Stokes-Einstein equation dynamics

The final section (Section 5.8) introduces dynamic light scattering with a particular focus on determination of diffusion coefficients (self-diffusion as well as mutual diffusion), particle size (using the Stokes-Einstein equation for the diffusion coefficient), and size distribution. 

In dynamic light scattering (DLS), or photon correlation spectroscopy, temporal fluctuations of the intensity of scattered light are measured and this is related to the dynamics of the solution. In dilute micellar solutions, DLS provides the z-average of the translational diffusion coefficient. The hydrodynamic radius, Rh, of the scattering particles can then be obtained from the Stokes-Einstein equation (eqn 1.2).The intensity fraction as a function of apparent hydrodynamic radius is shown for a triblock solution in Fig. 3.4. The peak with the smaller value of apparent hydrodynamic radius, RH.aPP corresponds to molecules and that at large / Hs,Pp to micelles. 

Dynamically raised processes in the dispersion, such as Brownian molecular motion, cause variations in the intensities of the scattered light with time, which is measured by PCS. Smaller the particle, higher the fluctuations by Brownian motion. Thus, a correlation between the different intensities measured is only possible for short time intervals. In a monodisperse system following first-order kinetics, the autocorrelation function decreases rather fast. In a half logarithmic plot of the auto correlation function, the slope of the graph enables the calculation of the hydrodynamic radius by the Stokes-Einstein equation. With the commercial PCS devices the z-average is determined, which corresponds to the hydrodynamic radius. 

The values of l) >,n — the diffusivity for the Brownian motion of aerosol — are calculated from the Stokes-Einstein equation. For spherical particulates with the effective radius rp, in a gas with the dynamic viscosity p2 (nearly constant for pressures about and less than one bar), the formula is ... 

To measure the droplet size distribution of the primary emulsion (W/O in W/O/W or O/W in O/W/O) that has a micron range (with an average radius of 0.5-1.0 pm), a dynamic light-scattering technique (also referred to as photon correlation spectroscopy PCS) can be apphed. Details of this method are described in Chapter 19. Basically, the intensity fluctuation of scattered light by the droplets as they undergo Brownian diffusion is measured from this, the diffusion coefficient of the droplets can be determined, and in turn the radius can be obtained by using the Stokes-Einstein equation. 

The effect of wQ on the enzyme was examined by comparing the EPR spectra of lysine-labeled tryptophanase. Although the spin label is nonspecific, it can still provide molecular-level information about the enzyme under different conditions. EPR spectra of the spin-labeled enzyme in bulk water and in reverse micelles are shown in Figure 6. Much broader spectra were obtained in reverse micelles, and the calculated rotational correlation time of the attached label (101 increased with w0. Thus, the enzyme-bound spin label became more constrained as the water content of the reverse micelle decreased. Since the rotational correlation time of the entire protein in bulk water calculated from the Stokes-Einstein equation is about 200 nsec, the motion of the spin label was still rapid relative to the tumbling rate of the enzyme. Therefore, broadening of the spectrum was apparently caused by a change in the local dynamics of tryptophanase rather than by a decrease in the enzyme s overall rotation rate. The tumbling rate of the enzyme could have decreased as well, however. 

Stokes formula for a Stokes-Einstein equation - Einsteins general laminar flow of spheres the basis of dynamic light formula for difFusion scattering particle sisdng... 

Dynamic quenching is diffusion dependent. The coefficient of diffusion D of a molecule is given by the Stokes-Einstein equation ... 

A method for measuring the size of aggregates in aqueous environments is dynamic light scattering (DLS). This technique uses scattered light to measure diffusion rates (Brownian motion) of particles in stable suspensions to determine a size based on the Stokes-Einstein equation ... 

DLS (dynamic light scattering)—in dynamic light scattering laser light is scattered by the nanoparticles. Due to the Brownian motion of the particles, a time-dependent fluctuation is imparted to the scattered light intensity. Analysis of the signal intensity yields information about the diffusional motion of the particles, which is in turn related to the hydrodynamic size via the Stoke-Einstein equation. 

The Stokes-Einstein equation relates the cooperative diffusion coefficient (Dc) to the bare dynamic correlation length ( d ) defined in terms of the temperature-dependentviscosity ofthe solvent (ri )in Eq. 12 ... 

Another method for determining 5h is to apply dynamic light scattering, referred to as photon correlation spectroscopy (PCS). For this purpose, dilute monodisperse particles must be used. From measurements of the intensity fluctuations of scattered light by the particles as they undergo Brownian diffusion, one can obtain the diffusion coefficient D, which can be used to obtain the hydrodynamic radius by using the Stokes-Einstein equation (equation (20.19)). By measuring D for the particles, both with and without the polymer layer, one can obtain / h and / , respectively. One should make sure that the bare particles are sufficiently stable 8 is then equal to (/ h — / ) ... 

In this chapter we examine some issues in mass transfer. The reader has already been introduced to some of the key aspects. In Chapter 3 (Section 7), flocculation kinetics of colloidal particles is considered. It shows the importance of diffusivity in the rate process, and in Equation 3.72, the Stokes-Einstein equation, the effect of particle size on diffusivity is observed, leading to the need to study sizes, shapes, and charges on colloidal particles, which is taken up in Chapter 3 (Section 4). Similarly some of the key studies in mass transfe in surfactant systems— dynamic surface tension, smface elasticity, contacting and solubilization kinetics—are considered in Chapter 6 (Sections 6, 7, 10, and 12 with some related issues considered in Sections 11 and 13). These emphasize the roles played by different phases, which are characterized by molecular aggregation of different kinds. In anticipation of this, the microstructures are discussed in detail in Chapter 4 (Sections 2,4, and 7). Section 2 also includes some discussion on micellization-demicellization kinetics. 

According to the Stokes-Einstein equation, diffusion coefficients have a reciprocal relationship to the dynamic viscosity of the host fluid (rj) and the hydrodynamic radius of the diffusing species (R). 

In all the above studies, the bulk chain motions are approximated by those of a rigid rotor. The validity of this approximation is underlined by a study of lithium polystyrene carboxylate in micellar form, undertaken by Raby et al [23]. They showed by electron microscopy that the micelles had a uniform diameter of 6.0 nm, and that near to room temperature their librational (segmental) motions were largely frozen, even though the micelles were still moving freely in the acetone solvent. They then converted the NMR determined value of Tr to a dynamic micellar diameter using the Stokes-Einstein equation, and thus obtained a value of 5.7 nm. This is satisfyingly close to the microscopy value. 

Polymer molecules tend to spread out to a larger hydro-dynamic size in a good solvent than they do in a poor solvent, which results in a reduced diffusion coefficient D as the Stoke-Einstein equation shows ... 

Brownian motion is the random thermal motion of a particle suspended in a fluid. This motion results from collisions between fluid molecules and suspended particles. For time intervals At much larger than the particle inertial response time, the dynamics of Brownian motion are independent of inertial parameters such as particle and fluid density. The Brownian diffusion coefficient D is given by the Stokes-Einstein equation as... 

Dynamic light scattering (photon correlation spectroscopy, PCS) can also be applied to obtain the hydrodynamic radius of the micelle. By measuring the intensity fluctuation of scattered light by the micelles (when these undergo Brownian diffusion), one can obtain the diffusion coefficient of the micelles D, from which the hydrodynamic radius R can be obtained using the Stokes-Einstein equation ... 

In a simple hydrodynamic approach, transport parameters such as the ionic conductivity <7 , the diffusion coefficient D , and the electrochemical mobility Ui of ionic/atomic species i in Uquid/glassy systems are linked to the dynamic viscosity t] by the Stokes-Einstein equation ... 

For all known systems, the material-dependent parameter B, in Eq. (12.14) does not necessarily have the same value as the parameter B, in Eq. (12.18a). In a system with a higher probability of ionic charge motions than for viscous flow events, the strict coupling between dynamic viscosity and conductivity via the Stokes-Einstein equation does not hold [33, 34]. By introducing another independent material parameter, B B, Eq. (12.18a) can be rewritten as... 

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