Von Smoluchowski equation

The temperature dependence of a diffusion-controlled rate constant is very small. Actually, it is just the temperature coefficient of the diffusion coefficient, as we see from the von Smoluchowski equation. Typically, Ea for diffusion is about 8-14 kJ mol"1 (2-4 kcal mol-1) in solvents of ordinary viscosity. 

The von Smoluchowski equation must be corrected when the partners are ions to account for attractive or repulsive forces. They can be approximated by an electrostatic model. The quantity by which Eq. (9-10) or (9-13) is to be multiplied is... 

In columns with thin double layers typical of dilute buffer solution, the electroosmotic flow, ueo, can be expressed by the following relationship based on the von Smoluchowski equation [36] ... 

The Helmholtz—Von Smoluchowski equation relates the electroosmotic velocity f eof to the zeta potential in the following way ... 

The electroosmotic velocity n od in CEC can be defined from the von Smoluchowski equation ... 

The dependence of the velocity of the EOF (Ve,) on the zeta potential is expressed by the Helmholtz-von Smoluchowski equation [13] ... 

The Helmholtz-von Smoluchowski equation indicates that under constant composition of the electrolyte solution, the EOF depends on the magnitude of the zeta potential, which is determined by various factors inhuencing the formation of the electric double layer, discussed above. Each of these factors depends on several variables, such as pH, specihc adsorption of ionic species in the compact region of the double layer, ionic strength, and temperature. 

Using the SI units, the velocity of the EOF is expressed in meters/second (m s ) and the electric held in volts/meter (V m ). Consequently, the electroosmotic mobility has the dimension of m V s. Since electroosmotic and electrophoretic mobility are converse manifestations of the same underlying phenomena, the Helmholtz-von Smoluchowski equation applies to electroosmosis, as well as to electrophoresis (see below). In fact, it describes the motion of a solution in contact with a charged surface or the motion of ions relative to a solution, both under the action of an electric held, in the case of electroosmosis and electrophoresis, respectively. 

It has been pointed out above that electroosmotic and electrophoretic mobilities are converse manifestations of the same underlying phenomena therefore the Helmholtz-von Smoluchowski equation based on the Debye-Huckel theory developed for electroosmosis applies to electrophoresis as well. In the case of electrophoresis, is the potential at the plane of share between a single ion and its counterions and the surrounding solution. 

There is essentially a single modeling approach that has been developed, referred to here as the von Smoluchowski approach, and this method will be presented first. The von Smoluchowski approach requires analytical expressions to represent particle collision rates, to calculate collision efficiencies, and to dictate aggregate structure formation. These individual components are discussed in the subsequent sections, followed by analytical and numerical techniques of solving the von Smoluchowski equation. 

Various analytical solutions to the von Smoluchowski equation set have been developed over the years, originating with the solution presented by von Smoluchowski coinciding with the expression of coagulation theory in 1917 [1]. No general analytical solutions to the von Smoluchowski equation are available, but many expressions have been developed with simplifying assumptions. This section will review the available analytical solutions and discuss the uses and limitations of the solutions. 

FIG. 4 Approximate analytical solution to the von Smoluchowski equation set normalized to the total initial particle concentration. 

Hunt [37] developed self-similar solutions to the von Smoluchowski equation based on dimensional considerations. Three assumptions that were required are as follows. 

On closer inspection, the combination rate constants are about 1/4 of the estimated diffusion-controlled rate constant. For acetonitrile, for example, fcjj - 2.9 X 10 L mol" s from the von Smoluchowski equation wiA a diffusion coefficient from a modified version of the Stokes-Einstein relation, D - fcT/4jiT r. Owing to the restriction to singlet state recombination, an experimental rate constant 1 /4 of is quite reasonable. On the other hand, for these heavy metals, the spin restriction may not apply, in which case one would argue that the geometrical and orientational requirements of these large species could well give recombination rates somewhat below the theoretical maximum. 

Using SI units, the velocity of the electro-osmotic flow is expressed in meters per second (m/s) and the electric field in volts per meter (V/m). Consequently, in analogy to the electrophoretic mobility, the electro-osmotic mobility has the dimension square meters per volt per second. Because electro-osmotic and electrophoretic mobilities are converse manifestations of the same underlying phenomenon, the Hehnholtz-von Smoluchowski equation applies to electro-osmosis as well as to electrophoresis. In fact, when an electric field is applied to an ion, this moves relative to the electrolyte solution, whereas in the case of electro-osmosis, it is the mobile diffuse layer that moves under an appUed electric field, carrying the electrolyte solution with it. 

In a capillary tube, the applied electric field E is expressed by the ratio VILj, where V is the potential difference in volts across the capillary tube of length Lj (in meters). The velocity of the electro-osmotic flow, Veo (in meters per second), can be evaluated from the migration time t of (in seconds) of an electrically neutral marker substance and the distance L, (in meters) from the end of the capillary where the samples are introduced to the detection windows (effective length of the capillary). This indicates that, experimentally, the electro-osmotic mobility can be easily calculated using the Helmholtz-von Smoluchowski equation in the following form ... 

The von Smoluchowski scheme based on Equations 5.326 and 5.327 has found numerous applications. An example for biochemical application is the study of the kinetics of flocculation of latex particles caused by human gamma globulin in the presence of specific key-lock interactions. The infinite set of von Smoluchowski equations (Equation 5.319) was solved by Bak and Heilmann in the particular case when the aggregates cannot grow larger than a given size an explicit analytical solution was obtained by these authors. 

Here a is the drop radius, k a specific solubilization rate determined experimentally, c, the concentration of surfactant in micelles, and 0g and 9 the ratios of concentrations of the soluble species in the bulk and at the interface to the equilibrium solubilization capacity at c,. This equation for interfacially controlled transport is the counterpart to the well-known von Smoluchowski equation for diffusion-controlled transport ... 

For solute molecules that are comparable in size to the solvent molecules or even smaller, Equation 2.29 yields values for kd that are too low, because small molecules can slip between the larger solvent molecules (see Section 2.2.5 on dioxygen). For small molecules, the von Smoluchowski Equation 2.27 gives more accurate predictions than the simplified Equation 2.29, provided that diffusion coefficients and reaction distances r can be determined independently.58... 

Proton transfer dynamics of photoacids to the solvent have thus, being reversible in nature, been modelled using the Debye-von Smoluchowski equation for diffusion-assisted reaction dynamics in a large body of experimental work on HPTS [84—87] and naphthols [88-92], with additional studies on the temperature dependence [93-98], and the pressure dependence [99-101], as well as the effects of special media such as reverse micelles [102] or chiral environments [103]. Moreover, results modelled with the Debye-von Smoluchowski approach have also been reported for proton acceptors triggered by optical excitation (photobases) [104, 105], and for molecular compounds with both photoacid and photobase functionalities, such as lO-hydroxycamptothecin [106] and coumarin 4 [107]. It can be expected that proton diffusion also plays a role in hydroxyquinoline compounds [108-112]. Finally, proton diffusion has been suggested in the long time dynamics of green fluorescent protein [113], where the chromophore functions as a photoacid [23,114], with an initial proton release on a 3-20 ps time scale [115,116]. 

Figure 14.4 Semi-logarithmic plot of normalized fluorescence decay of excited HPTS. Points are experimental data = 375 nm, = 420 nm) in water acidified by HCIO, after lifetime correction. The geminate recombination data (pH = 6) is fitted by a numerical solution ofthe Debye-von Smoluchowski equation convoluted with the instrument response function after lifetime correction. (Adapted from Ref [125].)...

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