INTERFACE SCIENCE 6, 259–264 (1998) c  1998 Kluwer Academic Publishers. Manufactured in The Netherlands. Monte Carlo Simulation of the Concentration Dependence of Segregation at Vicinal Grain Boundaries D. UDLER AND D.N. SEIDMAN Department of Materials Science and Engineering, R.R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208-3108 d seidman@nwu.edu Abstract. Using the Metropolis algorithm Monte Carlo technique solute-atom segregation is studied at two vicinal grain boundaries (GBs)—the  = 5/(002)/θ = 36.89 ◦ symmetrical twist and the  = 5/(310)/θ = 53.13 ◦ symmetrical tilt—at 850 K on the Ni-rich side of the Ni-Pt phase diagram, over the concentration range 0–10 at.% Pt. Unlike the Pt-rich side of the phase diagram the structures of both GBs remain stable in this concentration range. The dilute limit behavior for most GB sites extends to at least 0.1 at.%. At higher concentrations the effective segregation energies steadily decrease with increasing solute concentrations, due to solute-solute interactions between segregated atoms, until saturation occurs. It is argued that simple statistical mechanical models, e.g., the Fowler-Guggenheim model do not work well, even in the case of simple vicinal GBs. Keywords: grain boundaries, twist and tilt boundaries, solute-atom segregation, Metropolis Monte Carlo, simu- lation, nickel-platinum alloys, Langmuir-McLean and Fowler-Guggenheim segregation models 1. Introduction The chemical composition in alloys near crystalline imperfections is, in general, different from that in the bulk and this phenomenon is denoted solute-atom seg- regation. Solute-atom segregation at grain boundaries (GBs) is important for both scientific and technologi- cal reasons, as it affects material properties [1, 2]. The driving force for GB segregation is provided for by a decrease of a GB’s interfacial free energy, according to the Gibbs relation (for a binary alloy) Ŵ 2 =−  ∂γ ∂µ 2  T , P,specific geometric variables ; (1) where Ŵ 2 is the Gibbsian interfacial excess of solute, γ is the GB interfacial free energy, µ 2 is the chemical potential of the solute—which is uniquely related to the bulk concentration of solute, C bulk 2 . The sign of Ŵ 2 can be either positive or negative, corresponding to a net enhancement or depletion of solute at a GB. Besides the conventional thermodynamic degrees of freedom (DOFs), temperature (T ) and pressure ( P ), GB seg- regation also depends on geometrical DOFs [3]. The five macroscopic DOFs define the misorientation of the two grains and the plane of the interface [4]. While the three microscopic DOFs define the relative rigid-body displacement vector between the two grains [4]. The detailed exploration of the right-hand side of Eq. (1) is beyond the realm of classical thermodynam- ics and requires a microscopic investigation of the phys- ical mechanisms of segregation. In addition, for many applications, the spatial distribution of solute atoms near a GB is important, not just the value of Ŵ 2 . GBs are known to possess well-defined crystalline struc- tures [5], determined by their geometrical degrees of freedom and energetics. At a GB atoms sit on a lat- tice with two-dimensional periodicity which, together with symmetry, induces a partition of all GB sites into types, so that each individual site belongs to a certain type of crystallographically equivalent site. In the case of substitutional solute segregation the same GB sites, ignoring small local lattice relaxations, can be occu- pied either by a solute or solvent atom. The change