Journal of Nanoparticle Research 2: 333–344, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. Invited paper Forces that drive nanoscale self-assembly on solid surfaces Z. Suo and W. Lu Department of Mechanical and Aerospace Engineering, Princeton Materials Institute, Princeton University, Suite D404, Eng. Quadrangle, Princeton, NJ 08544, USA (Tel.: 609-258-0250; Fax: 609-258-5877; E-mail: suo@princeton.edu) Received 17 August 2000; accepted in revised form 11 October 2000 Key words: nanostructure, epitaxial film, self-assembly, surface stress, phase separation, nanoparticles Abstract 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. Sometimes the phases select sizes about 10 nm, and order into an array of stripes or disks. This paper reviews a model that accounts for these behaviors. Attention is focused on thermodynamic forces that drive the self-assembly. 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. Numerical simulations reveal rich dynamics of the pattern formation process. 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 is suitably broken. Introduction In the solid state, atoms can diffuse from one site to another, giving rise to conspicuous changes over time. In some circumstances, atoms may self-assemble into a periodic structure, such as an array of stripes or dots. The feature size may be of nanoscale, small compared to bulk structures, but large compared to individual atoms. In this intermediate size range, new phenomena appear. Why do atoms self-assemble? What sets the feature size? The answers differ for different material systems. A unifying concept, however, can be identi- fied. For many reasons the free energy of a material sys- tem depends on its configuration (e.g., the composition of the phases and their spatial arrangement). When the configuration changes, the free energy also changes. This defines thermodynamic forces that drive the con- figuration change. The change is effected by mass transport processes, such as diffusion. To assemble a nanostructure, some of the forces must act over the scale comparable to the feature size, and are there- fore much longer ranging than atomic bond length. The long-range forces have various physical origins, includ- ing elasticity, electrostatics, magnetostatics, photon dispersion, and electron confinement (Ng and Vander- bilt, 1995; Murray et al., 2000; Suo, 2000). They lead to self-assembly in diverse material systems (e.g., Chen & Khachaturyan, 1993; Seul & Andelman, 1995; BaIl, 1999; B ¨ ohringer, 1999). To discuss configurational forces in action, this paper focuses on a particular phenomenon: nanoscale phase patterns on solid surfaces. Studies of nanoscopic activ- ities on solid surfaces have surged after the invention of the Scanning Tunneling Microscopy (STM). Figure 1a is a schematic of an observed pattern. Kern et al. (1991) exposed a copper surface to gaseous oxygen of low