9 Experimental part
9.1 Characterization methods (basics)
9.1.1 Dynamic light scattering
Dynamic light scattering (DLS) is an analytical technique, allowing to determine the size of particles or macromolecules in the sub-micron region. The information about the size can be derived from the fluctuation of scattered light, which is constantly measured during the experiment. This fluctuation is caused by Brownian motion of the solute molecules, a random movement triggered by interaction with surrounding solvent molecules. Thus, the movement of solvent molecules is correlated with the solvent´s viscosity, making an accurate temperature control necessary for DLS measurements. To analyze the fluctuation of the scattered light, a formalism called autocorrelation function g2 can be applied.[221–223] In this formalism, the intensity I at the time t is multiplied with the intensity of the time t+τ. This operation is repeated for various times t and τ´s (time delays) and a mean value is formed thereof. One obtains an exponentially decaying curve g2(τ):
𝑔
2(𝜏) =
〈𝐼(𝑡)∙𝐼(𝑡+𝜏)〉〈𝐼(𝑡)〉2 (7.1)
with: I: intensity τ: delay time
At short delay times τ, the correlation is high, because the particles have not enough time to move long distances from their initial starting points. However, the correlation between two signals is decaying exponentially as the delay time is increasing (see Figure 133).
Figure 133: Left: fluctuation of the intensity of the scattered light. Right: Calculated second order autocorrelation function.[224]
The exponential decay is related to the diffusion coefficient of the particles. If the sample is monodisperse, the decay is a single exponential. Using the Siegert equation, a transformation of second-order autocorrelation function g2(t) to the first-order autocorrelation function g1(τ) can be achieved:
𝑔2(𝜏) = 𝐵[1 + |𝑔1(𝜏)|2]
(7.2) with: B: correction factor (=1, in case of spherical particles)
For monodisperse, highly diluted systems containing only spherical particles and assuming no ion-ion interactions, no collisions between particles, the following approach to link the autocorrelation function with the translational diffusion coefficient D is applicable[225]:
𝑔1(𝜏) = 𝑒(−𝛤𝜏) (7.3)
with: Γ: decay rate
𝛤 = 𝑞2𝐷 (7.4)
and
𝑞 =
4𝜋𝑛0𝜆
sin (
𝛳2
)
(7.5)with: 𝜆: wavelength of the laser n0: refractive index of the sample
𝛳: angle between detector and sample cell D: diffusion coefficient
q: wave vector
In most cases, however, the examined sample is polydisperse. Thus, the decay of the autocorrelation function is a sum of exponential decays[225]:
𝑔
1(𝑞, 𝜏) = ∫ 𝐺(𝐷)𝑒
0∞ (−𝑞2𝐷𝜏)𝑑𝐷
(7.6) with: G(D): distribution factor (proportional to relative scattering of each species)G(D) depends on the amount, the mass and the form of each species and, thus, is angle dependent.
Therefore, the determined translational diffusion coefficient is an apparent diffusion coefficient Dapp:
𝐷
𝑎𝑝𝑝=
∑ 𝑚𝑖𝑀𝑖2𝐺𝑖(𝑞)𝐷𝑖∑ 𝑚𝑖𝑀𝑖2𝐺𝑖(𝑞) (7.7)
with: i: species M: molar mass m: mass fraction
By extrapolation of the apparent diffusion coefficients to 0 (form factor G(q)=1), it is possible to obtain the so-called “z-averaged” diffusion coefficient Dz. From this translational coefficient, the hydrodynamic radius Rh of a spherical solute molecule can be obtained from the Stokes-Einstein relation:
𝑅
ℎ=
𝑘𝑏𝑇3𝜋𝜂𝐷
(7.8)
with: Rh: hydrodynamic radius kb: Boltzmann´s constant
D: translational diffusion coefficient T: temperature
9.1.2 Holography
In holographic experiments a laser beam is split into two coherent laser beams, namely the object and the reference beam, which are superimposed within a photoactive material. The two beams are incident from different angles on the holographic media (off-axis). If the angle between the two laser beams ranges between 0° and 90°, a transmission hologram, for larger angles, a reflection hologram is generated. The object beam can, but must not, carry information. Is the object beam not carrying any information, a simple plane wave or hologram of no object experiment is performed.[226] In this case the two beams are creating a sinusoidal interference pattern or grating (see Figure 134).
Figure 134: Scheme of an inscription process of a holographic grating into a photosensitive material in a holographic experiment. The grating constant Λ is the distance from maximum to maximum of the gratings.
Referring to[217].
The distance between the maxima (or minima) of the intensity grating is called grating constant Λ and is defined as:
𝛬 =
𝜆2𝑛0𝑠𝑖𝑛𝛳
(7.9)
with: Λ: grating constant 𝜆: writing wavelength
𝛳: half the angle between incident laser beams n0: refractive index
These gratings can either be inscribed on the surface of the material due to mass transport (surface relief gratings), or in the volume due to orientation of the azobenzenes (volume holograms).
Depending on which kind of photosensitive material is used, either a change in the absorption coefficient (amplitude hologram), or a local change in the refractive index (phase hologram) is generated in the photosensitive material. For phase holograms, mainly azobenzene-containing material or photopolymers are used. In this thesis, off-axis transmission volume holograms inscribed with plane waves are investigated only. Azobenzene-containing materials are expected to be affected mostly in the bright areas or regions of the intensity grating. Thus, the intensity grating leads to a change of the refractive index n.[226]
In the holographic experiments performed in this thesis, the inscription of gratings is done with a He-Ne-laser at 488 nm. The reading process had to be performed at a wavelength outside the absorption band of the dye (685 nm). Thus, in the reading process the reading beam is diffracted at the inscribed
grating without affecting the inscribed hologram. The quotient of the intensity of the diffracted reading beam into the first order I1 to its intensity at incident I0 is called diffraction efficiency η[226]:
𝜂 =
𝐼1𝐼0 (7.10)
When reading out a holographic grating, it must be distinguished if the hologram is thick or thin. Thin gratings diffract light in the Raman-Nath regime with multiple diffraction orders, and their theoretical maximum diffraction efficiency is 33.9 %. In thick gratings, all the light is diffracted into one order, for none of the others is fulfilling the Bragg condition. The theoretical diffraction limit is 100 %.[226] For thin gratings, η can be calculated according to the scalar theory for thin volume holograms by of Magnussen and Gaylord[227], while the diffraction efficiency of thick gratings is determined by Kogelnick´s[228]
coupled wave theory. For small diffraction efficiencies, and if the light is incident at the Bragg angle, the diffraction efficiencies of both kinds of gratings are equal to:
𝜂 = (
𝜋𝑛1𝑑0𝜆𝑐𝑜𝑠𝛳
)
2 (7.11)For the sake of comparability of different materials, one has to calculate the refractive index material n1, which, in contrast to the diffraction efficiency, is independent of the thickness of the material. The refractive index modulation n1 can be calculated from the diffraction efficiency, that is obtained in the holographic experiment[229,230]:
𝑛
1=
𝜆𝑐𝑜𝑠𝛳√𝜂𝜋𝑑0 (7.12)