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# Computation of Coefficient of Thermal Expansion

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### What is Thermal Expansion

Thermal expansion is the phenomenon can be observed in different forms of objects including solids, liquids, and gases. In this process, an object or body expands on the application of heat (temperature). Thermal expansion defines the tendency of an object to change its dimension either in length, density, area, or volume due to heat. When the substance is heated, it is often accompanied by an increase in kinetic energy. Thermal expansion can be divided of three types: Linear expansion, area expansion and volume expansion.

### What is Coefficient of Thermal Expansion (CTE)

The change ability of the material is expressed by the change in length, area or volume caused by unit temperature change under equal pressure, that is, the coefficient of thermal expansion. The essence of thermal expansion is that the average distance between the crystal lattice structure changes with temperature. The thermal expansion of materials is usually expressed in terms of linear expansion coefficient or volume expansion coefficient. CTE is one of the main physical properties of the material, and it is an important index to measure the thermal stability of the material.

The CTE generally varies with temperature and there are mainly three types of coefficient of thermal expansion: Llinear expansion coefficient α, surface expansion coefficient β, and volume expansion coefficient γ.
Linear expansion coefficient α=ΔL/(L*ΔT)
Surface expansion coefficient β=ΔS/(S*ΔT)
Volume expansion coefficient γ=ΔV/(V*ΔT)

Where ΔL is the change in the length of the object under the given temperature change ΔT, L is the initial length; ΔS is the change in the area of the object under the given temperature change ΔT, S is the initial area; ΔV is the change in the volume of the object under the given temperature change ΔT, V is the initial volume.

### Factors that Affect the Coefficient of Thermal Expansion

• Chemical mineral composition

The CTE is related to the chemical composition, crystalline state, crystal structure, and bond strength of the material.

• Phase change

When a material undergoes a phase change, its CTE also changes.

• Alloying elements

The CTE of a single-phase uniform solid solution alloy composed of simple metals and non-ferromagnetic metals is between the expansion coefficients of the internal components.

• Texture

There is a texture in single crystal or polycrystal, which leads to differences in the arrangement density of atoms in each crystal direction, resulting in anisotropy of thermal expansion.

• Internal cracks and defects will also affect CTE.

Figure 1. Thermal cycling of Ni-YSZ powder examining the variation of the lattice parameter within Ni for five thermal cycles from low temperature (100 °C) to operating temperature (600 °C) in a forming gas environment. (Heenan, T.; et al. 2018)

### Workflow of Computation of CTE

At Alfa Chemistry, we have developed a mature workflow for calculating the CTE. Our reliable thermal expansion coefficient calculation process is as follows:

• Import the polymer structure
• Anneal the polymer

We determine the volume change caused by increasing of the temperature for the calculation of the thermal expansion coefficient. The data points will be sampled within a defined temperature range. Before the actual sampling, a short simulated annealing sequence is found using molecular dynamics (MD) simulation.

• Extract strain vs. temperature profiles

We extract the strain and temperatures from the trajectory for post-processing.

• Calculation of the thermal expansion coefficient

Our experts use various graph plotting software to plot the strain. Then, the results file is imported into the software of choice and we plot two column volume against column temperature. Next, we perform a regression analysis and determine the slope. Finally, the CTE can be calculated from the volume of the first data point and the slope of the linear fit curve.

### Our Capabilities

• Ab-initio density functional theory (DFT) calculations

We use this method to study alloying effects on different metals and investigate the alloy solid solution strengthening represented by bond strength enhancement in thermal expansion coefficients,

• Electron localized function (ELF)

We apply ELF to evaluate the chemical bonds between different metal atoms.

Our computation of coefficient of thermal expansion services remarkably reduce the cost, promote further experiments, and enhance the understanding of chemical process for customers worldwide. Our personalized and all-around services will satisfy your innovative study demands. If you are interested in our services, please don't hesitate to contact us. We are glad to cooperate with you and witness your success!

Reference

• Heenan, T.; et al. Understanding the thermo-mechanical behaviour of solid oxide fuel cell anodes using synchrotron X-ray diffraction. Solid State Ionics. 2018, 314: 156-164.

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