The specific surface area of some structures (e.g. nanotubular structures) can be estimated by the calculation of the geometrical surface area (see Figure 3.4). The specific surface area of a nanotube (S) can be calculated as a sum of the external (Sext) and internal (Sint) surface areas divided by the mass of the nanotube using the formula:

P Vtube where p and Vtube are the density and the volume of the nanotube, respectively. For a cylindrical geometry, it is possible to express surface area and volume using the formulae:

Sext + Sint _ 2%rL + 2p(r + h)L _ 2p(2r + h)L (3.2)

Where L and r are the the length and internal radius of the nanotube, respectively, and h is the thickness of the nanotube wall. Combining Equation

Figure 3.4 Diagram showing the geometrical characteristics of a nanotube for the calculation of specific surface area and pore volume: r is the radius of the nanotube, h is the wall thickness, L is the length of the nanotube, p is the density ofthe nanotubular material, V is the volume, and Sext and Sint are the external and internal surface areas, respectively.

Figure 3.4 Diagram showing the geometrical characteristics of a nanotube for the calculation of specific surface area and pore volume: r is the radius of the nanotube, h is the wall thickness, L is the length of the nanotube, p is the density ofthe nanotubular material, V is the volume, and Sext and Sint are the external and internal surface areas, respectively.

(3.1) with Equations (3.2) and (3.3), the specific surface area of the nanotube surface can be estimated as:

This value for the specific surface area of nanotubes does not depend on the internal or external diameter of nanotubes, but only on the wall thickness and density of the nanotubular materials. In practice, however, the nanotubes do not have an ideal cylindrical geometry and Equation (3.4) can only be used for guidance purposes.

The experimental surface area of materials can be determined using various adsorption methods, in which an adsorbate (probe species) with well characterised sorption properties is adsorbed on the surface of the material to be studied. By varying the adsorbate concentration (pressure) and temperature, and measuring the amount of probe material adsorbed, it is possible to determine the specific surface area, pore size distribution, heat or activation energy of adsorption, as well as the distribution of energy centres. One of the standard quantitative methods for the characterisation of a specific surface area is the BET (Brunauer-Emmett-Teller ) surface area, obtained from the isotherm of nitrogen adsorption on the surface of porous materials at -195 °C.

A typical isotherm for nitrogen adsorption on the surface of titanate nanotube is shown in Figure 3.5. According to IUPAC recommendations,35 the

type of hysteresis loop for the N2 isotherms is intermediate between H1 (at 0.5 < p/p0<0.8) and H3 (at p/p0 > 0.8). The H1 type is characteristic of uniform pores inside aggregates of particles. The observed hysteresis extended to p/p0 e 1, indicates the presence of large pores which are not filled. Taking into account the morphology of the material observed by microscopy, the smaller pores may correspond to the pores inside the nanotubes, with the diameter of these pores being equal to the internal diameter of these nanotubes. The larger pores may correspond to the pores between the nanotubes. The hysteresis loop is relatively broad, indicating a wide distribution of pore sizes. This precludes speculation concerning the shape of the pores in the range of high relative pressures.

The BET surface area of titanate nanotubes determined from the nitrogen adsorption isotherm, typically varies between 200 and 300m2g_1, depending on the preparation method and the effectiveness of washing for the removal of sodium ions.36 A similar range of specific surface areas can be obtained using

Equation (3.4), with a nanotube wall thickness, h, of 2 to 3 nm and a density, r (determined by a helium pycnometer), of 3.12 gcm~3. The density of titanate nanotubes can also be estimated from the parameters of a unit cell of H2Ti3O7 (a = 1.602 nm, b = 0.375 nm, c = 0.919 nm and ß = 101.5°, see Table 3.1), and taking into account that four "molecules" of trititanate occupy one unit cell. In this case, the value for density can be estimated as 3.16 g cm~3, which is close to that measured by helium adsorption.

For TiO2 nanotube arrays obtained by the anodising method, it is difficult to obtain a BET surface area experimentally using nitrogen adsorption, since the impurities of metallic titanium distort the data. The value for the specific surface area, however, can be estimated from geometric considerations using Equation (3.4). For nanotubes having a typical wall thickness of 15 nm and the density of anatase being approximately 3.9 g cm 3, the value for the specific surface area can be estimated at ca. 35m2g_1, which is approximately one order of magnitude smaller than the surface area of titanate nanotubes. The experimental value for the anodic TiO2 surface area of nanotubes may be higher due to the porous structure of the walls.37

The specific surface area of titanate nanofibres produced by the alkaline hydrothermal treatment of TiO2 at elevated temperatures, is usually smaller than that of titanate nanotubes and is ca. 20m2g-1.36

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