# Reflectance of nanotubes

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1. Feb 25, 2013

### ralden

hi guys, I'm working about nanotechnology, and my current research is about nanotubes. Based on my observation, as the nanotubes length increased the reflectance also increased, one of my possible explanation is that; since dielectric constant is dependent on the dimension of the material, and refractive index is directly proportional to dielectric constant, and since reflectance may expressed in terms of refractive indices, therefore, as dielectric constant increased reflectance also increased, but i must also considered the absorption of that material, what happen to the absorption when nanotubes length increased? is it also affect the reflectance? please shared your knowledge and opinion. thank you

2. Feb 25, 2013

### DrDu

Maybe you could be a little bit more specific: Are you observing single nanotubes, solutions or bulk material? About what length scale are you taking, specifically larger or smaller than the wavelength of light? What kind and and of which material are the nanotubes?

3. Feb 25, 2013

### ralden

A bulk material, having a length range from 300nm-1700nm. I'm studying titanium dioxide nanotubes,

4. Feb 25, 2013

### DrDu

Ok.
Generally, absorption also affects reflectivity. I would not consider this important for TiO2 in the visible. What wavelengths are you interested in?

5. Feb 25, 2013

### ralden

actually i'm interested on the reflectance of increasing length of TiO2 nanotubes. since tiO2 have no reflection at 400nm-500nm wavelength because of its band gap that absorbed those em waves, then i focused on the wavelengths around 600nm-1100nm that gives refletance, what you think?

6. Feb 25, 2013

### DrDu

I would be astonished if TiO2 had strong absorptions in this range. After all, it is used as a white pigment, so it can't be absorptive in the visible (say 800 to 400 nm). I also see no mechanism for absorption in the near IR. Absorption would have to be rather strong to change reflectivity.

7. Feb 25, 2013

### ralden

ok. so what is the relationship between the reflectance and dielectric constant of the sample? is it varies with length of nanotubes, same to the absorption that varies also with thickness?

8. Feb 26, 2013

### DrDu

Reflectance is given by the Fresnel equations http://en.wikipedia.org/wiki/Fresnel_equations
and $n=\sqrt{\epsilon}$, so the first part of your question should be clear.
The real part of the dielectric constant n and the absorbance κ are related by Kramers Kronig relations http://en.wikipedia.org/wiki/Kramers-Kronig_relation, which is rather involved.
My first guess why epsilon increases with increasing length would be an increase in sample density. After all, a single slab of TiO2 has a larger density than two close packed slabs of half the length.

9. Mar 16, 2013

### Alkim

Ralden, this is a case of collective scattering and its treatment is quite complex. I know it from experience since I have worked with scattering by metallic spherical nanoparticles. Each fiber can be treated as a cylinder for which solving Maxwell equations is relatively easy but the problem comes when you have to consider the interaction between the waves scattered by each cylinder. The problem can be treated analytically using so called effective medium theories, but from my experience they are not very precise. You can think of an effective refractive index which increases with volume fraction of the high refractive index medium and therefore reflectance too (you just substitute the effective index in Fresnel equations already cited). The other possibility is EM numerical simulation, using e.g. FDTD, which is the best method since it solves "exactly" Maxwell equations. In fact, exactness depends only on your computer power available.

10. Mar 16, 2013

### ralden

Hi Alkim, may you give me links of articles and studies about the theoretical simulation of the collective scattering, or the FDTD? thanks :)

11. Mar 16, 2013

### Alkim

Hi Ralden, there is a lot of bibliography, you can start with wikipedia and articles, books and codes cited therein:

http://en.wikipedia.org/wiki/Scattering
http://en.wikipedia.org/wiki/Light_scattering_by_particles
http://en.wikipedia.org/wiki/Effective_medium_approximations
http://en.wikipedia.org/wiki/Finite-difference_time-domain_method

Also, a quick search about scattering by cylinders rendered some interesting links, see e.g.: http://inis.jinr.ru/sl/vol2/Physics...DBOOK_of_OPTICS/HANDBOOK_of_OPTICS/v1ch06.pdf This guy is a capacity in the field.

this may also give you some orientation:
http://eos.wdcb.ru/transl/izva/9404/pap06.htm

And this too: