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A Possible universal definition for the nanophase

A possible universal definition for a nanostructured material based purely on experimental data available in literature for a wide variety of physical phenomena is proposed. It is suggested that for values of the ratio of sample volume, V to that of the unit cell volume, Vc, 10^5- 10^6 the samples behave as nanostructures. This is mirrored in the ratio of the number of particles in sample, N to that in the unit cell, Nc. It is further proposed that the transition to the nanophase from bulk is a phase transition in all cases investigated. The nanophase should therefore be treated as a distinct phase of matter, entirely in the quantum mechanical domain, and treated appropriately.

Over the last decade or so the preparation of nano-structured materials, theoretical investigations thereof and applications of these have evoked wide interest. Intuitively designed nano-materials based on the currently available tools have been realised and their physico-chemical properties investigated [1]. Intuition demands that the volume of a nano-material should be small. However, the limitations on the size of sample to be defined as a nano-material are not apriori known; nor do universal theoretical calculations exist that quantify this phenomenon. Phenomena based model calculations are available in some cases, however, a universal quantified definition of the nanophase is still not forthcoming. In this paper, we address this from a phenomenological point of view and base our conclusions strictly on the large body of experimental evidence available in literature.
The first underlying assumption made is that the laws of quantum mechanics strictly govern the nanophase, and therefore the phenomena observed should be explainable only from quantum mechanics. This should be contrasted with quantum field theory effects such as superconductivity, ferromagnetism and superfluidity that are ground state phenomena obeyed in the bulk. In other words, we strictly restrict ourselves to the quantum mechanical domain without invoking the infinite volume limit. Since size is an important factor, the ratio of surface area to volume is large for small systems. i.e.
becomes smaller and smaller as V grows. The surface effects, therefore, dominate in the nanophase in contrast to the bulk state. So the question that emerges is how small should the system be to be defined as a nano-phased material. A similar question arises in crystallography. A phenomenological approach considered by us recently [2], came to 2 the conclusion that for V/V c < 10 6 the transition to the nanophase is made where V is the volume of crystallites in the solid and V c the unit cell volume. A similar analysis will be used here. In fact all second order transitions occur only when N and V both tend to infinity but N/V is finite. However, this is only a theoretical limit. The approach used here to define the limiting volume of the nanophase interestingly answers the other question, i.e. the volume tending to infinity limit. This seems to apply even to nonequillibrium phase transitions such as lasers.
The nanophase of a material need not be unique. Several possible structures allowed by quantum mechanics are possible. Furthermore, one can go to the nanophase from solid, liquid or gas. It is clear that a criterion for the transition can only emerge from experiments that are able to cause a gradual decrease from bulk-like sizes until a critical size is reached beyond which behaviour is completely governed by quantum mechanics.  Table I.

From these wide variety of examples it emerges that if the experimentally demonstrated
sample size is used as the limit then nano-phase is observed upto the upper bound of V/V c <10 6 . This volume ratio can be converted in to a ratio for the number of particles by 3 substituting the number of atoms per unit cell. In Bravais lattices, it is well-known that this number rarely exceeds four. Therefore it leads to the ratio of N/N c < 10 4 . Indeed this seems to be obeyed by Bose-Einstein condensates. The condensates occur only when N ≥ 10 4 . This is also true for lasers where population inversion occurs only when N>10 7 for a He-Ne laser. This number is slightly higher due to the dissipation effects in the cavity. It should be noted that no particular solutions of Euler equations or their like should be found in nanostructures. These correspond to the ground states of bulk systems and therefore vortices, dislocations, defects and Bloch walls etc. cannot occur in nanostructures. This is in accordance with experiments wherein nanostructures have been shown to exhibit greater hardness than their bulk forms.
The continuum limit, where the transition is from lattice quantisation to field theory, relies on the modes whose frequencies go to a continuous set of frequencies. In practice no system is infinite and therefore it is difficult to evolve a mathematical criterion that can set the limit of transition from finite to infinite volume based on a physical parameter.
With the emergence of nanomaterials, this question becomes more important since nonrelativistic quantum mechanics is operable in this regime while field theory and the grand canonical ensembles are operable only in the V and N tending to infinity limit. The N/N c ratio can be taken as the ratio that defines the transition to bulk behaviour such that for N/N c > 10 5 the solid behaves as bulk.
Some examples where the bulk to nanophase transition can be seen are presented in Table   I. For instance, one of the oldest known examples of such a transition is the size dependent ferromagnetic to superparamagnetic transition [3]. The superparamagnetic behaviour is completely quantum mechanical in nature while classical paramagnetism is 4 not. In Bose-Einstein condensation, only when the number of particles in the cloud is 10 5 is the superfluid behaviour observed [4]. Since all transitions go to a different phase in their physical characteristic described by a different phenomenon from that determining bulk behaviour, we propose that the nanobulk transition is a phase transition. Also, recent experiments suggest that the nanophase, as it should, is retained as temperature is raised while the bulk undergoes a phase 5 transition from solid to liquid, the nanophase does not do so [14]. Experiments performed on nano-water show that from 10 o K to 300 o K the large mean square displacement of hydrogen, the physical parameter used by them, does not undergo any abrupt changes, while in the bulk, ice transforms in to water and this physical parameter changes abruptly.
Recent experiments on carbon nanotubes show that the additional Raman lines, that are not present in the bulk phase, appear in the nanophase. To a first approximation it seems that the V/V c even in this case is 10 6 [15]. This further confirms our conclusion that nano is indeed a new phase, distinct from the solid, liquid and gas phases that appear in the bulk. If V/V c is greater than 10 5 -10 6 depending on the temperature it will go to one of the phases. The chemical potential, µ , is zero for the nanophase while it is non-zero for the bulk form. The Gibbs free will therefore abruptly change when V/V c exceeds the limit of 10 6 . This seems to be the limit above, which the system should be, considered a Grand Canonical Ensemble. The Gibbs free energy in general is a function of Temperature (T), Pressure (P) and average number of particles (N). The chemical potential, µ, per particle is non-zero for a grand canonical ensemble. For a nanostructure which is governed by quantum mechanics only and has a finite number of particles, µ is zero. Therefore, there is an abrupt change in the Gibbs function as a function of N when the limit N c is exceeded. These arguments originate from general considerations of thermodynamics.
Calculations, in particular cases, will only give the numerical values for that system. 6 The Gibbs free energy is In order to fit into Gibbs's classification the θ function can be replaced by the equivalent smooth function. Since there is an abrupt change in the Gibbs free energy, this is a new kind of phase transition that does not fit into Gibbs classification.
In short we propose that besides solid, liquid and gaseous states the nanophase should be considered as a distinct state of matter and treated appropriately. The signal for the onset of the nanophase is the phase transition.