Lattice-walk simulations

The shape factor given by equations 40 through 42 applies to steady-state diffusion. The results are compared to F obtained from the lattice walk simulation of transient diffusion in a cubic pore in Figure 9. An effective pore outlet angle was defined for the cubic pore 0 = tan 1 (w/W) where w is the width of the pore outlet and W is the width of the pore body (recall Figure 7). F values for the cubic pore are the reciprocal of the relative rate shown in Figure 8. 

Figure 7.11 summarizes the results obtained from (7.42) (lines) and compares it with random walk simulations on the Peano basin up to order Q = 10 (open circles) and OCNs (full circles). In the simulations, all the walkers were initially on the left side of the lattice and the front advanced to the right. A logistic growth rp(l — p) was introduced at every site at every time step to simulate the reaction process. For the OCNs, we averaged over 10 different 200 x 200 networks. 

For a two-dimensional square lattice (z=4), FH theory gives o>h = 1 104. The upper bound can be estimated by using ice model [22] to be h = (4/3) / = 1.5396. Numerical simulation evaluates o>h = 1.38. The estimate of the lower bound is possible by using the model of the Manhattan walk [23]. The Manhattan walk is a Hamilton walk on the directed lattice. Walks have to follow the arrows on the edges, which are alternately up/down and left/right, as the traffic regulation in Manhattan downtown. 

Lattice computer simulation also verified the scaling relationship. Figure 4.20 shows the scaling plot for four different chain lengths of self-avoiding walks on the cubic lattice. The data plotted as a function of reduced concentration 4>/4> are on a single master plot given by... 

The characteristic pore geometry in porous polymers is frequently too complicated to permit an analytical solution of the conduction equations. Therefore, we have examined the dynamics of lattice random walks in three-dimensional pores (as is Figure 7) [52] by an extension of the lattice walk method [62]. The simulation was performed on an NxNxN lattice each lattice represented a coordinate site for potential molecule occupation. Boundary conditions were Imposed by extending the lattice one unit in each of the six edge directions i.e. if the Interior of the cube was represented by the points [1...N,1...N,1...N], the cube with boundaries was represented by the points [0...N+1,0...N+1,0...N+1j. The extra points were used to describe the edges of the cube. 

In this chapter, techniques for describing pore connectivity and conductivity were reviewed. These methods provide an framework understanding problems of transport in porous polymers. For example, although the basic concepts in percolation theory are relatively simple, they provide a powerful tool for understanding cluster behavior in porous systems. Likewise, simple simulation lattice walk techniques can provide considerable insight into the microscopic determinants of fluid or solute transport in constricted pores. 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

The local conformational preferences of a PE chain are described by more complicated torsion potential energy functions than those in a random walk. The simulation must not only establish the coordinates on the 2nnd lattice of every second carbon atom in the initial configurations of the PE chains, but must also describe the intramolecular short range interactions of these carbon atoms, as well as the contributions to the short-range interactions from that... 

The whole construction procedure resembles a non-lattice-like self-avoiding random walk used in many MC- and MD-simulations. 

Simulation of the random walks on a site lattice is presented in Figs 2.13 and 2.14 they show that stochastic trajectories deviate systematically from the stationary solution [16]. Alongside those which correspond to the damping oscillations, above mentioned catastrophes are also observed and characterized by A b = 0, and Aa - oo. These results demonstrate indirectly... 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

The mathematical model called diffusion-limited aggregation (DLA) was introduced by Witten and Sander in 1981 [46]. The model starts with a particle at the origin of a lattice. Another particle is allowed to walk at random (simulating Brownian motion) until it arrives at a site adjacent to the seed particle. At each time step, the traveling particle moves from one site to... 

Evans, in a numerical simulation exploring primitive paths for random walks which interpeiKtrate an obstade lattice, also found an expoi ntial distribution of surplus sequence lengths A term similar to Eq. 48 appears in tte Doi-Edwards expression for the distribution of chain along a tube defined by a quadratk confinii wtential (Eq. A. 6, Ref. 3). 

The bond fluctuation model [72] is used to simulate the motion of the polymer chains on the lattice. In this model, each segment occupies eight lattice sites of a unit cell, and each site can be a part of only one segment (self-avoiding walk condition). This condition is necessary to account for the excluded volume of the polymer chains. For a given chain, the bond length between two successive seg-... 

---