Universal size/shape-dependent law for characteristic temperatures, phase transformation in nanoparticles,

The Nov. 09 issue of Materials World page 5, has captured my attention for some time. [1]

It's main feature [ref 1]was on multiscale materials modelling (MMM) upon which I finally managed to post recently (9 Feb 2010)[ref 2].

Against the above computationally intensive approach (bottom-up) and on the same page (paper edition only) was a small news snippet relating the claim by physicists Gregory Guisbiers and Lionel Buchaillot, based at the Institute of Electronics, Microelectronics and Nanotechnology (IEMN) Univ of Lille, Villeneuve d’Ascq, to a discovery of a Universal Law for characteristic temperature of nanosized-structures.

The equation, I learned from MW [1] was found by analysing and comparing how the size of nanoparticles affects the temperature at which they melt, become ferromagnetic or become superconductors. It is based upon the surface-to-volume ratio of the nanostructure and the spin of the particles which constitute a material. Surface to volume effect over multiple scales and from melting to superconduction sounded as though I ought to brush up my knowledge.


The description of different effects observed in nature by only one general equation is the “Holy Grail” for all physicists say the authors [best known examples are perhaps Maxwells Unifying Eqns of the Laws of electricity, magnetism and fluid dynamics and Einstein’s famous E=mc^2] Dr Guisbiers considers that this goal has been achieved for characteristic temperatures through a top-down approach that they present as opposed to the bottom-up approach. [3]

To investigate nanomaterials properties, two approaches are available: bottom-up and top-down. The first approach (bottom-up) makes use of computational methods (computer intensive_ fat) like molecular dynamics as in Multiscale Materials Modelling-MMM (earlier post) whereas the second (top-down) relies on classical thermodynamics, (computer lean).

Molecular dynamics in MMM often consider fewer than 100 000 atoms, in order to keep calculation time within reasonable values. This factor limits the nanostructure size modelled to a maximum size of tens of nanometres. Therefore, the top-down approach (computer lean) where one can consider bigger particles emerges as a simple complementary method that may provide useful insights in nanotechnology.[4]

Guisbiers and Buchaillot’s general equation is based only on the surface area to volume ratio of nanostructures and statistics (Fermi–Dirac or Bose–Einstein) followed by the particles involved in the considered phenomena (melting, ferromagnetism, vibration and superconduction). From the distinction between fermions and bosons, this equation indicates the universal behaviour of size and shape effects. Theoretical predictions show satisfactory agreement with experimental data taken from literature.[3]

Not having access, as yet, to their paper, I turned to two other summaries 1) by Nanowerk’s Michael Berger [5] and 2) by IOP, Physics World and Nanotechweb’s Belle Dumé. [6] as well
back-up earlier work on dimensional analysis, nano-thermodynamics and use in estimating binary phase diagrams for nanoparticles and calculation approximate melting point depression tendency at very small size (several nanometres) due to Wautelet et al. from the Belgium School at the Univ of Mons-Hainaut, in Mons, Belgium with whom Dr. Guisbiers worked in close collaboration. [7,8 ] (some background information is given below as well as further references to start the readers own enquiries)

Guisbiers and Buchaillot’s general equation is based only on the surface area to volume ratio of nanostructures and statistics (Fermi–Dirac or Bose–Einstein) followed by the particles involved in the considered phenomena (melting, ferromagnetism, vibration and superconduction). From the distinction between fermions and bosons, this equation indicates the universal behaviour of size and shape effects. Theoretical predictions show satisfactory agreement with experimental data taken from literature. [5,6]



The equation (TX/TX,∞ = [1–αshape/D](1/2S)) is based on the diameter of the nanostructure (D); a parameter (αshape) that is related to the surface-to-volume ratio; and the spin (S) of the particles involved in the considered material property. S equals 1/2 or 1 depending on whether the particles are fermions (particles with half-integer spin) or bosons (particles with integer spin). Tx stands for melting, Debye, Curie or superconducting temperature and Tx,∞ is that temperature in a macroscopic sample of the material. The equation has no adjustable parameters and works for all materials, says Guisbiers. In general, the characteristic temperatures decrease as particles become smaller. [6 ]

• The melting temperature is the highest temperature where the solid phase exists under thermodynamical equilibrium.
• The superconductive temperature is the highest temperature for which the material looses all resistance to the flow of electricity and tend to expel any magnetic fields inside it.
• The Curie temperature is the highest temperature for which the ferromagnetic phase is stable.
• The Debye temperature is the temperature corresponding to the maximal energy which can excite lattice vibrations. [ Nanowerk ref5]

Melting and ferromagnetism (which obey Fermi–Dirac statistics laws) are different from superconductivity and lattice vibrations (which follow Bose–Einstein statistics). This difference in behaviour is intimately related to the spin of the particles involved: melting is a solid-liquid phase transition and it occurs when inter-atomic bonds thermally break and a broken bond results in unpaired electrons, each characterized by a half-integer spin. Ferromagnetism, in turn, is a net magnetic moment that appears in the absence of an external magnetic field, and occurs thanks to partially filled shells of electrons. It is again characterized by a half-integer spin. [6 ]

Spin

Lattice vibrations are described by phonons, which, on the other hand, have integer spin. Superconductivity is a state in which conduction electrons are ordered into pairs of electrons, called Cooper pairs, also characterized by integer spin.

Predictions obtained from the equation agree very well with experimental data on melting and superconducting behaviour of nanoparticles – of silicon or lead, for example. The theory agrees fairly well with experimental results for ferromagnetism and lattice vibrations – the discrepancy is less than 10%. "This is an acceptable value because the model is quite simple and only requires knowledge of the size, shape and spin situation of the particles involved," adds Guisbiers” [ 6]


"The work shows that 19th century physics can still provide useful insights into 21st century nanotechnology, and all this can be done with just a pencil and paper – no supercomputers involved!" [6 ]



Back-ground material: [7,8,9, 10]
Understanding how materials behave at tiny length scales is crucial for developing future nanotechnologies and continues to be a great challenge for both theoretical and experimental physicists alike. Wautelet likes to reminds us that particles in the range 1-100 nm occupy an intermediary state between solid and molecular states. These nanoparticles range from particles made-up from a few atoms to so called ‘clusters up to the 100 nm range.(ref.x) from When the number of atoms in the particle lies in the range of 1000 or more, (factor of 10 or more) properties evolve from molecular to bulk solid in nature. Such particles are characterised by the fact that the ratio of the number of surface to volume atoms is not small. Eg. For a particle containing about 4000 atoms of radius R = 2nm approx., the ratio S/V =0.3 approx. ie. One third of the atoms are surface atoms. Therefore it is to be expected that surface effects will greatly influence cohesive properties of particles and must not be neglected. Theoretical work, on size dependant melting point depression goes as far back as 1909 to Pawlow (1909)[ref x]. Wautelet [ ] or in more detail P. Cheyssac [10 ] point out that for inorganic particles, it has been show experimentally in several studies notably by Buffet and Borel Phys Rev A 13 2287., 1976. that the melting point temperature (Tmpt.) decreases with decreasing particle radius (Tmpt decreases linearly with R^-1 (1/R).

Wautelet et al[ 7] outline the classical thermodynamics approach and extensions required to describe nanosystems with example published on binary systems such as (Se-Ge) first assuming particles are spherical (close packed) and non-spherical, [Wautelet et al. 8 ] They consider that the thermodynamical treatment remains valid for nanoparticles > 3nm. Further Wautelet et al [8 ] show that these size dependent effects are always larger for non-spherical shapes than for the spherical hypothesis, owing to the fact that the determining factor is the ratio of numbers of surface to volume atoms.

Note on high temperature superconductors for future reference:

The universal scaling law in magnetic phase diagram of high temperature superconductors (HTSC) K. Kitazawa, J. Shimoyama, H. Ikuta, T. Sasagawa and K. Kishio, Physica C: Superconductivity Volumes 282-287, Part 1, August 1997, Pages 335-338 online March 1999.
NB. Phase changes, Characteristic temperatures, T/Tc...

Abstract
An empirical scaling law γ2B = F(T/Tc) is proposed to successfully describe the three phase boundary lines; flux lattice melting , decoupling, and irreversibility lines in the magnetic phase diagram of the high temperature superconductors. The characteristic field of the three boundary lines changes by orders of magnitude, depending upon the material system; YBCO, LSCO and Bi2212. However, it was found that all the three lines could be scaled by a single material parameter, the electromagnetic anisotropy factor γ2 of each material. It is proposed that the three phase boundaries can be predicted for any HTSC system provided that its anisotropy factor is given.

Superconductors on Wikipedia

High-temperature superconductivity

Explaining high-Tc superconductors, M. Rice,Institute for Theoretical Physics, ETH Zürich in Physics World LINK Sign in to Physics World for free access to aricles and news letters.

References.

1. Model behaviour of ceramic and intermetallic alloys
(NB*** also IOM3 runs a Member-get-member scheme.
If you like my blog and references to the IOM3 and its resources, let me introduce you to the scheme. Visit the link at no cost to readers.)

2. Multiscale modelling of materials,MMM - Introduction and Explanatory Notes; Refs.,Images, on a Hot Interdisciplinary field

3. Universal size/shape-dependent law for characteristic temperatures, Abstract,
G. Guisbiers and L. Buchaillot,Physics Letters A Volume 374, Issue 2, 28 December 2009, Pages 305-308. doi:10.1016/j.physleta.2009.10.054

4. “Investigation on the nanomaterials properties”, Gregory Guisbiers,Seminar of The Microsystems Chair of the Louvain School of Engineering, 27 Nov. 2008. [LINK]

5. A universal law for characteristic temperatures at the nanoscale in Nanowerk 3 Nov 09

6. 'Universal' equation describes how materials behave at nanoscale,Physics World 5Nov09

7. Wautelet et al., Phase diagrams of small particles of binary systems: a theoretical approach.
Nanotechnology 11 (2000) 6-9. [LINK]

8. Wautelet et al, On the phase diagram of non-spherical nanoparticles, IOP pub, J. Phys.:Condens Matter 15 (2003) 3651-3655. [LINK].

9. Wautelet,M. Phase stability of electronically excited Si nanoparticles 2004 J. Phys.: Condens. Matter 16 L163-L166, download pdf free online from IOP Institute of Physics.

10 Phase transformation of metallic nanoparticles, P. Cheyssac, CNRS, Cargese Workshop Mai 2003 [LINK]