Theoretical background

In strongly scattering samples light is scattered many times before it leaves the sample again. This results in a turbid and soft appearance, like e.g. that of a cloud in the sky. If light is scattered often enough in a disordered material, each light path can be regarded as a random walk. One can say that light is diffusing.

Mathematically the transport of light can in this case be described by the diffusion equation, which makes the theoretical treatment of those systems possible.

The foundations of DWS go back to the eighties, when Maret and Wolf [107] investigated multiply scat-tered light in dense samples of randomly distributed interacting particles under Brownian motion. They showed that the diffusion approximation is well suited to describe the temporal correlations measured in the fluctuating intensity of speckles originating from a coherent light beam going through the sample.

Since then, DWS has been established and has become a standard tool to retrieve information about the dynamics of many optically dense systems like particle suspensions, emulsions, foams, polymers, gran-ular matter etc. [108]. Some of the theoretical ideas behind DWS are given in the following. We will partly follow the explanations in [109] and [110], which are good reviews including detailed derivations together with important applications.

7.1.1 General form of the autocorrelation function

In DWS we want to see a signal from the motion of the scatterers in a highly scattering turbid sample. Like in single dynamic light scattering (DLS), it is retrieved from measuring the auto correlation function𝑔₁(𝑡) of the electric field𝐸(𝑡) =∑𝑁

𝑗=1𝐸_𝑗𝑒^{−𝑖𝜙(𝑡)}, which is produced by all𝑁interfering plain waves registered in the detector:

𝑔₁(𝑡)≡ ⟨𝐸^∗(𝑡)𝐸(0)⟩

⟨|𝐸²|⟩ = 1

⟨𝐼⟩

⟨ _𝑁

∑

𝑗,𝑘=1

𝐸_𝑗^∗𝑒^{−𝑖𝜙𝑗(𝑡)}𝐸_𝑘𝑒^{𝑖𝜙𝑘(0)}

⟩

=

∑𝑁 𝑗=1

⟨𝐼_𝑗⟩

⟨𝐼⟩

⟨𝑒^{−𝑖Δ𝜙𝑗(𝑡)}⟩

(7.1) Each of the𝑁waves corresponds to one of the light paths through the sample. For the last equality we define𝐼_𝑗 =𝐸_𝑗^∗𝐸_𝑗and use that the field vectors𝐸_𝑗and the phaseΦ_𝑗 are not correlated. Their ensemble averages ⟨…⟩can thus be computed individually. Further we assume that the fields of different light paths are uncorrelated, so that only summands with𝑗 =𝑘remain. This approximation obviously works well for independent particles. But it was also proven to be valid for interacting particles, as long as their interactions are short-ranged compared to mean free path of light [111]. For single scattering, the phase shift for each wave due to the movement of the scattering particle isΔ𝜙_𝑗(𝑡) =𝑞⃗⋅Δ⃗𝑟_𝑗. In the case of multiple scattering, many scatterers contribute to the phase shift of each light path. As indicated in Figure7.1(to a very good approximation [109]) the total phase shift for one path is the following sum of the phase shifts from all scattering sites:

Δ𝜙_𝑗(𝑡) =

∑𝑛(𝑗) 𝑖=1

⃗

𝑞_𝑖⋅Δ⃗𝑟_𝑖(𝑡) (7.2)

Here𝑛(𝑗)is the number of scatterers contributing to path number𝑗. The𝑞⃗_𝑖are the individual scattering vectors for each scattering particle andΔ⃗𝑟_𝑖is the displacement of that particle. In order to simplify our

7.1 Theoretical background

Figure 7.1:A selection of detected light paths in a multiply scattering medium. Light comes in from the left and is detected on the right hand side. Paths in blue correspond to𝑡= 0and paths in light blue to a later time𝑡. The total phase shift for each light path due to the motion of the scatterers can be computed from the individual phase shift𝑞⃗_𝑖⋅Δ⃗𝑟_𝑖(𝑡)at each scatterer. While some correlation remains after time𝑡 for each individual path, among themselves the paths are uncorrelated.

expression for𝑔₁(𝑡)further, we rewrite the sum over all𝑁different light paths as a sum over the possible number of scatterers𝑛: which is the relative contribution of all light paths with𝑛scatterers to the total intensity. Now the average is not only an ensemble average over particle positions but also an average over the possible individual scattering vectors𝑞⃗_𝑖taking part in paths with𝑛scatterers. To obtain the second equality in7.3, we assume that the successive phase shifts𝑞⃗_𝑖Δ𝑟_𝑖(𝑡)are uncorrelated. This is a safe assumption in most disordered samples. Hence the average over all𝑞⃗_𝑖is reduced to the averaging over the scattering vector in a single scattering event.

In the following we consider a sample that consists of particles in a solvent or a gas that does not contribute to the scattering. To go further we need to assume that the particle displacementsΔ⃗𝑟_𝑖obey a Gaussian distribution. As discussed in AppendixBthis holds for free Brownian motion and also for Brownian motion in a harmonic potential. In the previous chapter (6.2.4) we have seen that there is some deviation from the Gaussian distribution in glassy systems. However, this deviation is relatively small, so the Gaussian approximation should be sufficiently good. The scalar product𝑞⃗_𝑖 ⋅Δ⃗𝑟_𝑖 then equally obeys a Gaussian distribution. By using that only moments of even order are nonzero, we obtain

⟨

For the last equality we need the assumption that scattering vectors𝑞⃗_𝑖and corresponding displacements Δ⃗𝑟_𝑖 are not correlated, the factor ¹₃ comes in because we are in 3D. As a consequence, hydrodynamic interactions are neglected from here on. The ensemble average over particle positions is restricted to

⟨Δ𝑟²(𝑡)⟩, so that it gives the mean squared displacement (MSD). An average over absolute values of the scattering vector𝑞 remains to be done. As long as the MSD is small compared to𝑞⁻², which is always true for small lag times𝑡, one can restrict the calculation to the first term of a cumulant expansion [110]:

⟨ Now the𝑞average is restricted to the term⟨𝑞²⟩. As mentioned above, we only need the distribution of𝑞

for a single scattering event. This is given by the product𝑃(𝑞)⋅𝑆(𝑞)of the form factor and the structure factor, which would be the result of a static single light scattering experiment (cf. Equ.1.26). Then we obtain Here we use that2𝑘₀𝑛_{ef f}is the largest possible scattering vector that occurs for backscattering. The wave vector 𝑘₀ = (2𝜋)∕𝜆corresponds to the wavelength of light𝜆outside the sample,𝑛_{ef f} is the refractive index in the sample. For the second equality in 7.6we use the relation 𝑞 = 2𝑘₀𝑛_{ef f}sin(𝜃∕2)between scattering vector𝑞and scattering angle𝜃.

The average 𝑔 = ⟨1 − cos𝜃⟩is known as the anisotropy factor. For 𝑔 = 1the scattering is isotropic:

The same intensity is scattered to𝜃and to𝜋−𝜃, so that the correlation between incoming and outgoing direction is lost. This is the case if the structure factor can be neglected, i.e. 𝑆(𝑞) ≃ 1, and the diameter𝜎 of the particles is small compared to the wavelength𝜆(Rayleigh scattering for𝜎 ≪ 𝜆in dilute systems).

Anisotropic scattering is indicated by𝑔 <1. It occurs in dense particle systems where𝑆(𝑞) ≠1and/or for particles with𝜎 ≳ 𝜆which must be described as Mie scatterers [112]. They usually scatter significant proportion of the light in forward direction to small angles𝜃.

To transfer the description into a picture of diffusing light waves, we need know the average distance between two scatterers in a light path. It is given as the mean free path𝑙.

𝑙 = 1 It decreases with an increasing density of the scatterers𝜌and an increasing scattering cross section𝜎. A calculation of𝜎[112] involves the form factor𝐹(𝜑, 𝜃)or𝐹(𝑞)and additionally the structure factor𝑆(𝑞) in dense systems. Since the structure peaks of𝑆(𝑞)increase with increasing density𝜌, the mean free path𝑙 is no more inversely proportional to𝜌in dense systems. For systems with anisotropic scattering 𝑔 < 1it makes sense to define an average length𝑙^∗, the so called transport mean free path, after which light has travelled long enough so that the correlation of its direction with the incoming direction is lost.

A natural definition is to divide𝑙by the anisotropy factor𝑔[110]:

𝑙^∗ = 𝑙 For the second equality one uses Equation7.6. This makes clear that𝑙^∗depends primarily on the values of𝐹(𝑞)and𝑆(𝑞)for large𝑞 close to the backscattering direction𝑞 = 2𝑘₀𝑛_{ef f}.

Now we can rewrite the autocorrelation function 𝑔₁(𝑡) in terms of path lengths𝑠 = 𝑛𝑙 instead of the number of scatterers𝑛. We combine Equation7.3with7.4,7.5,7.6and7.8to get:

𝑔₁(𝑡) =∑

Here, eventually we go to the continuum limit and write the sum over all occurring path lengths𝑠as an integral over the distribution𝑃(𝑠). One should note that this integration implies anincoherentsum in a sense that each light path contributes independently to the decay of𝑔₁(𝑡) [109]. Hence the correlation function in Equ.7.9is comparable to the incoherent intermediate scattering function𝐹_𝑠(𝑞, 𝑡)for single light scattering, which is equally a function of the MSD (in the Gaussian approximation, cf. Equ.1.23).

7.1 Theoretical background It has been found that coherent parts make up for a fraction of𝑐⁻¹with𝑐≈ (𝑘₀𝑙^∗)(𝑘₀𝐿)[109], where𝐿is the minimum distance that light must travel through the sample. Thus for𝑙^∗ ≫ 𝜆coherent contributions to the decay of𝑔₁(𝑡)are negligible.

It is also possible to rewrite𝑔₁(𝑡) in terms of a diffusion coefficient𝐷 = ⟨Δ𝑟²(𝑡)⟩∕(6𝑡). Now one can also include the dependence of𝐷on particle interactions by taking𝐷(𝑞) =𝐷_𝑠ℎ(𝑞)∕𝑆(𝑞), whereℎ(𝑞) = 𝐻(𝑞)∕𝐻(∞)describes the hydrodynamic interactions and𝐷_𝑠 is the short time diffusion coefficient. In this case one can omit the approximation applied in Equ.7.4, that requiredΔ⃗𝑟and𝑞⃗to be independent:

𝑔₁(𝑡) = With the use of visible light, the latter approximation is valid in systems with big particles (𝜎 >2µm) [109]. For those one can equally use Equation7.9, that directly connects𝑔₁(𝑡)to the MSD. For particles of a size𝜎= 412nm the approximation leads to a deviation of about20 %[109] in the decay time. Thus, one should have in mind that the error might be even larger for smaller particles.

It is convenient to hide the specialties of the particle system in a characteristic decay time𝜏₀: 𝜏₀ =[

(𝑘₀𝑛_{ef f})²𝐷_𝑠]−1

= 6𝑡⋅[

(𝑘₀𝑛_{ef f})²⟨Δ𝑟²(𝑡)⟩]−1

, (7.11)

Then one obtains a more clear and catchy form for the autocorrelation function:

𝑔₁(𝑡) = the correlation decreases by a factor ofexp(−2𝑡∕𝜏₀). Note thatexp(−2𝑡∕𝜏₀) is exactly the correlation function for single scattering (DLS) at an angle of90^◦(𝑞=^√2𝑘₀𝑛_{ef f}).

Summary of approximations

Most important to obtain an explicit expression for𝑔₁(𝑡)is the assumption of independent light paths, which should be well-fulfilled for𝑙^∗ ≫ 𝜆. Somehow connected to this is the assumed independence of successive scattering events in each light path. If𝑙 ≫ 𝜆and if the correlation between particle positions (seen e.g. in𝑔(𝑟)) has decayed at a distance of 𝑙, this should be valid as well. Possibly problematic for glassy systems could be the assumption of a Gaussian distribution for the displacementsΔ⃗𝑟_𝑖of the scatterers. To obtain Equation7.9, the scattering vector𝑞in a single scattering event must be uncorrelated to the displacement of that scatterer. Collective movements, that occur e.g. in hydrodynamic interactions, are therefore ignored. By usage of Equation7.10one could at least approximately include hydrodynamic interactions, but this goes beyond the scope of this thesis. Another approximation is done in Equation7.5, where the lag time𝑡must to be small enough, so that ⟨Δ𝑟²(𝑡)⟩ ≪ 𝜆². However, in most of the cases, the average path length is much larger than𝑙^∗ so that the factor𝑠∕𝑙^∗ in the exponent of Equations7.9 and7.10leads to a decay of𝑔₁(𝑡)long before the MSD becomes comparable to the wavelength𝜆.

7.1.2 Diffusion equation and path length distribution𝑃(𝑠)

In the diffusion approximation (for length scales larger than𝑙^∗) one can describe the energy density𝑈(⃗𝑟, 𝑡) of light with the diffusion equation

𝜕𝑈(⃗𝑟, 𝑡)

𝜕𝑡 =𝐷_𝑙∇²𝑈(⃗𝑟, 𝑡) with 𝐷_𝑙= 𝑐𝑙^∗

3 . (7.13)

The diffusion coefficient𝐷_𝑙contains the speed of light in the medium𝑐 =𝑐₀∕𝑛_{ef f}and the transport mean free path𝑙^∗, the distance after which the direction of light is randomized, the division by3accounts for the number of space dimensions. For a solution of Equ.7.13one also needs initial and boundary conditions.

The idea is to compute the outgoing flux𝐽⃗_out(⃗𝑟_out, 𝑡)at a position⃗𝑟_out at the boundary of the sample after an instantaneous light pulse has started to diffuse in the sample at time 𝑡 = 0. Since light needs the time𝑡=𝑠∕𝑐to reach the position of the detector at⃗𝑟_out, the distribution𝑃(𝑠) is directly proportional to 𝐽⃗_out(⃗𝑟_out, 𝑡=𝑠∕𝑐). Because the light path is only randomized after a distance of𝑙^∗the starting point for the light pulse is not directly at the border of the sample. Instead one defines it to start at a distance𝑧₀ ≈𝑙^∗ inside the sample. Therefore, as an example, the initial condition for a sample with a planar boundary at 𝑧= 0and a parallel beam coming from𝑧 <0is given by:

𝑈(𝑧, 𝑡= 0) =𝑈₀𝛿(𝑧−𝑧₀, 𝑡) (7.14)

Boundary conditions are usually obtained by requiring that the incoming net flux into the sample is zero [110]. This results in the condition

( 𝑈+ 2

3𝑙^∗ ̂⃗𝑛⋅∇𝑈) |

||^{⃗𝑟∈𝜕𝑉} = 0, (7.15)

where𝜕𝑉 means the surface of the sample and̂⃗𝑛is a unit vector which is always orthogonal to that surface and directing out of the sample. Other boundary conditions, e.g. absorption of light at the surface, were investigated, but Equ.7.15gave the best agreement with experiments [109]. With a solution𝑈(⃗𝑟, 𝑡)for the full diffusion problem, the path length distribution is given by

𝑃(𝑠) ∝|||𝐽⃗(⃗𝑟_out, 𝑡)|||=𝐷_𝑙|||̂⃗𝑛⋅∇𝑈(⃗𝑟_out, 𝑡)|||= 𝑐

2𝑈(⃗𝑟_out, 𝑡=𝑠∕𝑐) (7.16) Here one makes use of Fick’s law of diffusion and the boundary condition in Equation7.15. Since the autocorrelation function𝑔₁(𝑡)is basically a Laplace transform of𝑃(𝑠)(cf. equs.7.9and7.10) it is often convenient to solve the diffusion problem in Laplace space and obtain directly an expression for𝑔₁(𝑡) [109].

7.1.3 Transmission geometry

The detailed derivation of the path length distributions for arbitrary geometries and the final computation of𝑔₁ uses nontrivial mathematics and goes beyond the scope of this description. Only the formulas for the two most important scattering geometries (see Fig.7.2), that were also used in this work, are presented here including comments.

In transmission geometry, a parallel beam of coherent laser light uniformly illuminates one side of a slab with a thickness𝐿in𝑧direction and an infinite extent in the other two directions. In experiments this idealized situation is well approximated using a beam width much larger than𝑙^∗and a sample that is larger than the beam width by more than10𝑙^∗. As mentioned in the preceding paragraph, light is considered to

7.1 Theoretical background

Figure 7.2: Idealized transmission and backscattering geometry for a slab of thickness𝐿with infinite extent perpendicular to the incoming beam. The coherent parallel light is considered to be homogeneous with an infinite width, like an idealized plane wave. Precise angles of the detector with respect to the incoming beam are arbitrary. Sketches of typically detected light paths are given in red.

start diffusing from a plane at𝑧₀≈𝑙^∗. However, the exact value of𝑧₀is not very important since𝐿 ≫ 𝑙^∗ is required anyway. At least𝐿 >10𝑙^∗is necessary for the diffusion approximation to be valid. Using the characteristic decay time of the particle system𝜏₀, one obtains the autocorrelation function [109]:

𝑔₁(𝑡) =

As the approximation suggests, the shape of this function somewhat resembles an exponential decay. But particularly for large𝑡, deviations from exponential behaviour are observed. The characteristic decay time is𝜏₀(𝑙^∗∕𝐿)². Recalling that𝑔₁(𝑡) decreases by a factorexp(−2𝑡∕𝜏₀)for each step of the random walk, one infers that the mean number of steps is⟨𝑛^∗⟩= ¹

2(𝐿∕𝑙^∗)². This corresponds to an average path length of⟨𝑠⟩ = 𝐿²∕(2𝑙^∗). An easy way to derive this is to consider the growing mean squared displacement 6𝐷_𝑙𝑡 of the photons diffusing in the sample: In order to diffuse an average distance of𝐿, which is the sample thickness, light needs the time𝐿²∕(6𝐷_𝑙) =𝐿²∕(2𝑐 𝑙^∗). A division by the speed of light𝑐yields the average length⟨𝑠⟩of a light path. Since all light paths must travel the distance𝐿in𝑧direction, the distribution𝑃(𝑠)has a sharp maximum at the average path length⟨𝑠⟩.

From the definition of𝜏₀in Equation7.11one can see that each scatterer moves an average distance of Δ𝑟₀ =√ transmis-sion geometry is very sensitive to the dynamics in the sample. An average translation of the scatterers by only a fraction𝑙^∗∕𝐿of the wavelength is enough to destroy the correlation. By adjusting the thick-ness of the sample cell the sensitivity can be adjusted, for instance starting from a decay for an average displacement of∼ 50nm in a thin cell down to several nanometers in a thick cell.

7.1.4 Backscattering geometry

The path length distribution𝑃(𝑠)for the backscattering geometry is a lot broader than that for the trans-mission case. Both very short light paths with a length of a few𝑙^∗ and very long paths are possible.

Incoming beam and detector are on the same side of the sample (see Fig.7.2). The exact angle at which light is collected is not important. But it is substantial that the incoming parallel beam is expanded much

wider than𝑙^∗. The detector needs to focus at a spot near the centre of the illuminated area. If the latter two conditions are not fulfilled, the path length distribution is qualitatively different. For instance, if the detector is focused to the border of the illuminated area, the contribution of short paths becomes smaller while that of large paths becomes longer.

For a computation of𝑔₁(𝑡)one again assumes that the diffusing light starts from a plane at𝑧=𝑧₀ =𝛾𝑙^∗ in the sample. The exact value of𝛾 ≈ 2(see [110]) depends on the sample and the sample cell (e.g.

on the particle sizes, thickness of glass walls, reflectivities etc.). Additionally, there is a dependence of𝛾 on the polarization of incoming and detected light, which is measurable by putting a polarizer in front of the detector. If the polarizations of incoming and detected light are perpendicular the very short paths coming from one single backscattering are not detected, because single scattering preserves the polarization. This yields an increase in𝛾. On the other hand, if incoming and detected light have mainly parallel polarizations,𝛾decreases compared to a measurement without a polarizer.

For the longer paths𝑠 ≫ 𝑙^∗, one can show that𝑃(𝑠) ∝𝑠^−3∕2. Therefore,𝑃(𝑠)only decays algebraically for𝑠 →∞. This means that also longer paths contribute to the decay of𝑔₁, not only the more frequent short paths. Since many scatterers contribute to the phase shift in long paths, very small displacements Δ⃗𝑟_𝑖are enough to destroy their correlation. Thus, the long paths are responsible for the decay of𝑔₁(𝑡) at small lag times 𝑡. On the other hand, the longer lag times are dominated by the correlations of the short paths.

For a sample of thickness𝐿, the exact solution of the diffusion equation yields [109]:

𝑔₁(𝑡) = accu-rate normalization𝑔₁(𝑡 = 0) = 1. Since paths of many different lengths𝑠contribute, the decay upon increasing the lag time𝑡is slower than exponential. By comparison ofexp(−𝛾√

6𝑡∕𝜏₀)with the decay exp(−2𝑡∕𝜏₀)for90^◦ single scattering, one comes to the conclusion that for𝑡 >3∕2𝛾²𝜏₀the correlation for backscattering DWS is actually larger than that for single scattering. It is however clear that correla-tions of multiply scattered light must decay faster. Therefore, the result given in Equation7.19can only be valid for small lag times𝑡 ≪ 𝜏₀. This is not only true for the approximation, but also for the exact solution of the diffusion equation. The reason is that for𝑡 > 𝜏₀ the correlations mainly stem from the shortest paths. But for short paths, many of the approximations mentioned above (see7.1.1), including the diffusion approximation, are not valid.

Still DWS in backscattering geometry is a very useful tool especially for glassy samples. It allows to probe the dynamics at many length scales at once. Small movements of the particles contribute to the decay of𝑔₁(𝑡)at short lag times. But even quite large displacements do not yield a total decay, because the dominating short light paths are still correlated. Using the definition of𝜏₀in Equation7.11together with the approximated𝑔₁(𝑡)from above, we see that𝑔₁(𝑡)has decayed to1∕𝑒with an average displacement of the scatterers by:

Δ𝑟₀= (𝛾𝑘₀𝑛_{ef f})⁻¹ =𝜆⋅(2𝜋𝑛_{ef f}𝛾)⁻¹ (7.20) This is only a factor∼𝛾earlier than in a single scattering experiment. A total decay requires an average displacement by about one wavelength𝜆.

Im Dokument Glass transition in charged colloidal suspensions (Seite 194-200)