Surface diffusion is a general process involving the motion of adatoms, molecules, and atomic clusters (adparticles) at solid material surfaces.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 325 The process can generally be thought of in terms of particles jumping between adjacent adsorption sites on a surface, as in figure 1. Just as in bulk diffusion, this motion is typically a thermally promoted process with rates increasing with increasing temperature. Many systems display diffusion behavior that deviates from the conventional model of nearest-neighbor jumps.Antczak, Ehrlich 2007, p.39 Tunneling diffusion is a particularly interesting example of an unconventional mechanism wherein hydrogen has been shown to diffuse on clean metal surfaces via the quantum tunneling effect.
Various analytical tools may be used to elucidate surface diffusion mechanisms and rates, the most important of which are field ion microscopy and scanning tunneling microscopy.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 349 While in principle the process can occur on a variety of materials, most experiments are performed on crystalline metal surfaces. Due to experimental constraints most studies of surface diffusion are limited to well below the melting point of the substrate, and much has yet to be discovered regarding how these processes take place at higher temperatures.Antczak, Ehrlich 2007, p. 50, 59
Surface diffusion rates and mechanisms are affected by a variety of factors including the strength of the surface-adparticle bond, orientation of the surface lattice, attraction and repulsion between surface species and chemical potential gradients. It is an important concept in surface phase formation, epitaxial growth, heterogeneous catalysis, and other topics in surface science.Shustorovich 1991, p. 109 As such, the principles of surface diffusion are critical for the chemical production and semiconductor industries. Real-world applications relying heavily on these phenomena include catalytic converters, integrated circuits used in electronic devices, and silver halide salts used in photographic film.
Kinetics
Surface diffusion kinetics can be thought of in terms of adatoms residing at adsorption sites on a 2D lattice, moving between adjacent (nearest-neighbor) adsorption sites by a jumping process.Shustorovich 1991, p. 109-111 The jump rate is characterized by an attempt frequency and a thermodynamic factor that dictates the probability of an attempt resulting in a successful jump. The attempt frequency ν is typically taken to be simply the vibrational frequency of the adatom, while the thermodynamic factor is a Boltzmann factor dependent on temperature and Ediff, the potential energy barrier to diffusion. Equation 1 describes the relationship:
Where ν and Ediff are as described above, Γ is the jump or hopping rate, T is temperature, and kB is the Boltzmann constant. Ediff must be smaller than the energy of desorption for diffusion to occur, otherwise desorption processes would dominate. Importantly, equation 1 tells us how strongly the jump rate varies with temperature. The manner in which diffusion takes place is dependent on the relationship between Ediff and kBT as is given in the thermodynamic factor: when Ediff < kBT the thermodynamic factor approaches unity and Ediff ceases to be a meaningful barrier to diffusion. This case, known as mobile diffusion, is relatively uncommon and has only been observed in a few systems.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 327 For the phenomena described throughout this article, it is assumed that Ediff >> kBT and therefore Γ << ν. In the case of Fickian diffusion it is possible to extract both the ν and Ediff from an Arrhenius plot of the logarithm of the diffusion coefficient, D, versus 1/T. For cases where more than one diffusion mechanism is present (see below), there may be more than one Ediff such that the relative distribution between the different processes would change with temperature.
Random walk statistics describe the mean squared displacement of diffusing species in terms of the number of jumps N and the distance per jump a. The number of successful jumps is simply Γ multiplied by the time allowed for diffusion, t. In the most basic model only nearest-neighbor jumps are considered and a corresponds to the spacing between nearest-neighbor adsorption sites. The root mean squared displacement goes as:
The diffusion coefficient is given as:
where for 1D diffusion as would be the case for in-channel diffusion, for 2D diffusion, and for 3D diffusion.Structure and Dynamics of Surfaces II (Topics in Current Physics), W. Schommers, P. Von Blanckenhagen, . Chapter 3.2, p. 75
Regimes
There are four different general schemes in which diffusion may take place.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 330-333 Tracer diffusion and chemical diffusion differ in the level of adsorbate coverage at the surface, while intrinsic diffusion and mass transfer diffusion differ in the nature of the diffusion environment. Tracer diffusion and intrinsic diffusion both refer to systems where adparticles experience a relatively homogeneous environment, whereas in chemical and mass transfer diffusion adparticles are more strongly affected by their surroundings. Tracer diffusion describes the motion of individual adparticles on a surface at relatively low coverage levels. At these low levels (< 0.01 monolayer), particle interaction is low and each particle can be considered to move independently of the others. The single atom diffusing in figure 1 is a nice example of tracer diffusion. Chemical diffusion describes the process at higher level of coverage where the effects of attraction or repulsion between adatoms becomes important. These interactions serve to alter the mobility of adatoms. In a crude way, figure 3 serves to show how adatoms may interact at higher coverage levels. The adatoms have no "choice" but to move to the right at first, and adjacent adatoms may block adsorption sites from one another. Intrinsic diffusion occurs on a uniform surface (e.g. lacking steps or vacancies) such as a single terrace, where no adatom traps or sources are present. This regime is often studied using field ion microscopy, wherein the terrace is a sharp sample tip on which an adparticle diffuses. Even in the case of a clean terrace the process may be influenced by non-uniformity near the edges of the terrace. Mass transfer diffusion takes place in the case where adparticle sources and traps such as kinks, steps, and vacancies are present. Instead of being dependent only on the jump potential barrier Ediff, diffusion in this regime is now also dependent on the formation energy of mobile adparticles. The exact nature of the diffusion environment therefore plays a role in dictating the diffusion rate, since the formation energy of an adparticle is different for each type of surface feature as is described in the Terrace Ledge Kink model.
Anisotropy Orientational anisotropy takes the form of a difference in both diffusion rates and mechanisms at the various surface orientations of a given material. For a given crystalline material each Miller Index plane may display unique diffusion phenomena. Close packed surfaces such as the fcc (111) tend to have higher diffusion rates than the correspondingly more "open" faces of the same material such as fcc (100).Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 333Shustorovich 1991, p. 114-115
Directional anisotropy refers to a difference in diffusion mechanism or rate in a particular direction on a given crystallographic plane. These differences may be a result of either anisotropy in the surface lattice (e.g. a rectangular lattice) or the presence of steps on a surface. One of the more dramatic examples of directional anisotropy is the diffusion of adatoms on channeled surfaces such as fcc (110), where diffusion along the channel is much faster than diffusion across the channel.
Mechanisms
Adatom diffusion Diffusion of adatoms may occur by a variety of mechanisms. The manner in which they diffuse is important as it may dictate the kinetics of movement, temperature dependence, and overall mobility of surface species, among other parameters. The following is a summary of the most important of these processes:Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 336-340 Hopping or jumping is conceptually the most basic mechanism for diffusion of adatoms. In this model, the adatoms reside on adsorption sites on the surface lattice. Motion occurs through successive jumps to adjacent sites, the number of which depends on the nature of the surface lattice. Figures 1 and 3 both display adatoms undergoing diffusion via the hopping process. Studies have shown the presence of metastable transition states between adsorption sites wherein it may be possible for adatoms to temporarily reside.Shustorovich 1991, p. 111 Atomic exchange involves exchange between an adatom and an adjacent atom within the surface lattice. As shown in figure 4, after an atomic exchange event the adatom has taken the place of a surface atom and the surface atom has been displaced and has now become an adatom. This process may take place in both heterodiffusion (e.g. Pt adatoms on Ni) and self-diffusion (e.g. Pt adatoms on Pt). It is still unclear from a theoretical point of view why the atomic exchange mechanism is more predominant in some systems than in others. Current theory points towards multiple possibilities, including tensile surface stresses, surface relaxation about the adatom, and increased stability of the intermediate due to the fact that both atoms involved maintain high levels of coordination throughout the process.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 338Antczak, Ehrlich 2007, p. 48 Tunneling diffusion is a physical manifestation of the quantum tunneling effect involving particles tunneling across diffusion barriers. It can occur in the case of low diffusing particle mass and low Ediff, and has been observed in the case of hydrogen diffusion on tungsten and copper surfaces.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 338-340 The phenomenon is unique in that in the regime where the tunneling mechanism dominates, the diffusion rate is nearly temperature-independent.Shustorovich 1991, p. 115 Vacancy diffusion can occur as the predominant method of surface diffusion at high coverage levels approaching complete coverage. This process is akin to the manner in which pieces slide around in a "sliding puzzle". It is very difficult to directly observe vacancy diffusion due to the typically high diffusion rates and low vacancy concentration.Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 340-341 Figure 5 shows the basic theme of this mechanism in an albeit oversimplified manner.
Recent theoretical work as well as experimental work performed since the late 1970s has brought to light a remarkable variety of surface diffusion phenomena both with regard to kinetics as well as to mechanisms. Following is a summary of some of the more notable phenomena: Long jumps consist of adatom displacement to a non-nearest-neighbor adsorption site. They may include double, triple, and longer jumps in the same direction as a nearest-neighbor jump would travel, or they may be in entirely different directions as shown in figure 6. They have been predicted by theory to exist in many different systems, and have been shown by experiment to take place at temperatures as low as 0.1 Tm (melting temperature). In some cases data indicate long jumps dominating the diffusion process over single jumps at elevated temperatures; the phenomena of variable jump lengths is expressed in different characteristic distributions of atomic displacement over time (see figure 7).Antczak, Ehrlich 2007, p. 51