The theory of percolation (flow) is the most general approach to describing transport processes in disordered systems. It is used to consider the probabilities of cluster formation from particles touching each other and to predict both the percolation thresholds and the properties 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 meets conductivityg = g H and with probability (1- p) - conductivityg = g D , where g H is the electrical conductivity of the filler,g D is the electrical conductivity of the dielectric. The percolation threshold in this case is equal to the minimum fraction of space x C occupied by conducting regions, at which the system is still conducting. Thus, at the critical value of the probability 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 at p=x C first appears in the systeminfinite cluster . And finally, at high p non-conductive regions will be isolated from each other.

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

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

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

Relationship between exponents (critical indices):

Q=t(1/S-1)

Probably the only exact 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 at x D \u003d x H \u003d 0.5 replacement of a metal by a dielectric does not statistically change the structure. This makes it possible 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 \u003d 0.62, q 3 \u003d 1, t 3 \u003d 1.6.

One of the most important parameters of the percolation theory is the percolation threshold x C. This parameter is more sensitive to structure changes than critical indexes. For two-dimensional systems, it varies within 0.30-0.50 with the average theoretical x C\u003d 0.45, and for three-dimensional - within 0.05-0.60 s x C=0.15. These variations are associated with a variety of types of structures of composite materials, since in real systems the critical concentration is largely determined by the technological mode of obtaining a mixture: the nature of the powder dispersion, the method of spraying, the modes of pressing, heat treatment, etc. Therefore, it is most expedient to determine the percolation threshold experimentally from 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, from the shape of the filler particles, the type of matrix.

For structuredcomposite materials the nature of the electrical conductivity and the type of dependenceg (x) do not qualitatively differ from similar dependences for statistical systems, however, the percolation threshold is shifted towards lower concentrations. Structuring can be due to the interaction of the matrix and the filler, or be carried out in a forced way, for example, under the action of electric or magnetic fields.

Also percolation threshold depends on the shape of the filler particles. For elongated and scaly particles, the percolation threshold is lower than for spherical particles. This is due to the fact that a significant length of the electrically conductive sections, due to 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 received wide application for three-component and more complexcomposite materials .

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

Introduction

1. Theory of percolation

2.1 Gelling processes

Conclusion

Percolation theories have been around for over fifty years. Hundreds of articles are published annually in the West, devoted both to the theoretical questions of percolation and its applications.

Percolation theory deals with the formation of connected objects in disordered environments. From the point of view of a mathematician, the theory of percolation should be attributed to the theory of probability in graphs. From the point of view of physics, percolation is a geometric phase transition. From the programmer's point of view, this is the widest field for the development of new algorithms. From the point of view of practice, it is a simple but powerful tool that allows you to solve a wide variety of life tasks in a single approach.

This work will be devoted to the main provisions of the theory of percolation. I will consider the theoretical foundations of percolation, give examples that explain the phenomenon of percolation. The main applications of percolation theory will also be considered.

The theory of percolation (flow) is a theory that describes the emergence of infinite connected structures (clusters) consisting of individual elements. Representing the environment as a discrete lattice, we formulate two simple types of problems. It is possible to selectively color (open) the nodes of the lattice in a random way, considering the proportion of colored nodes as the main independent parameter and assuming that two colored nodes belong to the same cluster if they can be connected by a continuous chain of neighboring colored nodes.

Questions such as the average number of nodes in a cluster, the size distribution of clusters, the appearance of an infinite cluster, and the proportion of colored nodes in it constitute the content of the node problem. It is also possible to selectively color (open) links between neighboring nodes and consider that nodes connected by chains of open links belong to the same cluster. Then the same questions about the average number of nodes in a cluster, etc. constitute the content of the connection problem. When all nodes (or all connections) are closed, the lattice is an insulator model. When they are all open and current can flow through the conductive bonds through the open nodes, then the lattice models the metal. At some critical value, a percolation transition will occur, which is a geometric analog of the metal-insulator transition.

The theory of percolation is important precisely in the vicinity of the transition. Far from the transition, it suffices to approximate the effective medium. The percolation transition is analogous to a second-order phase transition.

The phenomenon of percolation (or the flow of a medium) is determined by:

The environment in which this phenomenon is observed;

An external source that provides flow in this environment;

The way in which a medium flows, which depends on an external source.

As the simplest example, we can consider a model of flow (for example, electrical breakdown) in a two-dimensional square lattice consisting of nodes that can be conductive or non-conductive. At the initial moment of time, all grid nodes are non-conductive. Over time, the source replaces non-conductive nodes with conductive nodes, and the number of conductive nodes gradually increases. In this case, the nodes are replaced randomly, that is, the choice of any of the nodes for replacement is equally probable for the entire surface of the lattice.

Percolation is the moment when such a state of the lattice appears, in which there is at least one continuous path through neighboring conductive nodes from one to the opposite edge. Obviously, with an increase in the number of conducting nodes, this moment will come before the entire surface of the lattice will consist exclusively of conducting nodes.

Let us denote the non-conductive and conductive states of the nodes by zeros and ones, respectively. In the two-dimensional case, the medium will correspond to a binary matrix. The sequence of replacing matrix zeros with ones will correspond to the source of leakage.

At the initial moment of time, the matrix consists entirely of non-conductive elements:

percolation gelation gas sensitive cluster

As the number of conductive nodes increases, there comes a critical moment when percolation occurs, as shown below:

It can be seen that from the left to the right border of the last matrix there is a chain of elements that ensures the flow of current through the conductive nodes (units) that continuously follow each other.

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 can be the passage of a liquid through a bulky porous sample (for example, water through a sponge made of foaming material), in which bubbles are gradually inflated until their size is sufficient for the liquid to seep from one edge of the sample to another.

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

Leakage threshold

The set of elements through which the flow occurs is called a percolation cluster. Being a connected random graph by its nature, depending on the specific implementation, it can have a different form. Therefore, it is customary to characterize its overall size. The percolation threshold is the number of elements of a percolation cluster related to 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 final 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 region narrows to a point.

2. Scope of application of the theory of percolation

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

2.1 Gelling processes

Although the processes of gelation were the first problems where the percolation approach was applied, this area is far from being exhausted. The process of gelation is the fusion of molecules. When aggregates appear in the system, extending through the entire system, it is said that a sol-gel transition has occurred. It is usually believed that the 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 bond formation. Thus, the process of gelation can be considered as a mixed problem of percolation theory. It is noteworthy that this approach is also used to describe magnetic systems. There is an interesting direction for the development of this approach. The task of gelation of the albumin protein is important for medical diagnosis.

There is an interesting direction for the development of this approach. The task of gelation of the albumin protein is important for medical diagnosis. It is known that protein molecules have an elongated shape. When a protein solution passes into the gel phase, not only the temperature but also the presence of impurities in the solution or on the surface of the protein itself has a significant effect. 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 the Nakamura problem. Determining the percolation threshold in a mixed problem for anisotropic objects is a new problem in percolation theory. 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 and is the transition from the antiferromagnetic to the paramagnetic state already with a slight deviation from stoichiometry. Disappearance of the long-range order occurs at an excess concentration of holes in the plane , while the short-range antiferromagnetic order is preserved in a wide range of concentrations x up to the superconducting phase.

Qualitatively, the phenomenon is explained as follows. When doped, holes appear on oxygen atoms, which leads to the appearance of a competing ferromagnetic interaction between spins and suppression of antiferromagnetism. A sharp decrease in the Néel temperature is also facilitated by the motion of a hole, which leads to the destruction of the antiferromagnetic order.

On the other hand, the quantitative results differ sharply from the values ​​of the percolation threshold for a square lattice, within which it is possible to describe the phase transition in isostructural materials. The problem arises to modify 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 there is one localized hole for each copper atom, 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. On a qualitative level, this result is easy to understand 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.

It can be assumed that about half of the copper atoms have one hole each, and the remaining atoms have either none or two holes. Alternative interpretation: the hole spends only half of its time on copper atoms. Antiferromagnetic ordering arises when the nearest copper atoms have one hole each. In addition, it is necessary that the oxygen atom between these copper atoms either does not have a hole or has two holes in order to exclude the occurrence of a ferromagnetic interaction. In this case, it does not matter whether we consider the instantaneous configuration of holes or one or the components of the wave function of the ground state.

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

At least two points sharply distinguish the problem from the standard percolation theory: 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), whose properties vary depending on the localization of the hole; secondly, the standard theory considers two nodes to be 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 the blocking of nodes and the breaking of links occur.

Thus, the problem is reduced to finding the percolation threshold on a square lattice for combining the knot and bond problem.

2.3 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 been widely used in nanotechnology. At all stages of the sol-gel processes, various reactions occur that affect the final composition and structure of the xerogel. At the stage of sol synthesis and maturation, fractal aggregates arise, the evolution of which depends on the composition of precursors, their concentration, mixing order, pH value of the medium, reaction temperature and time, atmosphere composition, etc. The products of sol-gel technology in microelectronics, as a rule, are the layers to which the requirements of smoothness, continuity and uniformity in composition are imposed. For gas sensitive sensors of the new generation, technological methods for obtaining porous nanocomposite layers with controlled and reproducible pore sizes are of greater interest. In this case, the nanocomposites must contain a phase to improve adhesion and one or more phases of semiconductor metal oxides of n-type electrical conductivity to provide gas sensitivity. The principle of operation of semiconductor gas sensors based on the percolation structures of metal oxide layers (for example, tin dioxide) is to change the electrical properties during the adsorption of charged oxygen species and desorption of the products of their reactions with molecules of reducing gases. It follows from the concepts of semiconductor physics that if the transverse dimensions of the conductive branches of percolation nanocomposites are commensurate with the value of the characteristic Debye screening length, 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 the branches that are several times greater than the values ​​of the screening length and depend on the conditions of fractal formation.

The branches of the network structures represent a silicon dioxide matrix (or a mixed matrix of tin and silicon dioxides) with tin dioxide crystallites included in it (which is confirmed by the simulation results), which form a conducting constricting percolation cluster at a SnO2 content of more than 50%. Thus, it is possible to qualitatively explain the increase in the value of the percolation threshold 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 seems to be more complex. Numerous experiments on the analysis of the layer structure by AFM methods near the assumed value of the percolation transition threshold did not allow us to obtain 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 the growth of fractal aggregates in the SnO2 - SnO2 system qualitatively describe only the initial stages of sol evolution.

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 region of contacts, etc. take place in structures with a hierarchy of pores. ) are applicable only for understanding the prevailing average role of one or another phenomenon. To deepen the study of the physical features of the mechanisms of gas sensitivity, it was necessary to create a special laboratory setup that makes it possible to record the time dependences of the change 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 revealed, which are observed when porous nanostructures based on metal oxides are exposed to an atmosphere of reducing gases.

It follows from the proposed model of gas sensitive structures with a pore hierarchy that, in order to increase the sensitivity of adsorption semiconductor sensor layers, it is fundamentally possible to provide a relatively high sample resistance 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 nanosized pores with a high distribution density in the grains, which ensures effective modulation of current flow processes in percolation network structures. This was realized by the targeted introduction of indium oxide into a system based on tin and silicon dioxide.

Conclusion

The theory of percolation is a fairly new and not fully understood phenomenon. Every year, discoveries are made in the field of percolation theory, algorithms are written, 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 the theory of percolation 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 random process problems;

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

Bibliography

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2. Shabalin V.N., Shatokhina S.N. Morphology of human biological fluids. - M.: Chrysostom, 2001. - 340 p.: ill.
3. Plakida NM High-temperature superconductors. - M.: International Education Program, 1996.
4. Physical properties of high-temperature superconductors / Under. Ed. D. M. Ginzberg.- M.: Mir, 1990.
5. Prosandeev S.A., Tarasevich Yu.Yu. Influence of correlation effects on the band structure, low-energy electronic excitations and response functions in layered copper oxides. // UFZh 36(3), 434-440 (1991).
6. 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 Emery's Hamiltonian. // ZhETF 99(1), 237-248 (1991).
7. Moshnikov V.A. Mesh gas sensitive nanocomponents based on tin and silicon dioxides. - Ryazan, "Vestnik RGGTU", - 2007.

Many degrees of freedom

Hierarchy of architecture/structure at different scale levels

V in conventional materials, inhomogeneity manifests itself at atomic dimensions, and physics

phenomena has a quantum mechanical nature. Speaking of artificial media - polymeric CMs, we mean mixtures composed of such ordinary substances and having both a regular and a random, disordered structure. The main attention will be focused on the phenomena associated with such secondary inhomogeneity. This means that the scale of the inhomogeneity of artificial media is large enough to ensure that at each point the usual local material equations are satisfied, which are inherent in the substance filling the volume around this point. Although most of the results are also true for the case of a smooth change in material parameters, the simplest model of a composite material will be assumed - a matrix filled with some kind of inclusions.

Structure of polymeric CM

In the manufacture of CMs for structural purposes, the main purpose of filling is to obtain a reinforced polymeric material, i.e. material with an improved complex of physical and mechanical properties. It is achieved both by introducing fibrous reinforcing fillers and finely dispersed fillers, chopped fiberglass, aerosil, etc. When creating CM with special properties, fillers are usually introduced in order to give the material not mechanical, but the desired electrophysical, thermal, sensory, etc. . properties. In this case, the filler particles are distributed in one way or another in the polymer matrix.

By the nature of the distribution of components, composites can be divided into matrix systems, random mixtures, and structured compositions. In matrix (regular) systems, filler particles are located at the nodes of a regular lattice (a). In statistical systems, the components are randomly distributed and do not form regular structures (b). Structured composites include systems in which the components form chain, flat or bulk structures (c, d). On fig. 1 shows the typical structures of composites and the distribution of the filler in the matrix.

Rice. 1 Structures of composites and distribution of fillers in the matrix

Topology of heterogeneous systems (composites)

The topology of the CM is understood as the shape of the particles of the dispersed phase, their sizes, as well as the distribution of the dispersed phase over the volume of the dispersion medium. This also includes the size of the inclusions, the distance between them, the coordinates of the centers of the inclusions, the angle of orientation in space of non-isomeric inclusions (i.e., inclusions whose size in one or two selected directions is much larger than the size in other directions, for example, fibers, plates).

Composite materials based on uniaxially oriented continuous fibers or fabrics (Fig. 2) are easy to analyze. In the direction along the fibers (in

Wiener) (Fig. 3). Here σ f and σ m are the electrical conductivity of the filler and matrix, p is the volume fraction of the filler. These expressions are of a general nature, since they correspond to the effective conductivity of a two-phase system with sequential and parallel action of the phases and are optimal provided that only the volume fractions of each phase are known. It is easy to show that for layered composite materials, the longitudinal conductivity σ 1 is always higher than the conductivity σ 3 in the direction perpendicular to the layers. Indeed, for a pack of layers with thickness d i and conductivity σ i, the longitudinal conductivity is equal to σ 1 = Σd i σ i , and the transverse conductivity is 1/σ 3 = Σd i /σ i . Average longitudinal conductivity σ eff ,1 = σ 1 /Σd i . Average transverse conductivity 1/σ eff ,3 = Σd i /σ 3 . Using the Cauchy–Bunyakovsky inequality, we obtain that σ eff ,3< σ eff ,1 .

Rice. 2. Two extreme cases of filler laying microgeometry. The electrical conductivity in the direction parallel to the layers is determined by the Wiener upper bound; electrical conductivity perpendicular to the layers - the lower Wiener boundary.

Rice. Fig. 3. Dependence of the effective electrical conductivity of the composite σ eff / σ m on the filler concentration for the upper and lower Wiener boundaries in the case of σ f / σ m = 10.

The upper and lower Wiener boundaries determine the range of values ​​of the electrical conductivity of the CM for a given ratio of the parameters of the matrix and filler, regardless of the shape of the particles and the method of preparation of the CM. In fact, the Wiener boundaries give a too rough estimate of the conductivity, since they do not take into account the topology of the composite, contacts between filler particles, and other factors, but they allow one to estimate the range of changes in the conductivity and other transport characteristics (for example, thermal conductivity) for a particular pair of CM components.

Some topological characteristics of a number of frequently occurring structures of composite materials are given in the following Table.

Geometric structure of heterogeneous systems

Theory of percolation (flow)

The term percolation was originally used to contrast diffusion: if in the case of diffusion we are dealing with a random walk of a particle in a regular medium, then in the case of percolation we are talking about a regular movement (for example, a flow of liquid or current) in a random medium. Consider a 3x3 square grid. Let's paint some of the squares with black. In our case, there are 3 of them. The proportion of filled squares is p = 1/3. You can choose squares randomly and independently; You can enter any rules. In the first case, one speaks of random percolation (mathematicians also call it Bernoulli percolation), in the second, of correlated percolation. One of the main questions that percolation theory tries to answer is, at what fraction of p from filled squares does a chain of black squares appear that connects the top and bottom sides of our grid? It is easy to see that, for a grid of finite size, such chains can appear at different concentrations (Fig. 4). However, if the grid size L tends to infinity, then the critical concentration becomes quite definite (Fig. 5). This has been rigorously proven. This critical concentration is called percolation threshold.

In the case of an electrically conductive filler, until there is a chain of conductive sections connecting the top and bottom of the sample, it will be an insulator. If we consider the black squares as molecules, then the formation of a chain of molecules penetrating the entire system corresponds to the formation of a gel. If the black squares are microcracks, then the formation of a chain of such cracks will lead to destruction, splitting of the sample. So, the theory of percolation makes it possible to describe processes of a very different nature, when, with a smooth change in one of the parameters of the system (concentration of something), the properties of the system change abruptly. Even such a simple model turns out to be sufficient to describe, for example, the paramagnet-ferromagnet phase transition, the process of spreading an epidemic or a forest fire.

Rice. 4. Various grating filling options.

Rice. 5. Probability of occurrence of percolation P depending on the proportion of filled sites p. The smooth curve corresponds to a lattice of finite size. stepped - infinitely large lattice.

The tasks of the percolation theory are to describe the correlations between the corresponding physical and geometric characteristics of the analyzed media. The simplest and, accordingly, the most studied are the structures based on regular lattices. For them, one usually considers the problem of nodes and the problem of connections that arise when describing the physical properties (for definiteness, we will talk about electrical conductivity) of lattices from which a certain proportion (1 p) of randomly selected nodes (together with the bonds outgoing from them) or a proportion of randomly selected way of connections. In the problem of connections, they are looking for an answer to the question: what proportion of connections must be removed (cut) so that the grid breaks into two parts? In the problem of nodes, nodes are blocked (the node is removed, all links entering the node are cut) and they are searched for at what fraction of blocked nodes the mesh will fall apart. The square grid is only one of the possible patterns. It is possible to consider percolation on triangular, hexagonal grids, trees, three-dimensional lattices, for example, cubic ones, in space with dimension greater than 3. The grid need not be regular. Processes on random lattices are also considered.

Node problem (left) and connection problem (right) on a square lattice.

A chain of connected objects, such as black squares, is called a cluster in percolation theory (cluster - English - bunch). A cluster connecting two opposite sides of a system is called percolating, infinite, spanning, or connecting.

The percolation transition is a geometric phase transition. The percolation threshold or critical concentration separates two phases: in one phase there are finite clusters, in the other there is one infinite cluster.

To describe the electrical properties of CMs, the most adequate is the percolation problem formulated for a continuous medium. According to this problem, each point in space with probability p=v f corresponds to conductivity σ=σ f and with probability 1 p conductivity σ=σ m . Here the index f denotes the filler, and the index m the matrix. The percolation threshold (v f * ) in this case is equal to the minimum fraction of space occupied by conductive regions at which the system is still conductive. When vf varies from 0 to 1, the electrical conductivity of the composite increases from σ m to σ f , which is usually 20 orders of magnitude. ), which makes it possible to speak of a dielectric-to-metal transition or, as it is also called, a percolation transition, at vf equal to the percolation threshold. This transition is a second order phase transition.

Fig.6. Dependence of the electrical conductivity of CM polypropylene + aluminum obtained by various methods on the volume content of aluminum: 1 mixing of components in the form of powders with subsequent pressing, 2 polymerization filling, 3 mixing on rollers.

Let us consider the distribution of conductivities in the system at different filler contents v f . At small vf, all conducting particles are combined into clusters of finite size, isolated from each other. As v f increases, the average size of clusters increases, and at v f = v f * a significant part of isolated clusters merge into the so-called. an infinite cluster penetrating the entire system: a conduction channel appears. A further increase in v f leads to a sharp increase in the volume of an infinite cluster. It grows by absorbing finite clusters, and first of all the largest of them. As a result, the average size of the final clusters decreases.

Studying the topology of an infinite cluster, the researchers came to the conclusion that its main part is concentrated in chains ending in dead ends. These chains contribute to the density of the infinite cluster and to the permittivity, but do not contribute to the conductivity. Such chains are called "dead ends". An infinite cluster with no dead ends has been called the skeleton of an infinite cluster. The first model of the skeleton of an infinite cluster was the Shklovsky De Gennes model. It is an irregular lattice with an average distance between nodes depending on the proximity of the filler concentration to the percolation threshold.

Near the percolation threshold, the conductivity σ of a two-component mixture with a binomial distribution of particles is:

Qualitatively, the nature of the change in conductivity is shown in the following figure.

In the case of anisotropic fillers, the conductive phase may consist of randomly oriented anisometric particles (fibers, cylinders); the conductivity of such a material is always isotropic; or the conductive phase may consist of randomly oriented particles with anisotropic intrinsic conductivity. The percolation threshold for such fillers is usually much lower than for particles of a spherical or spheroid shape, which is easily seen from the figure: in the first case, a smaller number of particles is sufficient to cover the distance between opposite faces of the sample. It also shows the dependence of the percolation threshold on the shape factor of the filler particles – the ratio of the length l to the diameter d , l/d .

Another model for calculating the properties of composite materials is the effective medium theory, which uses the principle of a self-consistent field. It consists in the fact that when calculating the field inside a microscopic element

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, pores and voids can act as a filler.

Description

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

The percolation process can be visually considered by the example of the flow of an electric current in a two-dimensional square lattice consisting of electrically conductive and non-conductive regions. Metal contacts are soldered to two opposite sides of the grid, which are connected to a power source. At a certain critical value of the proportion of conductive elements arranged randomly, the circuit closes (Fig.).

In 2010, "for the proof of the conformal invariance of percolation and the Ising model in statistical physics" Stanislav Smirnov, a native of St. Petersburg, won the Fields Prize in Mathematics - the equivalent of the Nobel Prize.

Illustrations

FLOW THEORY(percolation theory, from lat. percolatio - percolation; percolation theory) - mat. theory, which is used to study processes occurring in inhomogeneous media with random properties, but fixed in space and unchanged in time. Arose in 1957 as a result of the work of J. Hammersley (J. Hammersley). In P. T., a distinction is made between lattice problems of P. T., continuum problems, and so-called. tasks on random nodes. Lattice problems, in turn, are divided into so-called. tasks of nodes and tasks of connections between them.

Communication tasks. Let connections be edges connecting neighboring nodes of an infinite periodic table. gratings (Fig., o). It is assumed that links between nodes can be of two types: intact or broken (blocked). The distribution of integer and blocked bonds in the lattice is random; the probability that this relationship is integer 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 integer links. A collection of nodes connected to each other. cluster. For small values x whole links are, as a rule, far from each other and clusters of a small number of nodes dominate, however, with an increase in x cluster sizes increase sharply. Percolation threshold ( x c) called such a value X, for which a cluster of an infinite number of nodes arises for the first time. P. t. allows you to calculate threshold values x s, as well as to 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-conductive elements. For example, if we assume that whole bonds conduct electricity. current, and the blocked ones do not conduct, it turns out that when X< х с beats the electrical conductivity of the lattice is equal to 0, and at x > x c it is different from 0.

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

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

Percolation thresholds for various gratings

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

Tasks on random nodes. Let the nodes do not form a regular lattice, but are randomly distributed in space. Two nodes are considered connected if the distance between them does not exceed a fixed value If is small compared to cf. 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 rum critical. meaning there is an infinite cluster. Computer simulation shows that in the three-dimensional case 0.86, where N- concentration of nodes. Problems on random nodes and their decomp. generalizations play an important role in the theory hopping conduction.

The effects described by P. t. refer to critical events, characterized by critical point, near which the system breaks up into blocks, and the size of the otd. blocks grows indefinitely when approaching critical. point. The emergence of an infinite cluster in problems of P. T. is in many respects analogous to phase transition second kind. For math. descriptions of these phenomena are introduced order parameter, which in the case of lattice problems is the fraction P(x) lattice nodes belonging to an infinite cluster. Near the flow threshold P(x) has the form

where - numerical coefficient, b - critical. the index of the order parameter. A similar f-la describes the behavior of beats. electrical conductivity s(x) near the percolation threshold:

where IN 2- numerical coefficient, s(1) - beats. electrical conductivity at c= 1, f - critical. 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.

Percolation thresholds essentially depend on the type of problems of P. t., but critical. indexes are the same for diff. problems and are determined only by the dimension of the space d(universality). Representations borrowed from the theory of phase transitions of the 2nd kind, make it possible to obtain relations relating various critical. indexes. Approximation self-consistent field applicable to the tasks of P. t. d> 6. In this approximation, the critical indices 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 inhomogeneous 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, Moscow, 1979; 3 a y-man D. M., Models of disorder, trans. from English, M., 1982; Efros A. L., Physics and geometry of disorder, Moscow, 1982; Sokolov I. M., Dimensions and other geometric critical exponents in the theory of percolation, "UFN", 1986, vol. 150 p. 221. A. L. Efros.