Energy characteristics of carbon clusters with passivated bonds V. V. Rotkin* ) and R. A. Suris A. F. Ioffe Physicotechnical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia Fiz. Tverd. Tela St. Petersburg 41, 809–812 May 1999 A modified phenomenological model is proposed for calculating the formation energy of carbon nanoclusters which makes it possible to analyze the regions of existence of clusters of various forms. A new parameter of the model, which corresponds to passivation of broken carbon bonds, affects the shape of the equilibrium optimum clusters, i.e., those having a minimum energy for a fixed number of atoms. Analytic dependences of equilibrium-configuration states determining the existence of spheroidal closed clusters, nanopipes, and fragments of a graphite plane, on the broken-bond energy parameter obtained in this model are presented. © 1999 American Institute of Physics. S1063-78349901305-2 Theoretical studies of the synthesis of carbon nanoclus- ters in electric arcs or by laser ablation of graphite, in mo- lecular and atomic beams, by igniting hydrocarbons, or by other methods are difficult since the conditions for synthesis Cluster production techniques have been reviewed by Smol- ley et al. 1  are utterly different and the set of synthesized clusters is usually diverse and hard to classify. At present it has been established firmly that a large number of small car- bon clusters are present in the synthesis products: presum- ably linear chains carbene type or with free bonds or frag- ments of a monolayer of a graphite plane graphene fragments. More compact nanoclusters have also been found: with cylindrical and spherical shapes or unclosed fragments of these with characteristic radii of a few nm. Conditions have been found 2 for synthesizing rather long several  m cylindrical clusters, i.e., nanopipes, with vari- ous diameters. Fragments of conical surfaces, multilayer clusters, etc., have been observed among the clusters. The theoretical description of the synthesis of carbon nanoclus- ters is also complicated because, up to now, no final conclu- sion has been reached as to how much this process is deter- mined by the reaction kinetics, or by an energy or entropy factor. Little is known about the reaction kinetics for forma- tion of the various clusters, while the formation energies of a large number of isomers C N have been calculated by various methods, ranging from phenomenological to first-principles calculations. We have proposed 3 a unified approach to the energy characteristics of the formation of carbon nanoclus- ters with a curved surface like graphite. It allows us to com- pare the formation energy of fullerenes with different shapes, so that it is possible to determine the most energetically fa- vorable clusters i.e., the clusters having the minimum for- mation energy for a fixed number of atoms in a continuum approximation. These calculations allow us to judge the probability of forming clusters of a given shape for an equi- librium synthesis process while including the entropy factor does not significantly change the free energy, as well as whether a given isomer is in equilibrium. Note that an en- ergetically nonequilibrium state of the cluster C N is not nec- essarily unstable. Examining the stability of a state requires a detailed study of the kinetics of a specific transition by the cluster from one state of configuration space to another. Our method has been formulated in earlier papers 4 and is based on expanding the total formation energy of a cluster into independent in our approximation terms corresponding to: 1 the ‘‘seed’’ formation energy of graphene this con- stant term determines the reference level and will not be included in the following calculations; 2 the energy of curvature of the cluster surface, which is analogous to the elastic deformation energy of a plane; 3 the energy associ- ated with pentagonal defects that are not characteristic of graphene; and, 4 the energy of broken bonds. This paper is devoted to accounting for the passivation of broken bonds, which in terms of the model corresponds to varying the broken-bond energy parameter. Here we show that this modification of the model leads to a change in the results on the energy equilibrium in the configuration space of clusters having different shapes. Specifically, it has been shown 5 that ‘‘softening’’ the bonds changes the energy diagram for the phases relationship between flat fragments of graphene and spheres and nanopipes in favor of the uncoiled fragments. In the first part of the paper, we formulate a model and study the effect of the magnitude of the broken bond energy on the shape of an optimum cluster with nanopipes as an example. The energy diagram for the coexistence of nan- opipes and flat graphene fragments is constructed in the sec- ond part. We also illustrate the change in the diagram when the bonds are ‘‘softened.’’ The third part is devoted to cal- culating the critical value of the bond ‘‘softness,’’ which is defined as the value at which the equilibrium positions of the different states in configuration space undergo a change. 1. EFFECT OF THE ‘‘SOFTNESS’’ OF BROKEN BONDS ON THE OPTIMUM CLUSTER SHAPE We shall specify the formation energy of a cluster through its geometric dimensions and shape. The greater the curvature of the cluster surface, the higher the energy asso- ciated with bond deformation. The first parameter of the model is a phenomenological parameter that specifies the characteristic deformation energy of a single bond for unit curvature and has been chosen equal to E c 0.9 eV. 6 The PHYSICS OF THE SOLID STATE VOLUME 41, NUMBER 5 MAY 1999 729 1063-7834/99/41(5)/4/$15.00 © 1999 American Institute of Physics