 10.1039/b502142cView ArticleGoogle Scholar
- Richman EK, Hutchison JE: ACS Nano. 2009, 3: 2441. COI number [1:CAS:528:DC%2BD1MXhtFGlsrnK] COI number [1:CAS:528:DC%2BD1MXhtFGlsrnK] 10.1021/nn901112pView ArticleGoogle Scholar
- Campbell TA: Nano Today. 2009, 4: 380. COI number [1:CAS:528:DC%2BC3cXltFGmsLo%3D] COI number [1:CAS:528:DC%2BC3cXltFGmsLo%3D] 10.1016/j.nantod.2009.06.012View ArticleGoogle Scholar
- Holyst R, Litniewski M: Phys. Rev. Lett.. 2008, 100: 055701. Bibcode number [2008PhRvL.100e5701H] Bibcode number [2008PhRvL.100e5701H] 10.1103/PhysRevLett.100.055701View ArticleGoogle Scholar
- Qi WH: Physica B. 2005, 368: 46. COI number [1:CAS:528:DC%2BD2MXhtFSmtr%2FM]; Bibcode number [2005PhyB..368...46Q] COI number [1:CAS:528:DC%2BD2MXhtFSmtr%2FM]; Bibcode number [2005PhyB..368...46Q] 10.1016/j.physb.2005.06.035View ArticleGoogle Scholar
- Halperin WP: Rev. Mod. Phy.. 1986, 58: 533. COI number [1:CAS:528:DyaL28Xltl2ktLg%3D]; Bibcode number [1986RvMP...58..533H] COI number [1:CAS:528:DyaL28Xltl2ktLg%3D]; Bibcode number [1986RvMP...58..533H] 10.1103/RevModPhys.58.533View ArticleGoogle Scholar
- Wautelet M, Duvivier D: Eur. J. Phys.. 2007, 28: 953. COI number [1:CAS:528:DC%2BD2sXhtFyjsLzN] COI number [1:CAS:528:DC%2BD2sXhtFyjsLzN] 10.1088/0143-0807/28/5/018View ArticleGoogle Scholar
- Guisbiers G, Buchaillot L: Phys. Lett. A. 2009, 374: 305. COI number [1:CAS:528:DC%2BD1MXhsVWru7fF]; Bibcode number [2009PhLA..374..305G] COI number [1:CAS:528:DC%2BD1MXhsVWru7fF]; Bibcode number [2009PhLA..374..305G] 10.1016/j.physleta.2009.10.054View ArticleGoogle Scholar
- Yang CC, Li S: Phys. Rev. B. 2007, 75: 165413. Bibcode number [2007PhRvB..75p5413Y] Bibcode number [2007PhRvB..75p5413Y] 10.1103/PhysRevB.75.165413View ArticleGoogle Scholar
- Shandiz MA: J. Phys.: Condens. Matter. 2008, 20: 325237. 10.1088/0953-8984/20/32/325237Google Scholar
- Vanithakumari SC, Nanda KK: Phys. Lett. A. 2008, 372: 6930. COI number [1:CAS:528:DC%2BD1cXhtlGisr%2FK]; Bibcode number [2008PhLA..372.6930V] COI number [1:CAS:528:DC%2BD1cXhtlGisr%2FK]; Bibcode number [2008PhLA..372.6930V] 10.1016/j.physleta.2008.09.050View ArticleGoogle Scholar
- Yang CC, Jiang Q: Acta Mater.. 2005, 53: 3305. COI number [1:CAS:528:DC%2BD2MXksV2iu7k%3D] COI number [1:CAS:528:DC%2BD2MXksV2iu7k%3D] 10.1016/j.actamat.2005.03.039View ArticleGoogle Scholar
- Guisbiers G, Buchaillot L: Nanotechnology. 2008, 19: 435701. 10.1088/0957-4484/19/43/435701View ArticleGoogle Scholar
- Balluffi RW: J. Nucl. Mater.. 1978, 69: 240. Bibcode number [1978JNuM...69..240B] Bibcode number [1978JNuM...69..240B] 10.1016/0022-3115(78)90247-7View ArticleGoogle Scholar
- Bollmann W, Uvarov NF, Hairetdinov EF: Cryst. Res. Technol.. 1989, 24: 421. COI number [1:CAS:528:DyaL1MXltVKju7w%3D] COI number [1:CAS:528:DyaL1MXltVKju7w%3D] 10.1002/crat.2170240418View ArticleGoogle Scholar
- Schaefer H-E: Physica Status Solidi A. 1987, 102: 47. COI number [1:CAS:528:DyaL2sXmt1KisL4%3D]; Bibcode number [1987PSSAR.102...47S] COI number [1:CAS:528:DyaL2sXmt1KisL4%3D]; Bibcode number [1987PSSAR.102...47S] 10.1002/pssa.2211020104View ArticleGoogle Scholar
- Tiwari GP, Patil RV: Scripta Metallurgica. 1975, 9: 833. COI number [1:CAS:528:DyaE28XnvVSguw%3D%3D] COI number [1:CAS:528:DyaE28XnvVSguw%3D%3D] 10.1016/0036-9748(75)90564-5View ArticleGoogle Scholar
- Qi WH, Wang MP: J. Mater. Sci.. 2004, 39: 2529. COI number [1:CAS:528:DC%2BD2cXitFWqt7w%3D]; Bibcode number [2004JMatS..39.2529Q] COI number [1:CAS:528:DC%2BD2cXitFWqt7w%3D]; Bibcode number [2004JMatS..39.2529Q] 10.1023/B:JMSC.0000020020.60857.6aView ArticleGoogle Scholar
- Qi WH, Wang MP, Zhou M, Hu WY: J. Phys. D Appl. Phys.. 2005, 38: 1429. COI number [1:CAS:528:DC%2BD2MXksFKntro%3D]; Bibcode number [2005JPhD...38.1429Q] COI number [1:CAS:528:DC%2BD2MXksFKntro%3D]; Bibcode number [2005JPhD...38.1429Q] 10.1088/0022-3727/38/9/016View ArticleGoogle Scholar
- Jiang Q, Li JC, Chi BQ: Chem. Phys. Lett.. 2002, 366: 551. COI number [1:CAS:528:DC%2BD38Xps1ahsLk%3D]; Bibcode number [2002CPL...366..551J] COI number [1:CAS:528:DC%2BD38Xps1ahsLk%3D]; Bibcode number [2002CPL...366..551J] 10.1016/S0009-2614(02)01641-XView ArticleGoogle Scholar
- Nanda KK, Sahu SN, Behera SN: Phys. Rev. A. 2002, 66: 013208. Bibcode number [2002PhRvA..66a3208N] Bibcode number [2002PhRvA..66a3208N] 10.1103/PhysRevA.66.013208View ArticleGoogle Scholar
- Kim HK, Huh SH, Park JW, Jeong JW, Lee GH: Chem. Phys. Lett.. 2002, 354: 165. COI number [1:CAS:528:DC%2BD38Xhsl2jsLc%3D]; Bibcode number [2002CPL...354..165K] COI number [1:CAS:528:DC%2BD38Xhsl2jsLc%3D]; Bibcode number [2002CPL...354..165K] 10.1016/S0009-2614(02)00146-XView ArticleGoogle Scholar
- Qi WH, Wang MP, Zhou M, Shen XQ, Zhang XF: J. Phys. Chem. Solids. 2006, 67: 851. COI number [1:CAS:528:DC%2BD28Xis1ektL8%3D]; Bibcode number [2006JPCS...67..851Q] COI number [1:CAS:528:DC%2BD28Xis1ektL8%3D]; Bibcode number [2006JPCS...67..851Q] 10.1016/j.jpcs.2005.12.003View ArticleGoogle Scholar
- Horváth J, Birringer R, Gleiter H: Solid State Commun.. 1987, 62: 319. Bibcode number [1987SSCom..62..319H] Bibcode number [1987SSCom..62..319H] 10.1016/0038-1098(87)90989-6View ArticleGoogle Scholar
- Guisbiers G, Kazan M, Van Overschelde O, Wautelet M, Pereira S: J. Phys. Chem. C. 2008, 112: 4097. COI number [1:CAS:528:DC%2BD1cXit1KjsLo%3D] COI number [1:CAS:528:DC%2BD1cXit1KjsLo%3D] 10.1021/jp077371nView ArticleGoogle Scholar