# softmatter:Block Copolymer

Diblock copolymer phase diagram as calculated using Mean-field Theory by Matsen and Bates, where fA is the Block fraction, χ is the Flory-Huggins Chi Parameter, and N is the length of the block; adapted from reference [1]. L = Lamella, H = Hexagonally Packed Cylindrical Micelles, $Q_{Im\bar{3}m}$ = BCC Cubically Ordered Spherical Micelles, $Q_{Ia\bar{3}d}$ = Double Gyroid, CPS = Close-packed Cubically Ordered Spherical Micelles.

Block copolymers are polymers made of two or more chemically distinct blocks that are covalently linked together. The simplest case of a block copolymer is a diblock copolymer, where only two distinct polymers are involved (an A-block and a B-block). Typically the polymers, or blocks, that make up a diblock copolymer are immiscible and will desire to phase separate. While binary mixtures can completely separate into a single A-rich domain and a single B-rich domain, the chemical bond between the two blocks in a diblock copolymer makes this impossible. As such, microphase separation occurs, resulting in complex morphologies, such as:

Higher order block copolymers, such as triblock where "3" blocks, can be created with various configurations: again two distinct blocks can be used creating an A-B-A triblock, or three distinct blocks may be used forming an A-B-C. For system with four or more blocks are utilized, the arrangement need not be linear--for instance, starblock copolymers have also been synthesized.

## Simulation

### Basic simulation model

In simulation, BCPs are typically modeled as bead spring chains made of 2 or more different beads (see figure to the right). In the melt, like species are treated with attractive potentials, such as the Lennard-Jones potential, and dislike species are treated with either a weaker attractive force or a purely repulsive potential, such as the Weeks-Chandler-Andersen potential.

Cartoon of 4-4, with a Block fraction fa = 0.5 bead spring representation of a block copolymer

For example, in the figure to the right a standard simulation may be set up as follows:

• blue beads would be interact with other blue beads via the Lennard-Jones potential
• yellow beads would interact with other yellow beads via the Lennard-Jones potential
• blue and yellow beads would interact via the Weeks-Chandler-Andersen potential
• particles would interact along the backbone with a spring, such as the FENE spring

This model is very similar to the standard way of simulating surfactant systems. Other potential forms and bond restrictions have been implemented.

## Examples

View all block copolymer examples on the MATDL Repository