Hydrodynamic particle radius

Figure 2. The hydrodynamic particle radius of 4mg/ml DODACl in MEGA-9 solution. O before dialysis, after dialysis (removal of MEGA-9 by dialysis above 60 °C for 20h) measurement temperature 22 °C.

Figure 8. The hydrodynamic particle radius of 4mg/ml DDdeCl in MEGA-9 solution, measurement temperature 22V.

Using static and dynamic light scattering it could be proved that in dispersion spherical poly(organosiloxane) particles with monomodal particle size distribution are present. Fig. 1 illustrates the asymptotic decrease of the hydrodynamic particle radius Rj, with increasing amounts of T units fi om approximate 50 nm to approximate 12 nm. 

As an example, we consider the motion of a spherical particle. C, is the potential at a distance 6 from the particle surface. The distance U I S is the hydrodynamic radius of the particle. It can be larger than the particle radius due to the binding of liquid molecules or ions. Up to this distance the surface charge density amounts to ag. The entire charge of the particle is Q = 47t(R + 5)2ag 4TtR2ag. An electric field of strength E causes a force QE. At constant drift velocity, v0, the force is compensated by the friction force... 

In essence, the sedimentation method gives a ratio of the hydrodynamic radius at a given pH to the hydrodynamic radius of the same particles at a reference pH. It is not an absolute method and requires an independent absolute measurement of particle radius. 

Being able to determine [r ] as a function of elution volume, one can now compare the hydrodynamic volumes Vh for different polymers. The hydrodynamic volume is, through Einstein s viscosity law, related to intrinsic viscosity and molar mass by Vh=[r ]M/2.5. Einstein s law is, strictly speaking, valid only for impenetrable spheres at infinitely low volume fractions of the solute (equivalent to concentration at very low values). However, it can be extended to particles of other shapes, defining the particle radius then as the radius of a hydrody-namically equivalent sphere. In this case Vjj is defined as the molar volume of impenetrable spheres which would have the same frictional properties or enhance viscosity to the same degree as the actual polymer in solution. 

The key link between ELS experiments and particle electrostatic properties is the theoretical model of colloidal electrohydrodynamics. The required model is considerably more complicated than the one needed in the interpretation of DLS data. DLS relies upon a relatively simple colloidal hydrodynamic model to relate the measured particle diffusivity to particle radius via the Stokes-Einstein Eq. (39). The colloidal electrohydrodynamic model for ELS must account for the complex physical/chemical/electrical structure of the particle surface as well as the distortion of the diffuse part of the electrostatic double layer due to the motion of the particle through the medium. 

FIG. 5 Sedimentation curve recorded for a latex particle (radius a = 1.5 /iin). r represents the particle height in the cell. One can observe the hydrodynamic slowing... 

From QELS the diffusion coefficient, and hence the particle radius, can be found, provided the particles are spherical and homodlsperse and the sol is dilute. Deconvolution is difficult when one of these premises does not apply. Dilution of the sol is a prerequisite to avoid multiple scattering and any hydrodynamic interaction between the particles. A variety of apparatus is nowadays commercially available, based on one of the above-mentioned techniques, or a combination of them, so that velocity and size are both obtained. With some of these instruments size and/or velocity distributions can be derived, but the caveat must be made that these tend to be based on software progreims in which a certain type of distribution is presupposed. 

For the UF of proteins, the concentration polarization model has been found to predict the filtration performance reasonably well [56]. However, this model is inherently weak in describing the two-dimensional mass transport mechanism during crossflow filtration and does not take into account the solute-solute interactions on mass transport that occur extensively in colloids, especially during MF [21,44,158,159]. The diffusion coefficient, which is inversely proportional to the particle radius, is low and underestimates the movement of particles away from the membrane [56]. This results to the well-known flux paradox problem where the predicted permeate flux is as much as two orders of magnitude lower than the observed flux during MF of colloidal suspensions [56,58,158]. This problem has then been underlined by the experimental finding of a critical flux for colloids, which demonstrates the specificity of colloidal suspension filtration wherein just a small variation in physicochemical or hydrodynamic conditions induces important changes in the way the process has to be operated [21]. 

A linear relationship exists between the ESA or CVP amplitude and the volume fraction of the suspended particles. At relatively high-volume fractions, hydrodynamic and electric double-layer interactions lead to a non-linear dependence of these two effects on volume fraction. Generally, non-linear behavior can be expected when the electric double-layer thickness is comparable to the interparticle spacing. In most aqueous systems, where the electric double layer is thin relative to the particle radius, the electro-acoustic signal will remain linear with respect to volume fraction up to 10% by volume. At volume-fractions that are even higher, particle-particle interactions lead to a reduction in the dynamic mobility. 

Hydrodynamic radius of a colloidal particle (radius u = 7 h assumed) Colloid diameter (d = 2a = 2/ assumed throughout)... 

Solutions of macromolecules are often sufficiently dilute that Eq. (13.5.21) applies. Moreover for large molecules can be computed from hydrodynamics. For a sphere with stick boundary conditions Cs = 6jirjas. Thus in dilute solutions D° and thereby as, the particle radius, can be determined (see Chapters 5 and 8). Since D° depends on the temperature and the solvent, it is important to report the data in a standardized manner. Usually the measurements are performed at room temperature and are extrapolated to inifinite dilution. Thus for example the notation D%0,a denotes the diffusion coefficient of the solute at 20°C in the solvent H20 extrapolated to infinite dilution. For nonideal solutions... 

Since Nad 1/ p and Ecyi -- the dependence of Ecyi on the particle radius Up is weaker than in the case of negligible molecular and hydrodynamic forces. 

When the sphere is solvated, the radius, r sphere, iS replaced by the hydrodynamically effective radius, r. In addition, deviations from spherical shape can be described by an asymmetry factor fA =/d//sphere. Thus, the coefficient of friction of a solvated particle of any given shape is given by... 

---