Reactant density fluctuations

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

Despite the fact that (5.3.3) reveals the same asymptotic behaviour (nA(t) oc f-1/2, as t —> oo), the relative concentration is always smaller than predicted by the Smoluchowski theory for nA — l/2nA(0), the discrepancy is 9%. In other words, the Smoluchowski approach slightly underestimates a real reaction rate due to its neglect of reactant density fluctuations stimulating. (Note that in the case of second reaction, A + A — A, the reaction rate K(t) in the Smoluchowski approach has to be corrected by a factor 1/2... 

As was shown in Section 2.1, in some cases thermal fluctuations of reactant densities affect the reaction kinetics. However, the equations of the formal chemical kinetics are not suited well enough to describe these fluctuations in fact they are introduced ad hoc through the initial conditions to equations. The role of fluctuations and different methods for incorporating them into formal kinetics equations were discussed more than once. 

In practical implementation of QM/MM-ER, the procedures (PI) and (P2) would be sufficient to compute the free energies with substantial accuracy. As was demonstrated in the previous paper [60], delocalization of electron distribution in space significantly affects the energetics of solvation. The effect of the electron density fluctuation can be safely neglected when one computes the free energy differences between reactants and products in chemical reactions in solution since the cancellations of the effect will take place. 

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

The general equations for chemical reaction in a turbulent medium are easy to write if not to solve (2). In addition to those for velocities (U = U + uJ and concentrations (Cj = Cj + Cj), balance equations for q = A u, the segregation ( , and the dissipations e and eg can be written (3). Whatever the shape of the reactor under consideration (usually a tube or a stirred tank), the solution of these equations poses difficult problems of closure, as u S, 5 cj, cj, and also c cj, c Cj in the reaction terms have to be evaluated. The situation is even more complicated when the temperature and the density of the reacting mixture are also fluctuating. Partial solutions to this problem have been proposed. In the case of instantaneous reactions (t << Tg) the "e-quilibrium assumption" applies the mixed reactants are immediately converted and the apparent rate of reaction is simply that of the decrease of segregation, with Corrsin s time constant xs. For instance, with a stoichiometric proportion of reactants, the extent of reaction X is given by 1 - /T ( 2), a simple result which can also be found by application of the IEM model (see (33)). 

Although the previous equation signifies the importance of the diffusion characteristics of the reactant species, it cannot be used to describe adequately the rate of the reaction. The reason is that the concept of global concentrations for the riA and ng molecules is meaningless, since a unit volume cannot be conceived due to the local fluctuations of concentrations. Hence, the local concentrations of the reactants determine the rate of the reaction for diffusion-limited reactions. Accordingly, local density functions with different diffusion coefficients for the reactant species are used to describe the diffusion component of reaction-diffusion equations describing the kinetics of diffusion-limited reactions. 

The position of the electronic state of the reactant in the energy scale depends on a series of parameters, and in the following we will analyse them. Its occupation probability indicates if the electron transfer from (into) the electrode has occurred and to which extent. In the course of solvent fluctuations they may get energetically closer to the Fermi level or further away, and their density of states (DOS) changes accordingly. 

At t = 0, the mean densities and Mqb of reactants A and B are identical (denoted no) and spatially homogeneous. However, the randomness embodied in the distribution of A and B allows for fluctuations in the number of particles within a volume V to an order (uoV) - After a time t, the particles have had a chance to diffuse and mutually anhiliate other particles within a volume (DtY - around themselves. Therefore, after the elapse of the time t, the volume (DtY - retains only the initial imbalance resulting from fluctuations. This corresponds to the presence of [no(F>t) 2]V2 particles in the volume (DiY, and therefore a density [no/(DtY Y, yielding... 

In other words, for each value of the external parameter X, which is a combination of the concentrations of the major reactants and the kinetic constants, there is exactly one steady state this state is globally stable. Thus the genetic model (35) does not display any transition for deterministic external constraints. Assume now that the concentrations of A and B fluctuate rapidly, which in turn implies that X fluctuates. In the Gaussian white noise idealization, we obtain from (33) the following equation for the extrema of the stationary probability density ... 

The next term reflects the existence of fluctuations in the system (a is the fluctuation strength, hj (x) are elements of the Nxn diffusion matrix). The third term uses a coalescence-dispersion mechanism (for details, see [1,4]) to model turbulent mixing (23 is the inverse characteristic mixing time). The last term describes the flow of reactants into and out of the CSTR (a is the inverse residence time p (2c,t) is the probability density for input reactant stream concentrations). 

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