Size Information Obtained by Using Static Light Scattering Technique

Detailed investigation of static light scattering $(SLS)$ has been attempted in this work using dilute water dispersions of homogenous spherical particles, poly($N$-isopropylacrylamide) microgels and simulated data. When Rayleigh-Gan-Debye approximation is valid, for large particles, the simple size information, the static radius $R_{s}$ and distribution $G(R_{s})$, can be accurately obtained from SLS. For small particles, the root mean-square radius of gyration $_{Zimm}^{1/2}$ and the molar mass of particles measured using the Zimm plot are discussed. The results show that the molar mass measured using the Zimm plot over the average molar mass of particles is a function of the size distribution. With the assistance of simulated data, the effects of the reflected light and noises have been investigated in detail. Measuring the static radius from the SLS data provides one method to avoid the stringent requirements for the sample quantity and the instrument capability at small scattering angles

Investigating Diffusion Coefficient Using Dynamic Light Scattering Technique

In this work, the Z-average, effective, apparent diffusion coefficients and their poly-dispersity indexes were investigated for dilute poly-disperse homogeneous spherical particles in dispersion where the Rayleigh-Gans-Debye approximation is valid. The results reveal that the values of the apparent and effective diffusion coefficients at a scattering angle investigated are consistent and the difference between the effective and Z-average diffusion coefficients is a function of the mean particle size, size distribution and scattering angle. For the small particles with narrow size distributions, the Z-average diffusion coefficient can be got directly at any scattering angle. For the small particles with wide size distributions, the Z-average diffusion coefficient should be measured at a small scattering angle. For large particles, in order to obtain a good approximate value of Z-average diffusion coefficient, the wider the particle size distribution, the smaller the scattering angle that the DLS data are measured. The poly-dispersity index of the effective diffusion coefficient at a scattering angle investigated is consistent with that of the Z-average diffusion coefficient and without considering the influences of noises, the difference between the poly-dispersity indexes of the Z-average and apparent diffusion coefficients is determined by the mean particle size, size distribution and scattering angle together.Comment: 6 figures 6 table

New Interpretation for Laser Light Scattering Technique

The new method proposed in this work not only measures the particle size distribution and the average molar mass accurately using the static light scattering (SLS) technique when the Rayleigh-Gans-Debye approximation is valid for dilute poly-disperse homogenous spherical particles in dispersion, but also enables us to have insight into the theoretical analysis of the dimensionless shape parameter $\rho$. With the method, a new size, static radii $R_{s}$, can be measured. Based on the new static particle size information, detailed investigation of the normalized time auto-correlation function of the scattered light intensity $g^{(2)}(\tau)$ reveals that there exist three different particle sizes: the static radius, hydrodynamic radius and apparent hydrodynamic radius that is the hydrodynamic radius obtained using the cumulants method. With a simple assumption that the hydrodynamic radius $R_{h}$ is in proportion to the static radius $R_{s}$, the expected values of $g^{(2)}(\tau)$ calculated based on the static and commercial particle size information are consistent with the experimental data. With the assistance of simulated data, the apparent hydrodynamic radius is discussed. The results show that the apparent hydrodynamic radius is different from the mean hydrodynamic radius and is determined by the optical, hydrodynamic characteristics and size distribution of particles and scattering vector. The analysis also reveals that $\rho$ is determined by not only the structure of particles but also the relationship between the optical and hydrodynamic characteristics of particles even for mono-disperse model.Comment: 9 figures, 9 table

Discussing the Relationship between the Static and Dynamic Light Scattering

Both the static $(SLS)$ and dynamic $(DLS)$ light scattering techniques are used to obtain the size information from the scattered intensity, but the static radius $R_{s}$ and the apparent hydrodynamic radius $R_{h,app}$ are different. In this paper, the relationship between SLS and DLS is discussed using dilute water dispersions of two different homogenous spherical particles, polystyrene latexes and poly($N$-isopropylacrylamide) microgels, with a simple assumption that the hydrodynamic radius $R_{h}$ is in proportion to the static radius $R_{s}$, when Rayleigh-Gans-Debye approximation is valid. With the assistance of the simulated data, the apparent hydrodynamic radius $R_{h,app}$ has been discussed. The results show that the apparent hydrodynamic radius is different with the mean hydrodynamic radius of particles and is a composite size obtained from averaging the term $\exp (-q^{2}D\tau)$ in the static size distribution $G(R_{s})$ with the weight $R_{s}^{6}P(q,R_{s})$

Different Particle Sizes Obtained from Static and Dynamic Laser Light Scattering

Detailed investigation of static and dynamic laser light scattering has been attempted in this work both theoretically and experimentally based on dilute water dispersions of two different homogenous spherical particles, polystyrene latexes and poly($N$-isopropylacrylamide) microgels. When Rayleigh-Gans-Debye approximation is valid, a new radius $R_{s}$, referred to as a static radius, can be obtained from the static light scattering $(SLS)$. If the absolute magnitude of the scattered intensity and some constants that are related to the instrument and samples are known, the average molar mass for large particles can be measured. The size information obtained from SLS is purely related to the optical properties of particles, i.e., to $R_{s}$ and its distribution $G(R_{s})$. The size information obtained from dynamic light scattering $(DLS)$ is more complicated, the size distribution of which is a composite distribution that is not only related to the optical properties of particles, but also related to the hydrodynamic properties and the scattering vector. Strictly speaking, an apparent hydrodynamic radius $R_{h,app}$ is a composite size obtained from averaging the term $\exp (-q^{2}D\tau)$ in the static size distribution $G(R_{s})$, with the weight $R_{s}^{6}P(q,R_{s})$ that is also a function of both $R_{s}$ and the scattering vector $q$.Comment: 7 pages, 3 figure

Investigating Auto-correlation of Scattered Light of Mixed Particles

In this work, the normalized time auto-correlation function of the electric field of the light $g^{(1)}(\tau)$ that is scattered by the two kinds of particles in dispersion is investigated. The results show that the logarithm of $g^{(1)}(\tau)$ can be consistent with a line and many reasons can cause the deviations between an exponentiality and plots of $g^{(1)}(\tau)$ as a function of delay time $\tau$. The nonexponentiality of $g^{(1)}(\tau)$ is not only determined by the particle size distribution and scattering angle but also greatly influenced by the relationship between the concentrations, mass densities and the values that the refractive index of the material expands as a function of the concentration of the two kinds of particles.Comment: 8 figure

Investigation of Volume Phase Transition from the Different Properties of Particles

In this work, three different particle sizes: the static radius $R_{s}$, hydrodynamic radius $R_{h}$ and apparent hydrodynamic radius $R_{h,app}$ obtained using the light scattering technique, are investigated for dilute poly-disperse homogenous spherical particles with a simple assumption that the hydrodynamic radius is in proportion to the static radius, when the Rayleigh-Gans-Debye approximation is valid. The results show that the expected values of the normalized time auto-correlation function of the scattered light intensity $g^{(2)}(\tau)$ calculated based on the static particle size information are consistent with the experimental data. The volume phase transition is thus investigated using the equilibrium swelling ratios of static radii and apparent hydrodynamic radii respectively. The changes of the static particle size information and apparent hydrodynamic radius as a function of temperature $T$ show the effects of the volume phase transition on optical properties and the total influences of the volume phase transition on the optical, hydrodynamic characteristics and size distribution of particles, respectively. The effects of cross-linker on the volume phase transition are also discussed.Comment: 6 figure

Auto-correlation Function Study of Scattered Light Intensity

In this work, the particle size distribution measured using the dynamic light scattering (DLS) technique is compared with that obtained from the static light scattering (SLS) technique or provided by the supplier measured using the Transmission Electron Microscopy (TEM) technique for dilute Poly($N$-isopropylacrylamide) microgel and standard polystyrene latex samples in dispersion respectively. The results show that the narrow particle size distribution that can be measured accurately using the SLS technique is not suited to the determination by the DLS technique and the particle size distribution obtained from the DLS technique is different from the value provided by the supplier. With the assistance of the simulated data of the normalized time auto-correlation function of the scattered light intensity $g^{(2)}(\tau)$, the effects of the particle size distribution on the nonexponentiality of $g^{(2)}(\tau)$ measured at a scattering angle of 30$^\mathrm o$ are investigated. The analysis reveals that the influences of the particle size distribution are small on the nonexponentiality of $g^{(2)}(\tau)$ and very large on the initial slope of the logarithm of $g^{(2)}(\tau)$. The values of the apparent hydrodynamic radius are also largely influenced by the particle size distribution and the difference between the distributions of the apparent hydrodynamic radius and hydrodynamic radius of particles is determined by the method of cumulants.Comment: 6 figures, 7 table

A remark on a Bernstein type theorem for entire Willmore graphs in R^3

In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.Comment: In this note we answer a question proposed by Jingyi Chen and Tobias Lam

RKKY interaction of magnetic impurities in multi-Weyl semimetals

We have systematically investigated the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities in Weyl semimetals with arbitrary monopole charge $Q$. We find that the RKKY interaction becomes intrinsically anisotropic for $Q\geq2$, and its dependence on Fermi energy and impurity separation is directly controlled by the monopole charge. With the increase of $Q$, the RKKY interaction becomes more long-ranged and more anisotropic, which makes interesting magnetic orders easier to form and thus may have important applications in spintronics.Comment: 6 pages, 2 figure