Pairwise brownian dynamics

The basic theoretical approach for the analysis of diffusion controlled reactions is due to Smoluchowski [9] who developed it for the analysis of diffusion limited aggregation of colloidal particles. We discuss the generalization of this approach to the case of rodlike molecules here. The computational method best suited for the simulation of the polymerization of rodlike molecules is Brownian dynamics. We discuss in this review both multiparticle Brownian dynamics and pairwise Brownian dynamics the latter is a hybrid method combining Smoluchowski s [9] theory and Brownian... 

The contents of the review are as follows. The dynamics of rodlike polymers are reviewed in Section 2 followed by a review of previous experimental results of the polymerization kinetics of rodlike molecules in Section 3. Theoretical analyses of the problem following Smoluchowski s approach are discussed next (Section 4), and this is followed by a review of computational studies based on multiparticle Brownian dynamics in Section 5. The pairwise Brownian dynamics method is discussed in some detail in Section 6, and the conclusions of the review are given in Section 7. 

The pairwise Brownian dynamics method is a combination of Brownian dynamics and the Smoluchowski [9] approach, and the effective rate constant is obtained from the reaction probability of a single molecule undergoingdiffusive motion in the neighbourhood of a stationary test molecule, so that only a pair of molecules is considered at a time. The method was first proposed by Northrup et al. [58], and the basis of the method is to obtain the steady state reaction flux (y) as the product of the first visit flux (Jq) to a surface (spherical) which envelopes the reaction zone and the probability (/ ) that a molecule starting from the surface reacts rather than escaping to the far field, that is, j = The first visit flux (Jq) is obtained analytically whereas... 

Pairwise Brownian dynamics has been primarily used for the analysis of diffusion controlled reactions involving the reaction between isotropic molecules with complex reactive sites. Since its introduction by Northrup et al. [58], the pairwise Brownian dynamics method has been considerably refined and modified. Some of the developments include the use of variable time steps to reduce computational times [61], efficient calculation methods for charge effects [63], and incorporation of finite rates of reaction [58,61,62]. We review in the following sections, application of the method to two example problems involving isotropic translational diffusion reaction of isotropic molecules with a spherical reaction surface containing reactive patches and the reaction between rodlike molecules in dilute solution. 

Figure 17 Variation of reaction probability with the orientation angle (0) and angular position (9,) for pairwise Brownian dynamics simulation of rodlike molecules for anisotropic translational diffusion.

The pairwise Brownian dynamics method has several advantages over numerical methods based on Smoluchowski s [9] approach (e.g., finite element method), and we discuss these here. The primary advantage of the method is the ease of mathematical formulation even for cases involving complex reaction site geometries, hydrodynamic interactions, charge effects, anisotropic diffusion and flow fields. Furthermore the method obviates the need to solve complex diffusion equations to obtain the concentration field from which the rate constant is calculated in the Smoluchowski method. In contrast, the rate constant is obtained directly in the pairwise Brownian dynamics method. The effective rate constants for different reaction conditions may be obtained from a single simulation this is not possible using the finite element method. 

From a computational viewpoint, the method does not require the inversion of large matrices, and thus computer memory requirements are small. Typical diffusion controlled reactions often produce sharp gradients in the concentration field [47]. Grid refinement to take these into account in three dimensions is difficult. The analogous problem for pairwise Brownian dynamics, which is the optimal location of the initiation points for the trajectories on the spherical initiation surface is much simpler to accomplish. Furthermore, the computations can easily be performed in parallel, since the result from each trajectory is independent of the rest. This also allows for sequential refining of... 

Dynamic simulation approaches to model kinetic percolation are difficult to implement because of the inherent complexity of the problem, which requires intensive computation. As with any kinetic modd, the duration of the simulation must be commensruate with the critical timescales of the experiments. An early study to investigate the effeas of interactions on the percolation threshold was conducted by Bug et al. Here, a continuum Monte Carlo algorithm was used to modd a small system of 500 spherical particles undergoing Brownian motion. More recently, advanced simulation approaches such as Dissipative Particle Dynamics (DPD) have been applied to study kinetic percolation in composite sys-tems. " DPD is an off-lattice simulation technique similar to molecular dynamics, but applied to the supramolecular scale. Here, the larger-scale dynamics of a system are studied by monitoring the motion of particle clusters in response to pairwise, dissipative, and random forces. ... 

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