Surface diffusion and mass transport are key processes in surface science, influencing reactions and material growth. These phenomena involve the movement of atoms or molecules across surfaces, driven by energy differences and concentration gradients. Understanding these mechanisms is crucial for optimizing catalysis, thin film growth, and nanostructure formation.

Kinetics of surface processes are deeply intertwined with diffusion. The rates of surface reactions, adsorption, and desorption are often controlled by how quickly species can move across the surface. This interplay shapes the overall behavior of surface systems, from catalytic efficiency to the morphology of growing crystals.

Surface Diffusion Mechanisms

Fundamental Processes and Energy Landscape

• Surface diffusion involves the movement of atoms, molecules, or clusters across a surface
  • Primary mechanisms are hopping between adjacent adsorption sites (terrace diffusion) and exchange with surface atoms (exchange diffusion)
• Diffusion rates are influenced by the strength of the interaction between the adsorbate and the surface (adsorption energy) and the activation energy barrier for hopping or exchange
  • Stronger adsorption energy leads to slower diffusion rates
  • Higher activation energy barriers result in slower diffusion kinetics
• The potential energy surface (PES) describes the variation in the adsorption energy as a function of the position on the surface
  • Minima in the PES correspond to stable adsorption sites
  • Maxima represent transition states for diffusion
  • The shape of the PES determines the preferred diffusion pathways and barriers

Diffusion Coefficients and Anisotropy

• The diffusion coefficient, D, quantifies the rate of surface diffusion
  • Related to the jump frequency and jump distance
  • Follows an Arrhenius-type temperature dependence: $D = D_0 \exp(-E_a/k_BT)$
  • $D_0$ is the pre-exponential factor, $E_a$ is the activation energy for diffusion, $k_B$ is the Boltzmann constant, and $T$ is the temperature
• Surface diffusion can be isotropic (equal in all directions) or anisotropic (direction-dependent)
  • Depends on the symmetry of the surface and the PES
  • Anisotropic diffusion can lead to the formation of elongated islands or anisotropic growth patterns (nanowires, nanoridges)
  • Examples of anisotropic diffusion: diffusion along step edges, reconstruction-mediated diffusion on semiconductor surfaces (Si(001), GaAs(001))

Surface Diffusion in Reactions

Role in Heterogeneous Catalysis

• Surface diffusion plays a crucial role in the kinetics of surface reactions by enabling reactants to encounter each other and form products
  • The rate of diffusion can be the limiting step in the overall reaction kinetics
  • Rapid diffusion allows for efficient mixing and increases the probability of reactive collisions
• In heterogeneous catalysis, surface diffusion allows reactants to reach active sites and facilitates the formation and desorption of products
  • The interplay between diffusion and reaction rates determines the overall catalytic performance
  • Examples: CO oxidation on platinum catalysts, ammonia synthesis on iron catalysts, hydrocarbon reforming on metal surfaces

Influence on Crystal Growth and Morphology

• Surface diffusion is a key process in crystal growth, enabling adatoms to reach step edges and kink sites for incorporation into the growing crystal lattice
• The balance between the rates of adatom arrival (via deposition or surface diffusion) and attachment at step edges determines the growth mode and resulting surface morphology
  • Layer-by-layer growth (Frank-van der Merwe): diffusion is fast relative to the deposition rate, allowing adatoms to reach step edges before forming new islands
  • Island growth (Volmer-Weber): diffusion is slow relative to the deposition rate, leading to the formation of 3D islands
  • Multilayer growth (Stranski-Krastanov): initial layer-by-layer growth followed by island formation due to strain or change in surface energy
• Diffusion-limited aggregation (DLA) models describe the formation of fractal-like structures when diffusion is the rate-limiting step in crystal growth or particle aggregation on surfaces
  • Examples: dendritic growth of snowflakes, electrodeposition of metals, growth of nanowires and nanoparticles

Surface Defects and Diffusion

Influence of Defects and Steps

• Surface defects, such as vacancies, adatoms, and impurities, can act as traps or barriers for diffusing species, altering the local diffusion rates and pathways
  • Vacancies can provide low-energy sites for adatom diffusion and nucleation
  • Adatoms can act as nucleation centers for island formation or as mobile species that facilitate mass transport
  • Impurities can block diffusion pathways or create local strain fields that influence diffusion
• Steps and kinks on surfaces provide low-coordination sites that can act as preferential adsorption sites and diffusion channels
  • Diffusion along step edges (1D diffusion) can be faster than on terraces (2D diffusion)
  • Kinks act as incorporation sites for adatoms during crystal growth
• The Ehrlich-Schwoebel barrier is an additional energy barrier for diffusion across step edges, arising from the reduced coordination of atoms at the step
  • Can lead to the formation of mounds or instabilities during crystal growth
  • Influences the interlayer mass transport and the resulting surface morphology

Grain Boundary and Surfactant Effects

• Grain boundaries in polycrystalline materials serve as high-diffusivity paths for mass transport due to their disordered structure and excess free volume
  • Grain boundary diffusion can dominate over bulk diffusion at low temperatures
  • Plays a role in sintering, creep, and grain growth processes
• Surfactants or adsorbates can modify the surface diffusion behavior by altering the PES, creating new diffusion barriers, or providing alternative diffusion pathways
  • Surfactants can promote layer-by-layer growth by reducing the Ehrlich-Schwoebel barrier and enhancing interlayer mass transport
  • Examples: Sb as a surfactant in the growth of Ge on Si(111), Bi as a surfactant in the growth of InGaAs on GaAs(001)

Modeling Surface Transport

Fick's Laws and Diffusion Equation

• Fick's laws of diffusion provide a mathematical framework for describing mass transport driven by concentration gradients
  • Fick's first law relates the diffusive flux ($J$) to the concentration gradient ($\nabla C$): $J = -D \nabla C$
  • Fick's second law describes the time evolution of the concentration profile: $\partial C/\partial t = D \nabla^2 C$
• The diffusion equation, derived from Fick's second law, can be solved analytically or numerically to predict the spatial and temporal distribution of diffusing species on surfaces or in materials
  • Solutions depend on the initial and boundary conditions, geometry, and dimensionality of the system
  • Examples: diffusion in semi-infinite media, diffusion in thin films, diffusion from a constant source

Kinetic Models and Simulations

• The Langmuir-Hinshelwood mechanism is a common model for surface reactions, assuming that the reaction rate is proportional to the surface coverages of the adsorbed reactants
  • Accounts for the interplay between adsorption, desorption, and surface diffusion
  • Provides insights into the reaction order, rate-limiting steps, and coverage-dependent kinetics
• Monte Carlo simulations and kinetic Monte Carlo (KMC) methods are powerful computational tools for modeling surface diffusion and reaction processes at the atomic scale
  • Capture the stochastic nature of diffusion events and provide insights into the evolution of surface morphology and composition
  • Can incorporate realistic interatomic potentials, lattice structures, and event rates based on experimental or theoretical input
  • Examples: modeling of island nucleation and growth, simulation of catalytic reactions, study of pattern formation during epitaxial growth

Continuum Models

• Continuum models describe the evolution of surface morphology during crystal growth by considering the interplay between surface diffusion, attachment/detachment kinetics at step edges, and step-step interactions
• The Burton-Cabrera-Frank (BCF) theory is a classic continuum model for crystal growth
  • Describes the evolution of step positions and surface height profile
  • Accounts for surface diffusion, step velocity, and step-step interactions (repulsive or attractive)
  • Predicts the formation of step bunches, meandering instabilities, and other growth morphologies
• Extensions of the BCF theory include:
  • Incorporation of elastic effects and strain relaxation in heteroepitaxial growth
  • Coupling with deposition fluxes and evaporation processes
  • Consideration of anisotropic diffusion and attachment kinetics
  • Modeling of multicomponent systems and alloy growth
• Phase-field models and level-set methods are alternative continuum approaches for simulating surface evolution and morphological instabilities
  • Describe the surface as a continuous field variable evolving under the influence of thermodynamic driving forces and kinetic processes
  • Can capture complex geometries, topological changes, and faceting transitions