The Smoluchowski equation

The American who was the first to discover Columbus made a bad discovery. 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate a(r) which is a function of coordinates and the parameter D = Da characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A -F B B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

We can then relate the charge density, p, to the electrostatic potential nsing the one-dimensional Poisson-Boltzmann eqnation, 

Since d //dx = 0 when dVJdx = 0, the integration constant, Ci, mnst be eqnal to zero and a second integration. 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

Consider the Brownian particles dissolved in a solution. The Brownian particles will diffuse from the higher concentration region to the lower concentration region. For simplicity, we consider one-dimensional diffusion. Let C x,t) be the concentration at position x and time t. The diffusion process is phenomenologically described by Tick s law, which says that the 

Let us assume D is constant then Eq. (3.1), together with the continuity equation 

If an external potential field V(x) is applied to the particle. Pick s law needs to be modified. As the potential V(x) exerts a force 

Prom Eq. (3.6), an important relation can be obtained as in the following In the equilibrium state, the concentration C x,t) follows the Boltzmann distribution 

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Using W2 = 17jP2, (A3.2.81 and (A3.2.9) may be used to satisfy the Smoluchowski equation, (A3.2.2). another necessary property for a stationary process. Thus u(t) is an example of a stationary Gaussian-Markov... 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

In the general case, whether the e-ion pair is isolated or not, the probability density P(r, t) that an electron will remain extant at time t is given by the Smoluchowski equation... 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

The coagulation, flocculation, and adsorption processes were modeled mathematically using classical coagulation theory as a starting point. The Smoluchowski equation for orthokinetic coagulation in laminar flow is written (18)... 

This equation has been used by Sundstrom and coworkers [151] and adapted to the analysis of femtosecond spectral evolution as monitored by the bond-twisting events in barrierless isomerization in solution. The theoretical derivation of Aberg et al. establishes a link between the Smoluchowski equation with a sink and the Schrodinger equation of a solute coupled to a thermal bath. The reader is referred to this important work for further theoretical details and a thorough description of the experimental set up. It is sufficient to say here that the classical link is established via the Hamilton-Jacobi equation formalism. By using the standard ansatz Xn(X,t)= A(X,i)cxp(S(X,t)/i1l), where S(X,t) is the action of the dynamical system, and neglecting terms in once this... 

The electrophoretic mobility, /jl, can be converted to a zeta potential by using the Smoluchowski equation,... 

Time-resolved fluorescence experiments carried out with 1,2-benzanthracene quenched by CBr4 in propane-1,2-diol show a better fit with the Collins-Kimball equation than with the Smoluchowski equation. 

By comparing time-resolved and steady-state fluorescence parameters, Ross et alm> have shown that in oxytocin, a lactation and uterine contraction hormone in mammals, the internal disulfide bridge quenches the fluorescence of the single tyrosine by a static mechanism. The quenching complex was attributed to an interaction between one C — tyrosine rotamer and the disulfide bond. Swadesh et al.(()<>> have studied the dithiothreitol quenching of the six tyrosine residues in ribonuclease A. They carefully examined the steady-state criteria that are useful for distinguishing pure static from pure dynamic quenching by consideration of the Smoluchowski equation(70) for the diffusion-controlled bimolecular rate constant k0,... 

In addition to the partition coefficient, the bimolecular quenching constant (km) is obtained from quenching experiments. 1"1 "7-IIX i and, in principle, this can be used to obtain the lateral diffusion constant of the quencher by using the Smoluchowski equation ... 

Kruk and Kowalewski combined the theory allowing for the radial distribution with their Redfield-limit description of the electron spin relaxation (147). Including the g(r) in the theory led to a more complicated form of the function f(x) of Eq. (69), which becomes dependent on the g(r), as well as on the propagator P(ro, 0/ r, t). The rest of the theory remains unchanged with respect to the presentation in sections VII.A-VII.B. The propagator was computed using the Smoluchowski equation ... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

The mobility depends on both the particle properties (e.g., surface charge density and size) and solution properties (e.g., ionic strength, electric permittivity, and pH). For high ionic strengths, an approximate expression for the electrophoretic mobility, pc, is given by the Smoluchowski equation ... 

The central problem in the theory of geminate ion recombination is to describe the relative motion and reaction with each other of two oppositely charged particles initially separated by a distance ro- If we assume that the particles perform an ideal diffusive motion, the time evolution of the probability density, w(r,t), that the two species are separated by r at time t, may be described by the Smoluchowski equation [1,2]... 

In the preceding part of this section, we have concentrated on the electron escape probability, which is an important quantity in the geminate phase of recombination, and can be experimentally observed. However, modern experimental techniques also give us a possibility to observe the time-resolved kinetics of geminate recombination in some systems. Theoretically, the decay of the geminate ion pairs can be described by the pair survival probability, W t), defined by Eq. (4). One method of calculating W t) is to solve the Smoluchowski equation [Eq. (2)] for w r,t) and, then, to integrate the solution over the space variable. Another method [4] is to directly solve Eq. (7) under relevant conditions. 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

The geminate recombination in the presence of a scavenger can be described by the Smoluchowski equation [Eq. (2)] with an additional term representing the loss of the geminate pairs by scavenging reactions... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

In this result, the condition of small particles means that the actual size of the particles (which is often difficult to obtain) is not required. For reasons to be discussed later, we will call the potential obtained by this method the zeta potential (Q rather than the surface potential. In the following section we consider the alternative case of large colloidal particles, which leads to the Smoluchowski equation. 

This important result is called the Smoluchowski equation and, as before, the zeta potential is directly related to the mobility and does not depend on either the size of the particle or the electrolyte concentration. 

The rate coefficient (k2 enc) for the collision of two species is given by 8RT/3Z (where Z is the viscosity of the medium at the reaction temperature), the Smoluchowski equation. This is the maximum possible rate of reaction, which is controlled by the rate at which the two reacting species diffuse together. For nitration in >90% H2S04, where nitric acid is completely ionized, if exclusively the free base nitrates the rate coefficient (k2 fb) would equal k2 obs KJhx (where Ka is the ionization constant of the base, and hx the acidity function that it follows). Thus, if k2 fb> k2 enc free base nitration is precluded, but if... 

It is clear from Eqs. (1) and (2) that calculation of p and F is essentially a matter of obtaining averages of the forms and <[(/(ro) /(ri)]2>- These may sometimes be obtained without solving the Smoluchowski equation explicitly, by means of a method we have used previously.4 We now restate this method in the slightly more general form needed for our present purpose. 

Now we present the standard derivation of the Fokker-Planck equation for polymers in solution. (Terminology can often be confusing in the present instance, the equation of interest is also called the Smoluchowski equation, and may be regarded as a limiting case of a more general Fokker-Planck equation, or a Kramers equation.)... 

In the limit of small steps the M-equation of the one-step process (1.1a) reduces to the Smoluchowski equation... 

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