FRACTAL PROPERTIES OF THE DIFFUSION FRONT

In this work, the structure of the diffusion front on a two-dimensional square lattice is studied. The concepts of the boundary, skeleton and dead ends of the diffusion front are introduced. The fractal properties of these objects are considered. The change in all absolute and relative characteristics with time has been studied. It is shown that all characteristics depend only on the penetration depth of the diffusion front and do not depend on the characteristics of the diffusion process itself. Thus, the diffusion front is a purely geometric formation.

The concept of a diffusion front (shell) was introduced by Mandelbrot as an example of a fractal object. It can be introduced only when simulating the diffusion process. Diffusion was simulated in a strip with a width of 150 sites of a square lattice as a random walk of particles along the lattice sites. Each particle in one step jumps to one of the four nearest lattice sites, if it is free. If it is busy, the particle remains in place. To avoid any correlations in the movement of particles, the enumeration of particles at each step was performed in a random sequence.

The concept of a diffusion front or shell [1] is closely related to the theory of percolation. A busy node is considered to be associated with a source if percolation from the source to this node along the nearest occupied neighbors is possible. A free node is considered associated with an isolated edge of the sample if it is possible to flow from this edge to this node along the nearest and second free neighbors. Then the connected occupied sites adjacent to the connected unoccupied sites in at least one of the eight directions along the edges and diagonals of the lattice form a shell.

The shell x coordinate is defined as the average coordinate of all nodes belonging to the shell along the diffusion direction. The shell is a self-similar fractal. Its fractal dimension D was determined from the formula,where s is the width of the strip perpendicular to the x-axis with the central coordinate Round (x) (rounded to an integer x value), M (s) is the number of shell nodes in this strip. The scaling theory gives the exact value of the fractal dimension of the shell D = 7/4 = 1.75 [1]. It also leads to power-law dependences of the number of nodes in the shell and the width of the shell on time (number of steps):.

Here, the width is calculated as the corrected root mean square of the shell from the mean of the x coordinate. It is clear that the dependence x (N) should also be power-law, and the exponent ε≈0.5. The shell is an unstable formation and can be very distorted in one step. But its mean coordinate changes very slowly at sufficiently long times.

In the theory of percolation, the concept of the backbone (skeleton) and dead ends of a percolation cluster is introduced [1]. The skeleton is formed by those nodes through which the flow occurs, and the dead ends are the nodes into which the liquid can flow once, and can only flow out if it flows in the opposite direction. A similar concept can be introduced for the shell [2]. For this, it is necessary to consider the flow through the shell across the direction of diffusion. The shell of the shell has a number of features. Due to the asymmetry of the shell, all dead ends are located on one side of the shell in the direction of diffusion. The skeleton is also a fractal. It is also possible to build the upper border of the shell (with diffusion from top to bottom). On one side of this border, there are those occupied nodes that do not border on the connected unoccupied nodes. On the other (in the direction of diffusion) and at the very border, all connected occupied nodes border on connected unoccupied ones. The skeleton consists of a boundary and loops extending from it in the direction of diffusion. The structure of the shell is shown in Figure 1. The border is highlighted in black, the hinges of the skeleton are in dark gray, and the dead ends are in light gray.

Figure 1. Structure of the diffusion front

The border is also a fractal. It's pretty symmetrical. Its thickness is 1, the width is slightly less than the width of the shell. The isolation of the shell and the shell boundary made it possible to study the structure of the shell and its change over time. The following characteristics of the shell were considered: the number of nodes in the skeleton loops, the number of skeleton loops, the number of branches of the dead ends, the number of branching of the dead ends, and all possible relative characteristics. The very picture of the distribution of diffusing particles does not depend (statistically) on whether the boundaries are impenetrable or whether periodic boundary conditions are used. But the shell patterns in these cases are different near the edges of the strip. Near the edges of the strip, the boundary coordinate slightly increases for the case when the walls are considered impenetrable. But what's interesting is the average shell coordinate practically does not change (statistically average). Deviations are possible both towards large values of the coordinate, and less. This is due to the fact that when the border coordinate does not change, the number of dead ends can decrease, and then the average shell coordinate decreases. Taking these edge effects into account can make some sense for nanostructures. But in ordinary situations, this effect is purely computational. Therefore, to avoid this effect, the construction of the shell was carried out for the case of periodic boundary conditions. Taking these edge effects into account can make some sense for nanostructures. But in ordinary situations, this effect is purely computational. Therefore, to avoid this effect, the construction of the shell was carried out for the case of periodic boundary conditions. Taking these edge effects into account can make some sense for nanostructures. But in ordinary situations, this effect is purely computational. Therefore, in order to avoid this effect, the construction of the shell was carried out for the case of periodic boundary conditions.

Diffusion patterns were constructed at different temperatures with and without regard to correlations in the choice of the direction of hopping of diffusing particles. Correlation was taken into account in the following way. The probability of a jump attempt does not depend on the direction and is equal to, where n is the number of occupied nearest sites, e is the interaction energy of two diffusing particles in neighboring lattice sites, T is the temperature (all in dimensionless units). The meaning should be e <0.25 so that the numerator in the exponent does not vanish. The calculations were carried out at infinite temperature, at T = 1, e = 0, and at T = 1, e = 0.1.

Obviously, the macroscopic characteristics of the distribution of diffusing particles do not depend on temperature and correlations and are determined only by the diffusion length, i.e. the root-mean-square coordinate of diffusing particles along the diffusion direction. In our case, it is convenient to take the average shell coordinate as such a parameter. It is clear that this quantity is proportional to the diffusion length and, therefore, should increase as in all cases. The presence of correlations in the hopping probability can lead to correlations in the location of the nearest particles. It was assumed that this would somehow affect the characteristics of the shell structure at low temperatures. This has not been confirmed. The points of the graphs of all characteristics of the shell, depending on the average coordinate of the shell, lie on top of each other (on average, see all figures). Therefore, for greater accuracy, all results were pooled. It turned out 255 diffusion paintings. The simulation results confirmed with high accuracy the data obtained from the shell scaling theory (α = 0.2015, β = 0.2887, ε = 0.5215). An especially accurate result was obtained for the fractal dimension: D = 1.7496 ± 0.0066 with a confidence level of 0.99. The scatter of data for the border and the shell turned out to be greater than for the shell (Figure 2). The fractal dimension of the skeleton is D '= 1.7448 ± 0.0092, the boundaries are D″ = 1.7576 ± 0.0105, that is, in all cases it is ≈1.75.

All the values for the shell and the boundaries, which are similar to the corresponding values for the shell, are also described by power functions, but the exponents are different. The exponents for the boundary are close to the corresponding numbers for the skeleton. For the core, the following values were obtained: α′= 0.1149, β ′ = 0.3104, ε ′ = 0.5551, for the boundary α″=0.1094, β″=0.3179, ε″=0.5655. The numbers of nodes in the border and in the shell increase with time much more slowly than in the shell. Thus, over time, most of the shell knots are concentrated in the dead ends.

The average position of the border and the skeleton and their width are smaller than for the shell, but they grow faster. This is a rather strange result. The coordinate and width of the border and skeleton cannot exceed the corresponding values for the shell (Figure 2, 3).

Shell coordinate

Figure 2. Dependence of the mean coordinate of the shell on the mean coordinate of the shell, where 1 is the mean coordinate of the shell; 2 - average coordinate of the skeleton

Apparently, over time, these exponents should become equal. If we extrapolate the dependences of the coordinates of the boundary, skeleton, and shell for long times, it turns out that they become equal at the values of the mean coordinate of the shell, comparable to the width of the strip along which diffusion occurs. Therefore, it is possible that this difference in exponents is a manifestation of edge effects.

It turned out that some relative characteristics have a small scatter of data and are practically independent of time (Figure 4). This is the proportion of border nodes from the number of nodes in the skeleton (mean value 0.854 ± 0.005), the number of border nodes per one dead end branch (5.31 ± 0.06). The branching of dead ends (the ratio of the number of branches of dead ends to the number of nodes in them) increases with time. It is interesting that the ratio of branching to the number of nodes in the boundary or in the skeleton is also practically independent of time. The number of border nodes per one loop of the skeleton has a very large scatter of data and is practically independent of time (mean value 19.82 ± 0.57, Figure 4). Hence it follows that the number of nodes in the boundary, the number of nodes in the skeleton, the number of branches of the dead ends, the number of loops of the skeleton, the branching of the dead ends increase with time according to the same law.

The average number of nodes in one loop of the skeleton has a very large scatter of data, but shows a weak tendency to increase (mean value 3.29 ± 0.10, exponent 0.047, Figure 5). The number of nodes per one branch of the dead ends is also characterized by high variance, but it increases with time approximately according to a power law much faster (exponent 0.1987).

Shell coordinate

Figure 3. Dependence of the average shell and shell width on the average shell coordinate, where 1 is the average shell width; 2 - average width of the skeleton

Shell coordinate

Figure 4. Characteristics of the shell structure, independent of time, where 1 is the distance between the island loops along the border; 2 - the distance between the branches of the dead ends along the border; 3 - the proportion of nodes in the boundary of the total number of nodes about the skeleton

If we approximated the number of nodes in the skeleton and shell with a power function, then it would be wrong to approximate the number of nodes in the dead ends with it. But, since this approximation is still average, it may make sense (Figure 5b). Then the proportion of dead ends knots from the number of shell knots increases according to a power law with an exponent of 0.0948 (Figure 5a). The number of dead end branching also increases (exponent 0.4047, Figure 5b).

Since the characteristics of the shell structure do not depend on the conditions of its formation, it is a purely geometric object.

Figure 5. Dependence of some characteristics of the shell on its average coordinate.

Only the model on the basis of which it is built is physical. Therefore, it is possible that the characteristics of the shell structure can be deduced theoretically on the basis of scaling concepts.

References:

1. Feder E. Fractals. - M .: Mir, 1991 .-- 254 p.
2. Khaimzon BB Study of the distribution of atoms in the course of diffusion on a square lattice // Izv. universities, Physics. - 2002. - No. 8 (appendix), - S. 158-161.