2.9 Magnetic birefringence study of LC particles
3.1.4 Backscattering geometry
Another geometry which is often used in DWS experiments is the backscattering geometry. This geometry can be particularly useful in industrial process-monitoring applications where only one side of the sample is available. Another advantage of the backscattering geometry is that it does not require independent knowledge of the transport mean free path in order to interpret the autocorrelation function. Another feature of the backscattering geometry is that it probes many length scales at once.
Since backscattering involves a significant number of light paths whose length is comparable with l∗, the diffusion approximation must be used with caution.
In a backscattering geometry, the laser beam is usually expanded so that it uniformly illuminates an area of the incident face that is much wider thanl∗. Light is collected from a very small area near the center of the illuminated area. This ensures that the shape of the measured autocorrelation function is insensitive to the size of the illuminated area and to the precise position of the detected light within that area.
Like in the case of the transmission geometry, we again take the incident face to be at z = 0 and the source of diffusing intensity to be a distance z0 which is expected to be of order l∗ inside the sample. Solution of the diffusion equation for a sample
of infinite thickness is given by
g1(t) = exph−zl∗0q6tτi
1 + 23q6tτ (3.12)
The stretched exponential form of Eq.(3.12) originates from the wide distribution of decay times, which in turn results from the wide distribution of path lengths in the backscattering geometry. Long paths which decay quickly and probe relatively short length-scale motion, contribute to the initial decay of the autocorrelation function.
At longer times, after the contributions from long paths have decayed away, the decay of the autocorrelation function comes from short paths and probes short paths. However, the diffusion approximation and central limit theorem are valid only for long paths, and break down for short paths.
3.1. DWS. 91
3.1.5 Experimental results and discussion
Figure 3.1: Set-up for diffusing-wave spectroscopy experiments.
Experimental set-up for measuring the effect of rotational diffusion on the tempo-ral intensity autocorrelation function in multiple scattering includes super-conducting magnet, an argon ion laser with λ = 514 nm, a microscope objective, polarization maintaining fiber, detecting single mode fiber, sample holder with optical elements, photomultiplier, correlator and PC. Microscope objective is used to couple the laser beam in the single mode fiber. The second end of the fiber is fixed on the sample holder which can be moved inside the magnet core. The parts of the set-up are made of nonmagnetic materials. The set-up provides reproducibility of results of DWS for different samples. The set-up has been aligned only once in the beginning.
With the help of the small lens the sample is illuminated with a parallel laser beam 4 mm wide (which corresponds to the extended source). Single speckles of the dif-fuse light emanating from the sample are collected with a single-mode fiber and a small lens. In front of the lens there is a polarizer film. A single-mode fiber does not actually collect light from a single speckle but projects the collected light into a
single spatial mode. In transmission geometry the scattered light is collected from the central part of the light spot on the opposite side of the sample (Fig.3.1). In backscattering geometry the scattered light is collected from the central part of the light spot on the front side of the sample. The signal from PMT is amplified and is then fed to the digital correlator card AVL-5000 housed in a PC.
First we shall consider the diffusion of light in direction perpendicular to field B when ~kin⊥B. In Fig.3.2 the normalized temporal field autocorrelation functions~ g1(t) in backscattering geometry for magnetic fields from 0 to 7 Tesla for the sample with φ = 0.05 are shown. The autocorrelation function g1(t) have been measured with the step 0.25 Tesla from 0 to 2 Tesla and with the step 0.5 Tesla from 2 to 7 Tesla. From the Fig.3.2 one clearly sees that the autocorrelation functions decay faster with increased magnetic field. The field autocorrelation functions g1(t) have been fitted with expression (3.12) with decay time τ as free parameter. The results of the fits are presented in Fig.3.3 as decay rates, which is 1/τ. For all volume fractions the decay rate increases with the field in the range 0 - 3 Tesla and then tends to saturate at higher fields. For the sample with φ = 0.05 the rate increases from 392 s−1 to 420 s−1, for the sample with φ = 0.1 the rate increases from 324 s−1 to 377 s−1 and for the sample with φ = 0.15 the rate increases from 230 s−1 to 294 s−1. The decay rates decrease with the volume fraction of particles due to the hydrodynamic interaction between the colloidal particles.
The autocorrelation functions g1(t) in transmission geometry for magnetic fields from 0 to 7 Tesla for the sample with φ= 0.05 are shown in Fig.3.4. The autocor-relation functions decay slower with the increase of the fieldB. The reason for that is that in transmission geometry g1(t) depends strongly on the transport mean free path l∗ as it seen from Eq.(3.11). In order to determine this dependence quantita-tively we do the following: we take τ from backscattering measurements, put them into Eq.(3.11) and fit the autocorrelation functions in transmission withl∗ as a free parameter. The transport mean free path l∗ in this geometry we calll⊥∗ in order to distinguish from the case ~kin||B. The results of the fit are shown in Fig.3.5. The~ transport mean free paths increases with the field B in the range 0-3 Tesla and then saturates at higher fields. For the sample with φ = 0.05 l∗ increases from 69.3 µm to 84.5 µm, for the sample with φ = 0.1 the l∗ increases from 36.4 µm to 43.9 µm and for the sample with φ= 0.15 the l∗ increases from 26.7 µmto 33.2 µm.
3.1. DWS. 93
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.01
Figure 3.2: Normalized temporal field autocorrelation functions g1(t) in backscat-tering geometry for magnetic fields from 0 to 7 Tesla. Sample thickness L= 1mm.
λ= 514nm. Particle diameter 200 nm. Particles are nonpolymerized. ~kin⊥B.~
0 1 2 3 4 5 6 7 8
Figure 3.3: The decay rates 1/τ of the temporal field autocorrelation functions g1(t) in backscattering geometry for magnetic fields from 0 to 7 Tesla and volume fractions of the particles φ = 0.05−0.15. Particle diameter 200 nm. Particles are nonpolymerized. ~kin⊥B~.
10-7 10-6 10-5 10-4 0.0
0.2 0.4 0.6 0.8 1.0
g1
t, s
2.0x10-5 4.0x10-5 6.0x10-5 0.1
1 B=0 T B=1 T
B=2 T B=3 T B=4 T B=5 T B=6 T B=7 T
g 1
t, s
Figure 3.4: Normalized temporal field autocorrelation functionsg1(t)in transmission geometry for magnetic fields from 0 (red lines) to 7 Tesla (blue lines). Volume fractions of the particles φ = 0.05. Sample thickness L = 1mm. Particle diameter 200 nm. λ= 514nm. ~kin⊥B~
3.1. DWS. 95
0 1 2 3 4 5 6 7 8
10 20 30 40 50 60 70 80 90 100
I
*, ( m )
B, Tesla
Figure 3.5: Transport mean free paths l∗⊥ for magnetic fields from 0 to 7 Tesla and volume fractions of the particles φ = 0.05−0.15. Sample thickness L = 1mm.
Particle diameter 200 nm. λ = 514nm. ~kin⊥B.~
This means that the samples become more transparent with the field B in di-rection perpendicular to B. The measured values of l∗ at B = 0 for the samples with volume fractionφ = 0.05−0.15: 26.7 µm, 36.4 µm, 69.3µm correspondingly, are in very good agreement with theoretical values obtained from Mie calculations with Percus-Yevick structure factor for optically isotropic spheres withn = 1.55 (see Table 1). This shows that the procedure for determining l∗ and τ is consistent, at least atB = 0.
Volume fraction, % l∗, Experiment l∗, Theory P-Y l∗, Theory S(q)=1
5 69.3 69.16 73.03
10 36.4 34.58 38.80
15 26.7 23.05 27.69
Table 1. Comparison between measuredl∗ and calculated using theory atB = 0.
Now we consider the diffusion of light in direction parallel to field B when ~kin
is parallel to B~. In Fig.3.6 the normalized temporal field autocorrelation functions g1(t) in backscattering geometry for magnetic fields from 0 to 7 Tesla for the sample with φ= 0.05 are shown.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.01
0.1
1 B=0 T
B=1 T B=2 T B=3 T B=4 T B=5 T B=6 T B=7 T
g1
t1/2
,
s1/2Figure 3.6: Normalized temporal field autocorrelation functions g1 in reflection for magnetic fields from 0 to 7 Tesla in geometry~kin||B~. Volume fractions of the particles φ= 0.05. Particle diameter 200 nm.
0 1 2 3 4 5 6 7 8
80 120 160 200 240 280 320 360
=0.05 =0.1 =0.15
rate, s-1
B, Tesla
Figure 3.7: The decay rates of the temporal field autocorrelation functionsg1 in re-flection for magnetic fields from 0 to 7 Tesla in geometry~kin||B~ and volume fractions of the particles φ= 0.05−0.15. Sample thickness L= 1mm. Particle diameter 200 nm. λ= 514nm.
3.1. DWS. 97
1.0x10-5 2.0x10-5 3.0x10-5 4.0x10-5 0.1
1 B=0 T
B=1 T B=2 T B=3 T B=4 T B=5 T B=6 T B=7 T
g1
t, s
Figure 3.8: Normalized temporal field autocorrelation functions g1 in transmission for magnetic fields from 0 to 7 Tesla in geometry ~kin||B. Volume fractions of the~ particles φ = 0.05. Sample thickness L = 1mm. Particle diameter 200 nm. λ = 514nm.
0 1 2 3 4 5 6 7 8
0 10 20 30 40 50 60 70
I * , ( m )
=0.05 =0.1 =0.15
B, Tesla
Figure 3.9: Transport mean free paths l∗ for magnetic fields from 0 to 7 Tesla in geometry ~kin||B~ and volume fractions of the particles φ = 0.05−0.15. Sample thickness L= 1mm. Particle diameter 200 nm. λ= 514nm.
The g1(t) have been measured in 0.25 Tesla steps from 0 to 2 Tesla and with the step 0.5 Tesla from 2 to 7 Tesla. The autocorrelation functions decay slower with increasing magnetic field. The field autocorrelation functions g1(t) have been fitted with expression (3.12) with decay time τ as free parameter. The results of the fits are presented in Fig.3.7 as decay rates. For all volume fractions the rate decreases with the field in the range 0-3 Tesla and then tends to saturate at higher fields. For the sample with φ = 0.05 the rate decreases from 320.9 s−1 to 288.6 s−1, for the sample withφ = 0.1 the rate decreases from 229.8 s−1 to 196.2 s−1 and for the sample with φ = 0.15 the rate decreases from 192.4 s−1 to 159.2 s−1. The autocorrelation functions g1(t) in transmission geometry for magnetic fields from 0 to 7 Tesla for the sample with φ = 0.05 are shown in Fig.3.8. The autocorrelation functions decay faster with increasing magnetic field. The reason for that is that in transmission geometry g1(t) depends strongly on the transport mean free path l∗ as it seen from Eq.(3.11). In order to determine this dependence quantitatively we do the following: we take the τ from backscattering measurements (Fig. 3.7), put them into Eq.(3.11) and fit the autocorrelation functions in transmission with l∗ as a free parameter. The transport mean free path l∗ in this geometry we call l∗||. The results of the fit are shown in Fig.3.9. The transport mean free paths decreases with the field B in the range 0-3 Tesla and then saturates at higher fields. For the sample withφ= 0.05 thel∗ decreases from 58.7 µmto 49.0µm, for the sample with φ= 0.1 thel∗ decreases from 29.3 µmto 25.4 µmand for the sample with φ= 0.15 the l∗ decreases from 25.3 µm to 22.0 µm. This means that the samples becomes less transparent with the field B in direction parallel to B.
The analysis of the data in both experimental geometries~kin||B~ and~kin⊥B, shows~ that the change of the autocorrelation functiong1(t) is not due to the acceleration or slowing down of the rotational diffusion by a magnetic field, but is due to variation of static optical properties of the samples, namely due to variation of the transport mean free path l∗. While the data in backscattering geometry are fitted very well by Eq.(3.12), they are equally well fitted by the simpler formg1(t) = e−γ√
6t/τ. This expression is independent of the transport mean free pathl∗and should describeg1(t) for both geometries ~kin⊥B~ and ~kin||B~. Experimentally, however, we find that the decay rates differ qualitatively. The fact that in geometry~kin⊥B~ the autocorrelation functionsg1(t) in backscattering decay faster with the fieldB and in geometry~kin||B~ decay slower with the field tells that the Eq.(3.12), which follows from isotropic theory, can not be used to analyze DWS data from anisotropic media. Also the reason for geometry dependent behaviour of the field autocorrelation function can be due to surface effects.
Chapter 4
Anisotropic diffusion of light
Most objects in nature scatter light. That is why we are able to see the world.
Interacting with matter, light changes its initial direction of propagation and going into eye creates the image of the object. The character of this interaction can be different. Depending on the nature and structure of matter light is scattered and absorbed differently. That is why some objects look dark and some look bright, and usually all of them have their own color.
When solving various practical and scientific problems one often has to deal with scattering of electromagnetic radiation. Among such problems there are many prob-lems of astronomy, optics of Earth atmosphere, optics of the ocean, biophysics, spec-troscopy of colloidal suspensions. The phenomenon of light scattering becomes much more complicated when the number of scattering events becomes large. In this case the photons are scattered many times on different particles before they leave the medium. Such scattering of light is called multiple scattering. Multiple scattering of light has to be taken into account during the development of photographic mate-rials, textile, paper, paints and many other synthetic materials. In some media light propagation becomes anisotropic. For example in wood, textile, liquid crystalline materials and others. In this chapter we shall consider the problem of light diffusion in such media.
4.1 Radiative transfer theory and diffusion of light
Historically, the problem of wave propagation in random media has been investi-gated from two different points of view. One of them is”multiple scattering theory”
and the other is ”radiative transfer theory” [177].
The analytical theory or ”multiple scattering theory” starts with Maxwell equa-tions or the wave equation and delivers soluequa-tions of scattering and absorption for a
99
single particle, introduces interaction effects of many particles and then considers statistical averages. This theory is mathematically rigorous since all effects, such as diffraction, interference, multiple scattering can be included. However in practice it is impossible to obtain solutions which include all these effects. One has to use approximate solutions for a specific range of parameters.
Radiative transfer theory considers the propagation of intensities. It is a phemeno-logical theory and it was introduced for the first time by Schuster in 1905 in his study of radiation in foggy atmosphere. The basic equation in this theory is called radiative transfer equation, which is equivalent to Boltzmann’s equation in the ki-netic theory of gases and in neutron transport theory. Radiative transfer equation describes the transport of energy through a medium containing particles. Transport theory itself does not include diffraction effects. Also, transport theory does not consider correlations between fields and rather an addition of intensities takes place.
In this chapter we shall consider the main characteristics of the electromagnetic field and optical characteristics of elementary volume in the scattering media. We shall derive the radiative transfer equation which describes the propagation of radiation in multiply scattering media.