The coefficient of expansion, also known as the coefficient of thermal expansion, is a physical property of materials that describes how the volume, length, or some other dimension of a material changes due to a change in temperature. That is, it is a measure that indicates how much a material expands or contracts when it is heated or cooled.

When the temperature of a material is increased, its atoms or molecules tend to vibrate with greater energy, resulting in an increase in the space between them, which in turn causes the material to expand.

Similarly, when the material cools, the decrease in thermal energy causes the atoms or molecules to move closer together, causing a contraction.

The coefficient of expansion is generally denoted by the Greek letter "α" (alpha) and is measured in units of 1/°C (inverse of degrees Celsius) or 1/°K (inverse of kelvin). Depending on the type of thermal expansion (volumetric, linear or superficial), the coefficient of expansion can have different values.

## Types of Expansion Coefficients

There are three main types of coefficients of expansion, which are used to describe different types of expansion in materials. These coefficients are commonly denoted as:

### Coefficient of Linear Expansion (αl)

The coefficient of linear expansion refers to the change in the length of a material in a single dimension (for example, in a bar or tube) due to changes in temperature.

It is used when the material has anisotropic properties (that is, its properties differ in different directions).

### Coefficient of Volumetric Expansion (αv)

The coefficient of volumetric expansion describes how the volume of a material varies with changes in temperature.

It is calculated from the coefficients of linear expansion in three dimensions (in the case of isotropic materials). For isotropic materials (with equal properties in all directions), the coefficient of volumetric expansion is related to the coefficient of linear expansion (αl) by the equation:

αv = 3 * αl

### Coefficient of Surface Expansion (αs)

The coefficient of surface expansion applies to sheet or plate materials, where expansion occurs primarily in two dimensions (length and width) due to temperature changes.

This coefficient is related to the coefficient of linear expansion and is used in calculations where expansion in only two directions needs to be taken into account.

## How Is the Coefficient of Expansion of a Material Determined?

The coefficient of thermal expansion is determined experimentally by a process called dilatometric analysis. A general method for determining the coefficient of linear expansion of a material is described below:

A sample of the material to be analyzed is selected. The sample may be in the form of a bar, wire, sheet, or other geometry that facilitates measurements.

The initial length of the sample is measured using an accurate measuring instrument, such as a caliper or millimeter ruler. This will be the reference length (L0).

The sample is placed in a device called a dilatometer or interferometer, which makes it possible to accurately measure changes in length of the sample with respect to changes in temperature.

The sample is subjected to controlled heating in the temperature range of interest in a uniform and gradual manner.

As the temperature is increased, the length measurements of the sample are recorded at regular temperature intervals.

With the collected data, a thermal expansion graph is constructed, showing how the length of the sample changes as a function of temperature. The slope of this graph provides information about the coefficient of linear expansion of the material.

The coefficient of linear expansion (αl) is calculated from the graph of thermal expansion using the following formula:

αl = (ΔL / (L0 * ΔT))

Where:

αl = Coefficient of linear expansion

ΔL = Change in sample length

L0 = Initial length of the sample

ΔT = Temperature change experienced

## Examples of Coefficients of Expansion

Below is a table with the approximate coefficients of linear expansion for some prominent materials. Values are in units of microstrain per degree Celsius (µε/°C).

Material | Coefficient of Linear Expansion (µε/°C) | Description |

Steel (Carbon Steel) | 11 - 13 | Steel is an alloy of iron with carbon, widely used in construction, machinery, and other applications due to its high strength and malleability. Its moderate coefficient of expansion makes it suitable for various applications. |

Concrete (Concrete) | 8 - 12 | Concrete is a mixture of cement, aggregates, and water, used to build strong, durable structures. It has a lower coefficient of expansion compared to other materials, which makes it suitable for foundations and massive structures. |

Glass (Borosilicate glass) | 3. 4 | Borosilicate glass is a type of glass that is resistant to heat and chemicals. It is used in applications where transparency and thermal resistance are required, such as laboratory utensils and high-strength windows. |

Glass (Common glass) | 8 - 10 | Common glass is a material widely used in windows, containers, and architectural applications. Its coefficient of expansion is higher than borosilicate glass, making it more prone to expansion with changes in temperature. |

Aluminum | 22 - 24 | Aluminum is a strong, lightweight metal with a wide range of applications in industry, from transportation to electronics. Its high coefficient of expansion makes it ideal for applications that require low thermal inertia. |

Copper | 16 - 18 | Copper is an excellent conductor of electricity and heat, used in electrical and electronic applications. Its moderate coefficient of expansion makes it suitable for applications where good thermal conductivity is required. |

Brass | 19 - 20 | Brass is an alloy of copper and zinc, prized for its shine and malleability. It is used in applications where a combination of resistance, conductivity and aesthetics is required, such as accessories and decorative elements. |

Silver | 19 - 20 | Silver is a precious metal known for its high electrical and thermal conductivity. Its moderate coefficient of expansion makes it suitable for applications where good thermal conductivity with less thermal expansion is required. |

Iron | 11 - 12 | Iron is an abundant metal used in the construction and manufacture of various structures and industrial products. Its moderate coefficient of expansion makes it suitable for structural and engineering applications. |

Zinc | 30 - 32 | Zinc is a metal used in galvanizing to protect steel from corrosion. Its high coefficient of expansion makes it suitable for applications where greater thermal expansion is required. |

Ceramic (Alumina) | 7 - 8 | Alumina is a high strength ceramic used in high temperature and wear applications such as engine components and cutting tools. Its low coefficient of expansion makes it suitable for applications with high dimensional stability. |

Uranium | 13 - 14 | Uranium is a radioactive metal used in nuclear applications and as fuel in nuclear reactors. Its moderate coefficient of expansion makes it relevant in the design of reactor elements. |