| Experimental evidence has accumulated in the recent decade that nanoscale patterns can self-assemble on solid surfaces. A two-component monolayer grown on a solid surface may separate into distinct phases, forming periodic stripes, triangular lattice of dots, or other regular patterns. The size of the phases may be in the range 1–100 nm, and stable against coarsening on annealing. This thesis develops a thermodynamic framework to study the remarkable phenomena. The thermodynamic forces that drive the self-assembly are identified. A double-welled, composition-dependent free energy drives phase separation. The phase boundary energy drives phase coarsening. The concentration-dependent surface stress drives phase refining. It is the competition between the coarsening and the refining that leads to size selection and spatial ordering. These thermodynamic forces are embodied in a nonlinear diffusion equation. The linear stability analysis gives the stable condition for a uniform concentration field, suggests the fastest growth wavelength, and provides an estimation of the pattern sizes. The numerical techniques to simulate the pattern formation process are developed. The simulations reveal rich dynamics. It is relatively fast for the phases to separate and select a uniform size, but exceedingly slow to order over a long distance, unless the symmetry of the system is suitably broken. The effects of inhomogeneity and anisotropy are investigated. Coarse pre-patterns, which can be introduced in the form of initial conditions or spatial dependent chemical potentials, may guide the self-assembly process and help to generate desired fine patterns. Anisotropy introduces directional preference. By varying the degree of surface stress anisotropy, a variety of patterns, such as serpentine stripes, parallel stripes, triangular lattice of dots, and herringbone structures are obtained. In addition, a surprising second order transition between the parallel stripes and the herringbone structures is discovered. The simulations reveal that patterns on an elastically anisotropic substrate with a cubic structure tend to arrange along the compliant directions. Patterns like square lattice of dots are obtained in this way. |
