## ICIAM 2011: Phase Behavior of Complex Fluids

**March 18, 2012**

Figure 1. Critical nuclei of gyroid–cylinder (left) and cylinder–gyroid (right) nucleation in diblock copolymer blends. The blue surface separates the A-rich and B-rich domains, and the red surface indicates the boundary of the nucleus.

**Pingwen Zhang**

Systems whose physical and mechanical energy scales are comparable to that of thermal energy at room temperature and are thus easily deformed by thermodynamic driving forces are considered soft matter. Examples include polymers, colloids, liquid crystals, gels, and a number of biological materials. Complex fluids, a subset of multi-component soft matter, are materials that can flow and that display non-Newtonian rheology. Soft matter and complex fluids are ubiquitous in nature and have a number of important industrial applications. What makes complex fluids interesting is that their properties can be quite different from those of liquids or crystals made up of small molecules.

Many phenomena of soft matter and complex fluids are independent of the particular chemical structure of their molecular building blocks, as demonstrated by the universality of phase behavior among the different classes of materials. Liquid crystals and block copolymers all self-assemble into ordered phases, as a consequence of effective interactions that are repulsive at short distances and attractive at long distances. Consequently, the systems favor an intermediate scale, which becomes the scale for the ordered phases.

Discovery of the ordered phases of soft matter is accomplished by mean field theory (MFT, also known as self-consistent field theory). The main idea is to replace all interactions of any one body by an average, or effective interaction. In essence, this reduces any multi-body problem to a one-body problem.

A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another. Phase transitions often (but not always) take place between phases with different symmetries. Generally, we can speak of one phase in a phase transition as being more symmetric than the other. The transition from the more symmetric to the less symmetric phase is a symmetry-breaking process. When symmetry is broken, one or more extra variables (order parameters) must be introduced to describe the state of the system. An order parameter can take the form of a complex number, a vector, or even a tensor, the magnitude of which goes to zero at the phase transition.

When an ordered phase becomes unstable, the initial transition pathway depends on its thermodynamic stability. When the ordered phase is thermodynamically unstable, the phase transition proceeds via spinodal decomposition, a mechanism by which a system of one or more components separates into distinct phases uniformly throughout the material. When the ordered phase is metastable, the phase transition proceeds via nucleation and growth, which happens randomly at discrete sites in the material. In the context of large deviation theory, it can be shown that the most probable transition path between two stable phases is a minimum-energy path (MEP) associated with the free energy of the system. The nucleation of ordered phases could be investigated by examining the MEP, which can be computed by the string method (Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, "String Method for the Study of Rare Events," in

*Physical Reviews B*, 66, 052301, 2002) applied to the self-consistent field theory of polymers. The structure, shape, and size of critical nuclei of the different ordered phases are obtained from the saddle point on the MEP. Figure 1 shows the critical nuclei of gyroid–cylinder transitions in diblock copolymer blends.

One very interesting aspect of polymer or liquid crystal systems is that there are numerous ordered phases, many of which have intricate geometric structure. The phase transitions between these phases are both important and difficult issues in the study of soft matter. We have built an efficient platform, including remodeling, effective iterative methods, and strategies for choosing initial values, to discover the ordered phases and transition states.

Along with phase behavior and phase transitions, the dynamic behavior of soft matter and complex fluids is a source of challenging research problems. Study of these problems involves a combination of modeling, simulation, mathematical analysis, and physical predictions.

*Pingwen Zhang is a professor of mathematics at Peking University. He works on the modeling and simulation of complex fluids, and has also made contributions to the moving mesh method. He is currently the executive deputy director of the Center for Computational Science and Engineering at Peking University.*