A cellular automaton consists of a regular grid of cells, each in one of a finite number of states, such as on and off. For each cell, a set of cells called its neighborhood is defined relative to the specified cell. An initial state (time t = 0) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule.

The simplest such system is one with two distinct states (“dead” or “alive”) and which only considers its two direct neighbors in one dimension (i.e. to its left and right). Such systems are called One-Dimensional Elementary Automata and an example (rule 30) is shown above.

Each of the three cells, can take any of two states (“dead” or “alive”), resulting in a set of eight (23) configurations to describe a particular rule.

Since the outcome of each of these eight configurations can also take two states (“dead” or “alive”), there are 256 (28) different possible rules.

In higher dimensions, it is typical to describe cell neighborhoods according to either Manhanttan (Von Neumann) or Chessboard (Moore) distances of radius r. The Figure above shows in two dimensions (from left to right) a Von Neumann neighborhood of radius 1 and Moore neighborhoods of radii 1 and 2 respectively.

Based on the work by Ren and Hamley, a cell dynamics simulation is performed in three-dimensions to investigate the effects of simulation parameters on the ordering of diblock copolymers (BCPs).

Mean-field theory models are used to discretize the order parameter and the Moore Neighborhood used extends to next-next-nearest neighbors (NNNN) according to the iterative rule shown at the bottom. The isotropic Laplacian (<<φ(t)>>), is a weighted average of the Moore Neighborhood and N(0,η) is a noise parameter which is randomly distributed with a mean of zero and variance η.

Changing the simulation parameters, one can achieve impressively dissimilar ordered copolymers which range from (left to right in the image below) lamellar structures, randomly distributed cylinders and spheres.