SWCNHs were produced with a patented method [13
], able to selectively produce different morphologies of SWCNHs (dahlia-like, bud-like, and seed-like). Some dispersant is necessary to avoid aggregation of nanoparticles in water, and sodium n
-dodecyl sulfate (99%, Alfa Aesar, Ward Hill, MA, USA) was demonstrated to be the best dispersant for this kind of carbon nanostructure [14
]. For the present work, we used two SWCNH suspensions, with dahlia- and bud-like nanohorn morphologies, labeled in the following as D and B, respectively. Both suspensions had the same nanoparticle concentration (0.3 g/l) and the same surfactant concentration (1.8 g/l).
The spectral scattering albedo has been obtained from spectrophotometric measurements carried out by means of a double-beam spectrophotometer (Lambda 900, PerkinElmer, Waltham, MA, USA) equipped with an integrating sphere for the measurement of transmittance (
= 150 mm, radius of the input aperture: R
= 9.5 mm). A specially designed sample cell was manufactured. The cell dimensions (surface, 95 × 40 mm2
; thickness, L
= 5 mm) were chosen in such a way to provide enough internal volume to allow several additions of SWCNH suspensions to pure water and to provide low noise curves with several data points for the chosen experimental method (see below).
The method we propose consists of measuring the transmittance for different concentrations of SWCNH (six progressive additions to pure water of a known amount of the original undiluted concentration). For each concentration, the measurement is repeated with the cell at two different distances from the integrating sphere: with the cell 'far' at a distance (dfar = 160 mm) and with the cell 'near' the integrating sphere, in contact with the aperture (dnear = 0 mm). The two measurements differ for the different fractions of scattered received power. The scattering albedo is obtained from these measurements, making the assumption that the SWCNH particles are sufficiently small with respect to the wavelength, so their scattering function can be approximated with the Rayleigh scattering function.
The expression for the scattering albedo has been obtained starting from the power PR received by the integrating sphere, which is given by:
where P0 is the ballistic component, and PS is the fraction of scattered power that enters the integrating sphere. With reference to Figure , P0 is related to the impinging power Pe by:
where T(θ = 0) is the transmittance of the cell windows (that takes into account the losses due to Fresnel reflections for normal incidence), and μe is the extinction coefficient. We remind that μe is the sum of the scattering (μs) and absorption (μa) coefficients, and the scattering albedo ω is defined as the ratio ω = μs/μe. The extinction coefficient, being proportional to the particle concentration, can be expressed as μe = εeρ, where ρ is the concentration of SWCNH particles (in grams per liter) and εe, their specific extinction coefficient (per millimeter per (gram per liter)).
Measurements have been carried out for moderate values of the optical thickness τe = μeL
(< 2.5). For these values of τe
and for the low values expected for the scattering albedo (ω
< 0.1), the scattered power is dominated by the contribution PS1
due to single scattering, so PS PS1
is given by [15
where p(θ) is the scattering function, l(z,θ) = (L-z)/cosθ, and
The angle α is the largest value of θ for which photons can be detected after a single scattering event. It is determined by total reflection at the water-glass-air interface, and its values are αnear = 48.7° for the near position and αfar = 2.55° for the far one.
If it is possible to assume that
) becomes independent on μe
and consequently on the concentration ρ
. This approximation means that the attenuation after a scattering event at point z
on the optical axis, due to the path in the cell that exceeds the remaining path (L-z)
, can be disregarded. This hypothesis will be discussed in more detail below. Under this approximation,
The power received for a ρ concentration of SWCNH can be written as:
(low scattering regime and albedo < 0.1), we have
We measured the sample transmittance at six different concentrations. From Equation 7, it is possible to obtain the measured intrinsic extinction coefficient εe meas from the slope of lnPR(ρ,α) as a function of ρ:
Finally, the albedo can be obtained as:
where we assumed that εe εe means
). To obtain ω
, it is therefore necessary to assume a model for the scattering function in order to evaluate (α)
. As mentioned before, for the SWCNH particles, we considered the Rayleigh scattering function
Equation 9 has been obtained, making some assumptions that need to be summarized and discussed. They, enumerated in order of importance, are (1) the Rayleigh scattering function for the SWCNH particles, (2) (α)
independent on the extinction coefficient (Equation 5), (3) PS PS1
, (4) negligible effect of internal reflections, (5)
, and (6) εe εe means
As for hypothesis (1), the Rayleigh scattering is probably not strictly applicable to SWCNHs especially at short wavelengths. Anyway, it should be noticed that the Rayleigh scattering function is nearly isotropic, while different scattering functions become more and more forward-peaked as the size of particles increases. Therefore, the value of (α)
we obtained in hypothesis (1) represents the lower limit, and the resulting value of the scattering albedo can be overestimated, thus representing a higher limit for the albedo itself. To evaluate the error due to this approximation, we calculated (α)
using the Mie scattering function for a graphite sphere with a diameter of 100 nm immersed in water. The values we obtained in this case both for (αfar)
were 74%, 20%, and 9% higher than those of the Rayleigh scattering at λ
= 350, 600, and 850 nm, respectively. However, it should be noticed that actual nanoparticles, being aggregates of individual SWCNHs, are strongly nonhomogeneous, and the Mie theory neither is strictly applicable. Their morphology could suggest, as for the light-particle interaction, a sort of effective radius, smaller than the physical radius.
Assumption (2) has been evaluated comparing the approximated values of (α)
obtained from Equation 5 with the exact values obtained from Equation 4. Values calculated with Equation 5 are independent on μe
and equal to (αfar)
= 0.00075 and (αnear)
= 0.206. Equation 4 has been calculated for the six values of μe
due to progressive addition of SWCNHs: results for (αfar)
were almost identical to the values obtained using Equation 5 in the calculation, while for (αnear)
, we obtained a value range from 0.180 to 0.198 at λ
= 600 nm. In the worst case, this difference could lead to relative errors of 10% to 15% in the albedo, which is acceptable in the framework of the proposed estimation method aimed to assess a higher boundary.
Assumptions (3) and (4) have been investigated by means of Monte Carlo simulations. We calculated the transmittance for different values of τe
ranging from 0.2 to 2.5, using a code for photon migration through a three-layer slab in which multiple scattering and internal reflections are taken into account [16
]. For the turbid medium, we assumed the Rayleigh scattering function and ω
= 0.1, and we considered the same geometry as the experiment. The results do not differ if assumptions either (3), (4), or both are made.
As for hypotheses (5) and (6), they do not appreciably affect the results. In fact, numerical investigations showed that in all our experiments,
was always smaller than 0.025 (in fact, in writing
, we made the hypothesis
and that the difference between εe
and εe meas(αfar)
was smaller than 0.1%.