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. 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 very 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. 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 (eq. 2). The diffusion coefficient is given as D = Γa2/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.
Read more about this topic: Surface Diffusion