Microscopic approach to the kinetics of pattern formation of charged molecules on surfaces

Abstract

A microscopic formalism based on computing many-particle densities is applied to the analysis of the diffusion-controlled kinetics of pattern formation in oppositely charged molecules on surfaces or adsorbed at interfaces with competing long-range Coulomb and short-range Lennard-Jones interactions. Particular attention is paid to the proper molecular treatment of energetic interactions driving pattern formation in inhomogeneous systems. The reverse Monte Carlo method is used to visualize the spatial molecular distribution based on the calculated radial distribution functions (joint correlation functions). We show the formation of charge domains for certain combinations of temperature and dynamical interaction parameters. The charge segregation evolves into quasicrystalline clusters of charges, due to the competing long- and short-range interactions. The clusters initially co-exist with a gas phase of charges that eventually add to the clusters, generating "fingers" or line of charges of the same sign, very different than the nanopatterns expected by molecular dynamics in systems with competing interactions in two dimensions, such as strain or dipolar versus van der Waals interactions.

Statistical characterization of self‐assembled charged nanoparticle structures

“…The complete set of coupled kinetic equations for particle many‐body densities is similar to the Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy and given by where subscripts α and β denote different sorts of particles. These equations describe the particle diffusion, D , in potentials of mean forces W that are functionals of the correlation functions η ( r , t ) (CF) at distance r and time t .…”

“…The set of equations coincide with the first exact equations based on many‐body theory while effective potentials of mean forces are obtained approximately. We used two approximations: (i) Coulomb interaction contribution in the potentials of mean forces is found in a standard way for Coulomb systems by solution of Poisson equation, where charge density is written in the terms of CF . (ii) Contribution of short‐range interactions is found in the Kirkwood approximation known in the statistical physics of condensed systems.…”

“…We used two approximations: (i) Coulomb interaction contribution in the potentials of mean forces is found in a standard way for Coulomb systems by solution of Poisson equation, where charge density is written in the terms of CF . (ii) Contribution of short‐range interactions is found in the Kirkwood approximation known in the statistical physics of condensed systems. For equilibrium systems this approximation along with other hypernetted‐chain or Percus–Yevick type approximations defines the group of self‐consistent approximations used in development of semi‐quantitative theories.…”

“…To handle this problem, we demonstrate here the potential of the integrated approach that includes: (i) analytical part (introduction of kinetic self‐consistent equations for radial distribution functions), (ii) the numerical part (solution of these equations by obtaining radial distribution functions), and (iii) change of radial distribution representation to real space particle distribution (transformation of information). In the latter case, partial structure factors and characteristic system snapshots obtained by the reverse Monte Carlo (RMC) are determined through the radial distribution functions.…”

Pattern Formation Kinetics for Charged Molecules on Surfaces: Microscopic Correlation Function Analysis

“…' APPENDIX Detailed analysis of our method is presented in ref 10. The process kinetics is described by the following set of integrodifferential equations for the correlation functions of similar, X ν (r, t), and dissimilar, Y(r, t), molecules: Here W νν 0(r, t) are the effective interaction energies and D ν are mutual diffusion coefficients, D = D A + D B .…”