M. Wautelet

Scaling laws in the macro-, micro- and nanoworlds

The microworld and the nanoworld are now of interest to the public, scientists, engineers and industrialists. Effects negligible at the macroscopic level become important at the micrometre scale, and vice versa. Moreover, at the nanoscale, quantum effects may become dominant. Scaling laws are useful for an understanding of the origin of such differences and/or to generalize results obtained at various scales. It is the aim of this paper to examine scaling laws relevant to mechanics, fluids, electromagnetism, thermodynamics, optics and quantum mechanics. Examples related to biology, micromachines and nanotechnologies are treated to show the usefulness of the scaling laws. In recent years, interest in the domains of the microworld and the nanoworld has increased rapidly. The advent of so-called mesoscopic physics, the development of micromachines or micro-electrical mechanical systems (MEMS), the synthesis of nanotubes, the development of nanotechnology, etc… are among the most cited subjects. From both the fundamental and applied points of view, there are interesting questions about these micro- and nanoworlds. In particular, effects negligible at the macroscopic level are important at the micrometre scale, and vice versa. Moreover, when one goes from the macroworld to the nanoworld, one passes through two diffuse limits. When the characteristic dimensions of the elements decrease from the macroscopic to the micrometre size, the effects of gravity become negligible as compared with adhesive and friction effects. Surface tension dominates gravity; other examples are given later. This implies that our reasoning, based on our experience at the macroscopic level, is no longer valid. We have to modify our rules of thinking. The boundary between the macroscopic and microscopic levels is not sharp; it depends on the effect to be considered. When the characteristic size decreases further to attain the nanometre range, another limit is encountered. While the macroscopic properties of matter remain generally valid at the micrometre size, surface effects become dominant at the nanometre scale. Moreover, when one reaches the interatomic distance range, quantum effects appear. 0143-0807/01/060601+11

Size and segregation effects on the phase diagrams of nanoparticles of binary systems

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Phase diagrams of small particles of binary systems: a theoretical approach

The phase diagrams of small particles (with diameters in the nanometre range) are studied theoretically. In the limit where thermodynamical arguments remain valid, it is deduced that the phase diagram of small particles is a function of their size. This is discussed for the cases of eutectics and regular solutions. The effects of surface segregation are also treated and lead to further modifications of the phase diagrams.

XPS study of TiOx thin films prepared by d.c. magnetron sputtering in Ar–O2 gas mixtures

The phase diagrams of small particles (with diameters in the nm range) are studied theoretically. In the limit where thermodynamical arguments remain valid, it is deduced that the phase diagram of small particles is a function of their size. For the case of ideal solutions, it is shown that the lens-shaped solidus-liquidus curves are shifted to lower temperatures when the dimensions of the particle decrease. Additionally, at fixed temperatures between the highest bulk melting point and the lowest melting point of the particle, the relative concentrations of the solid and liquid phases are different in the particle and bulk material.

Theoretical investigation of size and shape effects on the melting temperature and energy bandgap of TiO2 nanostructures

Titanium oxide films have been deposited, at a pressure of 5 mTorr and a discharge current of 500 mA, on borosilicate glass substrates. We have studied in situ by XPS the dependence of film composition on oxygen partial pressure. All the experiments have been performed under high vacuum; there is no air contamination of film surfaces before XPS analysis. The ions produced in the plasma have been analysed by glow discharge mass spectrometry (GDMS). The stoichiometry of the films is compared to the plasma composition. The XPS results show that for an increasing oxygen partial pressure four regimes are observed. At very low oxygen concentration ( 15% (zone IV) a pure TiO 2 film is obtained. When zone IV starts, the Ti + mass spectrometric signal is still higher than the TiO + signal, showing that a pure TiO 2 phase is occurring at the substrate while the sputtering mode is still partially metallic.

Analysis of DC magnetron discharges in Ar- gas mixtures. Comparison of a collisional-radiative model with optical emission spectroscopy

In this letter, we report a theoretical investigation concerning the size effect on the melting temperature and energy bandgap of TiO2 nanostructures. Within the thermodynamical approach, we predict a structural phase transition from rutile to anatase for the sizes around 40, 29, and 48nm, respectively, in the cases of spherical nanoparticles, cylindrical nanowires, and nanotubes. For spherical nanoparticles, this means that the more stable phase is anatase for sizes smaller than ∼40nm and rutile for sizes larger than ∼40nm. The energy bandgap of these structures is also estimated.

Density and temperature in an inductively amplified magnetron discharge for titanium deposition

DC magnetron discharges in argon-nitrogen gas mixtures have been characterized by optical emission spectroscopy (OES). Optical lines from the cathode species (aluminium) and the gas mixture (argon, nitrogen) have been measured at constant total pressure, as a function of the gas mixture composition and electrical power. The experimental data are compared with results of a theoretical plasma model which solves self-consistently the Boltzmann equation for electrons and the kinetic equations for aluminium, argon, molecular and atomic nitrogen states. The emission intensity variations of plasma species have been analysed versus the nitrogen relative concentration and electrical power and compared with calculated populations of emitting species with a satisfactory agreement. The variation of aluminium density versus the nitrogen concentration has been deduced by using the line intensity and excitation rates given by the model.

Glow discharge mass spectrometry study of the deposition of TiO2 thin films by direct current reactive magnetron sputtering of a Ti target

In order to determine the titanium neutral density, a direct current (dc) plasma discharge, amplified by a radio-frequency (rf) coil, was studied by absorption spectrometry. The argon pressure varied from 5 to 40 mTorr. The dc and rf powers varied between 100 and 1500 W and 0 and 500 W, respectively. The plasma gas temperature necessary for the density calculation was evaluated by analyzing the N2 rotational spectrum in an Ar–N2 gas mixture. When increasing the rf power a decrease of titanium neutral density was found. This decrease is related to the increased titanium ion density. When using the rf coil, the titanium degree of ionization can be up to 90%.

Optical diagnostics of d.c. and r.f. argon magnetron discharges

The properties of titanium oxide thin films deposited by direct current magnetron sputtering of a Ti target are strongly dependent on the sputtering conditions. The aim of the present work is to investigate the discharge parameters such as plasma potential, discharge voltage, deposition rate, and ion composition of the discharge as a function of the oxygen partial pressure. The plasma potential, relative to the ground, is determined from the ion energy distribution. Working in the constant current discharge mode, we observe, with increasing oxygen partial pressure, a drop of the plasma potential, an increase of the discharge voltage, a drop of the deposition rate, and an inversion of the Ti+–TiO+ intensities. For a given discharge current and pressure, the drop of the plasma potential and the increase of the discharge voltage occur at the same gas composition while the drop of the deposition rate and the ion intensity inversion happen at an oxygen richer gas composition. Both transitions are linearly corr...

Size effects on the phase diagrams of nanoparticles of various shapes

Abstract The plasmas of d.c. and r.f. magnetron discharges have been analysed by emission spectroscopy for argon and metal atom radiative states and by optical absorption for the Ar(3P2, 3P0) metastable and Ar(3P1, 1P1) resonant atoms. In the 6–200 mTorr pressure range for W = 1–50 W in r.f. and I = 2–50 mA in d.c. the metastable and resonant atoms are three to five times more populous in r.f. than in d.c. The densities are saturated for I > 20 mA in d.c. and W > 5 W in r.f., reaching a maximum value n Ar ( 3 P 2 ) ≈ 5 × 10 10 cm −3 . The emission line intensities I(λ) vary as I(λ) = Wβ in r.f., with β ≈ 0.5 for neutral Ar lines and β ≈ 1 for Ar ion and metal (Al, Mg, Mn) atom lines. In d.c. discharges I(λ) = Iβ, with β ≈ 2 for Ar ion and metal atom lines. These results suggest an electron density variation as ne ∞ W0.5 in r.f. and an excitation mechanism proportional to ne2 for Ar ions and metal atoms in both d.c. and r.f. discharges.