Light scattering can be understood as reemission of the electromagnetic wave upon interaction within a non-homogeneous medium in which the wave is propagating [28]. The electromagnetic wave electric field interacts with the electron cloud of the molecules. The electronic cloud is disturbed: the oscillating electrons reemit the radiation with the same frequency as the incident light. This phenomenon is named light scattering (LS).
Light scattering is an extremely useful tool for characterization of particles properties such as size, shape and molecular weight. Depending on how the intensity is monitored, light scattering can be divided into static (SLS) and dynamic (DLS). In the former the intensity is measured as a function of scattering angle, the latter measures the temporal variation of the intensity described by an autocorrelation function. Static light scattering might be used to obtain information about the positional correlations and the average spatial arrangement of the particles (concentrated systems) as well as the shape of individual particles (diluted systems).
On the other hand dynamic light scattering provides information about the diffusion coefficient, size as well as size distribution. The combination of SLS with DLS can provide additional information such as molecular weight and shape [152, 153].
60 Static light scattering (SLS)
A schematic representation of a light scattering set-up is presented below;
Figure 2-4. Schematic representation of an LS experimental set-up.[154]
The laser emits light of wavelength 𝜆𝑖 and wave vector 𝜅⃗𝑖 which upon propagation through a polarizer is characterized by a determined polarization. The part of incident light that undergoes scattering is characterized by a wave factor 𝜅⃗𝑠. When the light arrives at detector the intensity of the light is monitored at the angle θ and distance R. The volume at the intersection of the incident and scattered light comprising a number N of particles is known as scattering volume 𝑉𝑠 [155].
The difference between the incident and scattered wave factors is defined as scattering vector;
𝑞⃗ = 𝜅⃗𝑖 − 𝜅⃗𝑠 Equation 19
In a case of elastic scattering, the wave factors of incident and scattered light have the magnitude 2𝜋𝜆 = 2𝜋𝑛𝜆
𝑜, the magnitude of scattering vector is given by (n is the solution refractive index);
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|𝑞⃗| =4𝜋 𝜆 𝑠𝑖𝑛 (𝜃
2) Equation 20
The reverse dependence of scattering vector is defined as spatial resolution. The form factor, P(θ) is a parameter that is defined as the ratio of scattered intensity at angle θ and at angle θ=0, and illustrates the strong angular dependence of scattered intensity by large particles [155].
Quantitative analysis requires reduced scattered intensity represented by the excess Rayleigh ratio; scattered light by a specimen and the pure solvent monitored at distance R.
In the case of identical particle excess Rayleigh ratio is described by;
∆𝑅(𝜃) = 𝐾𝑐𝑀𝑃(𝑞)𝑆(𝑞) Equation 22 Where, 𝐾 = 4𝜋2𝑛2(𝑑𝑛𝑑𝑐)2/(𝑁𝐴𝜆𝑜4), 𝑑𝑛𝑑𝑐 refractive index increment, c particle concentration, M molar mass of particles, P(q) particle form factor and 𝑆(𝑞) structure factor (inter-particle interaction potential). In the dilute regime and in the limit of low scattering angles, the form factor can be presented as;
𝑃(𝑄) = 1 −𝑞2𝑅𝑔2
3 + ⋯ Equation 23
Where Rg is the radius of gyration. Thus the Rayleigh ratio can be presented as[28];
𝐾𝑐
∆𝑅(𝜃)= 1
𝑀(1 +𝑞2𝑅𝑔2
3 + ⋯ ) (1 + 2𝐵𝑐𝑀 + ⋯ ) Equation 24
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Where B stands for the second viral coefficient. This equation provides the basis for Zimm plot analysis. An example of this approach is given in Figure 4. It requires conduction of two extrapolations, one to the zero scattering angle and the second to the zero concentration. The result of this extrapolation provides the 1/Mw value. The values of Rg and B can be obtained from the slopes of the angle and concentration extrapolations. The 𝑠𝑖𝑛2(𝜃2) is proportional to q2 , in a case of plotting 𝐾𝑐∆𝑅 versus 𝑠𝑖𝑛2(𝜃2) + 𝛾𝑐 (𝛾 being an arbitrary constant used to spread the data).
Figure 2-5. Zimm plot of methylcellulose in water[156]. (Figure adapted with permission of American Chemical Society)
One possible choice for the 𝛾 value is to set it around 1/∆c, where ∆c is the interval in the concentration series. Subsequently, linear regression of each concentration data set should be performed, 1) 𝐾𝑐𝑅
𝜃 fitted against 𝑠𝑖𝑛2(𝜃2), at zero angle intercept recorded, 2) 𝐾𝑐𝑅
𝜃 fitted against c, at zero concentration intercept recorded. Subsequently, by fitting of the zero angle intercepts versus concentration and the zero concentration versus 𝑠𝑖𝑛2(𝜃2) has to be performed. From
63 motion. Two main factors namely scattering angle (θ) and the monitoring time (t) determine the scattered light intensity. In case of SLS, just particles with the size comparable to the incident light wavelength can be reliably characterized (the minimum condition is λ/20), while in the DLS experiment, particles can be significantly smaller than the light wavelength. Due to the Brownian motion of particles the scattered intensity is time dependent. All particles encountered by the light beam can be treated as secondary light sources and due to their constant movement; the distance which light has to pass to enter the detector varies with time.
The intensity fluctuates in time, due to particle motion, which appears as “speckles” in space.
To ensure that these fluctuations are monitored, the photo detector spatial resolution must be suitable. The single speckle is known as the “coherence area” (Acoh) and is dependent on the separation between detector and object (R) as well as on the wavelength (λ)[157];
𝐴𝑐𝑜ℎ =𝜆2𝑅2 𝜋𝑥2
Equation 25
Where x stands for the radius of the scattering volume. To ensure high resolution of the measurement it is necessary to provide high value of the scattering vector, which can be achieved by decreasing the size of the detector aperture. The time of fluctuations rely on the diffusion coefficient. Needless to say that the large particles will diffuse slower than smaller, which is reflected in the slowly fluctuating intensity signal in comparison to the small particle which moves faster. Autocorrelation, known also as the photon correlation spectroscopy (PCS), provides the quantitative information about the fluctuations time scale. The scattered
64 correlated. The normalized autocorrelation function for a photo count complying with a Gaussian function is associated with the first-order correlation function of the electric field 𝑔1(𝜏) by the following relation [157];
𝑔1(𝜏) = [𝑔1(𝜏) − 1]0.5 Equation 27 The DLS basic principle concerns the analysis of the autocorrelation function in order to retrieve information about particle size. Autocorrelation function decays exponentially with the delay time for a monodisperese suspension of spherical particles subject to a Brownian motion and is represented by the following equation;
𝑔1(𝜏) = 𝐴 ∙ 𝑒−𝐷𝑞2𝜏 + 𝐵 Equation 28 Where A is the correlation function amplitude, B stands for the baseline, D is defined as the particle translational diffusion coefficient and q is the magnitude of the scattering vector. The hydrodynamic radius of spherical particles can be calculated from the Stokes-Einstein equation;
𝐷 = 𝑘𝑇/(6𝜋𝜂𝑅ℎ) Equation 29
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Where k is the Boltzmann constant, T is the temperature and η is the viscosity of the solvent [157]. Noteworthy, this equation is applicable for spherical particles in diluted systems with the size that does not induce angle dependence of the scattered intensity.
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