Lattice walks

Ishinabe T (1982) Critical exponents for surface interacting self-avoiding lattice walks. I. Three-dimensional lattices. J Chem Phys 76 5589-5594 Israelachvili JN (1985) Intermolecular and surface forces. Academic, London Joanny JF, Leibler L, de Gennes PG (1979) Effects of polymer solutions on colloid stability. J Polym Sci Polym Phys Ed 17 1073-1084... 

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. 

This section discusses approaches for evaluating F in three-dimensional pores. The first method was performed on computer it Is an extension of the lattice walk model [62]. As in the two-dimensional case, it is limited to pore geometries with intermediate constrictions (the ratio of pore exit width to pore body width > 0.10). An additional approach Is based on an analytical method for evaluating steady-state diffusion in spherical geometries [63]. There is no limit on the magnitude of the pore constriction in this model, but it Is a steady-state analysis. Together, the approaches quantitatively describe the shape factor for realistic geometries. 

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. 

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. 

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. 

Pig. 1.4 A more general lattice walk the jumps are not limited to taking values in —1,0,+1. The walk can cross the x-axis, or defect line, without touching it (in gray the contacts with the defect line). The renewal points to,ti,7, t3,T4,. .. are 0,4,5, 6,9,.... ... 

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. 

In a random walk on a square lattice the chain can cross itself. 

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 simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

In the simple sampling procedure of generating chain conformations all successfully generated walks have equal probabihty. Walks are grown purely stochastically. Each time an attempted new bond hits a site which is already occupied, one has to start at the very beginning. Otherwise different conformations would have different probabihties and this would introduce an effective attraction among the monomers [54]. With this method, each conformation is taken randomly out of the q q — 1) possible random paths which do not include direct back-folding. However, the total number of SAW on a lattice is known [26] to be ... 

An improvement of this method—the so-called biased sampling [55] (or inversely restrieted sampling)—suggests to look ahead at least one step in order to overcome the attrition. Consider a SAW of i steps on a -coordination number lattice. To add the / + 1st step one first checks which of the = q — neighboring sites are empty. If k qQ > k>0) sites are empty one takes one of these with equal probability 1 /A if A = 0 the walk is terminated and one starts from the beginning. This reduces the attrition dramatically. Now each A-step walk has a probability PAr( i ) = Ylf=i so that dense configurations are clearly more probable. To compensate for this bias, each chain does not count as 1 in the sample but with a weight... 

A. B. Harris. Self-avoiding walks on random lattices. Z Phys B 49 347-349, 1983. 

While a single vant performs little more than a pseudo random-walk, multiple-vant evolutions are ripe with many interesting (Conway Life-rule-like) patterns, particularly when the background lattice food color is shown along with the moving... 

Another simple example is the traiditional two-dimensional random-walk on a four-neighbor Euclidean lattice [toff89]. Despite the fact that the underlying lattice is symmetric only with respect to rotations that are multiples of 90 deg, the probability distribution p(s, y) for a particle that begins its random walk at the origin becomes circularly symmetric in the limit as time t —> oo p x,y,t) —> (see figure 12.12). 

Fig. 12.12. A circularly symmetric Gaussian probability distribution p x,y) describing a two-dimensional random walk emerges for large times on the macroscopic level, despite the fact that the underlying Euclidean lattice is anisotropic.

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

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... 

Figure 1 Two examples of random walks 10,000 steps on a cubic lattice. 

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