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Protokoll zum Versuch Static Light Scattering
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Physical Chemistry IIIStatic Light Scattering
University of Stuttgart
Date: 29. June 2020
Abstract: Using Static Light Scattering the form factors 𝑃(𝑞) and the structure factors 𝑆(𝑞)
could be determined from a colloidal latex suspension. The particle distances 𝑑 could be
calculated using the pair correlation function 𝑔(𝑟) and by using Bragg´s law. In general, the
particle distances were estimated to be larger via Bragg´s law than via the pair correlation
function 𝑔(𝑟), which is because the Bragg equation strictly speaking only applies to exact
crystalline structures. By plotting the natural logarithm of 𝐼Guninier against 𝑞 2 for two Guninier
samples the Guninier radius 𝑅𝐺 and the sphere radius 𝑅𝐾 could be determined. These were
𝑅𝐺,1 = (89.11 ± 1.18) nm, 𝑅K,1 = (115.05 ± 1.52) nm, 𝑅G,2 = (116.54 ± 0.85) nm and
𝑅𝐾,2 = (150.45 ± 1.10) nm. The distances increase with decreasing particle concentration.
,Table of contents
1 Theory................................................................................................................................................... 2
2 Experiment ........................................................................................................................................... 3
2.1 Task ................................................................................................................................................ 3
2.2 Implementation ............................................................................................................................. 3
3 Analysis ................................................................................................................................................. 4
3.1 Representation of the raw data .................................................................................................... 4
3.2 Plotting the intensity over the scattering vector .......................................................................... 5
3.3 Structure factor 𝑆(𝑞)..................................................................................................................... 6
3.4 Pair distribution function 𝑔(𝑅) ..................................................................................................... 7
3.5 Calculation of the particle distance 𝑑 using Bragg's law ............................................................... 8
3.6 Calculation of the Guninier radius 𝑅𝐺 and the sphere radius 𝑅𝐾 ................................................. 9
4 Conclusion .......................................................................................................................................... 10
1
,1 Theory
If light hits on dot-shaped particles, the scattered light spreads as spherical waves. The magnitude of
the scatter vector 𝑞 is defined as follows:
4𝜋𝜂
|𝑞⃗| = sin(𝜃) (1)
𝜆0
Where 𝑛 is the refractive index of the transmitted medium, 𝜃 is the scattering angle and 𝜆0 is the
vacuum wavelength of light. The scattering power of a single particle can be described by the form
factor 𝑃(𝑞⃗), which can be determined by the square of the amplitude.
|𝑃(𝑞⃗)|2 = |𝐴(𝑞⃗)|2 (2)
Equation (2) can be expressed using the Guinier radius 𝑅𝐺 if the angles are not too large:
2
𝑞²𝑅𝐺
|𝑃(𝑞⃗)|2 = 𝑒 − 3 (3)
The Guinier radius is directly related to the sphere radius 𝑅𝐾 via a constant factor:
3
𝑅𝐺 = √ 𝑅𝐾 (4)
5
In concentrated solutions there is intraparticle and interparticle scattering contribution. The following
equation applies to the scattered intensity:
𝐼(𝑞) = 𝑁𝑝 𝑃(𝑞)²𝑆(𝑞) (5)
Here 𝑁𝑝 is the number of scattering particles and 𝑆(𝑞) the structure factor. If the short-range order is
cancelled by uncommon ions, 𝑆(𝑞) can be calculated by equation (6).
𝐼(𝑞⃗)
𝑆(𝑞) = (6)
𝐼0
Here 𝐼(𝑞⃗) is the intensity of a sample with short-range order (using ion exchanger) and 𝐼0 is the
intensity of a sample without short-range order (using uncommon ions). In a sample with an ion
exchanger, a crystal structure forms to some extent. Thus, the maximum of the structure factor can be
regarded approximately as a Bragg diffraction. Therefore, Bragg´s law (7) can be used to determine the
particle distance 𝑑:
𝑚 ⋅ 𝜆0
𝑑= (7)
2 ⋅ 𝜂 ⋅ sin(𝜃max )
Here 𝜃max is the angle of the first intensity maximum, 𝑛 is the refractive index and 𝑚 = 1 is the
diffraction order.
2
,2 Experiment
2.1 Task
The mean particle distance, the structure factor 𝑆(𝑞), the Guninier radius 𝑅𝐺 and the sphere radius 𝑅𝐾
of a colloidal suspension of spherical latex particles are to be determined. For this purpose, the light
intensity of the samples is measured with and without ion exchanger.
2.2 Implementation
Pairs of samples (with and without ion exchanger) with dilution rates of 1/15, 1/25, 1/35 and 1/50
were analyzed using a detector. The angle was changed from 20° to 160° in 1° steps. The integration
time was set 1.0 s. A cuvette filled with water was analyzed within an angle from 20° to 160° in 1° steps
and an integration time of 4.0 s. The same procedure was performed for the samples Guinier 1 and
Guinier 2.
3
,3 Analysis
3.1 Representation of the raw data
In figure 1 the intensities 𝐼 of the four samples with ion exchanger are plotted against the angle 2𝛩
and in figure 2 the intensities of the same samples without ion exchanger are plotted against the angle.
0.024
1/15
1/25
0.024
1/35
Intensity I [a.u.]
1/50
0.024
0.024
0.024
0.024
18 38 58 78 98 118 138 158
angle 2𝛩 [°]
Figure 1: The intensities 𝐼 plotted against the angle 2𝛩 for the latex suspension with ion exchanger.
0.024
1/15
0.024
1/25
0.024
1/35
0.024
Intensity I [a.u.]
1/50
0.024
0.024
0.024
0.024
0.024
0.024
0.024
18 38 58 78 98 118 138 158
angle 2𝛩 [°]
Figure 2: The intensities 𝐼 plotted against the angle 2𝛩 for the latex suspension without ion exchanger.
From the figures it can be seen that there is a maximum of intensity when ion exchangers are added.
This is because the ion exchangers remove uncommon ions. Therefore, the Coulomb potential is of
long enough range and the short-range order is maintained. The maximum shifts towards smaller
angles for higher dilution rates. Furthermore, the intensity also decreases with increasing dilution
4
,rates, since fewer particles imply fewer scattering centers, and thus the intensities are lower. The
increase in intensity at the edges of the measuring range indicates diffuse scattering and reflection.
3.2 Plotting the intensity over the scattering vector
Equation (1) is used to calculate the scattering vector 𝑞. For this purpose, a sample calculation is
performed at 2𝛩 = 18.4°. The remaining values are calculated analogously. The wavelength of the
helium-neon-laser is 𝜆0 = 633 nm and the refractive index of water is 𝜂 = 1.33.
4𝜋𝜂 2𝛩 4𝜋𝜂 18.4° 1
𝑞= sin ( ) = sin ( ) = 4.22 ⋅ 10−3
𝜆0 2 633 nm 2 nm
Now the intensities can be plotted against the respective angles. The diagrams for the plot with ion
exchangers and without are shown in Figures 2 and 3, respectively.
24,1
1/15
24,08 1/25
1/35
Intensity I [a.u.]
24,06 1/50
24,04
24,02
24
0,004 0,009 0,014 0,019 0,024
scattering vector q [nm-1]
Figure 3: The intensities 𝐼 plotted against the scattering vectors q for the latex suspension with ion exchanger.
0.024 1/15
1/25
0.024
1/35
0.024 1/50
Intensity I [a.u.]
0.024
0.024
0.024
0.024
0.024
0.024
0,004 0,009 0,014 0,019 0,024
scattering vector q [nm-1]
Figure 4: The intensities 𝐼 plotted against the scattering vectors q for the latex suspension without ion exchanger.
As expected, the trend is the same as in section 2.1. With smaller dilution rates the scattering vector q
becomes smaller.
5
, 3.3 Structure factor 𝑆(𝑞)
The structure factor 𝑆(𝑞) can be calculated using equation (6). This is done exemplarily for the first
values of 𝐼exchanger and 𝐼without for the lowest dilution rate and the remaining values are determined
analogously.
𝐼exchanger 24.0290
𝑆(𝑞) = = = 1.000013
𝐼without 24.0287
Now the 𝑆(𝑞) values can be plotted against the scattering vectors 𝑞. The diagram for this is shown in
figure 5.
1,0032 1/15
1,0027 1/25
structure factor S(q) [a.u.]
1/35
1,0022
1/50
1,0017
1,0012
1,0007
1,0002
0,9997
0,004 0,009 0,014 0,019 0,024
scattering vector q [nm-1]
Figure 5: The structure factor 𝑆(𝑞) plotted against the scattering vectors 𝑞.
It can be seen again that the peaks move to the left at higher dilution rates to smaller scattering
vectors. Values above 1 result in a peak, which shows an order of the particles. Otherwise the values
go towards 1 which stands for a complete disorder of the particles.
6
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