Scattering techniques for polymer nanoparticles

The most common scattering techniques for analysis of polymer nanoparticles in solution are dynamic and static light scattering (DLS and SLS), small angle X-ray scattering (SAXS) and small angle neutron scattering (SANS). Typically, radiation of known wavelength (λ) is directed towards a sample solution and interacts with the scatterers within the solution which results in a deviation of trajectory (light

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scattering).97 The intensity of the scattered radiation is then detected at a given angle (θ). Changing θ and λ can alter the scattering vector (q) shown in Equation 1.4,

Equation 1.497

where n is the refractive index of solvent for light scattering and 1 for SANS and SAXS.97 The reciprocal of q is proportional to the length scale at which the matter is detected; large q values are related to small length scales and vice versa. By interpreting scattering data, particle size, shape and molecular weight can be provided as in the case of SLS, SANS, and SAXS. In addition, if the data is collected as a function of time, particle dynamics can be analyzed, as by DLS. Figure 1.16 shows the information which is obtained by different scattering techniques for a spherical particle.97

Figure 1.16 A schematic showing that the types of information which is obtained by different scattering techniques for a spherical particle, where Rc is the radius of core, RH is

the hydrodynamic radius and Rg is radius of gyration. 97

1.5.2.1 Dynamic light scattering

Particles in solution undergo Brownian motion and thus the particle positions change with time. As a consequence of the changes in position, the scattered light from all particles undergoes either constructive or destructive interference and the resultant scattered intensity fluctuates with time at a given angle. This fluctuation can be

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interpreted to give a time scale of the movement of particles, which can be described by the second order autocorrelation function generated from the intensity fluctuations as follows:

( ) ( ) ( ) _{( )} Equation 1.5112

where 𝑔 (𝑞 𝜏) is the autocorrelation function at a particular wave vector q and correlation time τ and I is intensity. The angular brackets < > donates the average value.

For monodisperse particles, the relation between the second-order autocorrelation function 𝑔 (𝑞 𝜏) and first-order autocorrelation function 𝑔 (𝑞 𝜏) and the equation for 𝑔 (𝑞 𝜏) itself are as follows:

( ) ( ) Equation 1.6112 ( ) _{Equation 1.7}112

where Г is the decay rate and:

Equation 1.8112

Where Dt is the translational diffusion coefficient at the given detection angle and

sample concentration.

As the diffusion coefficient (Dt) is related to particles’ hydrodynamic radius (RH) by

the Stokes-Einstein equation (Equation 1.9), RH can be obtained:

Equation 1.9112

where KB is Boltzmann’s constant, T the absolute temperature, and η the viscosity of

solvent.112 It should be noted that in this method the scatterer is assumed to be a hard sphere and therefore the value of RH obtained is not always equal to the radius of the

particle unless it is spherical. In addition, given that polymer nanoparticles are never perfectly monodisperse, the cumulant method is routinely applied, which assumes a monomodal distribution of relaxation times. In the case of a multimodal system, the

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CONTIN analysis is ideal, which fits the data with a constrained regularization method, which produces a continuous distribution of relaxation times and allows for analysis of disperse samples.97,112 However, the RH values obtained by these methods

are still relative values due to the assumption that the particles are spherical, although the information on sample polydispersity is still valuable.76

1.5.2.2 Static light scattering

In contrast to DLS which monitors the change in scattering intensity over time, SLS measures the mean intensity of the scattered light averaged over time. The average molecular weight of a particle (Mw) and it’s radius of gyration (Rg) can be calculated

using the Zimm equation (Equation 1.10, q is scattering vector shown in Equation 1.4) when performing SLS measurements at multiple angles (θ) and concentrations (c). The average intensity scattered by the sample (Isample) is measured in relation to

the average intensity of a solvent (Isolvent) and a standard (Istandard), which is used to

determine the Rayleigh ratio of the sample (Rθ) based on the known Rayleigh ratio of

a standard (Rθ, standard) (Equation 1.11). K is related to the wavelength of laser (λ), the

refractive index of the solvent (nsolvent), the refractive index increment of sample

(dn/dc) and Avogadro’s number (NA) (Equation 1.12). Plotting Kc/Rθ vs q2 at

multiple concentrations allows for determination of the weight average molecular weight of the scatterers and radius of gyration from the intercept and slope of the plot. In addition, by plotting each extrapolated Kc/Rθ vs concentration, the second

virial coefficient (A2) can be obtained from the slope. If A2 is negative, it means

there are attractive interactions between particles; if A2 is positive, it shows repulsive

interactions. Moreover, the aggregation number of a particle can be determined by comparing weight-average molecular weights of particles with those of polymers. It should be noted that use of the Zimm plot is only valid for q × Rg < 1.

38 ( )( ) Equation 1.10 Equation 1.11 97 ( ) _{Equation 1.12}97

By combining particle sizes determined from both DLS and SLS, information on particle topology can be obtained, namely the so-called ρ-ratio (Rg/RH). In general, if

the value of the ρ-ratio is 0.775, it indicates the formation of spheres. Hollow spheres are suggested to form if the value is 1, while cylinders or extended structures are most likely formed if the value is larger than 1 (Figure 1.17).112

Figure 1.17 Schematic depicting how different morphologies would display a different

Rg/RH ratio. 97

1.5.2.3 Small angle neutron and X-ray scattering

SANS and SAXS use neutrons and X-rays as the radiation source respectively, which possess much smaller wavelengths (typically 0.1 nm) and therefore they are able to probe much smaller length scales (1/q, where higher scattering vector q values as shown in Equation 1.4). As a consequence, information on the size, shape and organization of the amphiphilic blocks within the structure can be obtained. Thus SANS and SAXS measurements can probe the scatterers as a whole (q × Rg < 1) as

well as their interior structures (q × Rg > 1). In addition, neutrons are scattered by the

nuclei and the scattered intensity is related to the nuclear scattering length density (SLD), which is determined from the chemical formula of samples and density of the materials. As the SLD of hydrogen and deuterium-containing materials is different, good contrast can be obtained between materials containing hydrogen and

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deuterium. Therefore, SANS is very suitable to measure deuterated materials or non- deuterated materials in deuterated solvents. In comparison, X-rays are scattered by the electron cloud and the scattered intensity is dependent on the atomic number squared; therefore, SAXS is sensitive to heavier elements.113

However, the main disadvantages of SANS and SAXS are that they are not widely accessible and the interpretation of the raw data requires fitting with suitable models. When fitting with a model, it is advisable to input some known parameters to make the fitting process easier. However, for complex and unknown structures, the fitting process is difficult and the information obtained would be limited.97