Monte Carlo Random Flights Simulation

43 Monte Carlo Random Flights Simulation 4.3.1 Introduction 

The diffusive motion of each particle is a random process which can be described by the stochastic differential equation [5, 6] as 

Unfortunately, Eq. (4.1) is only exact when dt is infinitesimal, however it can be approximated by using the Ito interpretation of a Wiener process [7], such that the increment Wt — Ws is shown to have the property of a normally distributed random 

For reactions which are diffusion controlled, reaction occurs with certainty on encounter with the time of the reaction noted. If the product of recombination is unreactive then it is removed from further consideration, otherwise the newly formed product replaces the reactants in the simulation and the simulation resumes as normal. For reactions which are partially diffusion controlled, the probability of reaction on encounter is calculated depending on the reactivity of the boundary. If the encounter is found to be unreactive, the particle positions are modified to account for reflection and the simulation would proceed as normal. A more thorough analysis will be presented later in this chapter (Sect. 4.3.3). 

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It has been reported in the literature [48] that Smoluchowski s rate constant overestimates the rate of scavenging for a single target that can be hit multiple times (for example DNA). In their work, the authors found that Smoluchowski s rate constant overestimated the scavenging yield in comparison to Monte Carlo random flights simulation (which makes no assumptions on the rate of scavenging as they are explicitly treated). The authors have found that a modiflcation to Smoluchowski s rate constant is required in order to properly take the correlation of reaction times into account however, the independent pairs approximation is still made. 

Although the IRT algorithm is sufficiently developed to completely simulate the system under study, the results are nevertheless compared with full Monte Carlo random flights simulation to make sure the correct kinetics and spin dynamics are obtained with no source of bias introduced by the IRT approximation. 

The exact computation time required for Slice depends on the number of slices used. As the number of slices increases, the computation time also increases as essentially the simulation is performing a Monte Carlo random flights simulation on a lattice. 

Fig. 7.23 Simulated scavenging rate constant obtained for e + S using Monte Carlo random flights simulation (black) and compared with Smoluchowski s time dependent rate constant (red). Ion-pair separation distances used a 10 A, b 30 A, c 40A and d 80A using a scavenger concentration in the range 0.02-0.5M and an Onsager distance of 290A. Error bars have been omitted for clarity purposes. A mutual diffusion coefficient of 0.28 A ps was used...

The diffusive behaviour of particles inside a micelle and other confined systems has been extensively studied, both experimentally and theoretically [1-9], Most simulation methods to date use Monte Carlo random flights simulation to model the diffusive motion of radicals and their subsequent recombination kinetics in confined systems. In this chapter, the possibility of using the IRT simulation to model the complete recombination kinetics and scavenging is explored (i) inside the micelle (ii) on the surface of the micelle and (iii) reversible reactions involving the micelle (i.e. adsorption and escape of solvent particles from the surface of the micelle). 

The IRT model [1,2, 12, 15] is essentially a Monte Carlo algorithm which assumes the independence of reaction times (i.e. each reaction is independent of other such reactions and that the covariance of these reaction times is zero). Unlike the random flights simulation, the diffusive trajectories are not tracked but instead encounter times are generated by sampling from an appropriate probability density function conditioned on the initial separation of the pair. The first encounter takes place at the minimum of the key times generated min(h t2, fs...) and all subsequent reactions occur based on the minimum of surviving reaction times. Unlike random flights... 

Fig. 7.16 Yield of eS at five different ion pair encounter distances a 3 A, b 5 A, c 7 A, d 10 A and e 13 A, using a scavenger concentration of 0.5M and an Onsager distance of 190A. An initial distance of 20 A between the radical ion pair was used, together with a scavenging encounter radius of 4 A. Standard error calculated on the final yield to one standard deviation is a 0.00216 (MC and IRT) and 0.00213 [IRT (uncorrected)] b 0.00216 (MC and IRT) and 0.00211 [IRT (uncorrected)] c 0.00213 (MC and IRT) and 0.0021 [IRT (uncorrected)] d 0.002 (MC and IRT) e 0.002 (MC and IRT). Here Monte Carlo (MC) refers to random flights simulation. Units of x-axis are in A ...

The actual random flight mechanism behind VTC deserves more detailed consideration. During a typical experiment a molecule must experience some 104 free flights. It enables direct simulation of each flight and of the sojourn time by a Monte Carlo technique. Modem computer workstations make it easy. The advantage is that both the position and shape of the peak are obtained in a consistent straightforward way. Besides assuming the cosine law of reflection (see below), there is another substantial assumption which we will examine in Sect. 5.4.1 - the absence of any lateral diffusion. In other words, we imply localized adsorption. 

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