Chemically controlled pattern formation in phase-separating materials

Phase transition of a symmetric diblock copolymer induced by nanorods with different surface chemistry

We investigate the phase transition of a symmetric diblock copolymer induced by nanorods with different surface chemistry. The results demonstrate that the system occurs the phase transition from a disordered structure to ordered parallel lamellae and then to the tilted layered structure as the number of rods increases. The dynamic evolution of the domain size and the order parameter of the microstructure are also examined. Furthermore, the influence of rod property, rod-phase interaction, rod-rod interaction, rod length, and polymerization degree on the behavior of the polymer system is also investigated systematically. Moreover, longer amphiphilic nanorods tend to make the polymer system form the hexagonal structure. It transforms into a perpendicular lamellar structure as the polymerization degree increases. Our simulations provide an efficient method for determining how to obtain the ordered structure on the nanometer scales and design the functional materials with optical, electronic, and magnetic properties.

Rigorous embedding of cell dynamics simulations in the Cahn-Hilliard-Cook framework: Imposing stability and isotropy

We have rigorously analyzed the stability of the efficient cell dynamics simulations (CDS) method by making use of the special properties of the local averaging operator 〈〈*〉〉-* in matrix form. Besides resolving a theoretical issue that has puzzled many over the past three decades, this analysis has considerable practical value: It relates CDS directly to finite-difference approximations of the Cahn-Hilliard-Cook equations and provides a straightforward recipe for replacing the original two- or three-dimensional (2D or 3D) averaging operators in CDS by an equivalent (in terms of stability) discrete Laplacian with superior isotropy and scaling behavior. As such, we open up a route to suppress the unphysical reflection of the computational grid in CDS results (grid artifacts). We found that proper rescaling of discrete Laplacians, needed to employ them in CDS, is equivalent to introducing a well-chosen time step in CDS. In turn, our analysis provides stability conditions for phase-field simulations based on the Cahn-Hilliard-Cook equations. Subsequently, we have quantitatively compared the isotropy and scaling behavior of several discrete 2D or 3D Laplacians, thereby extending the significance of this work to general field-based methodology. We found that all considered discrete Laplacians have equivalent scaling behavior along the Cartesian directions. In addition, and somewhat surprisingly, known "isotropic" discrete Laplacians, i.e., isotropic up to fourth order in |k|, become quite anisotropic for larger wave vectors, whereas "less isotropic" discrete Laplacians (second order) are only slightly anisotropic on the whole |k| range. We identified a hard limit to the accuracy with which the discrete Laplacian can emulate the two important properties of the optimal (continuum) Laplacian, as an improvement of the isotropy, by introducing additional points to the stencil, will negatively affect the scaling behavior. Within this limitation, the discrete compact Laplacians in the DnQm class known from lattice hydrodynamics, D2Q9 in 2D and D3Q19 in 3D, are found to be optimal in terms of isotropy. However, by being only slightly anisotropic on the whole range and enabling larger time steps, the discrete Laplacians that relate to the local averaging operator of Oono and Puri (2D) and Shinozaki and Oono (3D) as well as the less familiar 3D discrete BvV Laplacian developed for dynamic density functional theory are valid alternatives.