Light Scattering From Large Particles

C H A P T E R

10 Light Scattering From Large Particles: Lorenz2Mie Theory In this chapter we move from small particles to large particles, i.e., particles that have a size with the same order of magnitude as the wavelength of light. This will allow the extraction of more information than for small particles. However, there are two major obstacles: the theory necessary to describe the scattering process is much more complicated, and multiple scattering can be an essential problem when studying such large particles. Static light scattering (SLS) experiments from large particles contain enough information for the determination of size, shape, and internal structure if their size is comparable with the wavelength of the light; i.e., the particles should be in the size regime of about 100 nm up to several micrometers. If the particles are larger than several micrometers, we are dealing with Fraunhofer diffraction,254 which probes only the silhouette of the particles. We shall see in the following that in the above size regime we cannot use Rayleigh
Debye
Gans (RDG) theory any more, even in the case of low refractive index ratios we have to apply Lorenz
Mie theory.255,256 The original work of Lorenz and Mie related to spherical particles only, but these names are now often used for arbitrarily shaped scatterers in this size. This theory describes the propagation of electromagnetic radiation in an inhomogeneous dielectric medium, and takes into account that the interaction of the electric field with the inhomogeneities becomes more complicated. We can no longer assume that the electric field strength is the same everywhere in the probe volume. In order to find this field distribution, we have to solve the Maxwell equations for the actual scattering problem, i.e., for a particle of a particular shape, size, refractive index, and orientation embedded in a homogeneous surrounding medium with a different refractive index. The solution of this problem is far from trivial, and cannot be found without complex mathematical methods. But there are at least modern numerical algorithms that allow computation of the scattering problem for simple regular particles like spheres and prolate and oblate ellipsoids, and cylinders.257 Nonspherical structures can be calculated in a series of different orientations, and the results are then averaged to take into account that colloidal particles are usually not oriented. Spherical particles with perfect symmetry

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may lead to sharp resonant scattering. This effect is utilized in surface-enhanced Raman scattering. For many colloids in chemistry (macromolecular complexes in water or organic solvent) we rarely find this perfect symmetry, and the refractive index ratio is also rather small, so that we are in the transition regime between RDG scattering and resonant scattering, which is sometimes called the preresonant regime. The Lorenz
Mie regime is the most difficult one in terms of theoretical description. The electromagnetic radiation interacts nonlinearly with the particles, and the strength of the electric field depends highly on the position (close to and inside the particle). Also, the different possible states of polarization of the wave are important. They can be described by the four Stokes parameters being components of the so-called Stokes vector. This holds for the incident and the scattered wave. The Stokes vector of the incident wave is related to the Stokes vector of the scattered wave by the Mu¨ller or Stokes matrix M.27,253,258,259 The 16 components of this matrix all depend on the scattering vector q, and the orientation of the particle. The one-one element (M11) can be regarded as the classical scattering function for nonpolarized experiments. All these matrix elements can be calculated by solving the Maxwell equations. New numerical methods, such as the T-matrix method, can solve this difficult problem for regular bodies within reasonable computing time.257 We will only discuss the classical scattering curves (M11), because all other elements, which can be measured in principle, are difficult to interpret, i.e., there are no solutions for the inverse scattering problem. This inverse problem is to find the structure of the scatterer from its scattering pattern. It is difficult enough to do this for the M11 element. With increasing size (above several micrometers) the field of the electromagnetic wave hardly penetrates into the particle, so the scattering process is approximated by the interaction of a planar wave with the cross-section, or aperture, of a particle.254 This process is called Fraunhofer diffraction if only the far-field solution is important (as it is for SLS experiments). This means that Fraunhofer diffraction cannot give information about the overall shape or the internal structure. The same is true for the RDG approach for another reason: in the RDG regime the particles are too small to scatter an essential part of the total scattering curve into the accessible regime of an SLS instrument. If we use an HeNe laser in aqueous systems and scattering angles from 10 degrees to 150 degrees, we get the regime 0.0023 # q # 0.0255 nm21. This regime may be modified by using a special smallangle instrument, or a decrease in the wavelength. These details will be discussed in the experimental section. In summary, for particles larger than 100 nm we cannot use RDG theory; Lorenz
Mie and the Fraunhofer regime allow particle sizing of polydisperse systems in the size regimes of about 100 nm up to a few micrometers, or starting from a few micrometers up to about 1000 μm. Only in the Lorenz
Mie regime is there a chance to determine the shape and structure of monodisperse systems. In the following text, we want to discuss the scattering problem in the Lorenz
Mie regime, i.e., to see the influence of increasing size and/or contrast on the scattering data, and to learn about the main features of the corresponding theory when compared to RDG theory. All the discussion in the next section relates to ideally diluted systems, i.e., we do not take into account any particle interaction. Here we find another important difference

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to small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS): due to the high scattering power of these large particles we run into the problem of multiple scattering with increasing concentration much earlier, before we see the influence of excluded volume effects. Particle interactions could only dominate for highly charged particles, or in systems with vanishing contrast (close to matched index). In all other cases we have to worry primarily about the problem of multiple scattering, and not so much about particle interactions. We have already mentioned that the solution of the scattering problem is a difficult task, especially if the particles are not spherical. So we can imagine that the solution of the inverse problem will be even more difficult. This situation may be illustrated best by the fact that the scattering curves will be different not only in their total intensity, but also in their analytical appearance (q dependence) if only the contrast is changed but size and shape is kept constant. In the following, we shall develop a procedure for the solution of the inverse problem, similar to the strategies used in small-angle scattering (SAS). We start with some basic definitions: the refractive index is in most cases real, i.e., the absorption of the solvent nvs and of the particles npv is negligible. If nvs is high, we cannot perform light scattering experiments because all light will be absorbed, the scattering curves are not highly dependent on npv, so this assumption is reasonable. We use the ratio m (or exactly m0 ): m0  m 5 np =n0  np =nS

(10.1)

because only this ratio (“contrast”) is essential. In addition, it is tradition and useful to define a dimensionless size parameter: α 5 ka 5 2πa=λ

(10.2)

This is the ratio between the circumference of a sphere with radius a relative to the wavelength. The definition α 5 ka indicates that α describes the phase shift of the wave penetrating into the center of the sphere, and it is easy to show that 2α(m
l) is the phase difference between rays passing through the sphere and through the solvent for the same distance 2a. It can be shown that the RDG approximation can only be used if the condition: 2αðm 2 1Þ{1

(10.3)

holds; in all other cases we have to solve the Maxwell equations (i.e., use Lorenz
Mie theory). It should be mentioned here that there is an improved extended RDG theory that assumes a constant electric field and a corresponding polarization everywhere in the particle, but takes into account the phase shift due to the different refractive index inside and outside the particle. We shall not use this method here; wherever RDG does not apply we will use full Lorenz
Mie theory. In some textbooks, one is told that RDG applies only for particles smaller than λ/20, which means a # π/20. This will certainly fulfill the condition in Eq. (10.3), but this condition can also be fulfilled for much larger particles, if m is small enough (Fig. 10.1).

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FIGURE 10.1 Range of applications for the different scattering methods as a function of particle size. For the following simulation examples in this chapter, we use λ 5 546 nm, i.e., a B 65α.

SOLUTION OF THE INVERSE SCATTERING PROBLEM FOR HOMOGENEOUS MONODISPERSE SYSTEMS In the following text we shall show that the IFT method (see Chapter 8: Numerical Methods) can also successfully be applied to static lightscattering data from large particles in the preresonant regime, i.e., at low refractive index ratios m. In this application, it is essential that IFT minimizes termination effects, as the q range accessible to the experiment is restricted to about 11/2 decades in reciprocal space. The scattering curve in Lorenz
Mie theory is no longer simply the Fourier transformation of the p(r) function of the particles. But we shall demonstrate that the p(r) function calculated from such scattering data by the IFT method gives useful information for the determination of shape, size, and internal structure, even though the application of this technique is strictly correct only for RDG data. The p(r) function, according to Eq. (1.39), relates to the spatially averaged correlation function of the refractive index profile inside the particle or, exactly speaking, the index difference to the surrounding medium.

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SPHERICAL PARTICLES

SPHERICAL PARTICLES We start with this “simple” case, but we should keep in mind that it was about 70 years after the pioneering work of Lorenz and Mie that it became possible to solve the scattering problem for structures different from spheres or spherical cylinders of infinite length. We can see clearly from Fig. 10.2 that the scattering intensity for one particle size and shape is highly dependent on the contrast, m. The zeros, typical for perfect spheres in RDG theory, are increasingly smeared out and slightly shifted to smaller q values, and the decline of the main maximum is steeper. All these changes increase the difficulties of shape determination from the scattering curves. The q regime shown in this figure corresponds to scattering angles of 10
150 degrees (qmin 5 0.0027 nm21, qmax 5 0.03 nm21). For the wavelength of a HeNe-laser, 10 degrees would give qmin 5 0.0023 nm21; some instruments allow scattering angles smaller than 10 degrees, but we see that in general we may expect a value of qmin $ 0.002 nm21.

log I (q)

4.0

α = 4.6

3.0

m = 1.3

2.0

m = 1.1

1.0

m = 1.2

m = 1.0 0.0

0.0

0.01

0.02 q (nm–1)

0.03

FIGURE 10.2 Normalized Mie-scattering functions for spheres with α 5 4.6 (D C 600 nm for λ0 5 546 nm) for different m values. The curve with m 5 1.0 corresponds to the RDG approximation, normalized to the same forward scattering intensity.

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For every arbitrary particle there is a pair
distance distribution function (PDDF) p(r), but this function is no longer directly related by Eq. (1.39) to the scattering intensity I(q) in the case of Lorenz
Mie theory. This PDDF, however, was very helpful in the solution of the inverse scattering problem in the regime of RDG theory. So it is obvious that we would like to find at least an approximation for the PDDF by applying the method of IFT to Mie-scattering data.260 The results of such calculations using the scattering data from Fig. 10.2 are illustrated in Fig. 10.3. Let us start the discussion of these results with the PDDF for m 5 1.0, the RDG case, where Eq. (1.39) is a correct description of the scattering problem. The oscillations with very small amplitude at r . D are due to the termination in reciprocal space. With increasing m value these oscillations increase, but the PDDF is nearly unchanged for r , D, and so this function allows a correct shape determination; only the crossing point with the abscissa is shifted from r 5 D to somewhat higher r values with increasing m values pretending larger particles. This effect is small, varies systematically with the contrast, and could be eliminated for known m values. So we see that this approximation can give a very useful result for Lorenz
Mie theory; the approximation is better, the smaller the m value. In the next example, we keep the contrast fixed with m 5 1.1 but vary the size parameter α from 2.2 to 7.0. The corresponding scattering functions and PDDF’s resulting from IFT calculations are shown in Fig. 10.4. The abscissa in real space is r/D to scale all particles to the same regime. In reciprocal space, we see the expected shift of the curves to smaller q values for increasing size. All scattering curves are normalized to the same forward scattering intensity. The corresponding p(r) functions show hardly any difference in the

α = 4.6

p (r)

4

m = 1.3 m = 1.25

3

m = 1.2 2

m = 1.15 m = 1.1

1

m = 1.05 m = 1.0

0 0

200

400

600

800

1000

r (nm)

FIGURE 10.3 PDDF calculated by indirect Fourier transformation from the scattering data shown in Fig. 10.2 with qmin 5 0.0032 and qmax 5 0.0296 nm21.

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SPHERICAL PARTICLES

α = 2.2 α = 4.6 α = 7.0

log I (q)

4.0

3.0

1.0

α = 2.2 α = 4.6 α = 7.0

p (r)

2.0

0.5 1.0

0.0 0.0

FIGURE 10.4

0.0 0.01

0.03 0.02 q (nm–1)

1/2

1

3/2 r/D

Scattering functions (left side) and PDDFs (right side) for spheres with m 5 1.1 and α varying

from 2.2 to 7.0.

regime r/D , 1.0, and again the oscillations around the abscissa for r/D . 1. These oscillations are the largest for α 5 2.2, due to the severe termination (cut-off at the first minimum). So we see that this IFT approximation is applicable at least up to size parameters α of 7.0 (D  910 nm) for m 5 1.1. We shall see in the following that the systematic deviations increase with size and contrast. The fact that the p(r) functions can be calculated with such low systematic deviations suggests the idea to try to deconvolute this PDDF into the radial refractive index profile n(r) (actually, it is again the difference Δn(r) to the surrounding medium). The results of such a deconvolution for simulated data of a sphere with D 5 600 nm (α 5 4.6) and m 5 1.15 are shown in Figs. 10.5 and 10.6. The solution with 10 equidistant steps in Δn(r) is not only close to the correct profile (Fig. 10.5), but gives also a good fit to the p(r) function (Fig. 10.6). The calculation of optimized single- or two-step models also gives a reasonably good fit. All three solutions show an apparent increase in size of the particle of about 5% due to the nonlinear interaction of the electric field with the particles. These deviations from the ideal step-function behavior increase with α and m. All calculations shown until now were made with simulated data, and it has to be shown that these ideas can also be applied to experimental data. The following two figures show the results of the first corresponding test. These experiments were performed on an old FICA 50 light scattering system (made by Sofica Co). This instrument was supplied with an Hg-vapor lamp, and was not designed to be used at scattering angles below 20 degrees. Even with some technical modifications, the experimental errors were rather

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1.2 1.0

Δn(r)

0.8 0.6 0.4 0.2 0.0 100

200

–0.2

300 r (nm)

500

FIGURE 10.5 Radial refractive index profile Δn(r) calculated from the p(r) function of a sphere with α 5 4.6 (D 5 600 nm) and m 5 1.15. Solutions with a single-step (


), two-step (---) model and for 10 equidistant steps of fixed spacing (sss).

 

5.0

p (r)

4.0 3.0 2.0 1.0 0.0 200 –1.0

400

600

800 1000 r (nm)

FIGURE 10.6 PDDF of a sphere with α 5 4.6 (D 5 600 nm) and m 5 1.15 (o o o), together with the fits of the deconvoluted profiles discussed in Fig. 10.5.

high at low q values. The measurements were performed with conical cuvettes to avoid internal reflections (Figs. 10.7 and 10.8). We can see that the results from experimental data are in very good agreement with the results from numerical simulations.

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SPHERICAL PARTICLES

4.5 4.0

log I(q)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.00 0.005 0.01 0.015 0.02 0.025 0.03 q (nm–1)

FIGURE 10.7 Experimental scattering data of a latex sample, nominal diameter 481 nm (o o o), together with the fitting curve (________) produced by the indirect Fourier transformation algorithm.

6 5

p (r)

6 3 2 1 0 100

200

300

400

500

700

–1

900 r (nm)

FIGURE 10.8 PDDF calculated from the scattering data shown in Fig. 10.7.

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NONSPHERICAL PARTICLES (SPHEROIDS WITH ELLIPTICAL CROSS-SECTION) The essential question now is whether we can apply this type of evaluation only to spherical systems, or also to other structures of arbitrary shape. In order to investigate this question we need the solution of the scattering problem for nonspherical structures. The scattering problem for spheres was already solved in 1908 by G. Mie, and even in the standard textbook of M. Kerker258 on elastic light scattering published in 1969 we mostly find spheres. An increasing number of papers on scattering of electromagnetic radiation by— ideally conducting—cylinders in the 1960s and 1970s were triggered by military research projects. However, efficient numerical algorithms to solve the scattering problem of dielectric spheroids (extended boundary condition method, T-matrix method) have existed only since 1975,261 i.e., it took more than 60 years to make the step from spheres to spheroids! So it is not astonishing that there is even less knowledge of the solution of the inverse problem of nonspherical structures. The availability of computer routines for the solution of the scattering problem of spheroids257 made it possible to test the evaluation techniques in real space for nonspherical structures.262 It should be mentioned that the routines for the solution of the scattering problem work with series expansions of spherical harmonics, and the convergence decreases dramatically with increasing axial ratio, which limited the following simulations to a maximum ratio of 1:5.

Prolate Spheroids The results for prolate spheroids in reciprocal and real space are shown in Figs. 10.9 and 10.10. The axial ratio is fixed to a/c 5 5, with half-axes 750 and 150 nm, and the m value varies between 1.0 and 1.3. The scattering curves (normalized to the same I(0)) show the dependence on the m value at higher q values, but these differences are difficult to interpret. The PDDFs are identical for higher r values and show weak oscillations due to the unavoidable termination effects. Essential differences for different m values are found only in the regime of the maximum that represents the cross-section of the spheroid.28 This is a strong indication that resonances need a three-dimensional volume element for their emergence. This is an important point for our discussion, because it would signal that it is not the overall size that is essential for deviations from RDG theory, but the maximum dimension of a three-dimensional volume element. This would result in much larger structures that can be evaluated with this technique. In order to investigate this question, we compare the p(r) functions of the spheroid with the p(r) function of a sphere, where the sphere is chosen in size so that the position of the maximum is the same as for the spheroid for m 5 1.0; in addition they are normalized to the same height of the maximum (see Fig. 10.11). If the m value is increased from m 5 1.0 (thin lines) to m 5 1.3 (thick lines), we see the same shift of the position of the maximum for the sphere (broken lines) and for the prolate spheroid (full lines). This shift is essentially the result of the Mie effect, as already shown in Fig. 10.10. This is clear evidence for the supposition that the resonance effects are

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197

Prolate spheroid k.a = 11.5 a/c = 5 m = 1.0005 m = 1.1 m = 1.2 m = 1.3

log I (q)

4.0

3.0

2.0

0.0

0.01

0.02 q (nm–1)

0.03

FIGURE 10.9 Scattering curves for prolate spheroids with an axial ratio a/c 5 5 and k.a 5 11.5 for different m values.

Spheroid k.a = 11.5 a/c = 5

4.0

p (r)

m = 1.0005 m = 1.1 m = 1.2 m = 1.3 2.0

0.0

0

500

1000

1500

2000

r (nm)

FIGURE 10.10

PDDFs calculated from the scattering functions for prolate spheroids shown in Fig. 10.9.

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p (r)

4.0

2.0

0.0

0

500

1000

1500

2000

r (nm)

FIGURE 10.11 Comparison of the p(r) functions of a prolate spheroid (full lines) with a sphere (dashed lines) for m 5 1.0 (thin lines) and m 5 1.3 (thick lines).

related to three-dimensional volume elements. This finding is very important for many applications in chemistry and biology, where the contrast is usually rather low (m around 1.1) and the structures are often not compact and globular, so that the interpretation of SLS data can be performed, in most cases, in real space without problems.

Oblate Spheroid Large oblate spheroids with k.a 5 11.2 (a 5 730 nm) and axial ratios c/a from 0.2 to 1.0 (sphere) are investigated next in the regime 1.0 # m # 1.3 (Fig. 10.12). The effects are small for low m values (m # 1.1); this holds also for the sphere even though the diameter is nearly 1.5 μm! We can clearly see the increase of the resonance effects for higher m values when progressing from the flat oblate spheroid (c/a 5 0.2) to the sphere (c/a 5 1). The PDDF of the sphere apparently transforms to the PDDF of a hollow sphere (m 5 1.2), to the PDDF of a inhomogeneous sphere (m 5 1.3, see also Figs. 3.19 and 3.20). This behavior can be understood because the electric field can no longer penetrate the sphere homogeneously. With increasing size and contrast, the electric field is displaced from the center and forced to the surface. In cases of real resonances there is nearly no electric field strength inside the particle, but an extremely high field strength at the surface. This effect is used in the field of Raman spectroscopy where it is called surface-enhanced Raman scattering. The molecules under investigation are deposited on the surface of a perfect sphere with the right size parameter k.a to guarantee resonance. In addition, the spheres can be positioned by the light pressure of focused laser beams (optical levitation). It should be noted that the extremely high field resonances on the surface are only found for perfect spheres. Any deviation from spherical symmetry leads to a dramatic decrease of the field strength on the surface.

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INHOMOGENEOUS PARTICLES

m = 1.0005 m = 1.1 m = 1.2 m = 1.3

2.0 c/a = 0.2

p (r)

p (r)

k.a = 11.2

1.0

0.0

2.0

c/a = 0.4

1.0

0

500

1000

1500

0.0

2000

r (nm)

0

500

1000

1500

2000

1000 1500 r (nm)

2000

4.0

p (r)

p (r)

3.0

2.0

2.0 c/a = 1

1.0

1.0 c/a = 0.8

0.0

0

500

1000 1500 r (nm)

2000

0.0

0

500

FIGURE 10.12 Dependence of the Mie-effects from the axial ratio c/a for oblate spheroids of size k.a 5 11.2 with varying m-values from 1.0 to 1.3.

INHOMOGENEOUS PARTICLES We have already shown for the example of a homogeneous sphere that we can calculate the radial refractive index profile Δn(r) for spherical structures by a deconvolution of the PDDF. Now we want to check if this procedure can be generally applied to inhomogeneous spherical structures. This test will be performed in a first step on a core
shell structure with perfect spherical symmetry. In a second step we discuss the influence of deviations from such symmetry. The following example shows simulated data for a core
shell model with a core size Rc 5 400 nm and a shell radius Rs 5 600 nm, the m values are chosen to be mc 5 0.9 and ms 5 1.1 (dashed lines in Figs. 10.13
10.15). The full lines in these figures correspond to the inverse structure with the same dimensions Rc and Rs, but higher refractive index in the core, i.e., mc 5 1.1 and ms 5 0.9. Both models would show the same scattering behavior according to the Babinet principle in the RDG approximation. They would only differ in the sign of the (unmeasurable) amplitude. However, this does not hold for Mie theory, as we can easily see from Fig. 10.13, i.e., the interaction of light with the particle is quite different if the electric field hits a higher or lower refractive index at the shell. The scattering curves differ not only in forward direction (I(0)) but in the whole q range. This can also be understood qualitatively

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6.0

log I (q)

5.0

4.0

3.0

2.0

1.0 0.0

0.01

0.02

0.03

q (nm

–1)

FIGURE 10.13 Scattering curves for two spherical core
shell models with Rc 5 400 nm and Rs 5 600 nm. The dashed line corresponds to mc 5 0.9 and ms 5 1.1, while the full line represents the inverse structure with mc 5 1.1 and ms 5 0.9.

p (r)

600

300

0 0

FIGURE 10.14

500

1000

1500 r (nm)

PDDFs calculated from the scattering curves shown in Fig. 10.13.

by classical optical ray-tracing arguments which can be used for much larger systems, like rain drops, etc. (The rays deviate towards the normal of the surface when entering from a less dense medium into a denser one, but they are refracted in the opposite sense when entering a less dense medium. So the results would be quite different.)

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INHOMOGENEOUS PARTICLES

Δn(r)

4 2 0 0

250

500

0

250

500

–2

750 r (nm)

–4

Δn(r)

4 2 0 –2

750 r (nm)

–4

FIGURE 10.15 Refractive index profiles Δn(r) calculated by deconvolution of the p(r) functions shown in Fig. 10.14. Results for the inverse model are shown on the lower part of the figure.

The PDDFs differ in their amplitudes, as the area under the p(r) function is proportional to I(0). Besides this general difference, we see that the main differences are in the regime of the outer diameter of the shell (second maximum of p(r)). The model with the higher refractive index in the shell (dashed line) appears to be slightly enlarged to the nominal diameter of 1200 nm, while the inverse model is apparently reduced in size. This effect is more obvious in the radial density profiles Δn(r) calculated by deconvolution from p(r), see Fig. 10.15. These profiles are calculated as step functions with 10 equidistant fixed steps, and as an optimized two-step model with variable step widths. The core
shell model with the higher refractive index in the shell shows an apparent size increase of about 4%, while the inverse model appears reduced in size by 5%; the dimensions of the core are recovered in both cases within 1%. To summarize, one can say that the technique gives reasonable profiles. We should, however, not forget that the deconvolution of the p(r) function needs a priori information about the overall sign for the radial refractive index profile, i.e., we cannot determine from the light scattering data if the higher index is located in the core or in the shell. The influence of moderate deviations from spherical symmetry has also been discussed based on a slightly oblate core
shell model with mc 5 0.9 and ms 5 1.1.262 The shell had a nearly constant thickness of about 200 nm, the axes of the core were 400 and 280 nm, and the axes of the shell were 600 and 480 nm. The resulting radial profiles were a good approximation in terms of a spherical average of the given model. The Mie effect

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(deviation from RDG theory) was smaller than for the perfect sphere, in full agreement with the findings for homogeneous particles. All the above results are obtained for monodisperse systems. Monodisperse systems are rare in this size regime, so we shall discuss polydisperse systems in detail in the next section. This means that the first part of this chapter is primarily important to understand the Mie effects in terms of influence of size and relative contrast, while applicability to real systems is low, because there are very few monodisperse systems in this size regime.

STATIC LIGHT SCATTERING FROM POLYDISPERSE SYSTEMS Many systems in the size range between 100 nm and several micrometers, like emulsions, show a distinct polydispersity in size, and one can hardly find strictly monodisperse particles in practical applications. Well-known exceptions are latex particles, which are used as calibration standards for several techniques like electron microscopy. When assuming a certain shape—like spheres for emulsions—one is interested in a determination of the size distribution. Such an investigation is known as particle sizing. The most common optical methods for particle sizing are dynamic light scattering (DLS)263 and Fraunhofer diffraction.254 There exist several commercially available instruments. In SLS experiments, the angular dependent intensity of scattered light from noninteracting colloidal systems is influenced by the size, the shape, and the optical contrast of the particles. The most important parameter is the m value (optical contrast).264 To gain as much information as possible, we present a general approach to the problem of deducing as many parameters as possible from experimental data. In particular, we investigate the possibility of simultaneously determining the size distribution and the relative refractive index of polydisperse colloidal particles from SLS data. In the first part, we deal with single-scattering curves; an extended version for application to multiplescattering data253 is given in the next section. The application to experimental data needs to take into account the intensity contributions from the glass walls of the cylindrical sample cell of the light scattering device. Light scattered, e.g., to an angle of 30 degrees is partially reflected and will be detected as an additional contribution at 150 degrees. A new parameter must be introduced that represents the reflectivity of the cylindrical glass surface in the particular solvent. Although there is no interest in the particular value of this parameter, it must be determined in order to obtain the true values of the other parameters. Neglecting the reflectivity would yield a residual systematic error that cannot be removed by adjusting other parameters (e.g., relative refractive index or size distribution). We cannot simply use Eqs. (2.54) to (2.56) as in RDG, as the forward scattering intensity is no longer simply proportional to R6, and we have to take into account the influence of the refractive index ratio m of the particles. For the following, we have to assume that all particles have the same composition, i.e., the same refractive index ratio m. The angulardependent scattering curves of polydisperse systems can be regarded as a linear combination of Lorenz
Mie form factors P(q, R, m), where q is again the length of the scattering vector, R is the radius, and m is the relative complex refractive index of the particles in the particular solvent. If we proceed and regard the coefficients of the linear combination

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of B-splines as a continuous function D(R), we immediately obtain the Fredholm integral equation in question265,264: ðN IðqÞ 5 Dn ðRÞ Pðq; m; RÞdR (10.4) IðqÞ 5

ðN

0

Dv ðRÞPðq; m; RÞPð0; m; RÞ21=2 dR

(10.5)

0

IðqÞ 5

ðN

Di ðRÞPðq; m; RÞPð0; m; RÞ21 dR

(10.6)

0

or: IðqÞ 5

ðN

Di ðRÞP0 ðq; m; RÞdR

(10.7)

o

where P(q, m, R) is the particle form factor (scattering intensity) of a sphere with radius R and refractive index m, and: P0 ðq; m; RÞ 5 Pðq; m; RÞ=Pð0; m; RÞ

(10.8)

is the form factor normalized to 1 at q 5 0. Dn is the number distribution, Dv is the volume or mass distribution, and Di is the intensity distribution, i.e., it describes how much scattering intensity is created by the particles in the size regime between R and R 1 dR. This function is very sensitive to a small number of large particles, while Dn is the distribution at which small particles will easily be found. The relation: Di ðRÞCR3 Dv ðRÞCR6 Dn ðRÞ

(10.9)

is valid exactly only for spheres in RDG theory, and the deviations from this relation will be the higher, the less we satisfy the condition in Eq. (10.3), i.e., the higher the values of R and m (see Fig. 10.16). In nearly all light scattering instruments, we can neglect instrumental broadening (geometrical smearing effects). In addition, only monochromatic light is used. It was shown that nonspherical particles give broadened but reasonable size distributions when evaluated with the form factors of spheres.267 Therefore, one can simply use the Lorenz
Mie form factors of spheres calculated by means of the S1 program of Barber and Hill.257 The integral equation in the regime of Fraunhofer diffraction reads: ð Rmax IðqÞ 5 DðRÞJ12 ð2πnRθ=λ0 Þ ðR=θÞ2 dR (10.10) Rmin

The properties of the resulting distribution functions cannot be presumed to be different from those mentioned for the RDG regime, because in both cases the kernel is a damped oscillating Bessel function. We see that the three regimes: RDG, Lorenz
Mie, and Fraunhofer diffraction, have very similar mathematical descriptions. The most complicated kernel of the integral equation is found in the Lorenz
Mie regime. We will only focus on this case in this chapter.

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108 I0 107

106

105

104

Forward scattered intensity of spheres 103 m = 1.0005 m = 1.1 m = 1.2 m = 1.3

102 k·a

0

2.5

5

7.5

10

12.5

15

FIGURE 10.16 Dependence of the forward scattered intensity I0 of spheres as a function of the size parameter α 5 k.a and of the refractive index ratio m.266

In SLS experiments one is mostly working with cylindrical cuvettes from which the reflected light of the glass wall must be taken into consideration. These contributions become apparent especially at high scattering angles (backward direction), where the scattered intensity is usually low. Fig. 10.17 schematically shows the scattering geometry that leads to the mathematical description of the scattering problem applied in this presentation. The light intensity I0 (q, m) arriving at the detector consists of the directly scattered light intensity I(q, m) and the reflected light intensity I(qr, m) from the glass wall of the cylindrical sample holder: I 0 ðq; mÞ 5 Iðq; mÞ 1 cr Rp Iðqr ; mÞ

(10.11)

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Θr Θ

205

Laser Θ = 0°

Detector

FIGURE 10.17

The scattering geometry which leads to the mathematical description of the reflected light contributions. In the center of the cylindrical glass cell, the laser light is scattered in all directions. The intensity arriving at the detector consists of the directly scattered intensity with scattering angle θ and the reflected intensity from the glass with scattering angle θr 5 180 2 θ.

qr is the length of the scattering vector for θr 5 180 2 θ. Rp is Fresnel’s reflectivity coefficient for light polarized parallel to the plane of incidence. We use the approximation for normal incidence:   n0 2ng 2 Rp 5 (10.12) n0 1ng where n0 and ng are the refractive indices of the solvent and of the glass, respectively. Rp is the percentage of the reflected light for unit incident intensity, and it becomes zero when the refractive indices n0 and ng become equal. Therefore, Eq. (10.11) can be used even when the absolute intensity is unknown. Theoretically, the coefficient cr in Eq. (10.11) equals 1.0 for ideal experimental conditions. In our computational procedure we let cr be adjustable, in order to compensate for all deviations from the ideal model. These deviations may come, e.g., from a not exactly known refractive index of the glass ng, from a slight deformation of the cylindrical glass cell, and from the extinction of the reflected light in the turbid and absorbing sample. The absolute value of cr is not of primary interest, but it must be determined in order to avoid further systematic errors. This means that neglecting the reflected light contributions cannot be falsely interpreted as a modified distribution of sizes, to yield a good fit. Therefore, if Rp is significantly higher than zero, the fitting procedure is sensitive enough to detect and separate the amount of reflected light. In cases where the refractive index of the solvent is close to that of the glass (Rp  0), the contributions of the reflected light to the signal is negligible, and therefore the value for cr cannot and need not be determined. The algorithm for the solution of the inverse problem, i.e., the calculation of D(R) from I(q), is the same as that for the IFT; the only difference is that we have to use the form factor P(q, m, R) instead of the simple form factor for spheres (Eq. 2.16) in RDG theory. Application of the RDG approximation would give completely wrong results, even if the m value is as low as 1.05. Such an application of RDG theory to Mie data leads to a large number of fictitious small particles in the distribution. Even

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Dn (R)

6.0

3.0

0.0

0

300

600

900

1200

R (nm)

FIGURE 10.18 Bimodal Gauss distribution of spheres with m 5 1.05. Evaluation of the simulated scattering data with RDG theory: without cut-off (


), and with cut-off at R . 135 nm (2 2 2). The full line shows the inversion of the simulated scattering data with Mie theory with the correct m value. This solution agrees with the given distribution within the line width.

 

a cut-off of the corresponding size regime, which is dangerous due to the fact that there could be real small particles, will reduce but not eliminate these problems. A corresponding example is given in Fig. 10.18, with a simulated bimodal Gaussian distribution (peaks centered at 400 and 600 nm, σ 5 70.7, and height ratio 2:1). The corresponding scattering curve cannot be fitted by the RDG approximation without systematic deviations.265 These results clearly show the importance of the correct estimation of the m value for the inversion of SLS data from polydisperse systems. This is shown again in the next example (Fig. 10.19), for the same bimodal distribution but an m value of 1.2. The evaluation is performed with Mie theory in the regime 135 , R , 1050 nm, and for three m values: 1.15, 1.2, and 1.25. The scattering data were simulated, with a standard deviation of 1%. We can see that also in this case the inversion gives excellent results if the m value is known. The solution is worse if the m value is too low. This can also be seen from the mean deviation, MD, of the fit in reciprocal space. So we get, for m 5 1.15, a value of MD 5 13.6, for m 5 1.2 we get MD 5 1.09, and for m 5 1.25 a value of MD 5 6.5. This dependence of the mean deviation on the choice of m gives us the first evidence that it might be possible to determine the m value directly from the scattering data. Such a procedure will be discussed later in this chapter. Here, it is also important to stress the point that particle sizing by SLS can only be applied in a rather limited regime of two decades or less (50 , R , 5000 nm), depending on the instrument used. However, in this regime the resolution is quite good. In the previous example, we have seen that it is possible to resolve a bimodal distribution with a size difference of 50%, and it can be shown that the resolution limit is about 25%. This is better than DLS by a factor of 10 which, however, covers a much larger size range. This technique will be described in the next chapter.

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Dn (R)

4.0

2.0

0.0 0

300

600

900

1200

R (nm)

FIGURE 10.19 Bimodal Gaussian size distribution as in Fig. 10.18, but scattering data simulated with m 5 1.2. The full line shows the evaluation with the correct m value, the dashed line is the result for m 5 1.25, and the dotted line shows results for m 5 1.15.

An important field of application for SLS is oil
water emulsions. The emulsified droplets of oil-in-water or water-in-oil are at least close to spherical shape for energetic reasons, and have a certain polydispersity. Their refractive index ratio m is usually known, or can easily be determined from the components. Such oil-in-water emulsions are, for instance, used for parenteral nutrition in hospitals. Large droplets must be avoided in such emulsions, as they reduce long-term stability and may give rise to embolic diseases. Fig. 10.20 shows three scattering curves out of a large series numbered with BL 46, BL 48, and BL 51. The curves for BL 46 and BL 51 are very similar, while that for BL 48 shows a somewhat steeper descent at low q values, which could be evidence for larger droplets in this emulsion. Fig. 10.21 shows the corresponding number distributions Dn(R), and Fig. 10.22 the intensity distribution Di(R). No significant difference can be found in the number distributions; sample BL 51 contains somewhat smaller droplets, but the differences between BL 46 and BL 48 are negligible, the maximum is at about 150 nm. The situation of the intensity distribution is quite different; it is weighted according to the scattering power of the particles. This shifts the maximum for BL 46 and BL 48 to about 300 nm, but even more important is the clear evidence of larger particles (R C 600 nm and more) in sample BL 48. The recognition of these large particles in the intensity distribution is possible due to their relatively high scattering intensity. Now let us come back to the problem of determining the refractive index ratio m and the back-reflection coefficient cr directly from the experimental data. An optimized regularization technique264 for the calculation is very similar to the IFT technique described in Chapter 8, Numerical Methods. The least-squares problem for the determination of the coefficients is again illconditioned, and has to be stabilized by regularization techniques. The employed procedure

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5.5

log /(q) (a.u.)

5.5

5.5

4.5

BL 46

3.5

BL 48 BL 51

2.5

0.0

0.01

0.03 0.02 q (nm–1)

FIGURE 10.20 Measured scattering curves (xxxx) and fit (_______) for three different oil-in-water emulsions measured in an angular regime of 12 , θ , 150 .

4

BL 46 BL 48

DN (R)

BL 51

2

0 0

300

600

900

1200

R (nm)

FIGURE 10.21

Number distribution calculated for the scattering curves in Fig. 10.20.

uses a stability parameter, the so-called “Lagrange multiplier,” which levels the ratio of a priori and a posteriori information about the system. The value of this Lagrange multiplier has to be determined by means of a stability plot. In any case, the values of the parameters (e.g., m or cr) and the Lagrange multiplier have to be determined simultaneously. This cannot be done in one step, but only with

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BL 46

10

BL 48

DI (R)

BL 51

5

0

0

300

600

900

1200

R (nm)

FIGURE 10.22

Intensity distributions calculated for the scattering curves in Fig. 10.20.

iterations, because of the nonlinear nature of the least-squares problem. However, there are only one or two parameters (m and cr) that make the least-squares problem nonlinear, and the calculation of the gradients for all parameters, including ci, would be a waste of time. A simple approach consists of solving the linear least-squares problem for a fixed set of values of m and/or cr. The ill-conditioned nature of the least-squares problem demands the application of a stability parameter. Thus, for each value of the stability parameter, a best fitting value of m and/or cr can be calculated, e.g., by quadratic interpolation of the mean deviation surface.267 It turns out that it is essential to introduce a nonnegativity constraint to the size distribution to allow determination of the optimum parameters, as seen in Fig. 10.23. The next example is the scattering function of a suspension of latex particles which are made of melamine
formaldehyde resin. These particles are spherical, and can be produced in a narrow size distribution. Fig. 10.24 shows the fit (full line) to the experimental data (111 ) that were sampled between 8.72 degrees and 150.0 degrees scattering angle. From the value of qmin 5 2.1996 3 1023 nm21, we can calculate that the inversion should be performed below Rmax 5 714 nm. Actually, the inversion was done above 900 nm, due to the size of the latex. Despite a severe violation of the sampling theorem, the inversion obviously succeeded because of the high information content of the scattering curve. The structure of the scattering function that shows the positions of the maxima and minima defines the particles’ properties well. That means that the position and the width of the main peak in the refractive index of the latex particles are reliable results, but that the peaks on right are certainly misleading (Fig. 10.25). With this example, we can show the determination of the size distribution with a simultaneous estimation of the refractive index ratio m.

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log (MD)

1.75

1.25

0.75

0.25

–0.25 1.00

1.10

1.20

1.30 m

1.40

1.50

FIGURE 10.23

Effect of the nonnegativity constraint: the strength of this constraint is not controlled by the Lagrange multiplier λ. The minima in the mean deviation (MD) curves grow to a limit, and then remain constant. The case without nonnegativity constraint (pluses) is included in this graph for comparison.

log (I (a.u.))

6.00

5.00

4.00

3.00

2.00 0.0

0.06

0.12

0.18

0.24

0.30

q × 10 (1/nm)

FIGURE 10.24

Scattering curve of a latex suspension, including an electron micrograph to demonstrate the monodispersity of this sample. The experimental data (pluses) and the fit with m 5 1.129 6 0.005 (full line), corresponds to a refractive index np of 1.645 6 0.007, taking into account that the solvent was a 67% sucrose solution with n0 5 1.457.

The applicability to broad size distributions is shown with the example of an oil
water emulsion that is usually used for parenteral nutrition. The sample contained water, 20wt% soybean oil, 1.2wt% lecithin, 2.5wt% glycerol, 0.03wt% oleanolic acid, and 0.003wt% NaOH. In order to prevent multiple scattering, the sample was diluted with water so that

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DV(R) (a.u.)

0.48

0.36

0.24

0.12

0.0 0.90

1.20

1.50

1.80

2.10

2.40

R × 0.001 (nm)

FIGURE 10.25 Application to latex spheres: the volume distribution Dv(R) simultaneously determined from the experimental data in Fig. 10.24. The error bars symbolize the statistical uncertainty.

the final concentration of soybean oil was 9.1 3 1024 mg/mL. The light scattering experiment was performed within an angular range of 10.47
150.0 degrees The refractive index of the solvent was n20 632:8 5 1.3320. The variation of the m value (without correction for the reflected-light contributions) did not yield a satisfying fit to the scattering curve. Especially in the domain of large angles, the deviations from the experimental curve were exceedingly high. The substantial amount of reflected light is due to the relatively large difference of the refractive indices of the solvent and the glass cell. The behavior of the mean deviation when varying cr instead of m is shown in Fig. 10.26. The minimum in Fig. 10.26 seems to be very flat, because of the general strong decrease of the mean deviation when the Lagrange multiplier is decreased. Indeed, the optimum cr value cannot be determined with the same accuracy that is possible for the m value, owing to the relatively flat MD minimum. The final best-fitting values were m 5 1.1080 6 0.0041 and cr 5 1.058 6 0.06. The refractive index of the oil could be calculated from this m value to be n20 632:8 5 1.4759 6 0.0055. This number is comparable with the result obtained from a refractive index measurement of the pure oil after separation of the phases. We obtained an index of refraction of n20 632:8 5 1.4736 6 0.0001, which is in good agreement with the value determined from the light scattering data. The fit to the data and the resulting volume distribution Dv(R) are depicted in Figs. 10.27 and 10.28. The increase of the intensity at large angles is due to the contribution of reflected light. Regarding the consistency of the size distribution, it should be mentioned that the shape of the distribution function does not change over a range of approximately three decades of λ values, when the optimum parameters for m and cr are applied. This means that the solution is very stable, and can be regarded as correct from the statistical point of view.

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log (MD)

3.20

2.40

1.60

0.80

0.0 0.50

0.70

0.90

1.10 cr

1.30

1.50

FIGURE 10.26 Application to a polydisperse oil
water emulsion: the MD plot for the variation of cr with m 5 1.07. The flat minimum of this mean-deviation surface allows the determination of the best fitting value of cr 5 1.058 6 0.06, which is close to the theoretically predicted value of cr 5 1.

log (I (a.u.))

6.40

5.60

4.80

4.00

3.20 0.0

0.06

0.12

0.18

0.24

0.30

q × 10 (1/nm)

FIGURE 10.27 Application to a polydisperse oil 2 water emulsion: the experimental LS data (pluses), and the fit with m 5 1.1080 6 0.0041 and cr 5 1.058 6 0.06 (full line).

MULTIPLE SCATTERING We have already discussed that large particles in the micrometer regime scatter light very strongly. For higher concentrations, this leads to the problem of multiple scattering, i.e., the light is scattered by several particles before it reaches the detector. This effect

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MULTIPLE SCATTERING

DV(R) (a.u.)

0.48

0.36

0.24

0.12

0.0 0.0

1.60

3.20

4.80

6.40

8.00

R × 0.01 (nm)

FIGURE 10.28 Application to a polydisperse oil
water emulsion: the volume distribution Dν(R) as determined from the experimental data (Fig. 10.27). The error bars symbolize the statistical uncertainly.

increases with concentration, size, and refractive index ratio. In addition, we have to keep in mind that the multiple scattering effect will increase with sample thickness. The effect of multiple scattering predominates the influence of the structure factor in most applications of SLS, i.e., multiple scattering is essential already at very small concentrations of a few percent, where the effect of the structure factor is still negligible if the particles are not highly charged. Multiple scattering leads to a smearing-out of the details of the scattering curve, so that the scattering curves become smoother and flatter. This is shown in Fig. 10.29 for a suspension of latex particles with increasing concentration. We have to expect to find considerable multiple scattering in nonabsorbing samples if the transmittance is less than 85%, and if the sample is turbid so that we can see a bright halo around the primary beam (Tyndall effect). The transmittance can be increased experimentally by changing the sample, either by dilution or by contrast reduction. Otherwise, we can approach the problem of multiple scattering in principle in two different ways: either we try to take multiply scattered light into consideration in the numerical evaluation procedure; or we try to reduce or even eliminate multiple scattering contributions in the experimental set-up. A treatment of multiple scattering in SAS was given by Schelten and Schmatz (1980).268 A first-order approximation for the inverse problem of multiple light scattering of polydisperse systems based on Hartel’s theory269 was developed.270 This rather simple first-order approximation can be used for the inversion of experimental data on a PC. The essential optical parameter for these calculations is the transmittance T of the scattering volume, which can be determined experimentally. Its value should be above 0.3 (optical thickness 1.2) to give results without artifacts.270 As an example, a latex suspension with particles of two different radii, 200 and 317 nm, was used. The suspensions were mixed with a volume ratio of 2:1, i.e., the volume content

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log10(I) (a.u.) 6.40 5.60 4.80 4.00

c c/2 c/4

3.20 2.40 1.60

c/1550

0.80 0.0

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.38

q.10 (nm

–1)

FIGURE 10.29

Scattering functions of a latex suspension with increasing concentration (nominal diameter of the sample is 481 nm).

of the 200 nm particles was twice the volume content of the 317 nm particles. The final concentrations were 2.0 3 1024, 5.0 3 1023, and 2.0 3 1022 and 5.0 3 1022 mg/mL (the density of the polymer is 1.055 mg/mL). The transmittances of these samples are 0.981, 0.791, 0.408, and 0.104. After subtraction of the blank curve (water), one obtains the differential scattering curves which are depicted in Fig. 10.30, where the left side contains the intensities displayed as obtained from the measurement. The full lines present the fit to the experimental data. The calculated solution coefficients can also be used to reconstruct the corresponding singlescattering curves. These reconstructed functions are presented on the right side for the second (pluses) and the third (triangles) concentration, together with the experimental single scattering curve (first concentration, full lines, normalization factors about 64 and 265). The corresponding size distributions are shown in Fig. 10.31. The left side shows the volume distributions (same symbols as in Fig. 10.30) under the assumption of single scattering, i.e., no correction for multiple scattering. An artificial peak appears at smaller radii. The results improve if multiple scattering corrections are applied, as can be seen on the right side. Only the highest concentration with a transmittance of about 0.1 cannot be corrected completely for multiple scattering effects.270 The concentration of the colloidal systems could be increased only if the sample thickness is reduced. We can summarize the above-mentioned situation concerning Mie scattering in general and specifically multiple Mie scattering as follows: Mie scattering or SLS in the preresonant regime is important when we are dealing with scattering objects big enough to show more than just an extended Guinier regime. This size range is dictated by the q range

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MULTIPLE SCATTERING

7 6

log(I(Θ (a.u.))

log(I(Θ (a.u.))

7

6

5

5 4 3

4 0

50

100

150

0

50

Angle (deg)

100

150

Angle (deg)

FIGURE 10.30 Left side: Differential scattering curves for a bimodal latex suspension with concentrations: 2 3 1024 (3), 5 3 1023 (1), 2 3 1022 (Δ), and 5 3 1022 (}), including fit by the inversion routine (full lines). Right side: Corresponding single scattering curves to the data from the right side. Full lines: normalized single scattering curve (lowest concentration).

1.0

Dv(R) (a.u.)

Dv(R) (a.u.)

1.0

0.5

0

0.0 100

5

200

300 R (nm)

400

100

200

300

400

R (nm)

FIGURE 10.31 Size distribution from data in Fig. 10.30. Left side: No correction for multiple scattering. Right side: With multiple scattering corrections applied.

accessible in a light scattering goniometer with usable scattering angles that typically range from 10 degrees or more, up to about 150 degrees. The different available laser wavelengths are between about 450 and 700 nm. Finally, the refractive index ratio is also an important parameter that defines the limits. We can say that for most applications we have to be careful about Mie scattering when the scattering objects have a size of several hundred nanometers or more. For monodisperse objects, we could show that the use of the IFT technique can give reasonable results as long as the refractive index ratio is not too high, and the particles have sizes in the submicrometer regime up to several micrometers. Then one can get a good estimate of the shape and overall size from the resulting PDDF. However, monodisperse systems are very rare in the discussed size regime. In the case of polydisperse

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systems, one has to assume a certain shape 2 mostly spherical, which is a good choice for emulsions 2 then one can calculate the corresponding model free size distribution with software packages based on Mie theory. On the other hand, when one studies systems in this size range, the scattering power is relatively high, as the latter scales roughly with the sixth order of the size. Correcting these multiple scattering contributions by numerical methods is, in principle, possible in a certain concentration range, as shown in the last section. However, many practical problems have such a high turbidity that a transmittance of 30% or higher cannot be achieved with standard light scattering cuvettes. In order to reduce multiple scattering, one has to design special instrumentation, which will be discussed in the following section.

EXPERIMENTAL SET-UP Most SLS experiments are performed on a goniometer. When they were pioneered, goniometers were mostly equipped with mercury-vapor lamps, chromatic filters, and polarizers to allow experiments with a well-defined wavelength by using only one line of the spectrum. Additional optical devices were needed to produce a well-defined nondiverging primary beam. All these problems do not exist when lasers are used. They emit a monochromatic polarized light beam with a polarization perpendicular to the experimental plane, i.e., perpendicular to q with low divergence. There are also lasers with random polarization for special applications. He
Ne lasers emit light at a wavelength of 632.8 nm; they are very stable and the output power of 5
70 mW is sufficient for most applications. Only for weak scatterers (small particles and low contrast) does one need higher laser power that is available from ion
gas lasers such as Ar1 lasers or diode-pumped solid-state lasers, up to several watts at wavelengths of 488
532 nm, where one also takes advantage of the λ
4 dependence of the scattering intensity (see Eq. 9.14). The goniometer depicted in Fig. 10.32 shows a system with two lasers (two colors) and a double-tilted path geometry to reduce the linear extension of the instrument. The corresponding mirrors could be omitted to end up with a large, linear arrangement. The primary beam may be attenuated by a filter, then it passes through some beam definition optics, including lenses, pinholes, and a polarizer, if needed, to illuminate the sample in the center of the goniometer with a well-defined beam. The sample is usually in a precise cylindrical cuvette of high optical quality, with an outer diameter of 10
25 mm. The larger the cell is, the better in terms of accessible angular range, but the more sample is needed and the danger of multiple scattering increases. Thinner cells have strong focusing/defocusing effects on the primary beam and have increasing internal reflections, so they drastically limit the useable angular regime at small scattering angles. The sample cell is surrounded by an index-matching bath to reduce reflections and scattering from the outer cell surface. Decalin is widely used as an index-matching fluid (the refractive indices at 20 C for 632.8 nm are 1.4732 for decalin and about 1.46 for glass, depending on its composition). This index-matching fluid is stored in a cylindrical glass container with a typical diameter of 80 mm. The container is sometimes equipped with a flat entrance and exit windows for

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FIGURE 10.32

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Schematic drawing of a two-color goniometer for static light scattering.

the primary beam. After its passage through the sample, the primary beam is removed by means of a nonreflecting beam stop (Rayleigh horn, etc.) inside or after the index bath. It is essential for the quality of the experiment, that every component is perfectly aligned relative to the center of rotation of the goniometer, i.e., the primary beam must pass through it, the sample cell and the index bath must be centered with an accuracy of better than 10 μm, and the optical axis of the detection optics on the goniometer arm must go through this center at constant height for any position of the arm. The light detection system has to be linear in a wide regime (several decades in intensity), and should have a low dark current. High-quality photodiodes or photomultipliers (PMTs) fulfill these conditions. However, PMTs working in the photon-counting mode are favorable, because they can also be used for DLS experiments (see next chapter). Every detected photon emits an electron; this finally results in an electron avalanche after several dynodes. This electric pulse is transformed in a well-defined square pulse with a duration of 10 ns. These pulses are counted by a fast electronic device. The quantum efficiency of such PMTs is between 10% and 30%; the dark current can be as low as 10 counts per second, and the PMTs can detect up to 106 pulses per second. The alignment of a goniometer can be tested by using a constant scatterer, i.e., a sample of which the scattering intensity is independent of the scattering angle, θ. Toluene is wellsuited for this purpose. The light is scattered by small local fluctuations of the polarizability in the fluid. They are small enough that they show no q dependence in the low-q regime of light scattering. Toluene can also be used as a calibration fluid for absolute calibration of light-scattering experiments. After subtraction of the dark current and multiplication by sinθ to compensate for the varying scattering volumes at different observation angles, one should measure an angle-independent scattering intensity. The detection optics guarantee that only light from a certain scattering volume scattered in an angle θ is detected by the PMT. The minimum scattering angle is mostly given by

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FIGURE 10.33 Intersection of the primary beam I0 and the scattered beam Is (“viewing channel” of the detector); both have a diameter d inside the cylindrical sample cell with diameter D.

the intersection of the primary beam I0 and the “viewing channel” of the detection optics registering the scattered light Is, as shown in Fig. 10.33. The intersection area must not overlap with the cell wall, i.e., with the area where the primary beam enters the sample. There will always be some stray light from the glass-index bath interface (area P). When the area of the viewing channel (area S) overlaps with the primary beam P, we cannot measure exact scattering intensities from the sample. This minimum angle increases with the diameter, d, of the beam, and decreases with the diameter, D, of the cell. In addition, we have to keep in mind that the primary beam is never perfectly collimated, especially not after the passage through the cylindrical cuvette, when the refractive index of the solvent is different from the refractive index of the cell, as for water in glass. So the minimum scattering angle for a cuvette with D 5 21 mm and a beam diameter of 1.0 mm would be about 10 degrees. In practice, the minimum usable scattering angle or vector can be determined with an experiment that uses a well-defined colloidal dispersion, like monodisperse latex spheres in water. We perform two experiments, we measure the solution and the pure solvent, and subtract the scattering of the solvent to take into account any “blank scattering” of the system. The resulting scattering curve should show a linear behavior in a Guinier plot (log(I) vs. q2, see also Fig. 8.1). We determine the minimum scattering angle of our goniometer as the point at which the data points start to deviate from this straight line. The Guinier plot for a dilute latex sample (nominal radius given by the producer, Interfacial Dynamics Corporation, is 1064 nm), after subtraction of the water file, is shown in Fig. 10.34. It clearly demonstrates that the data can be used down to a 4.5 degrees scattering angle (the vertical bars indicate the range of data points used for the calculation of

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FIGURE 10.34

Guinier plot for SLS data from a latex suspension with nominal radius of 1064 nm. {The Guinier limits are marked by vertical lines. r: Goniometer: c 5 0.002 g/L, d 5 22.6 mm, θmin(Guinier) 5 4.5 degrees, Rg51125 nm; O: FCLSI: c 5 2 g/L, d 5 35 μm,θmin(Guinier) 5 0.7 degrees, Rg 5 1121 nm.}

Rg). This resolution is, to a certain extent, a function of the scattering intensity of the sample. This intensity should be at least as high as the blank scattering of the instrument. The small-angle light scattering instrument FCLSI (to be discussed below) has a resolution limit of about 0.7
1.0 degrees scattering angle (Fig. 10.35). An alternative to reduce the minimum scattering angle is given by a flat-cell instrument (FCLSI).271 Such a system allows minimum scattering angles of 1 degree or less, but the maximum scattering angle is in any case below 60 degrees. The primary beam, sometimes enlarged by a beam expander, passes through a flat cell with parallel windows. This sample cell is of variable thickness (from a few millimeters down to about 12 μm), and is surrounded by a temperature-control jacket and followed by a large plano-convex lens (Fourier lens) with a focal length of 100 mm. Light scattered off the primary beam by the sample to a scattering angle θ will be focused on a ring in the detection plane. The intensity of the transmitted beam is measured after attenuation by a photodiode behind the detection plane. Linear photodiode arrays in the detection plane detect the scattered light in an angular regime from 1 to 60 degrees scattering angle. The integration time of the diode elements can be set differently according to the decreasing scattering intensity at higher angles. The total integration time for a complete scattering curve is typically 10 s, but it can be as short as 1 s to allow for time-resolved experiments.272 This system is favorable for the measurement of large particles, and allows the use of very thin sample cells (as thin as 12 μm), which is very helpful to reduce multiple scattering contributions. Sometimes these contributions become negligible if the transmittance of the sample is 85% or higher.

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FIGURE 10.35

Flat-cell instrument FCLSI with Fourier lens.

Scattering curves of monodisperse 1064 nm latex from the FCLSI (&, c 5 2 g/L, d 5 35 μm) and from the goniometer (x, c 5 2 3 1023 g/L, d 5 22.6 mm).

FIGURE 10.36

The scattering curves of the latex sample with 1064 nm radius measured in our goniometer and in the flat cell instrument can be seen in Fig. 10.36. The angular regime of the goniometer ranges from about 4.5 to 150 degrees, whereas the largest accessible angle in the FCLSI is limited to about 60 degrees. A comparison of the two scattering curves between a scattering angle of 1 and 60 degrees clearly shows the improvements made by

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the FCLSI at small scattering angles. The agreement of the scattering curves of both instruments is satisfying, despite a little broadening effect at large angles in the FCLSI. It is important that light scattering experiments are, in general, very sensitive to dust particles and micro air-bubbles in the solution, as well as to any dirt on the cells where the primary beam enters. Careful cleaning of the cells and filtration of the solvents with fine filters is therefore an absolute necessity.

CONCLUSION Particles in the size range from some hundred nanometers up to several micrometers would perfectly fit to the accessible q range of a light scattering goniometer to make a detailed structural analysis like in SAXS and SANS possible. However, we are confronted with two main difficulties: we have to use Lorenz
Mie theory, and the situation is often complicated by multiple scattering. In this chapter, it was shown that the solution of the inverse scattering problem can be found in a similar way to that of SAS, as long as the contrast ratio m is not too far from 1.0. In these cases, a low-resolution shape analysis is possible for monodisperse systems. The latter are, however, rare in prictical applications. Polydisperse systems can be analyzed in terms of size distributions of spheres. Here we must use Lorenz
Mie theory to get correct results. This also means that one has to know the m value of the system. High-quality experimental data allow the determination of this value from the measured data during the size analysis. Multiple scattering contributions can be reduced by contrast matching or by special experimental set-ups like thin, flat cells, or by using the three-dimensional-cross-correlation technique described in the next chapter.

OTHER BOOKS COVERING THIS FIELD The following books also offer detailed descriptions of different aspects of this chapter. M. Kerker: The Scattering of Light and other Electromagnetic Radiation258; H.C. van de Hulst: Light Scattering by Small Particles253; B. Chu: Laser Light Scattering247; C.F. Bohren and D.R. Huffman: Absorption and Scattering of Light by Small Particles27; L.P. Bayvel and A.R. Jones: Electromagnetic Scattering and its Applications254; P.W. Barber and R.K. Chang273; P.W. Barber and S.C. Hill: Light Scattering by Particles: Computational Methods.257

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