The structure formation upon spinodal decomposition of a two-dimensional model system, a Au adatom gas on a Au(111) surface, was observed in situ by scanning tunneling microscopy (STM). A thermodynamically unstable state was prepared by applying microsecond voltage pulses to the STM tip in an electrochemical system, causing the random dissolution of Au atoms from the uppermost monolayer. Interconnected, labyrinthine island patterns were formed at Au coverages between 0.4 and 0.9 monolayer with dominating length scales m of the order of a few nanometers. DOI: 10.1103/PhysRevLett.91.066101 PACS numbers: 64.75.+g, 05.70.-a, 68.37.Ef, 82.45.Qr The evolution of structure out of a homogeneous system undergoing phase separation is a common phenomenon, encountered upon the condensation of water droplets in clouds [1] as well as upon metal deposition on surfaces [2]. From experience such first order phase transitions are mostly governed by the formation of compact shapes with minimized phase boundary energy, as a consequence of the formation of stable nuclei and their subsequent growth. However, as already pointed out by Gibbs, upon penetrating deeply enough into the coexistence region, the activation barrier against the formation of stable nuclei vanishes and the system becomes unstable. Because of the absence of any activation barrier the instantaneous, i.e., spinodal, decomposition of the system proceeds very fast, with density fluctuations of a certain wavelength growing exponentially with time [3]. This wavelength is determined solely by the thermodynamic properties of the system [4]. When the volume fractions of the phases of the heterogeneous system are close to 50%, labyrinthine interconnected patterns evolve, distinctively different from the morphologies expected upon nucleation and growth.In this Letter we demonstrate how the thermodynamic stability of a surface system can be employed to steer the morphology of resulting patterns from growth of compact islands to labyrinthine interconnected structures on the nanoscale. Phase transitions in a two-dimensional (2D) model system, a gas of Au adatoms on a Au(111) surface, are induced by electrochemical, randomly distributed dissolution of Au atoms out of the topmost layer of a single crystal surface. This process occurs during a voltage pulse of only microsecond duration applied to the tip of an electrochemical scanning tunneling microscope (STM) facing the sample surface. This period is short enough to avoid significant diffusion and ordering already during the change of the state of the system. The resulting atomic-scale morphology is then imaged in situ by STM. The simplicity of the system justifies its description by a 2D lattice gas model with pairwise nearest neighbor interactions. Therefore, the system comes close to the models considered in the very first theoretical treatments of spinodal decomposition by Cahn and Hilliard [4,5] and the evolving structures can be directly compared to the theoretical predictions, derived explicitly for lattice model syst...