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Fractal structure of surface layers of metal. fractal structure




) — (from lat. fractus- fractional, broken) structure that has the property of self-similarity, i.e., consists of such fragments, the structural motif of which is repeated when the scale changes.

Description

The fractal structure is characterized by the value of the degree of filling the space with a structure (dimension), which is not an integer value. So, n-dimensional fractals occupy an intermediate position between n-dimensional and ( n+ 1)-dimensional objects. To construct regular fractal objects, recursive functions are used.

In natural fractal-like structures, unlike regular fractals, there is no fractional dimension, and self-similarity is observed only up to a certain scale. Natural examples of objects with a fractal-like structure are cumulus clouds, tree crowns, and lightning. For example, at the crown of a tree, each of the large branches is divided into at least two smaller branches, after which the division is repeated again and again (see Fig.). As a result, each of the branches can be considered as a separate repeating motif of the fractal structure.

The geometry of some nanosystems, such as molecules and fractals, is described with good accuracy using recursive functions, which makes it possible to model their micro- and macroscopic properties.

Illustrations


The authors

  • Shlyakhtin Oleg Alexandrovich
  • Streletsky Alexey Vladimirovich

Sources

  1. Feder E. Fractals. - M.: Mir, 1991. - 254 p.
  2. Tretyakov Yu. D. Dendrites, Fractals and Materials // Soros Educational Journal. 1998. No. 11. pp. 96–102.
  3. Paytgen H.-O., Richter P. H. The beauty of fractals. - M.: Mir, 1993. - 176 p.

Theoretical solid state physics has mainly considered equilibrium systems. Irreversible processes were considered only in a very simplified way - as small perturbations, for example, in the study of transport phenomena. It is known that the condensed state of matter can exist not only in the form of a dense continuous medium, but also in the form of highly loosened porous structures. Structures of this kind are formed, as a rule, as a result of condensation under complex nonequilibrium conditions, for example, when solid particles moving according to a certain law stick together or as a result of the interaction of dislocations during plastic deformation of metals. Such structures are called fractal aggregates. For the most part, they are disordered, difficult to study, and their macroscopic properties have been little studied. The fractal aggregate of each substance is formed under certain physical conditions, which are not fully understood. Nevertheless, what is already known makes it possible to use the laws of formation of fractal aggregates to create materials with unusual physical properties. Fractal solid media, formed under the conditions of energy dissipation in open systems and being self-organized structures, have a number of unusual properties that cannot be obtained using traditional methods of forming the structural state of matter. The driving force of self-organization in dissipative systems is the tendency of matter in open systems to reduce entropy. Characteristic features of fractal structures - self-similarity, scale invariance, structural hierarchy, nanoscale porosity and fractal dimension.

Solid-state fractal systems represent a new type of structural state of matter, characterized by unique physical properties. Fractal solid-state systems are formed from atoms or molecules, as well as from nanosized particles or clusters. Fractal micro- or macroscopic structures formed from such particles or clusters are of interest both for studying fundamental properties and for using them in new technologies. It has been experimentally established that a fractal structure formed from metal nanoparticles is capable of absorbing electromagnetic radiation in the light wavelength range. It is shown that the thermoelectric power of the fractal structure of carbon increases by almost an order of magnitude compared to graphite.

In many cases, the fractal structure of a solid provides high specific strength characteristics, low thermal conductivity, and sound transmission. Therefore, obtaining and studying substances with a certain fractal structure is an urgent task. A characteristic feature of fractal formations is that their structure appears only when several levels are jointly resolved, the difference in the scales of which makes it difficult to present a visual geometric image (such as a rugged coastline).



Although the observation of the multiscale structures themselves is difficult, their consistent description can only be achieved within the framework of a fractal ideology. This is due to the fact that such nonequilibrium systems are represented as superensembles consisting of hierarchically subordinate statistical ensembles, which, in turn, consist of a set of subensembles, and so on. Therefore, speaking about fractals in a condensed medium, one should keep in mind, first of all, the use of the concept, and not a literal description of the observed geometric image.

One of the most important characteristics of fractal structures, which determines their physical properties, is fractal dimension.

Mathematical definition of fractal dimension. The volume of the fractal in its nesting space is always zero. However, it can be different from zero in a space of lower dimension. To determine the dimension of this space D, let's break everything n-dimensional space into small cubes with edge length ε and volume ε n(Fig. 14.12).

Rice. 14.12 Definition of fractal dimension

Let N(ε) - the minimum number of cubes that together completely cover the fractal set, then by definition

This value is usually called Hausdorff or fractal dimension.



The existence of this limit means that the volume of the fractal is finite in D-dimensional space for small ε:

N(ε)≈ Vε – D , (14.104)

where V= const.

In this way, N(ε) is nothing but the number D-dimensional cubes covering in D-dimensional space volume V, because covering the fractal n-measuring cubes can be almost empty

D< n, (14.105)

and in contrast to the usual dimension D can be a fractional value, which it most often is for fractal sets. Obviously, for ordinary sets this definition leads to well-known results. So, for many N isolated points we have N(ε) = N and therefore

For a segment, a smooth line of length is sufficient L N(ε) = L/ε and therefore D= 1. For the site S two-dimensional surface N(ε) = S/ε 2 and D= 2 etc.

Initially, the fractal was introduced as a geometric object in ordinary physical space. Therefore, it is advisable to begin consideration of examples of fractals with the visual geometric constructions of Kantor and Koch. Their choice is due to the fact that in the first case the fractal dimension D less topological d, and in the second D > d.

Cantor set. Take a segment of length 1 . Dividing it into three equal parts, we exclude the middle part. With the remaining two segments, we will do the same procedure and as a result we will get 4 segments of 1/9 length each, etc. to infinity (Fig. 14.13).

Rice. 14.13. Construction of the Cantor set

The set of points that appeared after this procedure is Cantor set. It is easy to see that the length L this set is zero. Really,

Let us now find its Hausdorff or fractal dimension. To do this, we choose as a "standard" a segment with a length

The minimum number of such segments required to cover the set is

N(ε) = 2 n. (14.109)

Therefore, its fractal dimension

Snowflake Koch. An example of constructing this fractal is shown below in Fig. 14.14

Rice. 14.14 Koch snowflake

Snowflake Koch is a line of infinite length bounding a finite area. The first assertion is proved very simply. If we notice that at each step the number of sides of the polygon increases by 4 times, and the length of each side decreases only by 3 times. If we take the length of the side of the generating triangle as 1, then the length of the Koch snowflake:

The area under the curve, if we take the area of ​​the generating triangle as 1, is equal to

Here we took into account that each time the number of additional triangles increases by 4 times, and their side decreases by 3 times (respectively, their area decreases by 3 2 = 9 times). Eventually:

Thus, the area under the Koch snowflake is 1.6 times the area of ​​the triangle that forms it. Find the fractal dimension of the Koch snowflake. As we have already said, on n-step number of sides of triangles N(ε) = 3×4 n, and the side length ε = 1/3 n. That's why

Sierpinski Napkin. The first three steps in building this fractal ( Sierpinski's napkins) are shown in Fig. 14.15, and the fractal itself - in fig. 14.16.

Rice. 14.15. Construction of the Sierpinski Napkin

Rice. 14.16. Sierpinski Napkin

The number of triangular pores of smaller and smaller scale in it is infinite. The number of black triangles in this construction grows as 3 n, where n- step number, and the length of their side decreases as 2 - n. Therefore, the fractal dimension is:

It can be shown that the area of ​​the white spots is equal to the area of ​​the original triangle.

The examples of fractals considered above belong to the so-called exact fractals or deterministic. They are all built according to a well-defined geometric rule. In addition to exact fractals, there are also so-called random fractals. There is a certain amount of randomness in the arrangement of their elements.

Brownian motion. The simplest random fractal is the trajectory of a particle making Brownian motion(Fig. 14.17).

Rice. 14.17 Brownian particle trajectory

And although the trajectory itself has a very complex tortuous character, it is very easy to determine its fractal dimension. For this, we note that if the particle has diffused over a distance R, then the average number of "steps" she took

where l is the characteristic length of one step. That's why:

This means that the characteristic size of the diffusion trajectory on a given area is proportional to the size of this area. That is, the trajectory on the plane is quite “dense”. However, this does not mean that the area swept out by the diffusion curve itself is finite because of the many self-intersections. It can be shown that for a two-dimensional Brownian motion, the probability of returning to any arbitrarily small neighborhood of an arbitrarily chosen point is equal to 1. In the case of diffusion in three-dimensional space, the trajectory of a Brownian particle is, on the contrary, very loose (its fractal dimension is still equal to 2) and does not fill the entire volume provided to her. In this case, the return probability is less than one.

Fractal clusters. Another example of a random fractal, more complex, but just as common in nature, is obtained in the process of so-called diffusion-limited aggregation. It can be modeled as follows. On a sphere (a circle in the two-dimensional case) of a sufficiently large radius, on the surface of which particles appear from time to time in random places, which then diffuse into the sphere. In the center of the sphere is the so-called "embryo". When colliding with it, the diffusing particle "sticks" to it and no longer moves. Then the next particle released from the surface of the sphere collides with this formation, and so on ad infinitum. The flow of particles from the surface of the sphere will be considered sufficiently small, so that collisions of diffusing particles with each other can be neglected. As a result, a very porous structure is formed, in the two-dimensional case shown in Fig. 14.18.

Rice. 14.18. Fractal cluster obtained in the process of diffusion-limited aggregation

Large pores inside are "shielded" by processes of rather large length. As the structure grows, the number of pores and their sizes increase. In the two-dimensional case, the fractal dimension of such a cluster turns out to be close to the value D = 1,7.

In nature, such fractal clusters are very common. So, for example, crystals grow from a supersaturated solution, snowflakes, corals, tumors in living organisms, ordinary oven soot. In superionic conductors, such as AgBr, such clusters limit the time of their practical use. Since, with a sufficiently long passage of current, mobile silver ions, when combined, form a fractal cluster, which eventually closes the electrodes and disables the conductor sample.

An interesting example of a random fractal is the models of our universe.

Fournier universe. Imagine a sphere with a very large radius R(cosmic scale), inside which there is a very large number of stars N>> 1. It is clear that the number N should increase with increasing sphere radius. We are just interested in this dependence N(R). If stars, galaxies, clusters of galaxies were distributed uniformly in the Universe with some constant density, then the number of stars in a sphere of radius R would be proportional to the volume of this sphere, i.e.

Astronomical observations, however, show that

Where D» 1.23, (14.119)

those. the fractal (Hausdorff) dimension is much closer to 1 than to 3. This means that our Universe is almost one-dimensional! How can this be understood qualitatively? To do this, let's take an example Fournier universe. It was proposed in 1907 by the American science fiction writer Fournier. A fragment of its structure is shown in fig. 14.19.

a b

Rice. 14.19. Fournier universe. Radius ratio R 2 /R 1 = R 3 /R 2 = ... = 7

Each dot in this picture represents one galaxy. They are grouped into clusters of radius R 1 with 7 galaxies in each cluster (Fig. 14.19, b). On fig. 14.19, a only five of them are visible: the missing two are located symmetrically above and below the plane of the figure, on a straight line passing through the center of the cluster. In turn, seven such clusters are similarly united into one supercluster of radius R 2. Then, according to the same principle, one supersupercluster of radius is built from seven superclusters R 3 , and R 3 /R 2 = R 2 /R 1 etc. As a result of repeated repetition of such a process, a self-similar fractal structure arises. It is obvious from this figure that the number of stars in a cluster of radius R 7 times the number of stars in a cluster of radius R/7:

Assuming , we get D= 1. Thus, the Fournier universe is one-dimensional. The number 7, which has penetrated into this scheme, does not play a fundamental role. In its place could be any other number. It is also clear that by varying the ratio between the sizes of clusters and the number of elements in them, one can construct fractal models of the Universe with other dimensions close to 1 D. Note also that Fournier's universe is an exact fractal, which, of course, our universe is not. How and what regularities lead to the fractal structure of the Universe is not yet known. We will only mention in this connection the so-called rings of saturn, which have a very loose and heterogeneous structure with slits of various sizes, in which there are no asteroids, from the largest - the so-called Cassini section, to the smallest. Presumably, the structure of Saturn's rings is fractal. If so, then this would be a clear confirmation that gravity is capable of creating fractal structures in the distribution of matter in the universe.

Fractal properties of chaos. Fractal geometry and concepts appear naturally in non-linear Newtonian dynamics when the motion of the system is chaotic. This, for example, takes place in forced oscillations of an anharmonic oscillator, described by the simplest one-dimensional equation:

where is the power F(x) - non-linear displacement function x. In certain intervals of parameter values ​​γ, f 0 , Ω the motion is chaotic. If, say, we mark the states of the system on the phase plane x, at discrete times 0, 2π/Ω, 4π/Ω, ... , then for a chaotic signal x(t) the resulting set of points is Cantorian, i.e. is a fractal (Fig. 14.28). The Hausdorff dimension of a fractal naturally depends on the values ​​of the parameters and lies within 0<D<2. В настоящее время не существует аналитических методов решения подобных уравнений. Большинство результатов в этой области получено путем компьютерного моделирования. То же относится и к вычислению фрактальной размерности D. Yes, for Ueda attractor shown in fig. 14.20 numerical calculations give D ≈ 1,6.

Chaotic motion means that it is impossible to accurately predict it, despite the given initial conditions and the theorem on the uniqueness of the solution. Therefore, in fact, we can talk about calculating only the probability of detecting a system in one or another element of the phase volume. Such a statistical description of chaotic motion is not the result of our ignorance of motion or the imperfection of our computers. It reflects the deep inner properties of the movement itself. And one of these properties is fractal geometry phase trajectories.

Rice. 14.20. Ueda attractor for the equation:

More to say: deterministic chaos is always fractal, which determines the importance of fractal concepts in physics.

Fractal aggregates can also be obtained by changing the dislocation structure in the metal with ever-increasing degrees of deformation, leading to the creation of a cellular structure (Fig. 14.21). In the initial stage of plastic deformation, a significant number of dislocations are formed, uniformly distributed over the volume. At higher degrees of deformation, clusters are formed in the form of coils and loose cell walls. In the end, a well-defined cellular structure is formed.

Rice. 14.21. Schematic representation of the transformation of a homogeneous dislocation structure into a cellular one:

a– chaotic distribution of dislocations; b, in– formation of dislocation coils and loose walls; G– honeycomb structures

It is believed that the clusters of dislocations that form the cell walls are fractals, the dimension of which first increases from D= 1 (uniform distribution of dislocations) to 1<D<2 (рыхлые скопления) и затем достигает D= 2 (geometric cell walls). These examples show the possibility of creating fractal structures in solids, the compactness of which is close to equilibrium.

The simplest experimental method for determining the fractal dimension of two-dimensional flat formations is the grid method. A flat image of a fractal formation is divided into square cells (pixels) in the range of experimental sizes fractal aggregate. Object area S and its perimeter L determined by the number of pixels that cover S and cross L. The size of one pixel (grid cell) is determined by the resolution of the device in which the surface structure of the object is analyzed. In general, the ratio between S and L a two-dimensional object is represented as:

where D is the fractal dimension of the object; μ( D) is a quantity independent of L. Building dependency ln S from ln L when using at least ten grids of pixels, it allows obtaining the values ​​of the fractal dimension of flat fractal objects. When the object of study has a smooth outer boundary, D= 2 and S » L 2. Non-integer value (1< D < 2) является свидетельством плоской фрактальной структуры.

Gevorg Simonyan, Candidate of Chemical Sciences, Associate Professor

Yerevan State University, Armenia

Championship participant: National Research Analytics Championship - "Armenia";

Open European-Asian championship in research analytics;

The article gives a detailed explanation of the terms fractal, fractal dimension and dendrite. Numerous examples of dendritic and fractal structures of chemical processes and chemical compounds are given.

Keywords: fractal, dendrid, chemical compound.

The article provides detailed explanation of terms fractal, fractal dimension and the dendrite. Numerous examples of dendritic and fractal structures of chemical processes and chemical compounds are given.

keywords: fractal, dendrite, chemical compound.

The concept of a fractal was introduced into scientific use by Benoit Mandelbrot. Fractal - from the Latin word fractus, broken stone, splintered, irregular medium. This is essentially a non-Euclidean geometry - non-smooth, rough, jagged, corroded by passages and holes, rough and similar objects. Fractal objects are those objects that have the properties of self-similarity, or scale invariance. Self-similar can be some fragments of the system, the structures of which are repeated at different scales. It turned out that fractals have unusual properties. For example, the "Koch snowflake" has a perimeter of infinite length, although it limits a finite area. In addition, it is so “prickly” that it is impossible to draw a tangent to it at any point of the contour (Fig. 1).

Rice. 1. Koch snowflake

It is customary to distinguish between regular and irregular fractals, of which the former are a figment of the imagination, similar to the Koch curve, and the latter are a product of nature or human activity. Irregular fractals, unlike regular fractals, retain the ability to self-similarity within limited limits determined by the actual size of the system.

The fractal structure is characterized by a fractal fractional dimension. The fractal dimension (D) is a characteristic of the unstable, chaotic behavior of systems. The latter shows the degree to which space is filled with an object or structure. This dimension was introduced by F. Hausdorff. Unlike ordinary geometric images - a point, a line, a square, a cube, having an integer dimension (0, 1, 2 and 3, respectively), fractal structures have a non-integer dimension. So, for the Koch curve D \u003d lg 4 / lg 3 \u003d 1.2618. The fractal dimension of a snowflake is 1.71, that is, like the Koch curve, it occupies an intermediate position between one- and two-dimensional objects.

Before the appearance of the term "fractals" in mineralogy, and then in chemistry, the term "dendrite" and "dendritic forms" were used. A dendrite is a branching and diverging formation that occurs during accelerated or constrained crystallization under non-equilibrium conditions, when the crystal splits according to certain laws. They branch and grow in different directions, like a tree. The process of dendrite formation is commonly called dendritic growth. In the process of dendritic development of an object, the crystallographic regularity of the original crystal is lost as it grows. Dendrites can be three-dimensional volumetric (in open voids) or flat two-dimensional (if they grow in thin cracks in rocks). Examples of dendrites include ice patterns on window glass, snowflakes, and picturesque oxides of manganese that look like trees in landscape chalcedony and in thin cracks in pink rhodonite. In the oxidation zones of ore deposits, native copper, silver, and gold have branched dendritic forms, while native bismuth and a number of sulfides form lattice dendrites. For barite, malachite and many other minerals, for example, "cave flowers" of aragonite and calcite in karst caves, kidney-shaped or coral-like dendrites are known. Dendrites, as a specific product of crystallization from solutions, undoubtedly have fractal properties, although virtually any complex products of nature and human activity possess these properties. Thus, the paper shows that fractal self-similarity is also characteristic of objects of oil fields, containing reservoirs and the oil itself. When water is injected under pressure into an oil-bearing formation, viscous fingers are observed, which have a fractal structure. During flooding, asphaltenes aggregate into large clusters with a pronounced fractal structure.. Thus, at asphaltene concentrations from 0.1 g/L to 0.15 g/L, oligomers form from asphaltene monomers. At a concentration of 1–3 g/l, stacking structural nanocolloids with a size of 2–10 nm are obtained from oligomers, which consist of 4–6 monomers. Nanocolloids in the concentration range of 7-10 g/l pass into particles larger than 10 nm. Finally, at a concentration of 25-30 g/l, loose fractal structures are formed. We also show the features of the fractal structures of biopolymers, such as polysaccharides - glycogen and chitosan, proteins, DNA, and lignin. It is shown that the structure of glycogen-animal starch is dendritic. It has been established that in the presence of benzoic acid, chitosan forms a film, the clusters of which have a fractal dimension from 1.55 to 1.9. It is shown that the protein surface exhibits a two-level organization. The fractal dimension of the microlevel fluctuates around 2.1, while the fractal dimension for different protein families ranges from 2.2 to 2.8. It has been established that DNA forms a folded fractal globule, in which the chain is never tied into a knot. It has been shown that lignin macromolecules are fractal aggregates, the fractal dimension of which is ~2.5 in the case of growth by the cluster-particle mechanism and ~1.8 by the cluster-cluster mechanism. coniferyl alcohol , in DMSO lignin is in the form of a fractal globule.Thiswork is a discussion of the features of fractal structures of chemical processes and chemicals.

The paper shows that during the crystallization of alloys of silver bromide with potassium bromide, dendritic crystals are formed. Flat zinc dendrites with a fractal dimension of 1.7 were obtained by electrolysis of a ZnSO 4 solution at the interface with n-butyl acetate. Solid-state electrolysis of AgBr produced dendritic structures of silver. It should be noted that recently the concept of a dendrite has gone far beyond the field of crystal formation. As an example, a dendritic polyaryl ether, which is a highly branched analog of linear polyaryl ethers. A dendrimer, an inorganic supermolecular complex of 1090 atoms, has also been synthesized. , including 22 ruthenium ions. The introduction of pectin into a solution of ammonium chloride leads to the formation of giant dendrites, and a small admixture of urea contributes to the formation of crystals with rounded edges, called "dog's tooth". Ovchinnikov and colleagues. proposed a method for obtaining an aqueous system of branched fractal clusters based on L-cysteine ​​and silver nitrate, including mixing a solution of L-cysteine ​​and a solution of silver nitrate so that the initial concentration of L-cysteine ​​in the initial mixture was in the range from 1.14 10 -4 M to 1.17 10 -2 M, and the concentration of silver nitrate was 1.2÷2 times higher than the concentration of L-cysteine, keeping the resulting mixture in a thermostat protected from light at a temperature of 10÷60°C for 0.3-48 h When small amounts of dilute hydrochloric acid are introduced into the solution, spontaneous self-organization of the solution occurs with the formation of a gel structure.

Within the framework of the oil-water model, the kinetics of the reaction of water-soluble N-[tri(hydroxymethyl)methyl]acrylamide with fat-soluble decylamine in a two-phase water-heptane system in the absence and presence of a surfactant was studied. It is shown that the reaction product has a fractal structure.

In chemistry, there are many entertaining experiments on obtaining metal dendrites, such as the "Saturn tree", "Mercury tree" and "Dorfman tree".

"Saturn tree" is sometimes called the tree of Paracelsus, the alchemist doctor, the founder of pharmaceutical chemistry. While preparing one of his medicines by dissolving metallic lead in acetic acid, he decided to add mercury as well, and therefore he introduced pieces of zinc into the vessel (at that time, many chemical elements, including very common metals, had not yet been truly identified and were considered that zinc contains a lot of mercury, which is why it is so fusible). Having no time to continue the experiment, Paracelsus left the vessel for several days, and how much he was amazed when he saw shiny twigs of an unknown nature on pieces of zinc! The scientist considered that mercury, having hardened, came out of the pieces of zinc. Later, the beautiful “tree” was called “Saturnian” by the alchemical name for lead. To grow a “Saturnian tree”, an aqueous solution of 25–30 g of lead acetate in 100 ml of water is poured into a tall glass or glass cylinder and a plate cleaned with fine sandpaper is immersed in it. or zinc rod. Instead, you can hang a few pieces of zinc on a thread, also cleaned with sandpaper. Over time, branched and shiny lead crystals fused together grow on the zinc surface. Their appearance is caused by the reaction of the reduction of lead from salt by a more chemically active metal.

Zn + Pb (CH 3 COO) 2 \u003d Pb + Zn (CH 3 COO) 2.

Paracelsus is also credited with obtaining tin crystals on pieces of zinc - the “tree of Jupiter”. To grow such a "tree", an aqueous solution of 30 - 40 g of tin chloride SnCl 2 in 100 ml of water is poured into a tall glass vessel and a zinc plate is immersed.

Zn + SnCl 2 \u003d Sn + ZnCl 2.

A silver "Dorfman's tree" is obtained by pouring a 10% aqueous solution of silver nitrate AgNO 3 into a glass beaker with a drop of mercury at the bottom. First, mercury is covered with a gray film of silver amalgam (an alloy of mercury with silver), and after 5 - 10 seconds, shiny needle-shaped silver crystals quickly begin to grow on it. After a few minutes, the needles begin to branch, and an hour later a sparkling silver tree grows in the vessel. Here it is very important to strictly observe the recommended concentration of silver nitrate: at a lower content of AgNO 3, no growth of metallic silver crystals is observed, and at a higher content, silver crystallization proceeds without the formation of branched crystals.

Hg + 2AgNO 3 \u003d 2Ag + Hg (NO 3) 2

Interesting multi-colored silicate dendrites are obtained by mixing sodium silicate and salts of certain metals. So, a solution of commercial silicate glue (sodium silicate Na 2 SiO 3) diluted with an equal volume of water is poured into a glass. Crystals of chlorides are thrown to the bottom of the glass: calcium chloride CaCl 2, manganese chloride MpCl 2, cobalt chloride CoCl 2, nickel chloride NiCl 2 and other metals. After some time, dendrites of crystals of the corresponding sparingly soluble silicates begin to grow in the glass, resembling algae:

Na 2 SiO 3 + CaCl 2 → CaSiO 3 ↓ + 2NaCl

Na 2 SiO 3 + MpCl 2 → MnSiO 3 ↓ + 2NaCl

Na 2 SiO 3 + CoCl 2 → CoSiO 3 ↓ + 2NaCl

Na 2 SiO 3 + NiCl 2 → NiSiO 3 ↓ + 2NaСl

In the work, the values ​​of the fractality index of individual sections of artificial salt crystals were obtained. The effects of anisotropy of fractal characteristics are found. The studied surfaces are characterized by low values ​​of the fractal dimension (2.0-2.2), corresponding to a weak degree of roughness. The question of the correlation between fractal parameters and mechanical characteristics is considered.

If sodium chloride crystals grow when the solution evaporates from the surface of porous ceramics, they often take the form of fibers. In the case of evaporation of a salt solution from the surface of the paper, it was possible to obtain intergrowths of crystals in the form of branches - dendrites. It is very easy to conduct such an experiment. It is necessary to roll a rectangular piece of filter paper into a cylinder with a diameter of 2-3 cm and a height of 15-25 cm, place the cylinder vertically in a Petri dish and fix it on top. Sodium chloride is poured into the cup almost to the top, adding a little yellow blood salt K 4 (a quarter of a teaspoon), then stir and add water so that it wets the salt well and the solution begins to rise up the filter paper. The solution will gradually evaporate from the surface of the paper, and fresh portions will rise from the cup in its place (due to the capillary effect). As the solution evaporates, add water to the cup and add salt. Gradually, salt crystals will begin to grow on the surface of the paper, which in a few days will take the form of twigs (Fig. 2). The paper cylinder itself will look like white coral. The addition of yellow blood salt favors the formation of fibrous sodium chloride crystals. Without it, table salt simply forms a crust on the surface of the paper.

Rice. 2. Unusual salt crystals

NaCl·2H 2 O dihydrate crystals form in salt lakes in winter. When the temperature drops sufficiently, accumulations of this mineral, which is called hydrohalite, form.

Literature:

  • 1. Mandelbrot B. B. Les Objects Fractals: Forme, Hasard et Dimension. Paris: Flammarion, 1975, 192 p.
  • 2. Mandelbrot B. Fractal geometry of nature. M.: Institute of Computer Research, 2002, 656 p.
  • 3. Grigoriev D.P. On the difference between mineralogical terms: skeleton, dendrite and poikilite. //Izv. Universities, geol. and development 1965, No. 8, pp. 145-147.
  • 4. Simonyan G.S. Fractality of oil deposits and oil // Technology of oil and gas. 2015, No. 3, pp. 24-31.
  • 5. Simonyan G.S., Simonyan A.G. Fractality of biological systems. Fractality of biopolymers.// Successes of modern natural science. 2015, No. 11, pp. 93-97.
  • 6. Tretyakov Yu.D. Dendridy, fractals and materials. // Soros educational journal. 1998, No. 12, pp. 96-102.
  • 7. Shubnikov A. V., Pavrov V. F. Origin and growth of crystals. M.: Nauka, 1969, 73 p.
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  • 9. M. M. Ovchinnikov, S. D. Khizhnyak, and P. M. Pakhomov, Sb. “Physical Chemistry of Polymers”, Tver, 2008, Vol. 14, p. 186-194.
  • 10. Simonyan G.S. Michael reaction in a model two-phase oil-water system. A particular case in conditions of limitlessness: Earth in the vast Universe Materials digest of the LXXIV International Research and Practice Conferenceand III stage of the Championship in Earth and Spacesciences, physics, mathematics and chemistry sciences(London, December 19- December 24, 2013) Publisher and producer International Academy of Science and Higher Education.2014 p.60-62.
  • 11. Adamyan R., Kochikyan T., Simonyan G. Laboratory work in chemistry. Yerevan-2011, 164 pp. (in Armenian)
  • 12. V. N. Aptukov, V. Yu. Ser. Mechanics. Maths. Informatics. 2014, issue 4(27), p. 16-21.

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A fractal is an infinitely self-similar geometric figure, each fragment of which is repeated when zoomed out.

Multifractal is a complex fractal structure, which is obtained using several successive algorithms.

To describe a fractal, only three parameters are required: the fractal dimension D, the dimensions of the primary block (R t in) and the object as a whole.

The fractal dimension makes it possible to quantitatively describe various structures that are highly complex and contain a large number of point, line, surface, and bulk defects.

A regular fractal is a fractal that is characterized by exact self-similarity, and this is an ideal model, because a certain derogation is always accepted.

A fractal cluster is a chaotic fractal.

Fractality of defects in the structure of materials

New ideas about the form of real objects of nature, about structures in biology and materials science are based on the concept of fractals, which was first formulated by B. Mandelbrot. He introduced the concept of not only fractal, but also fractal geometry, which differs from Euclidean in fractional dimensions, and drew attention to the fact that the contours, surfaces and volumes of the objects around us are not as even, smooth and perfect as is commonly thought. In fact, upon careful examination, it turns out that they are uneven, rough, ulcerated with many holes of the most bizarre shape, riddled with cracks and pores, covered with a network of wrinkles, scratches, etc.

To quantify these deviations from ideality (contour tortuosity, surface wrinkling, volume fracturing and porosity), B. Mandelbrot will use fractional dimensions. This new quantitative estimate, the fractional dimension of Hausdorff-Bezekovich as applied to ideal objects of classical Euclidean geometry, gave the same numerical values ​​as the known topological dimension (equal to zero for a point, one for a smooth line, two for a figure and a surface, three for body and space) (see the topology line in Fig. "Elements of the real structure of materials").

But in the case of assessing the morphology of real structures, the new dimension had a finer sensitivity to all sorts of imperfections in real objects. So, a straight line segment, a segment of a sinusoid and the most complex meander are indistinguishable when using the topological dimension - they all have a topological dimension equal to one, while their dimension according to the Hausdorff-Bezekovich scale is different and allows the number to measure the degree of sinuosity of the line.

The Hausdorff-Bezekovich dimension increases as the line sinuosity or surface roughness increases. This change in dimension is not accompanied by jumps, as in topology, but smoothly changes its value as the defectiveness increases.

So, at the intersection of mathematics and physics, when studying the behavior of complex dynamic systems, they received their new birth fractals are objects with fractional (fractal) dimension.

Many natural fractals (fault surfaces of rocks and metals, clouds, turbulent flows, foam, gels, soot particles, etc.) are devoid of explicit geometric similarity, but stubbornly reproduce the statistical properties of the whole in each fragment. Such statistical similarity, or self-similarity on average, distinguishes fractals among many natural objects.

A real snowflake (six types) is a dendritic ice crystal. This is a typical self-similar fractal that occurs during the primary crystallization of all metals and alloys.

Description of a snowflake using fractal geometry will require only three parameters: the fractal dimension D, the dimensions of the primary block (R m in) and the snowflake as a whole (R m ax). The fractal dimension of the computer and real snowflakes is the same (D = 1.71).

Fractals in materials science

The central issue of modern materials science is the study of the structure of a material and the establishment of a relationship between structural parameters and material properties. The main quantitative relationships in the case of hardening in the dissolution of foreign atoms, in the precipitation of dispersed phases, in the grinding of grains form the paradigm of modern materials science from structural defects of materials to their properties.

Traditionally, the analysis of the structure of materials at macro-, meso- and microscopic levels is carried out by quantitative measurements of structural components using topological dimensions. In this case, significant conditional approximations of very complex, real structures to simple figures of Euclidean geometry are allowed.

The fractal dimension makes it possible to quantitatively describe various structures that are highly complex and contain a large number of point, line, surface, and bulk defects. Fractal geometry makes it possible to describe disordered morphology - rough surfaces, porous media, complex contours of excess phases, etc. Often such structures have the property of self-similarity.

The basic principle of fractal analysis provides for the determination of the fractal dimension of the structure under study with the widespread use of optical microscopy, electron scanning and transmission microscopy, and other methods of quantitative metallography.

The main paradigm of modern materials science: "From the real structure of the material to its physical and mechanical properties":

Upper row - examples of models of defects in the micro- and mesostructure of the material (from left to right) elastic deformation of the crystal lattice by dissolved, impurity atoms, deceleration of a moving dislocation by dispersed excess phases (particles), deceleration of dislocation pileups by grain boundaries;

The bottom row - examples reflecting the change in some physical and mechanical properties under the influence of structural defects in the top row.

Rice. 1.17. The dependence of the properties of materials on the structure is the main paradigm of modern materials science

The self-similarity of the structures is confirmed by a geometric analysis of the obtained patterns and their measurement at various magnification scales. To establish the fractality of the structure, it is necessary to verify the presence of self-similarity and calculate the fractal dimension.

Further determination of the connection between the properties of a material and its fractal dimension requires certain new fundamental approaches in the analysis of fractal structures.

Fractographic studies of fracture surfaces of materials by determining their fractal dimension are the most effective for assessing the nature of fracture under impact or fatigue loading.

In recent years, many studies of the fractal structure of surfaces have been published. Everything was declared fractal - from the molecular surfaces of proteins to the runways of airfields. These studies apply the full range of methods of chemistry and physics. Generally speaking, the observed fractal behavior does not cover wide (several orders of magnitude) ranges of spatial scales, and one can doubt the reliability of the fractal dimension estimates found. Nevertheless, a very interesting series of observations has been analyzed, and here we discuss some new results.

14.1. Observed surface topography

Sales and Thomas measured and analyzed the surface roughness of objects ranging from supertanker skins and concrete airstrips to joint surfaces and brushed metal surfaces.

The surface height was measured at various points along a certain direction. Having a large number of measurements over the entire available surface area, it is possible to calculate the surface roughness determined by the dispersion

Here the angle brackets denote the averaging over a series of measurements (sometimes multiple repeated ones) of the surface topography. The vertical reference point is chosen so that

An important measure of the statistical properties of a surface is the correlation function defined by the relation

For stationary surfaces, the correlation function can be expressed in terms of the power spectrum using the Fourier transform

(click to view scan)

The spatial frequency is related to the wavelength of surface irregularities X by the equality Physical systems have a finite extent and, accordingly, a minimum spatial frequency. Therefore, the correlation function can be rewritten in the form

Sales and Thomas suggest that the power spectrum is

and call the constant k "indentation". Under this assumption, the variance is

i.e. we get and the variance increases with surface size, as expected for Gaussian random processes.

On fig. 14.1 reproduced the results of this work. Value plotted as a function If equation (14.1) is true, then we would expect this plot to be a straight line with a slope of 2. Sales and Thomas found surprising convergence of results for 23 surface types spanning 8 decades of wavelength. These authors believe that the value of k uniquely determines the statistical geometric properties of the random components of an isotropic surface for this wavelength range!

It should be noted, however, that the approximation of the observed spectral density by dependence (14.1) determines k and, with the chosen normalization, this dependence takes the form a logarithmic plot, in which individual segments are shifted along the vertical y-axis so that they are as close as possible to the line. With this procedure, the approximation will look the better, the wider the range of the initial data.

Berry and Hanni observe that statistically isotropic surfaces, on which no scale is distinguished and whose level is well defined but non-differentiable, can indeed have a fractal-like spectrum:

As shown by Mandelbrot, the exponent is equal to the fractal codimension and is expressed in terms of the fractal dimension of the surface as follows

For Brownian surfaces, i.e., in the case of ordinary Gaussian statistics, we get the equality (14.1) used by Sayles and Thomas, since for such surfaces

Rice. 14.2. Histogram of the values ​​of the indicator a for 23 series of measurements presented in the previous figure.

However, for the parameter a, the value that provides the best approximation should be found, and it turns out to be in the range from 1.07 to 3.03, which corresponds to fractal dimension values ​​from 2 to 3. In response to this remark, Sales and Thomas made a new approximation of their data and built a histogram of estimates of the spectral parameter a, shown in Fig. 14.2. The resulting a values ​​cluster around a Gaussian value of 2, but are distributed over an allowable range of 1 to 3. This result seems reasonable, since ball bearing surfaces and runway surfaces can hardly be expected to have the same statistical properties. Nevertheless, Sales and Thomas got interesting results, and they should be critically checked against high quality data.

Fractal fault surfaces. When a metal body breaks, the resulting fracture surface is rough and irregular. Mandelbrot et al. investigated the fractal structure of such surfaces. They studied the fractures of samples of martensitic steel grade 300. The fractures were first nickel-plated and then ground parallel to the fracture plane. The result was "islands" of steel surrounded by nickel; with further polishing, the islands grew and merged with each other. The length of the "coastline", or the perimeter and area A of such islands, was measured using a "standard" with a length

Fractal surfaces, like fault surfaces, should be characterized by different similarity laws in and across the fault plane. Therefore, fault surfaces can be at best self-affine with a local fractal dimension. However, the intersection of such a self-affine surface with a plane gives

Rice. 14.3. The ratio of the perimeter and area for the fracture surface of martensitic steel grade 300. The straight line shows the approximation at

coastlines that are undoubtedly self-similar and have a fractal dimension. Therefore, the perimeter-area ratio (12.2) can be used, written as

On fig. 14.3 shows the results of Mandelbrot et al. Approximation by dependence (14.3) gives an estimate from which it follows that in a noticeable range of scales the fault surface has a fractal dimension. Mandelbrot and his co-authors checked the estimate of the fractal dimension by analyzing the profiles of the fault surface. To detect its profile, the surface was cut and the spectral density was calculated for the measured profiles. According to the relation (14.2), the value and then the fractal dimension of the surface were found

which is in good agreement with the previous estimate.

In another series of interesting experiments, Mandelbrot et al. subjected samples of martensitic steel grade 300 to heat treatment at various temperatures. Then the amount of energy that

Rice. 14.4. Relationship between the measured fractal dimension of the fracture surface and the energy required to fracture a series of samples of martensitic steel grade 300, hardened at different temperatures.

must be embedded in order to destroy the samples, and the fractal dimension of the fault surfaces was determined. On fig. 14.4 the received results are presented. It is clearly seen that the fractal dimensions contained within the limits approximately linearly depend on the input energy. The connection of this dependence with the nature of metallurgical processes is unclear, but after the discovery of the dependence of the fractal dimension of the fault on the input energy, at least an approach to the study of surface topography was outlined.