Abstract:
Existence of unique solution to periodic boundary value problems of differential equations with continuous or discontinuous right-hand side is considered by utilizing the method of lower and upper solutions and the monotone properties of the operator. This is subject to discussion in the present paper.

Abstract:
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 $q$.

Abstract:
In this work, the effects of the scattering angle on the nonexponentiality of the normalized time auto-correlation function of the scattered light intensity $g^{(2)}(\tau) $ are investigated using dilute Poly($N$-isopropylacrylamide) microgel and standard polystyrene latex samples in dispersion respectively. The results show that the influences of the scattering angle on the deviation between an exponentiality and $% g^{(2)}(\tau) $ are small. With the assistance of the simulated data of $g^{(2)}(\tau) $, the effects of the particle size distribution and scattering angle on the deviation between an exponentiality and $g^{(2)}(\tau) $ are explored. The analysis reveals that the nonexponentiality of $% g^{(2)}(\tau) $ is determined by the particle size distribution and scattering angle. In general, 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) $ and the effects of the scattering angle are determined by the particle size distribution and mean particle size. Under some conditions, the deviation between an exponentiality and $g^{(2)}(\tau) $ is greatly influenced by the scattering angle. The values of the apparent hydrodynamic radius are also determined by the particle size distribution and scattering angle. The apparent hydrodynamic radius and its distribution obtained using the cumulants method are different from the hydrodynamic radius and its distribution.

Abstract:
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.

Abstract:
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}) $.

Abstract:
The average scattered intensity is determined by the optical characteristics of particles in dispersion. The normalized time auto-correlation function of the scattered light intensity $g^{(2)}(\tau) $ includes both the optical and hydrodynamic information of particles. From the different characteristics of particles, the particle sizes can be obtained. In this paper, three sizes: the static radius $R_{s}$, the hydrodynamic radius $R_{h}$ and the apparent hydrodynamic radius $R_{h,app}$ are discussed using dilute water dispersions of homogenous spherical particles poly($N$-isopropylacrylamide) microgels, with a simple assumption that the hydrodynamic radius is in proportion to the static radius, when Rayleigh-Gans-Debye approximation is valid. Our results show that the expected values of the normalized time auto-correlation function of the scattered light intensity are consistent with the experimental data very well and the difference between the static radius and the apparent hydrodynamic radius is large and the difference between the hydrodynamic radius and the apparent hydrodynamic radius is influenced by the particle size distribution.

Abstract:
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.

Abstract:
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 $< R_{g}^{2}>_{Zimm}^{1/2}$ 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.

Abstract:
A new size, static radii $R_{s}$, can be measured accurately using Static Light Scattering (SLS) technique when the Rayleigh-Gans-Debye approximation is valid for dilute homogenous spherical particles in dispersion. The method proposed in this work not only can measures the particle size distribution and average molar mass accurately but also enables us to explore Dynamic Light Scattering (DLS) technique further. Detailed investigation of the normalized time auto-correlation function of the scattered light intensity $g^{2)}(\tau)$ shows that the measurements of DLS can be expected accurately and the static and hydrodynamic radii of nanoparticles are different. Only at some special conditions, the Z-average hydrodynamic radius can be measured accurately at a given scattering angle. The fact that the values of average hydrodynamic radius measured at different scattering angles are consistent or the values of polydispersity index are small does not mean the particle size distribution is narrow or monodisperse.

Abstract:
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.