Smoluchowski equation numerical solution

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

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 tumbling parameter X can be obtained by a numerical solution of the Smoluchowski equation, or analytically using the following simple approximate form (Stepanov 1983 -Kroger and Sellers 1995 Archer and Larson 1995) ... 

Nearly exact numerical solutions of the Smoluchowski equation show that for the Maier-Saupe potential, A < 1 when S = S2 > 0.524. For the Onsager potential, A < 1 for all values of the order parameter within the nematic range. Values of A for the Onsager potential are plotted in Fig. 11-18. 

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

Most numerical techniques employed for aggregation simulation are based on the equilibrium growth assumption and on the Smoluchowski theory. As shown in Meakin (1988, 1998), analytical solutions for the Smoluchowski equation have been obtained for a variety of different reaction kernels these kernels represent the rate of aggregation of clusters of sizes x and y. In most cases, these reaction kernels are based on heuristics or semi-empirical rules. 

Exact theory does not produce an analytical solution, and the mobility at a given potential is obtained as a result of successive approximations. Numerous equations give a good approximation to the exact theory over a broader range of Kfl than the Smoluchowski equation, for instance, the following Dukhin-Semenikhin equation ... 

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. 

Elminyawi IM, Gangopadhyay S, Sorensen CM. Numerical solutions to the Smoluchowski aggregation—fragmentation equation. J Colloid Interface Sci 1991 144 315-323. 

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].)...

Figure 8. Two solutions for the ordinary kinetic equations of the Eigen mechanism [3], eq 1 (dashed curves), as compared with the exact numerical solution for the Smoluchowski equation (full curves). Both models have the same /Cj and /Cr, but different values for the complex separation rate constant in the kinetic scheme were employed (a) giving the same area or (b) the same initial transient behavior as compared with the exact solution for the R OH decay [10b].

Delaire, J.A., Croc, E. and Cordier, P., 1981, Numerical solution of Smoluchowski equation, J.Phvs.Chem.. 85 1549. 

Pines E, Huppert D, Agmon N (1988) Geminate recombination in excited-stale proton-transfer reactions—numerical solution of the debye-smoluchowski equation with backreaction and comparison with experimental results. J Chem Phys 88 5620... 

Brownian dynamics is nothing but the numerical solution of the Smoluchowski equation. The method exploits the mathematical equivalence between a Fokker-Planck type of equation and the corresponding Langevin... 

The first step, which is described by back-reaction boundary condition with intrinsic rate constants and k, is followed by a diffusional second step in which the hydrated proton is removed from the parent molecule. Separation of a contact ion pair from the contact radius, a, to infinity is described by the transient numerical solution of the Debye-Smoluchowski equation (DSE). The asymptotic expression (the long time behavior) for the fluorescence of ROH (f) is derived as [160]... 

Odriozola, G., A. Schmitt, J. Callejas-Femandez, R. Martinez-Garcia, and R. Hidalgo-Alvarez. 1999. Dynamic scaling concepts applied to numerical solutions of Smoluchowski s rate equation. Journal of Chemical Physics 111 (16) 7657-7667. 

The majority of solutions of von Smoluchowski s equation have been numerical approximations due to the lack of general analytical solutions or satisfactory approximations. This section will review a number of approximations that have been generated numerically, and will discuss the difficulty associated with achieving an approximation for large aggregate structures. 

The fundamental rate expression to be considered is the Smoluchowski relation k = 4n iVDAB AB (Equation (2.1)). The derived expression ART/r] (Equation (2.3a)), is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous fluid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality Da 1 /rj is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1 /4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that slip conditions hold. If, however, one assumes instead that there is no slipping ( stick conditions), so that momentum is... 

---