Broadband and Crack-Free Antireflection Coatings by Self-

Assembled Moth Eye Patterns

F. Galeotti,*,a F. Trespidi,*,b G. Timò,b and M. Pasini a

a CNR Istituto per lo Studio delle Macromolecole (ISMAC), via E. Bassini 15, 20133 Milano, Italy. E-mail:

f.galeotti@ismac.cnr.it b Ricerca sul Sistema Energetico (RSE), Strada Torre della Razza, loc. Le Mose, 29122 Piacenza, Italy. E-mail:

franco.trespidi@rse-web.it.

Packing evaluation in breath figures films

For a quantitative analysis, packing of pores in a breath figure array can be analyzed by Voronoi polygons tessellation

[1]. A Voronoi polygon is the smallest convex polygon surrounding a point whose sides are perpendicular bisectors of

lines between a point and its neighbors.

The analysis of Voronoi construction is made in terms of the coordination number n of a polygon, which is the number

of sides of a Voronoi polygon and Pn, the fraction of the number of polygons having the coordination number n.

From this analysis the conformation entropy can be calculated by the formula S=-Pn ∑ lnPn

By applying the Voronoi tessellation to the SEM image above, which can be taken as representative example of the

packing obtainable in the conditions described for good performing coating, the following image is generated (we used

ImageJ software [2]).

We calculated that the probabilities of pores with five, six and seven nearest neighbours (P5, P6, and P7) are 0.16, 0.74,

and 0.09, respectively. This corresponds to conformal entropy of 0.78.

Considering that, for perfect hexagonal packing S=0, while for random packing S=1.71, we can considered our

structures as non-perfect hexagonal packing.

Calculation of the effective index of refraction profile in the presence of a hemispherical

nanostructure hexagonally packed

r ≡ radius of the nano-hemisphere

AT ≡ Considered coating surface

AP (x) ≡ Area occupied by the PDMS (at the different cross sections)

AA (x) ≡ Area occupied by the Air (at the different cross sections)

N ≡ Maximum number of hemispheres in AT

nP (l) ≡ index of refraction of the PDMS

b

x

Air

PDMS

r

nA ≡ index of refraction of the AIR

ne (x, l) ≡ effective index of refraction at different heights

In order to calculate the maximum number of hemispheres present in AT it is necessary to calculate the ratio between

the surface of the circle (base of the nanosphere) and the hexagon circumscribed:

Surface of the circle = π * r2

Surface of the hexagon = 2* √3 * r2

Surface of the circle / Surface of the hexagon = π / (2*√3) ; (equivalent to approximately 0.907)

Therefore:

N = Effective surface used by hemispheres / base area of a single hemispheres =

=AT * π / (2*√3) / (π * r2 ) 1)

From the theorem of Pythagoras: b2 = r2 - x2 hence:

AP = N * π * b2 = π *AT * (r2 - x2)/ (2* √3 * r2) 2)

If we define the fill factor as:

ff = AP / AT 3)

by substituting formula 2) into the formula 3) we have:

ff = AP / AT = π * (r2 - x2)/ (2* √3 * r2) 4)

The value of the effective index of refraction (ne) at different heights (x) can be calculated from the Effective Medium

Theory [3-5]:

ne = [ ff * nP q + (1 – ff ) * nAq ]1/q 5)

Optical set up used for characterization

Instrument setup. A) Optical configuration used to measure the spectral transmission of the coating: a white light

source produces a collimated probe beam which is sent to the sample; the light beam once crossed the sample is

collected by an integrating sphere which is coupled to a spectrometer by means of an optical fibre. B) Picture of the

experimental setup.

Measurement description

The measurement setup is composed by a white light source generating a quasi collimated white light beam of about

5mm in diameter. The light beam crosses the sample under test and is collected by an integrating sphere having a

9.5mm circular input aperture. The integrating sphere is properly coupled to the spectrometer by means of an optical

fibre kept in a fixed position to minimize possible modifications in the spectrum delivered to the spectrometer.

The measurement is performed in a relative way by measuring the change in the transmittance with respect to an

uncoated surface.

The nano-coating is deposited on a clean flat glass surface that acts like a movable support. A first spectral transmission

measurement, that will be used as reference, is performed by measuring the glass transmission in an area where the

nano-coating is not present. The glass support is then displaced to measure the transmission where the nano-coating is

present. The spectral ratio between the two measurements (“transmission with coating”/”transmission without coating”)

shows the effect of the nano-coating.

A ratio greater than 1 (or greater than 100% if the ratio is considered in percentage values) means that the coating is

improving the transmission with respect to the uncoated glass surface.

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