Surface diffusion

[
adatom diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to scale on a spatial or temporal basis.] 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 have been shown to display diffusion behavior which 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 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 which dictates the probability of an attempt resulting in a successful jump. The attempt frequency $u,!$ is typically taken to be simply the vibrational frequency of the adatom, while the thermodynamic factor is a Boltzmann factor dependent on temperature and E_diff, the potential energy barrier to diffusion. Equation 1 describes the relationship:

: $Gamma\; =\; u\; e^\{-E\_mathrm\{diff\}\; /k\_BT\}\; qquad\; ext\{(eq.\; 1)\}$

Where $u,!$ and E_diff are as described above, $Gamma,!$ is the jump or hopping rate, T is temperature, and k_B is the Boltzmann constant. E_diff must be smaller than the energy of desorption for diffusion to occur, otherwise desorption processes would dominate. Importantly, equation 1 tells us how very strongly the jump rate varies with temperature. The manner in which diffusion takes place is dependent on the relationship between E_diff and k_BT as is given in the thermodynamic factor: when E_diff is less than k_BT the thermodynamic factor approaches unity and E_diff 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, E_diff is assumed to be ≫ "k"_"B""T" and therefore $Gamma\; ll\; u\; ,!$. In the case of Fickian diffusion it is possible to extract both the $u,!$ and "E"_diff from an Arrhenius plot of the logarithm of the diffusion coefficient, "D", versus 1/"T". For cases in which more than one diffusion mechanism is present (see below), there may be more than one "E"_diff such that the relative distribution between the different processes would change with temperature.

Random walk statistics describe the mean square 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 $Gamma,!$ 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 square displacement goes as $sqrt\{langleDelta\; r^2\; angle\}\; =\; asqrt\{Gamma\; t\},!$ (eq. 2). The diffusion coefficient is given as $D\; =\; frac\{Gamma\; a^2\}\{z\},!$ (eq. 3), where "z" = 2 for 1D diffusion as would be the case for in-channel diffusion, z = 4 as in a square lattice, and "z" = 6 as in a hexagonal lattice. [Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 325-326]

Regimes

Fick’s law, flux is in the opposite direction of the concentration gradient, a purely statistical effect. The model is not intended to show repulsion or attraction, and is not to scale on a spatial or temporal basis.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 homogenous 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 E_diff, 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-step-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. 333] [Shustorovich 1991, p. 114-115]

Directional anistropy 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 Lattice (group)
rectangular lattice) or the presence of steps on a surface. One of the more dramatic examples of directional anistropy 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

*Concerted mechanisms are those that involve movement of either sections of the cluster or the entire cluster all at once. [Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 343-345]
**Dislocation diffusion occurs when adjacent sub-units of a cluster move in a row-by-row fashion through displacement of a dislocation. As shown in figure 11(a) the process begins with nucleation of the dislocation followed by what is essentially sequential displacement on a concerted basis.
**Glide diffusion refers to the concerted motion of an entire cluster all at once (see figure 11(b)).
**Reptation is a snake-like movement (hence the name) involving sequential motion of cluster sub-units (see figure 11(c)).
**Shearing is a concerted displacement of a sub-unit of atoms within a cluster (see figure 11(d)).
*Size-dependence: the rate of cluster diffusion has a strong dependence on the size of the cluster, with larger cluster size generally corresponding to slower diffusion. This is not, however, a universal trend and it has been shown in some systems that the diffusion rate takes on a periodic tendency wherein some larger clusters diffuse faster than those smaller than them. [Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 341-343]

urface diffusion and heterogeneous catalysis

Surface diffusion is a critically important concept in heterogeneous catalysis, as reaction rates are often dictated by the ability of reactants to "find" each other at a catalyst surface. With increased temperature adsorbed molecules, molecular fragments, atoms, and clusters tend to have much greater mobility (see equation 1). However, with increased temperature the lifetime of adsorption decreases as the factor k_BT becomes large enough for the adsorbed species to overcome the barrier to desorption, Q (see figure 2). Reaction thermodynamics aside, because of the interplay between increased rates of diffusion and decreased lifetime of adsorption, increased temperature may in some cases decrease the overall rate of the reaction.

Experimental

Surface diffusion may be studied by a variety of techniques, including both direct and indirect observations. Two experimental techniques that have proved very useful in this area of study are field ion microscopy and scanning tunneling microscopy. [Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 349] By visualizing the displacement of atoms or clusters over time, it is possible to extract useful information regarding the manner in which the relevant species diffuse-both mechanistic and rate-related information. In order to study surface diffusion on the atomistic scale it is unfortunately necessary to perform studies on rigorously clean surfaces and in UHV conditions or in the presence of small amounts of inert gas, as is the case when using He or Ne as imaging gas in FIM experiments.

ee also

*Surface engineering
*Surface science

References

Cited works

*G. Antczak, G. Ehrlich. "Surface Science Reports" 62 (2007), 39-61. (Review)
*cite book | first = K. | last = Oura | coauthors = V.G. Lifshits, A.A. Saranin, A.V. Zotov, M. Katayama | title = Surface Science: An Introduction | publisher = Springer-Verlag Berlin Heidelberg | year = 2003 | id = ISBN 3-540-00545-5
*cite book | first = E. | last = Shustorovich | title = Metal-Surface Reaction Energetics: Theory and Applications to Heterogeneous Catalysis, Chemisorption, and Surface Diffusion| publisher = VCH Publishers, Inc.| year = 1991 | id = ISBN 3-527-27938-5