Hydrodynamic correction

Bernard et al.[18] made the evaluation of this contribution from the excess internal energy [26]. We report their result here. In this case, the sum is made over three species instead of two. 

In this expression, the last two terms in this expression arise from the asymetry of size of the ions. 

---

Schindewolf and WUnschel [112] have studied solvated electron reactions in liquid ammonia and water with several univalent anions and divalent cations. Ions such as NO3, N02, and BrO in water showed diffusion-limited behaviour and the ions Cd2+, Ni2+, Co2+, and Zn2+ in water displayed diffusion-limited behaviour or faster. Schindewolf and WUnschel considered that reactions of none of these ions were quite diffusion-limited in liquid ammonia. Applying the hydrodynamic correction suggests that the anionic reaction with solvated electrons may just be diffusion-limited, but the cations reaction with solvated electrons remains slower than diffusion-limited. 

Domination of aggregation kinetics by differential settling implies more significant influences of hydrodynamics. Hydrodynamic corrections on the order of 10-4 were demonstrated by Han and Lawler [3], indicating a significant portion of the collision efficiency may be attributable to hydrodynamic forces. 

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 pm) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s 1, HA is 0.0535, log HA is — 1.27, and dfdj is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowsla s model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowski s rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

In other studies, Jonah et al. [116] measured the rate of reaction of the hydrated electron with Cd and Cu cations. They noted a decreasing rate coefficient with increasing ionic strength. In all cases, the rate was slower than that based on the Debye- Smoluchowski equation [68], eqn. (51), but greater than or equal to the corrected rate coefficient using the Bronsted- Bjerram correction [eqn. (58)]. In fact, Jonah et al. found that the rate coefficient for reaction of hydrated electrons with pure Cd(C104 )2 or Cu(C104 )2 follows that predicted by Coyle et al. [94] where no ionic atmosphere has developed around e q. Jonah et al. pointed out that such a situation was improbable (see Sect. 1.6). Furthermore, no hydrodynamic correction was made to the rate coefficient, which would lower the expected value by 20%. Jonah et al. [119] showed that the observed rate for reaction of e q with HsO was about one third of the expected Debye—Smoluchowski diffusion-limited rate (see the Debye... 

No hydrodynamic correction was made and the estimated value of 0.64 nm would be nearer 0.9 nm if this were made this value suggests that electron tunnelling occurs. 

TABLE 4.6. Hydrodynamic Correction Factors, G for Diffusion in Two-Particle or Partide-Snrface Systems... 

The term Tlik gives the hydrodynamic corrections on the moving ion Xt due to all ions Xk, hik= gik — 1 is the total correlation function, and r] is the solvent viscosity. 

For porous particles with small pores, the particle volume in Eq. (15) should be replaced with the envelope volume of the particle as if the particles were nonporous as shown in Fig. 2. This would be more hydrodynamically correct if the particle behavior in the flow field is of interest or if the bulk volume of the particles is to be estimated. For total weight estimation, then the skeleton density should be known. The skeleton density is defined as the mass of the particle divided by the skeletal volume of the particle. In practice, the pore volume rather than the skeletal volume is measured through gas adsorption, gas or water displacement, and mercury porosimetry. These techniques will be discussed in more detail later. There are also porous particles with open and closed pores. The closed pores are not accessible to the gas, water or mercury and thus their volume cannot be measured. In this case, the calculated skeleton density would include the volume of closed pores as shown in Fig. 2. For nonporous particles, the particle density is exactly equal to the skeleton density. For porous particles, the skeleton density will be larger than the particle density. 

The dominant forces that determine deviations from ideal behaviour of transport processes in electrolytes are the relaxation and electrophoretic forces [16]. The first of these forces was discussed by Debye [6, 17]. When the equilibrium ionic distribution is perturbed by some external force in an ionic solution, electrostatic forces appear, which will tend to restore the equilibrium distribution of the ions. There is also a hydrodynamic effect. It was first discussed by Onsager [2, 3]. Different ions in a solution will respond differently to external forces, and will thus tend to have different drift velocities The hydrodynamic (friction) forces, mediated by the solvent, will tend to equalize these velocities. The electrophoretic ( hydrodynamic) correction can be evaluated by means ofNavier-Stokes equation [18, 19]. Calculating the relaxation effect requires the evaluation of the electrostatic drag of the ions by their surroundings. The time lag of this effect is known as the Debye relaxation time. 

J. C. Crocker. Measurement of the hydrodynamic corrections to the Brownian motion of two colloidal spheres. J. Chem. Phys., 106 (1997), 2837-2840. 

---