Electric conductivity

Piece of a material with known section and length, equipped with electrical contacts at its ends.

The electrical conductivity (symbol σ) is the measure of the ability of a material or substance to allow electric current to pass through it. Conductivity depends of the atomic and molecular structure of the material. Metals are good conductors because they have a structure with many electrons with weak bonds, and this allows their movement. The conductivity also depends on other physical factors of the material itself, and on the temperature.

Conductivity is reverse of resistivity (symbol ρ); therefore, σ σ =1/ρ ρ {displaystyle sigma =1/rho }, and his unit is the S/m (seven per subway) or Ω⁻¹·m⁻¹. Usually, the magnitude of the conductivity is the proportionality between the electric field E{displaystyle mathbf {E} } and driving current density J{displaystyle mathbf {J} }:

J=σ σ E{displaystyle mathbf {J} =sigma mathbf {E} }

History

The English physicist Stephen Gray (1666-1736) mainly studied the electrical conductivity of bodies and, after many experiments, was the first in 1729 to transmit electricity through a conductor. In his experiments he discovered that for electricity, or "effluvia" to flow or "electrical virtue", as he called it, could circulate through the conductor, it had to be isolated from earth. Later he studied other forms of transmission and, together with the scientists G. Wheler and J. Godfrey, classified materials into conductors and insulators of electricity.

Electrical conductivity in different media

The mechanisms of conductivity differ between the three states of matter. For example, in solids, the atoms as such are not free to move and the conductivity is due to the electrons. In metals there are quasi-free electrons that can move freely throughout the volume, while in insulators, many of them are ionic solids.

Conductivity in liquid media

The electrolytic conductivity in liquid media is related to the presence of salts in solutions, whose dissociation generates positive and negative ions capable of transporting electrical energy if the liquid is subjected to an electric field. These ionic conductors are called electrolytes or electrolytic conductors.

Conductivity determinations are called conductometric determinations and have many applications, such as:

• in electrolysis, since the consumption of electricity in this process depends to a large extent on it;
• in laboratory studies to determine the salt content of various solutions during water evaporation (e.g. boiler water or condensed milk production);
• in the study of the basicities of acids, since they can be determined by conductivity measurements;
• to determine the solubility of low-soluble electrolytes and to find concentrations of electrolytes in degree solutions.

The basis of solubility determinations is that saturated solutions of sparingly soluble electrolytes can be considered infinitely dilute. By measuring the specific conductivity of such a solution and calculating the equivalent conductivity according to it, the concentration of the electrolyte is found, that is, its solubility.

An extremely important practical method is that of conductometric titration, that is, the determination of the concentration of an electrolyte in solution by measuring its conductivity during the titration. This method is especially valuable for turbid or strongly colored solutions that often cannot be titrated using indicators.

Electrical conductivity is used to determine the salinity (salt content) of soils and growing media, for which they are dissolved in water and the conductivity of the resulting liquid medium is measured. It is usually referenced to 25 °C and the value obtained must be corrected depending on the temperature. There are many conductivity expression units for this purpose, although the most commonly used are dS/m (deciSiemens per meter), mmhos/cm (millimhos per centimeter) and, according to European standards bodies, mS/m (milliSiemens per meter). The salt content of a soil or substrate can also be expressed by resistivity (it used to be expressed like this in France before the application of the INEN standards).

Conductivity in solid media

According to the theory of energy bands in crystalline solids, conductive materials are those in which the valence and conduction bands overlap, forming a "cloud" of free electrons that causes the current when subjecting the material to an electric field. These conductive media are called electrical conductors.

The International Electrotechnical Commission defined as a standard of electrical conductivity:

«A 1 metre long copper thread and a gram of mass, which gives a resistance of 0.15388 Ω to 20 °C» to which it assigned an electrical conductivity of 100% IACS (International Annealed Copper Standard, International Copper Standard. All copper alloy with a conductivity greater than 100% IACS is called high conductivity (H.C.) for its English acronyms.

Some electrical conductivities

Conductivity of metals

Electrical conductivity of pure metals at temperatures between 273 and 300K (10⁶ S⋅m^-1):

Conductivity in metals explained

Before the advent of quantum mechanics, the classical theory used to explain the conductivity of metals was the Drude-Lorentz model, where electrons move at an average speed that is approximately constant, which is the limit speed associated with the accelerator effect. of the electric field and the decelerating effect of the crystal lattice with which the electrons collide, producing the Joule effect.

However, the advent of quantum mechanics made it possible to build more refined theoretical models based on energy band theory that explain in detail the behavior of conducting materials.

Drude's Pattern

Representation of the Drude model: the electrons, in blue, are moved by the gradient of the electric field, and clash with the ions of the crystalline network, in red.

Phenomenologically, the interaction of the free electrons of metals with the crystal lattice is assimilated to a "viscous" force, like the one that exists in a fluid that has friction with the walls of the conduit through which it flows. The equation of movement of the electrons of a metal can therefore be approximated by an expression of the type:

medvdt=− − eE− − kv{displaystyle m_{e}{frac {dmathbf {v}}{dt}}}=-emathbf {E} -kmathbf {v} }}

Thus, the drag speed of the current is the one in which the accelerating effect of the electric field is equalized with the resistance due to the network, this speed is what satisfies:

− − eE− − kve=0,ve=− − eEk{displaystyle -emathbf {E} -kmathbf {v} _{e}=0,qquad mathbf {v} _{e}={frac {-emathbf {E} }{k}}}}}}

For a conductor that satisfies Ohm's law and with a number n of electrons per unit volume that move at the same speed, it can be written:

j=σ σ E,j=− − enve{displaystyle mathbf {j} =sigma mathbf {E}quad mathbf {j} =-enmathbf {v} _{e}}}}

Introducing relaxation time Δ Δ =me/k{displaystyle scriptstyle tau =m_{e}/k} and by comparing the latest expressions it becomes that conductivity can be expressed as:

σ σ =ne2Δ Δ me{displaystyle sigma ={frac {ne^{2}{m_{e}}}}}

From known values of σ σ {displaystyle scriptstyle sigma } you can estimate the time of relaxation Δ Δ {displaystyle scriptstyle tau } and compare it with the average time between electron impacts with the network. Assuming that each atom contributes to an electron and that n is the order of 10²⁸ electrons per m3 in most metals. Using also the values of the electron mass me{displaystyle scriptstyle m_{e}} and the load of the electron e{displaystyle scriptstyle e} relaxing time 10⁻¹⁴ s.

To judge whether that phenomenological model properly explains the law of Ohm and conductivity in metals should be interpreted the time of relaxation with the properties of the network. While the model cannot be theoretically correct because the movement of electrons in a metallic glass is governed by quantum mechanics, at least the magnitude orders predicted by the model are reasonable. For example, it is reasonable to relate the time of relaxation Δ Δ {displaystyle scriptstyle tau } with the average time between collisions of an electron with the crystalline network. Bearing in mind that the typical separation between atoms of the network is l = 5·10⁻⁹ and using the ideal gas theory applied to the free electrons the speed of the same would be ve=(3kT/me)1/2{displaystyle scriptstyle v_{e}=(3kT/m_{e})^{1/2}} = 10⁵ m/s, so Δ Δ ≈ ≈ l/v{displaystyle scriptstyle tau approx l/v} = 5·10⁻¹⁴ s, which is in good agreement with the values inferred for the same magnitude from the conductivity of the metals.

Quantum model

According to the Drude-Lorentz model the speed of the electrons should vary with the square root of the temperature, but when the time between collisions estimated by the Drude-Lorentz model is compared with the conductivity at low speeds, it is not they obtain consistent values, since those model predictions are only compatible with interionic distances much larger than the actual distances.

In the quantum model, the electrons are accelerated by the electric field, and also interact with the crystal lattice, transferring part of their energy to it and causing the Joule effect. However, to be scattered in a collision with the lattice, by the Pauli exclusion principle the electrons must end up after the collision with the linear momentum of a quantum state that was previously empty; that makes the most likely scattered electrons the most energetic. After being dispersed, they pass into quantum states with a negative momentum of lower energy; This continuous dispersion towards states of opposite momentum is what counteracts the accelerating effect of the field. In essence, this model shares with the classical Drude-Lorentz model the idea that it is the interaction with the crystal lattice that makes the electrons move at a stationary speed and do not accelerate beyond a certain limit. Although quantitatively the two models differ especially at low temperatures.

Within the quantum model, conductivity is given by an expression superficially similar to the classical Drude-Lorentz model:

σ σ =ne2Δ Δ m↓ ↓ {displaystyle sigma ={frac {ne^{2}{m^{*}}}}}

Where:

Δ Δ {displaystyle tau ,} is also called relaxation time and is inversely proportional to the probability of dispersal by the crystalline network.

m↓ ↓ {displaystyle m^{},} It is not now directly the mass of the electron but an effective mass that is related to the energy of Fermi of metal.

If by quantum reasoning we try to calculate the probability of dispersion we have:

Δ Δ =1Pdis=1ndis・ ・ vF{displaystyle tau ={frac {1}{P_{dis}}}}}{{frac {1}{n_{dis}{sigma v_{F}}}}}}}}}}}}

Where:

Pdis{displaystyle P_{dis},} is the probability of dispersal.

ndis{displaystyle n_{dis},} the number of scattered ions per volume unit.

・ ・ {displaystyle sigma ,} is the effective section of each disperser.

vF{displaystyle v_{F},} It's the speed of an electron with Fermi's energy.

According to quantum calculations, the effective section of the scatterers is proportional to the square of the amplitude of its thermal vibration, and as said square is proportional to the thermal energy, and this is proportional to the temperature T you have to at low temperatures:

Δ Δ 1T{displaystyle tau propto {frac {1}{T}}}}

This behavior correctly predicted by the model could not be explained by the classical Drude-Lorentz model, so this model is considered to be outperformed by the corresponding quantum model, especially for low temperatures.