Phase ordering dynamics of three dimensional lamellar patterns in block-copolymer melts

In collaboration with Denis Boyer (UNAM, Mexico) & Jorge Viñals (CSIT, FSU)
 

Spatial structures with a well-defined periodicity (or modulated phases) are likely to emerge spontaneously in many thermodynamical systems. Cases of special interest are convection rolls in Rayleigh-Bénard cells driven out of equilibrium [16], or the structures observed in block-copolymer melts [17]. Modulated phases are ubiquitous in nature, as they can form in biological systems, in chemical reactions with diffusion, or in magnetic compounds, magnets, superconductors [15]. A tendency towards periodic order generally results from a frustration due to the competition between interactions that are repulsive at large distances and attractive at short range.

Our general objective here is to understand theoretically the formation of these structures, their dynamics, and their properties in spatially extended systems where they have complex spatial distributions and numerous topological defects.

Our approach is to use methods of nonlinear analysis combined with statistical theories based on scaling approaches, and complement this theoretical considerations by extensive numerical computations.
 
 
 

(A) Disordered	(C) Lamellar with defects
(B) Transient

Figure:Evolution (A-B-C) of the order parameter representing the local density difference between the two monomers constituting the polymer.
 
 

To this last end, we solve numerically a time-dependent Ginzburg-Landau equation that describes the dynamics of layered patterns observed in three dimensional non-equilibrium systems. The equation describes ordering process of symmetric diblock copolymers melts towards lamellar phases, when quenched at a temperature sightly below the order-disorder transition temperature. Parallel computing allows us to deal with systems of large aspect ratio for the first time in three dimensions. During ordering at shallow quenches, we observed qualitative differences with well studied two-dimensional case in both dynamics and topological structure.

The initial calculations have been already undertaken and the results have been reported at APS March Meeting, 2001. The problem turned out to be really challenging from the computational point of view, mainly because of two reasons: very slow wavenumber selection and strong finite size effects, both compared with 2D case [18]. Thus our preliminary results indicate that two and three-dimensional systems may belong to different universality classes. The physical explanations of this phenomenon is a subject of further study.