Rayleigh-Gans approximation

In a droplet surrounded by a medium with an optical frequency dielectric constant ǫm, the part of the induced polarizationPs(rm) which causes scattering is given by:

Ps(rm) = ǫ0ǫm

hǫr(rm)−1ⁱE(rm) (2.20) where Estands for the local electric field and ǫ_r for the relative (optical frequency) dielectric tensor defined as the ratioǫ/ǫm between the local dielectric constant of the surrounding medium and dielectric constant of the particle [112]. In our uniaxial orientational order parameter approximation the relative dielectric tensor ǫr(r) is written in its local principal frame:

ǫr=

Starting with Maxwell’s equations one can derive the following exact expression for the scattered electric far field:

E_s =f(k,k^′)e^ikr

r (2.22)

where the scattering amplitude f(k,k^′), is given by f(k,k^′) =− 1

4π

Z

k×k^′ ×^nhǫr(rn)−1ⁱE(k,rn)^o×e^−ik′·rndV (2.23)

Unfortunately the internal field E(rm) is unknown. In the case where the condi-tion 2kR|nLC/nm−1| ≪1 is satisfied, the internal field can be well approximated by the undisturbed incident plane-wave field. The Rayleigh-Gans approximation is analogous to the well-known Born approximation in quantum mechanics. The scattered far field is thus the total of dipole radiation emerging from nonuniformly oriented molecules in the droplet. Each dipole is assumed to be excited only by the light from the external source, while the effect of the scattered light from the neighboring molecules and droplets is neglected.

Introducing i=k/k,i^′ =k^′/k, and the polarization vector e=E0/E0 we can write

Here unit vectors i_|| and i⊥ are orthogonal to the direction of the incoming beam.

The symbols || and ⊥ denote directions parallel and orthogonal to the scattering plane, respectively.

The distribution of the scattered light is usually represented by the differential cross section which is defined as

There are two common arrangements: VV type with incoming polarization paral-lel to the transmission direction of the analyzer so that the transmitted scattered intensity is

VH geometry where the analyzer is perpendicular to the direction of the polarization of the incoming beam. Here the scattered intensity is

IV H = 1 (kr)²

¯¯

¯S||⊥sin²α−S⊥||cos²(α) + (S||||−S⊥⊥)sin(α)cos(α)^¯¯¯² (2.26)

The scattered light from a nematic droplet depends on the impact angle θ and the angle of polarizationα (angle between polarization and the scattering plane) as well as on the scattering angles δ. The values of refractive indices have been chosen as n⊥ = 1.52 and n|| = 1.70 which are typical of cyanobiphenyl components known to form droplets in some polymeric materials. The refractive index of the polymer matrix has been chosen nm= 1.55.

In order to illustrate the effectiveness of nematic droplet dispersions in beam shut-ters, it is useful to calculate the total scattering cross section of a nematic droplet.

The angular dependence of the differential scattering cross section for different di-rector configurations, droplet sizes, and impact angles are shown in Fig.2.9.

2.6. LIGHT SCATTERING FROM SMALL PARTICLES 49

Figure 2.9: Dependence of the differential cross section of an optically uniaxial particle with uniform director on the scattering angle, for the case of a very high field. On the figure curve (a) represents the case θ = α = π/2 for kR = 0.5 and curve (b) the situation θ = α = 0 for the same kR. Curve (c) represents kR = 1.5 with θ = α= π/2 and curve (d) θ = α= 0 for the same kR. The parameters used are ξ = 0.04,η = 0.08, and γ = 0. (Figure from ref. [112])

Here the results for parallel director configuration are shown. This figure illus-trates several general features. Firstly, there appears little difference between the forward and backward scattering from small droplets whereas the difference becomes more pronounced for large droplets (kR≥1). When the size of the droplet becomes comparable to the wavelength of light there are phase shifts inside the droplet and the destructive interfere leads to a breaking of the symmetry between forward and backward scattering.

Secondly, the angular dependence appears to be strongly dependent on the director configuration inside the droplet, in particular for larger droplets.

This disparity between different director configurations is more pronounced at large scattering angles. It is interesting to point out that the angular dependence exhib-ited in Fig. 2.9 is identical to the case of an isotropic droplet. In Fig.2.9 the impact angle and polarizations are such that they give minimum scattering from aligned droplets.

Figure 2.10: Dependence of the differential cross section on the scattering angle for θ =π/2and α = 0for the case of a very high field with kR = 0.5 in curve (a) and kR= 1.5in curve (b); zero-field tangential surface alinement withkR= 0.5in curve (c) and kR = 1.5 in curve (d) ; zero-field normal surface alignment with kR = 0.5 in curve (e) and kR= 1.5in curve (f). The parameters used are ξ= 0.04, η= 0.08, and γ = 0. (Figure from ref. [112])

The impact angles and polarizations which give maximum scattering are those which exhibit the plots of Fig. 2.10. From this figure it is clear that droplets with star configuration are poor scatterers compared to those of the bipolar or aligned type. Parallel surface alignment is therefore desired in droplet dispersions used in electro-optic switching devices. The symmetry between forward and backward scattering for small kR is particulary evident in these plots.

2.6. LIGHT SCATTERING FROM SMALL PARTICLES 51

Figure 2.11: Dependence of the total scattering cross section on the incident angle θ for the case of high field with curve (a) forkR= 0.15and curve (e) for kR= 1.5;

external field for tangential surface alignement represented by curve (b) for kR = 0.15 and curve (g) to kR = 1.5; zero field for tangential surface alignement with curve (c) for kR = 0.15 and curve (h) forkR = 1.5. The polarization angle α0 = 0, ξ= 0.04, η= 0.08were used. (Figure from ref. [112])

Fig.2.11 shows the dependence of the total scattering cross section upon the orientation of a droplet relative to the direction of incident light in various field strength. Curves (a) and (e) at θ = 0 determine the transparency of a film in the presence of the field. With a perfect match between n_m and n_⊥ there would be no scattering at all for this condition and the film would be perfectly transparent.

Curves (c) and (h) give the cross section in the absence of a field. Normally there is a random orientation of droplets in a film. The opaqueness of the film is then determined by the help of this curve.

2.6.4 Polarized and depolarized light scattering in

ForB → ∞we have IV H = 0, which means that there is no depolarized scattering.

To test how good the Rayleigh-Gans approximation describes the form factors of colloidal liquid crystalline particles we used an experimental set-up which consists of argon-ion laser operating atλ= 514 nm, a ganiometer and a photomultiplier as a detector. The scattered intensity was measured as a function of the scattering angle in geometry VV (polarized scattering). The results are shown in Fig.2.12. There is a very good agreement between the theory and experiment.