Random deposition model

The random deposition model represented by Eq. (4.1) produces a mono tonic increase of surface width with time. 

Surface scaling parameters for a number of nonequilibrium atomistic models have also been established [6, 10]. Continuum equations for the surface motion have to be used to find a solution for discrete models. Thus, for ballistic deposition [14] and the Eden model [15] the inter ce saturates, resulting in a = 1/2 and P = 1/3 for Z>pop = 2, and a 0.35 and 0.21 for Z>top = 3. Conversely, from the random deposition model P - 1/2 and, since the correlation length is always zero, fire interface does not saturate and, therefore, a is not defined. Depending on the rules used in the simulations, for the atomistic model including surfece difrusion a = 3/2 and p = 3/7 [6], a = 3/2 and p =... 

There a particle reaches the surface as in the random deposition model, but then is allowed to diffuse on the surface. The diffusion continues until the particle finds the colunm of minimum height inside a domain of finite size around the initial contact. A schematic view of the model and an example of the resulting interface for d = 2 is shown in Fig. 31.9. The surface diffusion generates a nontrivial... 

With these preliminary definitions we can now discuss some simple models of growth. The purpose of these models is to describe in a qualitative manner the evolution of real surfaces and to determine the three exponents we introduced above actually only two values are needed, since they are related by Eq. (11.17). The simplest model consists of a uniform flux of atoms being deposited on the surface. Each atom sticks wherever it happens to fall on the surface. Eor this, the so called random deposition model, the evolution of the surface height is given by... 

Figure 11.14. Simulation of one-dimensional growth models. The highly irregular hnes correspond to the surface profile of the random deposition model. The smoother, thicker hnes correspond to a model which includes random deposition plus diffusion to next neighbor sites, if this reduces the surface curvature locally. The two sets of data correspond to the same time instances in the two models, i.e. the same amount of deposited material, as indicated by the average height. It is evident how the diffusion step leads to a much smoother surface profile.

For a two-dimensional surface of a three-dimensional crystal = 2 in the above equation ), the roughness exponent is a = 1, larger than in the EW model. Thus, the surface profile width in the WV model is more rough than in the EW model, that is, the surface diffusion is not as effective in reducing the roughness as the desorption/deposifion mechanism. However, the surface profile will still be quite a bit smoother than in the random deposition model (see Eig. 11.14 for an example). 

Taking into consideration the Si—O bonds within the glass surface, the difference between a strongly reacted layer and a highly polymerized network is difficult to define. However, with the above model the siloxane layer also contains partially polymerized structural units and/or hydrolysed remnants of the three-dimensional layer which would be expected from the random deposition of the hydrolysed APS. Consequently, some fragments may arise from pendant chains. Thus, the actual struture of the deposit will consist of a poly-siloxane probably chemically bonded to the glass surface every third silicon atom. 

The observed almost universal value of the surface fractal dimension ds 2.6 of furnace blacks can be traced back to the conditions of disordered surface growth during carbon black processing. It compares very well to the results evaluated within the an-isotropic KPZ-model as well as numerical simulations of surface growth found for random deposition with surface relaxation. This is demonstrated in some detail in [18]. 

This section proposes a mechanism for r that is consistent with the random coke deposition model, the microbalance and TPO results, and the metal/acid dichotomy discussed... 

If each of the catalytic reactions involves only one active site, the random coke deposition model gives the following activity-time relation for nC reforming... 

Figure 1 Comparison between a simple random walk model of particle deposition and an electrochemically deposited copper fractal. (A) 2000 random walks on a square grid, (B) the digitized image (256 x 256 pixels) of copper electrodeposited from 0.75 mol I copper sulfate and 1 moll sulfuric acid in an 11 cm Whatman 541 filter paper at 5V. Both fractals are displayed using Lotus for Windows.

Random deposition (RD) is the simplest (but most unrealistic) deposition model for surface growth. The particles fall vertically at a constant rate independently of one another and stick when they reach the top of a column of deposited particles. As there is no horizontal correlation between the neighboring columns, the surface is extremely rough and the surface structure is compact, i.e., Ds = d = 3 for the case of three dimensional simulatimis [34—37]. 

A similar realistic model for surface growth during c.b. processing is random deposition with surface diffusion ... 

It can be concluded that as-deposited carbon films are initially amorphous but become organized under the weakest stress. Because such films initially do not contain coherent scattering domains, they are amorphous and conform to a random network model similar to that of Polk [36,37]. The occurrence of a persistent halo near 0.2 A favors the hypothesis of sp bonds initially being present, possibly replaced by sp bonds when the metastable amorphous stage really... 

In both cases, deposition occurs only during NP-electrode contact, which lasts approximately 1-10 ms. A random walk model was adapted to investigate the contact time required to electro deposit a monolayer of metal onto an NP during collision and found this to be on the order of 10" s for a 45 nm radius NP in 10 mM solution of analyte. This provided confirmation that the thermally driven NP collision contacts were of a sufficient duration to allow an experimentally observable faradaic process to occur. A theoretical model was also developed for the oxidation of an NP contacting an electrode held at a suitably oxidizing potential. The model incorporates Brownian motion to account for the experimentally observed timescale of such reactions. ... 

The main disadvantage of the perfect sink model is that it can only be applied for irreversible deposition of particles the reversible adsorption of colloidal particles is outside the scope of this approach. Dahneke [95] has studied the resuspension of particles that are attached to surfaces. The escape of particles is a consequence of their random thermal (Brownian) motion. To this avail he used the one-dimensional Fokker-Planck equation... 

In the mixed potential theory (MPT) model, both partial reactions occur randomly on the surface, both with respect to time and space. However, given the catalytic nature of the reductant oxidation reaction, it may be contended that such a reaction would tend to favor active sites on the surface, especially at the onset of deposition, and especially on an insulator surface catalyzed with Pd nuclei. Since each reaction strives to reach its own equilibrium potential and impose this on the surface, a situation is achieved in which a compromise potential, known as the mixed potential (.Emp), is assumed by the surface. Spiro [27] has argued the mixed potential should more correctly be termed the mixture potential , since it is the potential adopted by the complete electroless solution which comprises a mixture of reducing agent and metal ions, along with other constituents. However, the term mixed potential is deeply entrenched in the literature relating to several systems, not just electroless deposition. 

---