Debye equation, light scattering

This result was published by Debye in 1946. Since we shall also encounter a light-scattering equation associated with his name, we shall refer to Eq. (2.56) and its variations as the Debye viscosity equation. 

Equation 3.27 forms the basis for determination of Molecular weight from light scattering data. Like Osmotic pressure measurements, it is essential to consider the non-ideality of solutions and the concentration dependence. Following Debye, eq. 3.27 gets modified to... 

Light scattering technique was used in determining the oil solubilization rate. Debye s equation ( ) was used in the interpretation. The basic principle involves the measurement of the surfactant aggregate size during the solubilization. As the oil goes into the surfactant micelle, the increased size will be reflected by the turbidity of the solution. 

Interpretation of Light Scattering. We used Debye s equation ( for micellar solution as a basis for the light scattering measurement ... 

Equation (67) shows clearly that i should be measured as a function of both concentration and angle of observation in order to take full advantage of the Debye theory. The light scattering photometer described in Section 5.4 is designed with this capability, so this requirement introduces no new experimental difficulties. The data collected then consist of an array of i/I0 values (i needs no subscript since it now applies to small and large particles) measured... 

In contrast to osmotic pressure, light-scattering measurements become easier as the particle size increases. For spherical particles the upper limit of applicability of the Debye equation is a particle diameter of c. A/20 (i.e. 20-25 nm for A0 600 nm or Awater 450 nm or a relative molecular mass of the order of 10 ). For asymmetric particles this upper limit is lower. However, by modification of the theory, much larger particles can also be studied by light scattering methods. For polydispersed systems a mass-average relative molecular mass is given. 

The application of static light scattering to polymers is based on the theoretical equations of Debye (1944, 1947) and the methodology of Zimm (1948). The principles apply equally to polysaccharides (Sorochan et al., 1971). In total intensity light scattering, monochromatic light (436 and 546 nm) at constant T passes through the dispersion and becomes plane polarized the horizontal beam is scattered in accordance with the equation (Hiemenz, 1986)... 

Equation (7.36) is known as the Debye plot. Figure 7.11a and Figure 7.11b show the excess Rayleigh scattering intensities of polyacrylamide from MALLSP (multi angle laser light scattering photometer) and LALLSP, respectively. 

It is well established that the excluded volume effect vanishes under a special condition of temperature or solvent, which is usually known as the Flory theta temperature or solvent. Thus, light scattering measurements performed on solutions under theta conditions can furnish direct knowledge of the unperturbed dimensions [see, for example, Outer, Carr and Zimm (207) Shultz (233) and Notley and Debye (207)]. Viscosity measurements, though less directly, can also furnish similar knowledge with the aid of the Flory-Fox equation (103,109), which may be written... 

Mie Scattering. For systems more complex than very small particles (Rayleigh) or small particles with low refractive indices (Rayleigh-Debye), the scattering from widely separated spherical particles requires solving Maxwells equations. The solution of these boundary-value problems for a plane wave incident upon a particle of arbitrary size, shape, orientation, and index of refraction has not been achieved mathematically, except for spheres via the Mie theory (12,13). Mie obtained a series expression in terms of spherical harmonics for the intensity of scattered light emergent from a sphere of arbitrary size and index of fraction. The coeflBcients of this series are functions of the relative refractive index m and the dimensionless size parameter a = ird/k. 

The Debye-Bueche equation is more general than just for light scattering and applies also to x-ray and neutron scattering, but in these cases, the p s and the K s will be different. 

In the classic light-scattering experiment one solves the Debye equation over a wide range of angles and concentrations for unfractionated polymer samples. The data are plotted in a rectilinear grid known as a Zimm plot in which the ordinate and abscissa are Kc/AR and sin 0/2 -i- kc, respectively, where k is an arbitrary constant used to adjust the spacing of the data points (27). The Zimm plot yields parallel lines of either equal concentration or angle. The slope... 

For time-averaged light-scattering in the Rayleigh-Debye approximation, equation (18) reduces to ... 

A useful experimental method, providing quantitative information for the micelle-micelle interactions (in not too concentrated solutions), is the static light scattering (SLS). This method is based on the measurement of the concentration dependence of the scattered light and the solution refractive index [402,403]. From the intensity of the scattered light, the Rayleigh ratio, i e is determined and the data are plotted in accordance with the Debye equation [402-404]... 

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