Department of Mathematics
Pattern Formation in Diblock Copolymers
Polymers are long molecules composed of many smaller units called monomers. Generally, polymers are flexible and can bend at the monomer junctions. Diblock copolymers are a type of polymer that is composed of a long chain of one monomer, followed by a long chain of another monomer. So if a diblock copolymer is composed of monomers A and B, its chain would look like A-A-...-A-B-...-B-B. The chemical process that is the basis for this research is the deposition of a Polystyrene-Polyethyleneoxide (referred to as PS-PEO from this point on) diblock copolymer solution (dissolved in chloroform) onto a water surface. The self-organized patterns that form at the air-water interface as a result are the subject of the modeling.
The PEO part of the polymer is hydrophilic so contact with the water surface decreases the free energy of the PEO. Thus the PEO part of the polymer organizes as a flat pancake at the interface. On the other side of the polymer, the PS is hydrophobic so water contact is not preferable for it. Thus the PS sticks out of the PEO pancake like a tail. So when the polymer solution is deposited onto the water surface, the molecules move around on the surface because of diffusion and surface tension effects. The PS tails get entangled during this movement and as the solvent evaporates, the whole system slows down. Soon, the patterns are fixed. When this happens, the whole system is compressed to a particular surface pressure in the Langmuir-Blodgett trough where the whole process takes place. Here, we never compress the system enough for the PEO pancakes to interact with each other. So brushes (3D structures caused by the subduction of the PEO) are never observed. This is thus a 2D problem. After the compression, the patterns are deposited on a substrate and imaged. Depending on the parameters, we can get dots, spaghetti or continents. Some example experimental images are in Figure 1.
The main effect responsible for the patterns is the entanglement of the PS tails. This is modeled by computing a function that describes the forces felt by a single PS-PEO polymer due to another PS-PEO polymer some distance away. This function is computed probabilistically and a plot of it for a particular polymer/solvent system appears in Figure 2. After all of the other effects are accounted for (diffusion, surface tension and the solvent evaporation), full simulations can be run. Starting with a constant (plus 1% random disturbance) concentration of solution, the system organizes itself into spaghetti which then breaks up into dots (a sample run is in Figure 3). In fact, it appears that the final state of the system is determined largely by the time it takes the system to anneal.
More details can be found in the full report linked above. Note that this report contains information as of August 2002. A lot of progress has been made since then, the results of which will soon be submitted for publication. Here are some pictures and animations from the new simulations:
Figure 1: Microscope images of patterns created by the deposition of PS-PEO. Dots, spaghetti and continents are shown here. More image are here.
Figure 2: Function that describes the force felt by a single polymer due to another polymer some distance away. Notice the repulsion for small separations and the attraction for larger separations. Larger image here .
Figure 3: A simulation showing a time evolution of the PS-PEO monolayer. We see the initial continent break up into spaghetti and then start breaking further into dots.