## Thermodynamic Equilibrium between the Nanotube and its Environment

The interaction of nanostructured materials with their surrounding environment occurs at their interface. In comparison with bulk materials, nanos-tructures have a reduced size and an increased surface area, which can lead to such interactions having a significant impact. This results in an alteration in the chemical and physico-chemical properties of nanostructures, and these properties may become size dependent.

The dissolution of nanostructured TiO2 and titanate in aqueous solutions can be described using three possible chemical reactions:

RSC Nanoscience & Nanotechnology No. 12

Titanate and Titania Nanotubes: Synthesis, Properties and Applications By Dmitry V. Bavykin and Frank C. Walsh © Dmitry V. Bavykin and Frank C. Walsh 2010 Published by the Royal Society of Chemistry, www.rsc.org

86 Chapter 4 TiO2(s) + 2OH~ <--> TiO^- + H2O (4.3)

The exact form of the Ti41 and TiO32 ions may be more sophisticated, including the formation of soluble hydroxides or polytitanic anions, however, for demonstration purposes their simplified form is used here. Depending on the pH of the solution, one of these reactions will dominate dissolution.

At thermodynamic equilibrium, the chemical potentials of each species, mi obey the following equation:

where ni values are the stochiometric coefficients in reactions (4.1), (4.2) and (4.3), which are positive for reactants and negative for products. The chemical potential of dissolved components can be expressed as:

where R is the Boltzmann constant, T is the temperature and Xi is the molar fraction of component i in solution, and mio is the chemical potential under standard conditions.

The chemical potential of solid TiO2 may be calculated by recalling the definition of chemical potential as a partial derivative of thermodynamic potential GTiOi over the amount of TiO2, i.e. nTiOf Since a change in the amount (number of moles) in solid TiO2 results in a variation in its surface area, A, the chemical potential will become equal to:

where mbTiO is the chemical potential of TiO2 without an interfacial boundary (infinite crystal), a is the excess surface energy in the TiO2-solution interface, Vm is the partial molar volume of TiO2, A and V are the surface area and volume of the TiO2 nanostructure, respectively. Let us consider the change in the surface area, A, and volume, V, during the dissolution-crystallization of nanotubes, providing that this process does not occur on the ends of nanotubes, but rather on the nanotube wall surfaces. Equation (4.6) is only valid if the surface energy per unit area, s, and the molar volume, Vm, are constant even when the nanostructures are small.

In nanotubes there are two types of surface, namely the internal concave surface, Aint, and the external convex surface, Aext. Deposition of materials on

both sides of the nanotubes, results in an incremental change in the internal radius, rint, and the external radius, rext, of the nanotubes (see Figure 4.1).

Providing this change is differentially small, it is possible to express the last part of Equation (4.6) as:

dV dVext dVint 2p Ltuber ext dr 2n L^nt dr where Ltube is the nanotube length. Inserting Equation (4.7) into (4.6) provides a formula for the chemical potential of solid nanotubes:

rext rint

Substitution of Equation (4.8) and (4.5) into Equation (4.4), followed by a rearrangement leads to:

rext rint i i i

Taking the exponential of Equation (4.9) and performing a rearrangement, results in a formula for the solubility product of TiO2 nanotubes:

where Ksp is the solubility product of bulk TiO2 with a flat surface (i.e., zero-curvature). The solubility of titanate nanotubes is size dependent.

The equilibrium concentration of titanium(iv) near an external (convex) surface is higher than that near an internal (concave) surface in accordance with Equation (4.10), resulting in a thermodynamic metastability of nanotubular morphology. Indeed, in the absence of activation barriers to dissolution-crystallisation, there would be a continuous re-crystallisation of solid materials from external convex to internal concave surfaces due to a difference in solubility, resulting in the collapse of internal hollow cavities leading to the formation of nanorods. The mechanism of this process is similar to that of the process of Ostwald ripening.1 It is possible that a layered structure of titanate nanotubes can provide a higher activation barrier towards dissolution from and crystallisation onto the walls of the nanotubes, resulting in a stabilisation of the metastable nanotubular state.

Topologically, nanotubes are characterised by a larger rext and a smaller rint, resulting in the overall nanotube solubility product, Ksp, as calculated from Equation (4.10), always being smaller than that of bulk Kbp.

In general, any first-order phase transformation process with a two-phase mixture composed of a dispersed second phase in a matrix, is characterised by an excess of free energy associated with the interface, as seen in Equation (4.6). As a result, many properties of the material become dependent on the size of the nanoparticles. Since underlying phenomena resulting in such size-dependence are common, the resulting equations have similar forms. For example, the vapour pressure, p, above the liquid, depends on the two principal radii of interface curvature, r1 and r2, according to the Kelvin equation:2

where p0 is the saturation vapour pressure at temperature T, ag-l is the interfacial tension at the gas-liquid interface, and Vm is the molar volume of the liquid. There is a similarity between Equations (4.10) and (4.11).

A decrease in the size of solid particles can also result in a melting-point depression phenomenon, when the melting temperature, Tm, of nanosized crystals becomes size dependent. Such a dependence can be described using the Gibbs-Thomson equation:3

where Tn is the melting temperature of an infinite bulk crystal with a flat surface, Hm is the bulk latent heat of fusion (melting), ps is the density of solid particles, a is the excess free surface energy per unit area, and d is the diameter of the spheroidal particles. Equation (4.12) demonstrates a similar dependence on 1/d as Equations (4.11) and (4.10), suggesting a common underlying thermodynamic effect of the interface on the processes of melting, evaporation and dissolution of the dispersed phases.

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