percolation otherwise leakage(English) - in materials science - the abrupt appearance of new properties in a material (electrical conductivity - for an insulator, gas permeability - for a gas-tight material, etc.) when it is filled with a “filler” that has this characteristic. In some cases, the filler may be pores and voids.

Description

Percolation occurs at a certain critical concentration of filler or pores (percolation threshold) as a result of the formation of a continuous network (channel) of filler particles (clusters) from one side of the material sample to the opposite side.

The percolation process can be visually examined using the example of the flow of electric current in a two-dimensional square lattice consisting of electrically conducting and non-conducting sections. Metal contacts are soldered to two opposite sides of the grille, which are connected to the power source. At a certain critical value of the proportion of conductive elements arranged randomly, the circuit is closed (Fig.).

In 2010, “for proving the conformal invariance of percolation and the Ising model in statistical physics,” Stanislav Smirnov, a native of St. Petersburg, became a laureate of the Fields Mathematical Prize, the equivalent of the Nobel Prize.

Illustrations

Percolation can be observed both in lattices and other geometric structures, including continuous ones, consisting of a large number of similar elements or continuous regions, respectively, which can be in one of two states. The corresponding mathematical models are called lattice or continuum.

An example of percolation in a continuous medium is the passage of a liquid through a voluminous porous sample (for example, water through a sponge made of foam-forming material), in which bubbles are gradually inflated until their size becomes sufficient for the liquid to percolate from one edge of the sample to another.

Inductively, the concept of percolation is transferred to any structures or materials that are called a percolation medium, for which an external source of flow must be determined, a method of flow and elements (fragments) of which can be in different states, one of which (primary) does not satisfy this method of flow , and the other satisfies. The method of flow also implies a certain sequence of occurrence of elements or a change in fragments of the medium into the state necessary for the flow, which is provided by the source. The source gradually transfers elements or fragments of the sample from one state to another until the moment of percolation occurs.

Leakage threshold

The set of elements through which flow occurs is called a percolation cluster. Being a connected random graph by nature, it can take different forms depending on the specific implementation. Therefore, it is customary to characterize its overall size. The percolation threshold is the number of elements of the percolation cluster divided by the total number of elements of the medium under consideration.

Due to the random nature of switching states of the elements of the environment, in the finite system there is no clearly defined threshold (the size of the critical cluster), but there is a so-called critical range of values, into which the percolation threshold values obtained as a result of various random implementations fall. As the size of the system increases, the area narrows to a point.

2. Scope of application of percolation theory

The applications of percolation theory are wide and varied. It is difficult to name an area in which the theory of percolation would not be applied. The formation of gels, hopping conductivity in semiconductors, the spread of epidemics, nuclear reactions, the formation of galactic structures, the properties of porous materials - this is not a complete list of the various applications of the percolation theory. It is not possible to give any complete overview of works on applications of percolation theory, so we will dwell on some of them.

2.1 Gelation processes

Although gelation processes were the first problems where the percolation approach was applied, this area is far from being exhausted. The process of gelation involves the fusion of molecules. When aggregates appear in a system, extending throughout the entire system, it is said that a sol-gel transition has occurred. It is usually believed that a system is described by three parameters - the concentration of molecules, the probability of formation of bonds between molecules and temperature. The last parameter affects the probability of forming connections. Thus, the gelation process can be considered as a mixed problem of the percolation theory. It is quite remarkable that this approach is also used to describe magnetic systems. There is an interesting direction for developing this approach. The problem of albumin protein gelation is important for medical diagnostics.

There is an interesting direction for developing this approach. The problem of albumin protein gelation is important for medical diagnostics. It is known that protein molecules have an elongated shape. When a protein solution passes into the gel phase, not only temperature has a significant influence, but also the presence of impurities in the solution or on the surface of the protein itself. Thus, in the mixed problem of percolation theory it is necessary to additionally take into account the anisotropy of molecules. In a certain sense, this brings the problem under consideration closer to the “needles” problem and Nakamura’s problem. Determining the percolation threshold in a mixed problem for anisotropic objects is a new problem in the theory of percolation. Although for the purposes of medical diagnostics it is sufficient to solve the problem for objects of the same type, it is of interest to study the problem for cases of objects of different anisotropy and even different shapes.

2.2 Application of percolation theory to describe magnetic phase transitions

One of the features of compounds based on i is the transition from an antiferromagnetic to a paramagnetic state even with a slight deviation from stoichiometry. The disappearance of long-range order occurs when there is an excess concentration of holes in the plane, while at the same time, short-range antiferromagnetic order is preserved in a wide range of concentrations x up to the superconducting phase.

At a qualitative level, the phenomenon is explained as follows. When doped, holes appear on oxygen atoms, which leads to the emergence of a competing ferromagnetic interaction between spins and suppression of antiferromagnetism. The sharp decrease in the Néel temperature is also facilitated by the movement of the hole, leading to the destruction of the antiferromagnetic order.

On the other hand, the quantitative results sharply disagree with the values of the percolation threshold for a square lattice, within which it is possible to describe the phase transition in isostructural materials. The task arises of modifying the percolation theory in such a way as to describe the phase transition in the layer within the framework.

When describing the layer, it is assumed that for each copper atom there is one localized hole, that is, it is assumed that all copper atoms are magnetic. However, the results of band and cluster calculations show that in the undoped state the occupation numbers of copper are 0.5 - 0.6, and for oxygen - 0.1-0.2. At a qualitative level, this result can be easily understood by analyzing the result of the exact diagonalization of the Hamiltonian for a cluster with periodic boundary conditions. The ground state of the cluster is a superposition of the antiferromagnetic state and states without antiferromagnetic ordering on copper atoms.

We can assume that approximately half of the copper atoms have one hole, and the remaining atoms have either none or two holes. An alternative interpretation is that the hole spends only half its time on copper atoms. Antiferromagnetic ordering occurs when the nearest copper atoms each have one hole. In addition, it is necessary that on the oxygen atom between these copper atoms there is either no hole or two holes in order to exclude the occurrence of ferromagnetic interaction. In this case, it does not matter whether we consider the instantaneous configuration of holes or one or components of the wave function of the ground state.

Using percolation theory terminology, we will call copper atoms with one hole unblocked sites, and oxygen atoms with one hole broken bonds. The transition from long-range ferromagnetic order to short-range ferromagnetic order in this case will correspond to the percolation threshold, that is, the appearance of a contracting cluster - an endless chain of unblocked nodes connected by unbroken bonds.

At least two points sharply distinguish the problem from the standard theory of percolation: firstly, the standard theory assumes the presence of atoms of two types, magnetic and non-magnetic, while we have only atoms of one type (copper), the properties of which change depending on the location of the hole; secondly, the standard theory considers two nodes connected if both of them are not blocked (magnetic) - the problem of nodes, or, if the connection between them is not broken - the problem of connections; in our case, both nodes are blocked and connections are broken.

Thus, the problem is reduced to finding the percolation threshold on a square lattice to combine the problem of nodes and connections.

.3 Application of the theory of percolation to the study of gas-sensitive sensors with a percolation structure

In recent years, sol-gel processes that are not thermodynamically equilibrium have found wide application in nanotechnology. At all stages of sol-gel processes, various reactions occur that affect the final composition and structure of the xerogel. At the stage of synthesis and maturation of the sol, fractal aggregates arise, the evolution of which depends on the composition of the precursors, their concentration, mixing order, pH value of the medium, temperature and reaction time, atmospheric composition, etc. Products of sol-gel technology in microelectronics, as a rule, are layers that are subject to the requirements of smoothness, continuity and uniformity in composition. For new generation gas-sensitive sensors, technological methods for producing porous nanocomposite layers with controlled and reproducible pore sizes are of greater interest. In this case, nanocomposites must contain a phase to improve adhesion and one or more phases of semiconductor metal oxides of n-type electrical conductivity to ensure gas sensitivity. The principle of operation of semiconductor gas sensors based on percolation structures of metal oxide layers (for example, tin dioxide) is to change the electrical properties during the adsorption of charged forms of oxygen and desorption of the products of their reactions with molecules of reducing gases. From the concepts of semiconductor physics it follows that if the transverse dimensions of the conducting branches of percolation nanocomposites are commensurate with the value of the characteristic length of Debye screening, the gas sensitivity of electronic sensors will increase by several orders of magnitude. However, the experimental material accumulated by the authors indicates a more complex nature of the occurrence of the effect of a sharp increase in gas sensitivity. A sharp increase in gas sensitivity can occur on network structures with geometric dimensions of branches several times greater than the screening length and depend on the conditions of fractal formation.

The branches of the network structures are a matrix of silicon dioxide (or a mixed matrix of tin and silicon dioxides) with tin dioxide crystallites included in it (which is confirmed by the modeling results), forming a conductive contracting percolation cluster with a SnO2 content of more than 50%. Thus, the increase in the percolation threshold value can be qualitatively explained due to the consumption of part of the SnO2 content into the mixed non-conducting phase. However, the nature of the formation of network structures appears to be more complex. Numerous experiments on analyzing the structure of layers using AFM methods near the expected value of the percolation transition threshold did not allow obtaining reliable documentary evidence of the evolution of the system with the formation of large pores according to the laws of percolation models. In other words, models of growth of fractal aggregates in the SnO2 - SnO2 system qualitatively describe only the initial stages of sol evolution.

In structures with a hierarchy of pores, complex processes of adsorption-desorption, recharging of surface states, relaxation phenomena at grain and pore boundaries, catalysis on the surface of layers and in the contact area, etc. occur. Simple model representations within the framework of the Langmuir and Brunauer-Emmett-Teller (BET) models ) are applicable only for understanding the predominant averaged role of a particular phenomenon. To deepen the study of the physical features of gas sensitivity mechanisms, it was necessary to create a special laboratory installation that would provide the ability to record the time dependences of changes in the analytical signal at different temperatures in the presence and absence of reducing gases of a given concentration. The creation of an experimental setup made it possible to automatically take and process 120 measurements per minute in the operating temperature range of 20 - 400 ºС.

For structures with a network percolation structure, new effects were identified that were observed when porous nanostructures based on metal oxides were exposed to an atmosphere of reducing gases.

From the proposed model of gas-sensitive structures with a hierarchy of pores, it follows that in order to increase the sensitivity of adsorption semiconductor sensor layers, it is fundamentally possible to ensure a relatively high resistance of the sample in air and a relatively low resistance of film nanostructures in the presence of a reagent gas. A practical technical solution can be implemented by creating a system of nano-sized pores with a high distribution density in the grains, providing effective modulation of current flow processes in percolation network structures. This was achieved through the targeted introduction of indium oxide into a system based on tin and silicon dioxides.

Conclusion

The theory of percolation is a fairly new and not fully studied phenomenon. Every year, discoveries are made in the field of percolation theory, algorithms are written, and papers are published.

The theory of percolation attracts the attention of various specialists for a number of reasons:

Easy and elegant formulations of problems in percolation theory are combined with the difficulty of solving them;

Solving percolation problems requires combining new ideas from geometry, analysis, and discrete mathematics;

Physical intuition can be very fruitful in solving percolation problems;

The technique developed for percolation theory has numerous applications in other problems of random processes;

The theory of percolation provides the key to understanding other physical processes.

Bibliography

Tarasevich Yu.Yu. Percolation: theory, applications, algorithms. - M.: URSS, 2002.

Shabalin V.N., Shatokhina S.N. Morphology of human biological fluids. - M.: Chrysostom, 2001. - 340 pp.: ill.

Plakida N. M. High-temperature superconductors. - M.: International Education Program, 1996.

Physical properties of high-temperature superconductors/ Pod. Ed. D. M. Ginsberg. - M.: Mir, 1990.

Prosandeev S.A., Tarasevich Yu.Yu. Influence of correlation effects on band structure, low-energy electronic excitations and response functions in layered copper oxides. // UFZH 36(3), 434-440 (1991).

Elsin V.F., Kashurnikov V.A., Openov L.A. Podlivaev A.I. Binding energy of electrons or holes in Cu - O clusters: exact diagonalization of the Emery Hamiltonian. // JETP 99(1), 237-248 (1991).

Moshnikov V.A. Mesh gas-sensitive nanocomponents based on tin and silicon dioxides. - Ryazan, "Bulletin of RGGTU", - 2007.

PERCEPTION THEORY(percolation theory, from Latin percolatio - straining; seepage theory) - math. a theory that is used to study processes occurring in inhomogeneous media with random properties, but fixed in space and unchanged in time. It arose in 1957 as a result of the work of J. Hammersley. In P. t., a distinction is made between lattice problems of P. t., continuum problems, and the so-called. tasks on random nodes. Lattice problems, in turn, are divided into so-called. tasks of nodes and problems of connections between them.

Communication tasks. Let connections be edges connecting neighboring nodes of an infinite periodic. gratings (Fig., o). It is assumed that connections between nodes can be of two types: intact or broken (blocked). The distribution of intact and blocked bonds in the lattice is random; the probability that a given connection is intact is equal to X. It is assumed that it does not depend on the state of neighboring bonds. Two lattice nodes are considered connected to each other if they are connected by a chain of entire bonds. A set of nodes connected to each other is called. cluster. At small values x entire connections, as a rule, are far from each other and clusters of a small number of nodes dominate, but with increasing x cluster sizes increase sharply. Threshold ( x c) called this meaning X, in which for the first time a cluster of an infinite number of nodes appears. P.t. allows you to calculate threshold values x s, and also study the topology of large-scale clusters near the threshold (see. Fractals C With the help of P. t. it is possible to describe the electrical conductivity of a system consisting of conductive and non-conducting elements. For example, if we assume that entire connections conduct electricity. current, but the blocked ones do not conduct, then it turns out that when X< х с beat the electrical conductivity of the lattice is O, and at x > x c it is different from 0.

Flow through the grid: A- connection problem (there is no flow path through the specified block); b - task of nodes (flow path shown).

Lattice knot problems differ from connection problems in that the blocked connections are not distributed individually on the lattice - all connections coming out of the block are blocked. node (Fig. b). The nodes blocked in this way are distributed randomly on the lattice, with probability 1 - X. It has been proven that the threshold x s for the problem of connections on any lattice does not exceed the threshold x s for the problem of nodes on the same lattice. For certain flat lattices exact values have been found x s. For example, for problems of connections on triangular and hexagonal lattices x s= 2sin(p/18) and x c = 1 - 2sin(p/18). For the problem of nodes on a square lattice x c = 0.5. For three-dimensional lattices the values x s found approximately using computer simulation (table).

Flow thresholds for various grids

Grate type	x s for the connection problem	x s For the node task
Flat gratings
hexagonal
square
triangular
Three-dimensional lattices
diamond type
simple cubic
body-centered cubic
face-centered cubic

Continuum tasks. In this case, instead of flowing through bonds and nodes, they are considered in a disordered continuous medium. A continuous random function of coordinates is specified throughout the entire space. Let us fix a certain value of the function and call the regions of space in which they are black. At sufficiently small values, these areas are rare and, as a rule, isolated from each other, and at sufficiently large values they occupy almost the entire space. You need to find the so-called. flow level - min. meaning when the black areas form a connected labyrinth of paths extending to an infinite distance. In the three-dimensional case, an exact solution to the continuum problem has not yet been found. However, computer simulation shows that for Gaussian random functions in three-dimensional space, the fraction of the volume occupied by black areas is roughly equal to 0.16. In the two-dimensional case, the fraction of area occupied by the black regions at is exactly 0.5.

Tasks on random nodes. Let the nodes not form a regular lattice, but randomly distributed in space. Two nodes are considered connected if the distance between them does not exceed a fixed value. Small compared to avg. distance between nodes, then clusters containing 2 or more nodes connected to each other are rare, but the number of such clusters increases sharply with increasing G and with some criticality. meaning an infinite cluster arises. Computer simulation shows that in the three-dimensional case 0.86, where N- concentration of nodes. Problems on random nodes and their various types. generalizations play an important role in theory hopping conduction.

The effects described by P. t. relate to critical events, characterized by critical point, near the cut the system breaks up into blocks, and the size of the parts. blocks increases indefinitely when approaching critical. point. The emergence of an infinite cluster in P.T. problems is in many ways similar phase transition of the second kind. For math. descriptions of these phenomena are introduced order parameter,Crimea in the case of lattice problems is the share P(x) lattice nodes belonging to an infinite cluster. Near the threshold of the function P(x) has the form

where - numerical coefficient, b - critical. order parameter index. A similar formula describes the behavior of the beat. electrical conductivity s(x)near the flow threshold:

Where AT 2- numerical coefficient, s(1) - spec. electrical conductivity at c= 1, f - critical. electrical conductivity index. The spatial dimensions of clusters are characterized by the correlation radius R(x), applying to

Here B 3 - numerical coefficient, A- lattice constant, v - critical. correlation radius index.

Thresholds of occurrence significantly depend on the type of problems of P. t., but critical. the indices are the same for different problems and are determined only by the dimension of space d(versatility). Concepts borrowed from the theory of phase transitions of the 2nd order make it possible to obtain relationships connecting various critical factors. indexes. Approximation self-consistent field applicable to P. t. problems with d> 6. In this approximation, critical. indexes do not depend on d; b = 1, = 1/2.

The results of P.T. are used in the study of electronic properties disordered systems, phase metal transitions - dielectric, ferromagnetism solid solutions, kinetic. phenomena in highly heterogeneous media, physical-chemical. processes in solids, etc.

Lit.: Mott N., Davis E., Electronic processes V non-crystalline substances, trans. from English, 2nd ed., vol. 1-2, M., 1982; Shklovsky B.I., Efros A.L., Electronic properties of doped materials, M., 1979; 3 and y-man D. M., Models of disorder, trans. from English, M., 1982; Efros A.L., Physics and geometry of disorder, M., 1982; Sokolov I.M., Dimensions and other geometric critical exponents in the theory of flow, "UFN", 1986, v. 150 p. 221. A. L. Efros.

The erromagnetic order is preserved in a wide range of concentrations x up to the superconducting phase.

Thus, the problem is reduced to finding the percolation threshold on a square lattice to combine the problem of nodes and connections.

3 Application of percolation theory to the study of gas-sensitive sensors with a percolation structure

For structures with a network percolation structure, new effects were identified that were observed when porous nanostructures based on metal oxides were exposed to an atmosphere of reducing gases.

Conclusion

The theory of percolation is a fairly new and not fully studied phenomenon. Every year, discoveries are made in the field of percolation theory, algorithms are written, and papers are published.

The theory of percolation attracts the attention of various specialists for a number of reasons:

Easy and elegant formulations of problems in percolation theory are combined with the difficulty of solving them;

Solving percolation problems requires combining new ideas from geometry, analysis, and discrete mathematics;

Physical intuition can be very fruitful in solving percolation problems;

The technique developed for percolation theory has numerous applications in other problems of random processes;

The theory of percolation provides the key to understanding other physical processes.

Bibliography

Tarasevich Yu.Yu. Percolation: theory, applications, algorithms. - M.: URSS, 2002.

Shabalin V.N., Shatokhina S.N. Morphology of human biological fluids. - M.: Chrysostom, 2001. - 340 pp.: ill.

Plakida N. M. High-temperature superconductors. - M.: International Education Program, 1996.

Physical properties of high-temperature superconductors/ Pod. Ed. D. M. Ginsberg. - M.: Mir, 1990.

Elsin V.F., Kashurnikov V.A., Openov L.A. Podlivaev A.I. Binding energy of electrons or holes in Cu - O clusters: exact diagonalization of the Emery Hamiltonian. // JETP 99(1), 237-248 (1991).

Moshnikov V.A. Mesh gas-sensitive nanocomponents based on tin and silicon dioxides. - Ryazan, "Bulletin of RGGTU", - 2007.

The theory of percolation (percolation) is the most general approach to describing transport processes in disordered systems. With its help, the probabilities of the formation of clusters from particles touching each other are considered and both the values of percolation thresholds and the properties of composites (electrical, mechanical, thermal, etc.).

The flow of electric current in composite materials is most adequate to the percolation problem formulated for a continuous medium. According to this problem, each point in space with probability p=x conductivity answersg = g N and with probability (1- p) – conductivityg = g D, where g N – electrical conductivity of the filler,g D – electrical conductivity of the dielectric. The leakage threshold in this case is equal to the minimum fraction of space xC occupied by conducting regions, in which the system is still conducting. Thus, at a critical probability value p=x C, a metal-insulator transition is observed in the system. At small p all conducting elements are contained in clusters of finite size, isolated from each other. As you increase p the average cluster size also increases with p=x C appears for the first time in the systeminfinite cluster . And finally, at high p Non-conducting areas will be isolated from each other.

The main result of the percolation theory is the power-law nature of the concentration behavior of conductivity in the critical region:

Where x– volume concentration of the conducting phase with conductivityg N ; x C– critical concentration (percolation threshold);g D – conductivity of the dielectric phase. Dependence (1)-(3) is shown in Fig. 1.

Rice. 1. Dependence of the conductivity of the composite material on the filler concentration

Relationship between exponents (critical indices):

Q=t(1/S-1)

Probably the only accurate result obtained in the theory of heterogeneous systems is the result for a two-dimensional two-phase metal-insulator system with such a structure that when x D = x N = 0.5 replacing metal with dielectric does not statistically change the structure. This allows us to determine the critical index S for two-dimensional systems: S 2 =0.5. Then from (1.17) q 2 =t 2 =1.3. For three-dimensional systems: S 3 =0.62, q 3 =1, t 3 =1.6.

One of the most important parameters of the percolation theory is the percolation threshold x S. This parameter is more sensitive to changes in structure than critical indices. For two-dimensional systems it varies within 0.30-0.50 with the theoretical average x C=0.45, and for three-dimensional – within 0.05-0.60 s x C=0.15. These variations are associated with the variety of types of structures of composite materials, since in real systems the critical concentration is largely determined by the technological regime for obtaining the mixture: the nature of the powder dispersion, the spraying method, pressing modes, heat treatment, etc. Therefore, it is most advisable to determine the percolation threshold experimentally using the concentration dependencesg (x), and not be considered a theoretical parameter.

The percolation threshold is determined by the nature of the distribution of the filler in the matrix, the shape of the filler particles, and the type of matrix.

For structuredcomposite materials nature of electrical conductivity and type of dependenceg (x) are not qualitatively different from similar dependencies for statistical systems, however, the percolation threshold shifts towards lower concentrations. Structuring can be caused by the interaction of the matrix and the filler, or it can be carried out in a forced manner, for example, under the influence of electric or magnetic fields.

Also leakage threshold depends on the shape of the filler particles. For elongated particles and flake-shaped particles, the percolation threshold is lower than for spherical particles. This is due to the fact that the significant extent of electrically conductive sections, determined by the geometry of the particles, increases the likelihood of creating a reliable contact and contributes to the formation of an infinite cluster at relatively low degrees of filling of the composite.

For fibers having the same length to diameter ratio, but introduced into different polymers, different values were obtained x C.

Despite significant progress, the theory of percolation has not been widely used for three-component and more complexcomposite materials .

It is also possible to combine percolation theory and other calculation methods to

Description

Illustrations

2.1 Gelation processes

2.2 Application of percolation theory to describe magnetic phase transitions

.3 Application of the theory of percolation to the study of gas-sensitive sensors with a percolation structure

Conclusion

Bibliography

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