Debye-Smoluchowski model

Marcus12 and others13 extended this model to include reactions in which electron transfer occurred during collisions between the donor and acceptor species, that is, between the short-lived Dn—Am complexes. In this context, electron transfer within transient precursor complexes ([Dn — A" in Scheme 1.1) resulted in the formation of short-lived successor complexes ([D(, + — A(m 1)] in Scheme 1.1). The Debye-Smoluchowski description of the diffusion-controlled collision frequency between D" and A " was retained. This has important implications for application of the Marcus model, particularly where—as is common in inorganic electron transfer reactions—charged donors or acceptors are involved. In these cases, use of the Marcus model to evaluate such reactions is only defensible if the collision rates between the reactants vary with ionic strength as required by the Debye-Smoluchowski model. The requirements of that model, and how electrolyte theory can be used to verify whether a reaction is a defensible candidate for evaluation using the Marcus model, are presented at the end of this section. 

The atbove-described Debye-Smoluchowski model is subject to severe limitations (Wilemski and Fixroam, 1973). First, the choice of the coordinate system is not self-evidently valid. Second, the mutual diffusion coefficient is assumed to be constant, even for short separation distances between A and B, where some vairiations are expected. Third, the diffusion ecjuation is only valid for low concentrations. A last limitation is due to the method of describing the reaction process in which it is... 

SCEATS - In your work you have considered the Kramers model for 1-dimensional motion in the high and low friction limits and the Smoluchowski model for 3-dimensional motion. In my paper at this conference, I will show that the 3-d motion can be reduced to a one-dimensional problem by use of the one dimensional potential V(R)-2kTlnR, and that application of Kramers theory to the dynamics on this potential yields the Smoluchowski-Debye result in the high friction limit. Hence unimolecular and bimolecular reactions can be compared using the same model. 

Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a two-dimensional lattice then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant Ax, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation... 

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)—(11) may be obtained from our fractional generalizations of the Fokker-Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate 4> (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

The basic concepts of linear response theory are best illustrated by considering the rotational diffusion model of an assembly of electric dipoles constrained to rotate in two dimensions due to Debye [14] which is governed by the Smoluchowski equation... 

Smoluchowski and Debye investigated the problem of diffusion controlled reactions between uniformly reactive spheres in the absence (1 ) and presence (2 ) of centrosymmetric Coulombic forces. Since these pioneering works, there has been a proliferation of theoretical studies based on more refined models. These have considered the inclusion of hydrodynamic interaction, (3-4) solvent... 

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]. 

The chapter then includes an introduction to models for collision rates between charged species in solution, and the effects on these of salts and ionic strength, which all predated the Marcus model, but upon which it is an extension. Collision rate and electrolyte models, such as those of Smoluchowski, Debye, Hiickel, and others, apply in ideal cases rarely met in practice. The assumptions of the models will be defined, and the common situations in which real reacting systems fail to comply with them will be highlighted. These models will be referred to extensively in the second half of the chapter, where the conditions that must be met in order to use the Marcus model properly, to avoid common pitfalls, and to evaluate situations where calculated values fail to agree with experimental ones will be clarified. 

Particle trajectories are determined by a combination of fluid-flow, electrophoresis and DEP. The fluid-flow is driven by electroosmosis for the case of interest here. For the thin Debye layer approximation, electroosmotic flow may be simply modeled with a slip velocity adjacent to the channel walls that is proportional to the tangential component of the local electric field, as shown by the Helmholtz-Smoluchowski equation 12). Here, the proportionality constant between the velocity and field is called the electroosmotic mobility, tigo- Fluid-flow in microchannels becomes even simpler for ideal flow conditions where the zeta potential, and hence /ieo, is uniform over all walls and where there are no pressure gradients. For these conditions, it can be shown that the fluid velocity at all points in the fluid domain is given by the product of the local electric field and// o(/i). 

PATE CONSTANTS OP REACTIONS IN THE LIQUID STATE PRESENT STATE OF KRAMERS AND SMOLUCHOWSKI-DEBYE MODELS... 

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