Colloidal dispersions diffusion coefficient

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Chu 1991 Schmitz 1990). For example, the dynamic version of the diffusing wave spectroscopy described in Vignette V is a form of DLS, although in diffusing wave spectroscopy the method of analysis is different in view of multiple scattering. Most of the advanced developments are beyond the scope of this book. However, DLS is currently a routine laboratory technique for measuring diffusion coefficients, particle size, and particle size distributions in colloidal dispersions, and our objective in this section is to present the most essential ideas behind the method and show how they are used for particle size and size distribution measurements. 

Pulsed field gradient NMR (PFG-NMR) is a powerful, nondestructive technique of measuring self-diffusion coefficients in a colloidal dispersion [69-71]. Molecules associated with an aggregate or a particle will diffuse more slowly than their free dissolving state. More specifically, when a water-soluble species is partially adsorbed onto an abrasive particle, the measured overall diffusion coefficient (D ) of the species is decreased. If the diffusion coefficient of the free dissolving species can be measured in the absence... 

Since most colloidal dispersions are stabilized by particle interactions, the use of equation (10.51) may lead to biased estimates of particle size that are often concentration dependent. The effect may be taken into account by expanding the diffusion coefficient to a concentration power series that, at low concentrations, gives ... 

Fig. 19 Reduced flow curves for a core-shell dispersion at an effective volume fraction of ij>eff = 0.580 data from [33], analysis from [86]. Here Rg denotes the hydrodynamic radius and Dq the self diffusion coefficient of the colloidal particles IcgT is the themial energy. The solid line (red) shows the result for the fitted fi -model with = 2.0. The fitted parameters are e = -0.00042, 7c = 0.14, V(j = VOfcaT/Rg, F = 8()/J,-,/A j, and = (),394A [i7 The dashed line shows the corresponding result for the A-formula. The dotted line shows the inflection tangent of the numerically determined flow curve with a slope of p = 0.12. The inset shows the corresponding results for the viscosity...

Equation (V. 10) is known as Fick s first law of diffusion. In this equation D is the diffusion coefficient in units of mV1. The diffusional flux,/d, represents the amount of substance that crosses a section of unit area, S, per second in the direction normal to that of diffusion. In the above equation dn/dz is the concentration gradient, which for steady-state diffusion is constant in time at any point within the system. The units of j and c should be consistent. We will further express the concentration as either c or n, where c is the number of moles of dispersed particles per unit volume (it is assumed that 1 mole contains 6.02xl023 colloidal particles), and n is the particle number concentration, i.e., the number of particles per unit volume. Consequently, d c and yd are expressed in mol m 2 s 1, and m"2 s"1, respectively. According... 

Thus we have seen that the rate of change in degree of dispersion in a colloidal system is governed by the solubility and the diffusion coefficient of dispersed substance and by the interfacial tension. The diffusion coefficient, D, in turn, depends to a significant extent on the aggregate state of dispersion medium (very small values of D are typical for solid dispersion media)and, to a lesser extent on the size of molecules of dispersed substance, and, as a rule, can not be altered by introducing any kind of admixtures into the system. At the same time, the presence of adsorption layers at the particle surface (particularly in concentrated dispersions, where such layers fill most of the... 

In colloidal systems where the dispersion medium is solid all processes aimed at changing the degree of dispersion are retarded due to high viscosity of dispersion medium and small diffusion coefficients of components. 

The most commonly used technique for determining 5 is photon correlation spectroscopy (PCS) [also known as quasi-elastic light scattering (QELS)]. PCS has become one of the standard tools of the trade for the colloid chemist. In this technique concentration fluctuations arising from the diffusive motion of the dispersion particles give rise to fluctuations in the dielectric constant of the medium are monitored photometrically. These fluctuations decay exponentially with a time constant related to the diffusion coefficient, Ds, of the scatterer, which can in turn be related to its hydrodynamic radius through the Stokes-Einstein equation ... 

Dynamic light-scattering, sometimes called quasi-elastic light scattering or photon correlation spectroscopy, can be used to measure the diffusion coefficients of polymer chains in solution and colloids, a kind of Doppler effect see Section 3.6.6. In a dilute dispersion of spherical particles, the diffusion coefficient D is related to the particle radius, a, through the Stokes-Einstein equation. 

In some adsorption systems, as an additional method, pulsed field gradient (PFG) diffusion measurements were applied to dilute colloidal dispersions, these methods are described in detail elsewhere [18, 19]. Briefly, by this method the self-diffusion coefficients for all spectrally resolvable liquid resonances can be obtained. Usually the method is applied to protons, and can in coated colloid dispersions monitor mobile species such as the solvent or free surfactants. 

The second assumption is the spherical beads particles are used in the experiment. These particles have a small diameter compared to the molecular dimensions. Hence, the translational diffusion coefficient (Dj-.D,y) of colloidal particles in dispersions and hydrodynamic radius (Rn) can be measured with DLS technique if Stokes- Einstein relation is applied to particle [62-64]. The translational diffusion coefficient is used to calculate the particle size. [60]. Stokes- Einstein is given in Eq. (16) [11, 60, 63, 65] ... 

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