Light scattering

Summary

The aim of this experiment is to determine the hydrodynamic radius of polystyrene latex spheres and silica nanoparticles (Ludox™) in aqueous dispersion, using dynamic light scattering. You will also learn how particle size polydispersity and high particle concentration affect the angular dependence of your result, the former caused by the particle form factor contributing to the weighted time correlation functions, the later by the effect of the static structure factor.

Learning goals:

• how to measure dynamic light scattering, in principle
• data evaluation (cumulant fitting of the time correlation function)
• treating polydisperse samples (bimodal sample, particle form factor contributions)
• treating concentrated samples (static structure factor contributions)
• the theoretical concept of light scattering
• the theoretical concept of Brownian motion, leading to the time correlation function (dynamic structure factor)
• the theoretical concepts of selfdiffusion coefficient (Stokes-Einstein-eq.), apparent diffusion coefficient (polydisperse samples), and cooperative diffusion coefficient (concentrated samples, static structure factor contribution)

Theoretical background

Static light scattering

Scattering of Single Molecules in the Gas Phase

Light, as an electromagnetic wave, interacts with molecules by periodically shifting the electrons versus the nuclei and thereby inducing an oscillating electric dipole m (Herz dipole), which serves as a source for secondary electromagnetic radiation of the same wavelength as the incident light (scattering). The amplitude (= electric field) of the scattered light in this case depends on the polarizability of the molecules \alpha, whereas the oscillating electric field of the incident light is given as:

\omega = 2\pi\nu = 2\pi c/\lambda is the frequency of the incident light of wavelength \lambda, and |\vec k|= 2\pi / \lambda is its wave vector. Note that in equation 1 for simplification we have assumed linearly polarized light spreading in x-direction. Figure 1 shows the corresponding scattered light which is spreading isotropic and perpendicular to the oscillation axis of the Hertz dipole.

The amplitude of the scattered light is given as:

Note that, according to the complex exponential function given in equation 2, the scattered amplitude is both periodic in time and space.

In a static light scattering experiment, not the amplitude (= electric field, see above), but the intensity of the scattered light is detected: I_\text s=\vec E_\text s \cdot \vec E_\text s^*=|\vec E_\text s|^2. The so-called Rayleigh scattering intensity of N molecules in the gas phase then is given as:

with I_0 the intensity of the incident light, and r_\text D the distance between the scattering sample and the detector.

Scattering of small molecules in solution

Pure solvents weakly scatter light due to thermal density fluctuations. For solutions, in contrast, the concentration fluctuations of the solute molecules are the main source of scattering. Therefore, the scattered light intensity is approximately given by these concentration fluctuations and the scattering contrast of an individual solute molecule, b:

In equation 4, we have used that the concentration fluctuations are directly related the thermal energy kT and the variation of the osmotic pressure with solute concentration \mathrm d \pi at given temperature, \left(\frac{\mathrm d \pi}{\mathrm d c}\right)_T

According to van’t Hoff, we find for ideally dilute solutions and for real solutions of finite concentrations, respectively:

(M is the molar mass of solute molecules, 𝐴_2 the second virial coefficient of osmotic pressure)

We therefore get for ideal (= highly dilute) solutions of small molecules the following simple expression for the intensity of scattered light:

In this case, the scattering contrast b^2 does not depend, as in case of molecules in the gas phase, on the absolute polarizability, but on the difference of the polarizabilities of solute and solvent molecules \Delta \alpha.

On the other hand, \Delta \alpha depends on the respective dielectric constants \varepsilon (solute) and \varepsilon_0 (solvent), and therefore on the refractive indices, as:

with n_\text D the refractive index of the solute, n_{\text{D},0} the refractive index of the solvent, and N/V the number of solute scattering particles within the illuminated and observed detection volume (= scattering volume).

To measure an absolute scattering intensity independent of the respective experimental setup (= sensitivity of the detector, size of the scattering volume, distance of the detector from the sample etc.), we define the absolute scattering intensity or Rayleigh ratio as:

with the so-called refractive index increment given as:

In experimental practice, one determines this Rayleigh ratio by calibration, typically using toluene as a scattering standard. In this case, first the scattering intensity of the pure solvent I_\text{LM} is subtracted from the scattering intensity of the solution I_\text s, and then this difference is renormalized with the ratio of the absolute intensity of the standard I_\text{std,abs} to the intensity measured for the standard I_\text{std} with a given setup:

Comparing equations 6 and 8 yields for the scattering contrast (b^2 or K):

Finally, in case of non-ideally dilute real solutions, interparticle interactions between solute and solvent molecules also have to be taken into account via the 2nd virial coefficient A_2 (see equation 5), and we finally get for the Rayleigh ratio 𝑅 in case of scattering particles of size < 10 nm in solution:

Scattering of solutions of nanoparticles of size > 10 nm

For very small scattering particles or molecules of size < 10 nm, the secondary light waves emitted from all Hertz dipoles within one particle (= intraparticular) all show perfect constructive interference, and therefore the detected Rayleigh ratio is isotropic and not depending on the scattering angle. R in this case only depends on the number concentration of scattering particles N and on the number of Hertz dipoles Z (= scattering centers) within a single particle squared (R\propto N\cdot Z^2 ). On the other hand, Z should be proportional to the mass of a scattering particle m, and therefore we find:

with N_\text A the Avogadro number, M the molar mass of the solute particles, n the molar concentration, and c the mass concentration. Note here that in equation (11), we added the term 1/N_\text A to the contrast factor, and therefore got R=K\cdot c\cdot M.

For solute scattering particles larger than 10 nm, on the other hand, the secondary light waves emitted from a single particle also may show partially destructive interference. Therefore, in this case the Rayleigh ration R is not any longer isotropic but becomes dependent on scattering angle. This important difference in the scattered intensity from small particles of size < 10 nm (also called point scatterers) and that from larger particles is also sketched in figure 2.

As described, the absolute scattered intensity for particles larger than 10 nm depends on the scattering angle. Therefore, we introduce the so-called scattering vector \vec q, which defines the optical resolution and length scale of the light scattering experiment (see figure 3).

As seen in figure 3, \vec q is given by the difference of the wave vectors of the detected scattered light and the incident light, i.e. \vec q = \vec k - \vec k_0. Since both wave vectors have the ⃗same magnitude |\vec k| = |\vec k_0|=2\pi n_\text D/\lambda (elastic scattering), the magnitude of the scattering vector is given as:

\theta is the scattering angle, \lambda is the wavelength of our light source, and n_\text D is the refractive index of the solvent. Therefore, \lambda / n_\text D is the wavelength within the sample. Practically speaking, the light scattering experiment resolves more details of the scattering particles at higher q values, since q directly corresponds to the reciprocal length scale of our experiment. This is illustrated for polymer coils in solution as an example in figure 4 and table 1:

q-range and length scales (= sample characteristics) for polymer coils; here R is the radius of gyration of a polymer chain.

Range	Sample resolution	Sample characteristics	Data evaluation
qR \ll 1	overall size	Mass, radius of gyration	Zimm plot
qR \lt 1	Particle shape	crude topology
qR \approx 1	details of particle shape	elongation of anisotropic particles etc.
qR \gt 1	inner structure of coil	chain structure (helix, coil, rod etc.)
qR \gg 1	chain segments	conformation, taxicity

Adding up all possible interferences of light waves originating from a pair of scattering centers within a single solute particle, we obtain for the q-dependence of the absolute scattered intensity:

If we consider that the scattering particles are randomly oriented in space, we can replace the distance vectors (\vec r_i - \vec r_j) with the absolute values r_{ij} (= spatial averaging). Also, we define the so-called particle form factor P(q) as R(q) normalized by particle number N number of scattering centers per particle Z^2, and contrast factor b^2. Spatial averaging and subsequent series expansion then leads to:

Finally, we can replace the Cartesian coordinates r_\text{ij} with coordinates based on the center off mass as the spatial center, and truncate the Taylor series expansion of P(q) at the 2nd order term:

with the radius of gyration s (or R_\text g) given as:

Note that for particles larger than 50 nm it is not allowed to truncate the Taylor series at the 2nd order term due to the comparably higher spatial resolution of the structure of the scattering particles in this case. For homogeneous spheres of radius R, for example, the particle form factor is given as:

Dynamic Light Scattering

Particles in solution undergo Brownian motion, causing in a light scattering experiment temporal fluctuations of the interparticle interferences, and therefore leading to a fluctuating scattered intensity I(q,t), as sketched in figure 6.

In real space the van-Hove-autocorrelation function describes the changes in particle position with time, whereas the signal detected in dynamic light scattering is given by its Fourier transform:

Here, the local particle number density n(\vec 0,t) or n(\vec r,t+\tau) either assumes the value 0 or 1, depending if a particle is found at a certain time at a given position vector within the sample. \vec r is the distance vector between the two positions which are correlated in time. The Brownian motion of a single particle (= “random walk”) can be described quantitatively by its mean squared displacement \langle\Delta R(\tau)^2\rangle and its selfdiffusion coefficient D_\text s according to the Einstein-Smoluchowski- and the Stokes-Einstein- equation:

D_\text s is given by the balance between thermal energy k T which drives the Brownian motion, and a friction coefficient f, which slows down the moving particles. f depends on the hydrodynamic particle radius R_\text H and on the solvent viscosity \eta.

In a dynamic light scattering experiment one determines the amplitude correlation function F_\text s(q,\tau) from the measured time-dependent scattered intensity I(q,t) and its time autocorrelations function \langle I(q,t)\cdot I(q,t+\tau)\rangle as following:

Using the Siegert relation (equation 22), the amplitude correlation can be calculated from the intensity correlation:

For samples with monodisperse solute particles at very low concentration, according to equation (22) \ln F_\text{s} (q,\tau) plotted versus correlation time \tau corresponds to a straight line. From the slope one determines the selfdiffusion coefficient D_\text s, which then is used to calculate the hydrodynamic particle radius by the Stokes-Einstein eq. (equation 21).

DLS data analysis for polydisperse samples:

For polydisperse samples, F_\text s(q,\tau) corresponds to a superposition of several exponentials:

Note that the amplitude of a given exponential in eq.(23) corresponds to n_i M_i^2 P_i(q), which besides the missing contrast factor K_i is exactly the average absolute scattered intensity R_i of particles of species i. Here, the contrast factor cancels since the considered particles only differ in size but not in chemical composition, and therefore all species should have identical contrast factors K_i = K.

Applying a Taylor series expansion to eq. (23) we get the so-called cumulant expression for \ln F_\text s(q,\tau):

The first cumulant \kappa_1 = \langle D_\text s\rangle q^2 directly yields the average selfdiffusion coefficient \langle D_\text s\rangle and therefore a corresponding average hydrodynamic radius. Note, however, that this average only is well-defined for particles of size < 10 nm, whereas it becomes 𝑞- dependent for larger particles, as will be explained in more detail below. The 2nd cumulant \kappa_2 here provides a quantitative measure of sample polydispersity, if the principle shape (e.g. Gaussian) of the size-distribution function is known.

Important: For samples of polydisperse particles which, on average, are larger than 10 nm in hydrodynamic radius, the diffusion coefficient depends on the scattering angle because of the 𝑞-dependent contributions of the particle form factors P_i(q) to the amplitudes of the exponentials (see eq.(23)). Therefore, this diffusion coefficient is also called apparent selfdiffusion coefficient D_\text{app}(q).

The series expansion shown in eq.(25) explains how, by interpolation of D_\text{app}(q) versus q\rightarrow 0 , the z-average is obtained as the intercept. This is to be expected, since irrespective of particle size all particle form factors P_i(q\rightarrow 0) = 1!

According to the cumulant series expansion (equation 24), the apparent selfdiffusion coefficient D_\text{app}(q) is given by the initial slope of the curves shown in figure 8. Again, please note that D_\text{app}(q) has to be plotted versus q^2 and interpolated towards q\rightarrow 0 to obtain a meaningful z-average value \langle D_\text s\rangle_z, if the particles are larger than 10 nm (see figure 9). Here, it also should be noted that D_\text{app}(q) only gives a straight line if the particles are smaller than 100 nm, since in this case the particle form factor can be approximated by a Taylor series expansion truncated after the 2nd term.

Dynamic light scattering for concentrated samples:

For concentrated samples, interparticle interactions of the scattering solute particles lead to structural ordering of particle positions as well as to a correlation of particle diffusion. Consequently, the particle trajectories can’t any longer be described as an independent random walk of individual solute particles, and the selfdiffusion coefficient has to be replaced by a so-called cooperative diffusion coefficient. The structural ordering leads to additional interferences, wherefore the normalized time-averaged scattered intensity due to interparticle interferences (= static structure factor S(q)) is no longer constant (= 1), but becomes dependent on scattering angle, as shown in figure 10:

If we ignore hydrodynamic coupling of interparticle motion, the q-dependence of the cooperative diffusion coefficient is given by the static structure factor as:

with D_0 the selfdiffusion coefficient for very dilute samples, in which case interparticle interactions can be neglected (S(q) = 1).

Experimental Setup

The next figure shows a photograph of the dynamic light scattering setup used in this practical course. The setup is not a standard commercially available equipment, but has been built, as a prototype, in October 2012 by Dr. Wolfgang Schupp/HS GmbH/Oberhilbersheim as specifically ordered by W.Schärtl.

Experimental procedure

Sample preparation

The samples are aqueous dispersions of polystyrene latex or silica nanoparticles, respectively, with different hydrodynamic radii. All samples had been cleaned from dust by careful filtration, and also the cuvettes had been flushed with filtered distilled water before the samples were added. Before each measurement is started, the respective sample cuvette is carefully inserted into the sample holder of the setup, as demonstrated by the supervisor at the beginning of the experiment.

Dynamic light scattering measurement procedure

Each sample is measured simultaneously at all four scattering angles. During the measurement (total duration: 120 seconds each sample), the digital correlator determines the normalized amplitude time correlation function F_\text s(q, \tau), which can be displayed and preliminary analyzed at the PC. Note that the final data analysis is conducted with Excel worksheets provided by the supervisor, wherefore the correlation functions are saved to the computer harddisk as *.txt data (5 columns separated by “;” with \log\left(\tau/\mathrm s\right) , and F_\text s(q, \tau) from 110° to 50° scattering angle (from left to right)) at the end of each measurement. To introduce the students to the measurement procedure, the supervisor will personally measure the first sample.

Important: maximum scattered intensity must not exceed 5000 counts !!!

The following samples, already prepared within 1 cm cylindrical light scattering cuvettes, are investigated by dynamic light scattering simultaneously at the four scattering angles 50°, 70°, 90°, and 110°:

After each measurement, the data will be stored on the desktop of the DLS measurement computer.

Data evaluation

The normalized amplitude time correlation functions F_\text s(q, \tau) are analyzed according to the cumulant method (see equation 24), using a special Excel worksheet with predefined mathematical operations: First, F_\text{s}(q, \tau) is plotted in semi-logarithmic scale. Next, data not fitting to the initial linear slope are manually truncated at longer correlation times, and finally the automated linear fitting routine yields the initial slope according to:

Using the Stokes-Einstein equation (see equation 21) the apparent hydrodynamic radius R_\text{H,app} is calculated.

After the measurement of all five samples, the supervisor will teach you how to use the special Excel worksheets for data import and evaluation. All worksheets can be downloaded from the seafile server folder of the advanced practical course.

On your written reports, you should discuss any systematic angular dependence of the determined hydrodynamic particle radii!

Hints for your discussion:

1. In case of sample 3, you have to consider the contributions of the respective particle form factors. Please note that the form factor for latex spheres II shows a minimum at scattering angle about 80°, whereas the form factor for latex spheres I steadily decreases with increasing scattering angle (see also figure 5).
2. In case of sample 5, you have to consider contributions of the static structure factor. Please note that the four scattering angles used in the experiment correspond to a q- range much smaller than q_\text{max} (see figure 10 and discussion in the figure caption).

Questions for the oral entrance examination

1. Give the definition of the absolute scattered intensity (= Rayleigh ratio) for particles of arbitrary size. The Rayleigh ratio depends on which experimental parameters and quantities?
2. Brownian motion: write down the equation of motion describing the random walk of a diffusing particle. How is the selfdiffusion coefficient defined according to the Stokes- Einstein-equation?
3. How is the time intensity correlation function defined mathematically? Give also graphic representations of the fluctuating scattered intensity and the correlation function, respectively.
4. What is the Siegert relation? Write down the respective formula how to calculate F_\text s(q,\tau) from the measured intensity correlation function.
5. What is the definition of the apparent diffusion coefficient measured for dilute dispersions of polydisperse particles?
6. What is the definition of the collective diffusion coefficient measured for concentrated dispersions? Sketch also the structure factor for ideally dilute, semi- dilute, and concentrated dispersions in one single graph.

Risk assessment of the experiment

The setup contains a diode laser with wavelength 532 nm and laser power upto 100 mW (class 3b). The laser is switched off automatically, however, if the lid of the apparatus is opened. Therefore, the apparatus itself, if used as instructed, is perfectly safe for the human eye (laser class 1).

Important and mandatory:

1. You may only use the equipment after you have been instructed by the supervisor of the light scattering experiment!
2. Never (!) touch the laser safety switch (at the edge of the sample holder) if the lid is open!!!

Literature

• B.J.Berne, R.Pecora: Dynamic Light Scattering: With Applications to Chemistry, Biology and Physics, Dover, Dover 2000
• W.Schaertl: Light Scattering from Polymer Solutions and Nanoparticle Dispersions, Springer, Berlin Heidelberg 2007