Figure 3.6a shows the differential pore-volume distributions obtained from the desorption of nitrogen curve using the BJH algorithm, for two titanate nano-tube samples characterised by different average tube diameters. The curves have a bell-like shape, and the maxima corresponds to pore sizes of 4 to 20 nm, relating this material to pores with a mesoporous range. In general, methods of nitrogen adsorption and, in particular, the BJH pore size distribution technique are suitable for characterising titanate nanotubes and determining their pore size distribution. A comparison of the pore size distribution of titanate nano-tubes with two differing average diameters, obtained from nitrogen adsorption data with distribution built up from histograms using TEM images, is shown in Figure 3.6. In order to compare the distribution in the diameter of nanotubes with that of pore volume, the number of particles, N, was multiplied by the square of diameter, d2, since the volume of nanotubes is proportional to d2. It is apparent that the two distributions are slightly different, as seen in Figures 3.6a and 3.6b. The distribution obtained from the adsorption data yields larger nanotube diameter. However, the relative positions of the distributions for samples 1 and 2 are identical for both methods. Thus, it is possible to use BJH pore volume distribution for a semi-quantitative characterisation of the morphology of nanotubular titanates.

The disagreement between pore size distributions obtained from microscopy and adsorption data is attributable to the small number (60) of nanotubes included in the analysis of electron microscopy images. Only the nanotubes

Comparison of a) pore volume distribution obtained from nitrogen desorption (data points) and b) nanotube internal diameter distribution obtained from TEM (histograms) for two samples of TiO2 nanotubes produced by the hydrothermal treatment of (1) 0.25 g and (2) 3g of ana-tase in 300 cm3 of NaOH (10 mol dm"3) at 140 °C. (Data are reproduced with kind permission from ref. 36).

Comparison of a) pore volume distribution obtained from nitrogen desorption (data points) and b) nanotube internal diameter distribution obtained from TEM (histograms) for two samples of TiO2 nanotubes produced by the hydrothermal treatment of (1) 0.25 g and (2) 3g of ana-tase in 300 cm3 of NaOH (10 mol dm"3) at 140 °C. (Data are reproduced with kind permission from ref. 36).

which featured on a single image were considered, whereas the adsorption technique measures the entire sample. In this particular case, the larger nanotubes (with pore diameters of up to 20 nm) that were present in some samples did not feature in the histogram calculations, thus the mean tube diameter was underestimated. Furthermore, the BJH pore volume distribution includes not only the internal pores of nanotubes, but the larger pores formed between nanotubes, leading to an overestimation of mean tube volume. The BJH method is based on the principle that the surface tension of liquid nitrogen is constant and does not depend on the radius of meniscus (the Kelvin equation).

This may not, however, be true for pores of very small diameter (less than 4nm). Hence, novel methods for determining pore size distribution should be applied, such as: a correction to the Kelvin equation,38 a t-function39 or DFT calculations.40

Pore volume distribution determined by the nitrogen adsorption method includes both internal (inside individual nanotubes) and external (cavities formed between nanotubes) pores. In order to distinguish each component, an ultrasonic treatment of samples of titanate nanotubes was used to break up the agglomerates into individual particles. From the SEM data shown in Figures 3.7c and 3.7d, it was found that two hours of ultrasonic treatment of an aqueous suspension of titanate nanotubes resulted in the destruction of the secondary

Figure 3.7 Pore-volume distribution (BJH desorption) of a) titanate nanotubes and b) its deconvolution into two Lorentzian curves. (O) initial sample, BET surface area of 199m2g_1, BJH desorption pore volume of 0.70 = 0.35 (peak I) + 0.35(peak II) cm3 g"1, (■) sample ultrasonically treated for 2 h, BET surface area of 198m2g_1, BJH desorption pore volume of 0.55 = 0.35 (peak I) + 0.20 (peak II) cm3 g"1. SEM images are shown for c) initial nanotubes and d) ultrasonically treated nanotubes. (Data are reproduced with kind permission from ref. 36).

Figure 3.7 Pore-volume distribution (BJH desorption) of a) titanate nanotubes and b) its deconvolution into two Lorentzian curves. (O) initial sample, BET surface area of 199m2g_1, BJH desorption pore volume of 0.70 = 0.35 (peak I) + 0.35(peak II) cm3 g"1, (■) sample ultrasonically treated for 2 h, BET surface area of 198m2g_1, BJH desorption pore volume of 0.55 = 0.35 (peak I) + 0.20 (peak II) cm3 g"1. SEM images are shown for c) initial nanotubes and d) ultrasonically treated nanotubes. (Data are reproduced with kind permission from ref. 36).

structure of the material and a decrease in the average nanotube length. As a result, small nanotubes tended to assemble into close packing with very narrow pores between the tubes. A comparison of the pore volume distribution of both samples (before and after ultrasonic treatment) is shown in Figure 3.7. It is clear that the pore volume distribution has at least two components, and can be presented as a sum of two Lorentzian curves. The first component exhibits a maximum pore volume at a pore diameter of 8.5 nm and an integral of this component of 0.35cm3g—1 It is proposed that this component corresponds to the internal pore volume distribution of the nanotubes, since it is not affected by the ultrasonic treatment.

The second component exhibits a maximum pore volume at a pore diameter of 20 nm and an integral of 0.35 cm3 g-1 for the initial sample, and 16 nm and 0.20cm3g—1 for the ultrasonically treated sample. It is likely that the second component corresponds to the pore volume distribution of pores formed between the nanotubes. Following the ultrasonic treatment, the agglomerates of titanate nanotubes were destroyed and some tubes were broken. This resulted in a separation of the individual nanotubes and the formation of a much more compact structure, with a smaller average distance between nanotubes compared with that in the initial sample. Consequently, the average pore diameter and total pore volume of external pores is also smaller. The ultrasonic treatment does not significantly change the BET surface area of the sample, since the increase in surface area due to an increasing of number of nanotube ends is negligible. In this fashion, it is possible to distinguish the contributions of internal pores and external pores to the total pore volume distribution function.

The specific volume of pores inside cylindrical nanotubes can be estimated, assuming a cylindrical nanotube geometry, using the following equation:

1 r2

Taking the values of both r = 3.5 nm and h = 2nm from TEM images, and a density of p = 3.12 gcm-3, the volume of the internal pores of titanate nanotubes can be estimated as 0.22 cm3 g—1, which is less than the value 0.35 cm3 g—1 obtained from Figure 3.7. This implies that the sample consists largely of nanotubes and the conversion of precursor anatase to titanate nanotubes during the alkali hydrothermal treatment is high. This conclusion is in agreement with the fact that no other phase was found on the TEM and SEM images.

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