Smoluchowski equation boundaries

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 solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

The first step is described by back-reaction boundary conditions with intrinsic rate constants Aj and k.d. This is followed by a diffusion second step in which the hydrated proton is removed from the parent molecule. TTiis latter step is described by the Debye-Smoluchowski equation (DSE). 

Exercise. Let W be the differential operator in the Smoluchowski equation (1.9), defined in the interval L y R on the space of functions P(y) that vanish at R and obey the reflecting boundary condition at L. Then the adjoint has the property that... 

These equations for / contain more information than those for 7i, t, 9 but are also more difficult to solve. In fact, solving them is equivalent to solving the original Smoluchowski equation with absorbing boundaries. They lack the simplification that makes the equations for 7i, t, useful. 

Exercise. Find for the general Smoluchowski equation (5.1) between two regular boundaries L, R the condition that two mixed boundary conditions at L and R are compatible. 

The upper boundary of the reaction rate is reached when every collision between substrate and enzyme molecules leads to reaction and thus to product. In this case, the Boltzmann factor, exp(-EJRT), is equal to lin the transition-state theory equations and the reaction is diffusion-limited or diffusion-controlled (owing to the difference in mass, the reaction is controlled only by the rate of diffusion of the substrate molecule). The reaction rate under diffusion control is limited by the number of collisions, the frequency Z of which can be calculated according to the Smoluchowski equation [Smoluchowski, 1915 Eq. (2.9)]. 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

A slip boundary condition is assumed to exist at the walls due to electroosmotic flow given by the Helmholtz-Smoluchowski equation... 

In Section III, we described how the Smoluchowski equation could be used in conjunction with boundary conditions or sink terms to describe chemical reactions. We now return to (9.11) and consider its relation to sink Smoluchowski equation. 

Extension to 2D and 3D Systems In the majority of microfluidic cases where 1/k is much smaller than the channel height, the Helmholtz-Smoluchowski equation provides a reasonable estimate of the flow velocity at the edge of the double layer field. As such when modeling two- and three-dimensional flow systems, it is common to apply this equation as a slip boundary condition on the bulk flow field. Since beyond the double layer by definition... 

Irreversibility of the Smoluchowski equation An important property of the Smoluchowski equation is that if C/( jc ) is independent of time and if there b no flux at the boundary, the distribution function P always approaches V equilibrium (Veq) ... 

The traditional analysis of proton transfer reactions [3, 5-7] is by a kinetic scheme e.g., eq 1, and the ensuing chemical rate equations describing the time dependence of bulk concentrations. This is insufficient for explaining the long time tail. The minimal level of complexity has to involve the spatial diffusion of the proton in the electrostatic field of the anion, as depicted by the time-dependent Smoluchowski equation [15], but with a boundary condition (the back-reaction boundary condition) which describes reversible reactions [10]. 

Figure 14. The bimolecular boundary condition for the Smoluchowski equation (dashed curves) compared with an exact simulation (full curves) [18a]. The simulation involves 100,000 realizations of unbiased (zero potential), one dimensional random walks involving 20 walkers on 100 lattice sites. The walkers are initially distributed randomly over all sites except the binding site. At later times one walker (at most) can occupy the binding site. The diffusion rate is equal to the rate of entering the site (/Cr), and both are set to unity. The rate for leaving the site (/c ) varies.

The discussion so far centered on proton diffusion in an infinite space. Hence, a spherically symmetric diffusion (Smoluchowski) equation in three dimensional space has been employed in the data analysis. An inner boundary condition (at the contact distance) has been imposed to describe reaction, but no outer boundary condition. Almost all of the interesting biological applications [4] involve proton diffusion in cavities and restricted geometries. These may include the inner volume of an organelle, the water layers between membranes or pores within a membrane. 

The Smoluchowski equation applies when the double layer is thin enough or R is large enough such that the motion of the diffuse part of the double layer can be considered to be nniform and parallel to a flat surface. The flow is taken to be laminar i.e. infinitesimal layers of liquid flow past each other. Within each layer, the electrical and viscous forces are balanced. By balancing these forces and nsing Poisson s equation (Eq. 3.6) with suitable boundary conditions, it can be shown that the mobility has a form similar to the Hiickel equation, althongh the numerical prefactor is different ... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

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