4.3 Thermal expansion

Thermal expansion is a crucial property of materials that affects their behavior under temperature changes. This phenomenon impacts everything from everyday objects to advanced engineering applications, making it essential for designers and engineers to understand.

At its core, thermal expansion describes how materials change size with temperature. By grasping the microscopic origins and factors influencing this property, we can better predict material behavior and create innovative solutions for temperature-sensitive applications.

Thermal expansion overview

• Thermal expansion is a fundamental property of materials that describes how their dimensions change with temperature
• Understanding thermal expansion is crucial for designing materials and devices that can withstand temperature variations without failure or loss of functionality
• Thermal expansion is closely related to other thermal properties such as specific heat capacity and thermal conductivity

Thermal expansion definition

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• Thermal expansion refers to the tendency of materials to change their dimensions (length, area, or volume) in response to a change in temperature
• The change in dimensions is typically an increase with increasing temperature, although some materials may exhibit negative thermal expansion (contraction with increasing temperature)
• Thermal expansion occurs due to the increase in the average distance between atoms as they vibrate more vigorously at higher temperatures

Thermal expansion coefficient

• The thermal expansion coefficient quantifies the extent to which a material expands or contracts with temperature change
• Linear thermal expansion coefficient ($\alpha$) describes the fractional change in length per unit temperature change: $\alpha = (1/L_0) (dL/dT)$, where $L_0$ is the initial length and $dL/dT$ is the rate of length change with temperature
• Volumetric thermal expansion coefficient ($\beta$) describes the fractional change in volume per unit temperature change: $\beta = (1/V_0) (dV/dT)$, where $V_0$ is the initial volume and $dV/dT$ is the rate of volume change with temperature

Linear vs volumetric expansion

• Linear thermal expansion refers to the change in one dimension (length) of a material with temperature, while volumetric thermal expansion refers to the change in the material's volume
• For isotropic materials, the volumetric thermal expansion coefficient is approximately three times the linear thermal expansion coefficient: $\beta \approx 3\alpha$
• Anisotropic materials may have different linear thermal expansion coefficients along different crystallographic directions, leading to more complex volumetric expansion behavior

Microscopic origin of thermal expansion

• Thermal expansion arises from the interplay between interatomic forces and thermal vibrations in a material
• The microscopic mechanisms responsible for thermal expansion provide insights into how to control and engineer this property for specific applications
• Understanding the microscopic origin of thermal expansion requires considering the potential energy landscape of the material and the anharmonicity of lattice vibrations

Asymmetric interatomic potential

• The interatomic potential energy curve is typically asymmetric, with a steeper slope at shorter interatomic distances compared to longer distances
• This asymmetry means that atoms experience a greater repulsive force when they approach each other than the attractive force when they move apart
• As temperature increases and atoms vibrate more, the average interatomic distance increases due to the asymmetric potential, leading to thermal expansion

Anharmonicity of lattice vibrations

• In a perfectly harmonic potential, the restoring force is proportional to the displacement, and the average position of atoms remains unchanged with temperature
• However, real materials exhibit anharmonic potentials, where the restoring force is not strictly proportional to the displacement
• Anharmonicity leads to the thermal expansion of materials, as the average interatomic distance increases with increasing temperature due to the asymmetric potential energy landscape

Quasi-harmonic approximation

• The quasi-harmonic approximation is a method to describe the thermal expansion of materials by considering the volume dependence of phonon frequencies
• In this approximation, the phonon frequencies are assumed to depend on the volume of the material, while the anharmonic effects are treated as a volume-dependent correction to the harmonic potential
• The quasi-harmonic approximation allows for the calculation of thermal expansion coefficients and other thermodynamic properties, providing a link between the microscopic and macroscopic descriptions of thermal expansion

Factors affecting thermal expansion

• Several factors influence the thermal expansion behavior of materials, including bonding strength, lattice structure, and composition
• Understanding these factors allows for the design and selection of materials with desired thermal expansion properties for specific applications
• By tuning these factors, it is possible to control the thermal expansion of materials and create materials with tailored thermal expansion coefficients

Bonding strength and thermal expansion

• The strength of the chemical bonds between atoms in a material plays a significant role in determining its thermal expansion behavior
• Materials with stronger interatomic bonds (covalent or ionic) generally have lower thermal expansion coefficients compared to those with weaker bonds (van der Waals or metallic)
• Stronger bonds require more energy to stretch or compress, leading to a smaller change in interatomic distances with temperature and, consequently, lower thermal expansion

Lattice structure and thermal expansion

• The arrangement of atoms in a material's crystal lattice influences its thermal expansion properties
• Different crystal structures (cubic, hexagonal, tetragonal, etc.) can exhibit varying degrees of thermal expansion anisotropy
• Open lattice structures (diamond cubic) typically have lower thermal expansion coefficients than close-packed structures (face-centered cubic) due to the greater space available for atomic vibrations

Composition and thermal expansion

• The chemical composition of a material, including the types of elements present and their relative proportions, affects its thermal expansion behavior
• Alloying elements can modify the bonding strength and lattice structure of a material, leading to changes in thermal expansion
• Composite materials, which combine two or more distinct materials, can be designed to have specific thermal expansion properties by controlling the composition and microstructure of the constituents

Thermal expansion in crystals

• Crystalline materials exhibit thermal expansion behavior that depends on their crystal structure and symmetry
• Understanding thermal expansion in crystals is crucial for applications involving single crystals or textured polycrystalline materials
• The thermal expansion of crystals can be described using tensorial quantities, which capture the anisotropic nature of the expansion

Isotropic vs anisotropic expansion

• Isotropic materials exhibit the same thermal expansion behavior in all directions, with a single linear thermal expansion coefficient
• Anisotropic materials have different thermal expansion coefficients along different crystallographic directions, leading to direction-dependent expansion
• The degree of thermal expansion anisotropy depends on the crystal structure and symmetry of the material (cubic crystals are isotropic, while hexagonal crystals are anisotropic)

Thermal expansion tensor

• The thermal expansion tensor is a second-rank tensor that describes the thermal expansion of a crystal in three dimensions
• The tensor components ($\alpha_{ij}$) relate the strain in the material to the temperature change: $\varepsilon_{ij} = \alpha_{ij} \Delta T$
• The thermal expansion tensor is symmetric ($\alpha_{ij} = \alpha_{ji}$) and depends on the crystal symmetry (higher symmetry leads to fewer independent tensor components)

Negative thermal expansion materials

• Some materials exhibit negative thermal expansion (NTE), where they contract along one or more dimensions with increasing temperature
• NTE can occur due to various mechanisms, such as transverse vibrational modes, phase transitions, or structural flexibility
• Examples of NTE materials include zirconium tungstate (ZrW2O8), beta-eucryptite (LiAlSiO4), and certain metal-organic frameworks (MOFs)

Thermal expansion measurements

• Measuring the thermal expansion of materials is essential for characterizing their behavior and selecting appropriate materials for specific applications
• Several techniques are available for measuring thermal expansion, each with its own advantages and limitations
• The choice of measurement technique depends on factors such as the material's properties, the temperature range of interest, and the required accuracy and resolution

Dilatometry techniques

• Dilatometry is a common method for measuring the linear thermal expansion of materials
• In a dilatometer, the sample is placed in a controlled temperature environment, and the change in its length is measured using a displacement sensor (capacitive, inductive, or optical)
• Dilatometry can be used to measure the thermal expansion of bulk materials, thin films, and powders over a wide temperature range

X-ray diffraction methods

• X-ray diffraction (XRD) techniques can be used to measure the thermal expansion of crystalline materials by tracking changes in the lattice parameters with temperature
• High-temperature XRD involves collecting diffraction patterns at various temperatures and analyzing the shifts in peak positions to determine the lattice parameters and thermal expansion coefficients
• XRD methods are particularly useful for studying the anisotropic thermal expansion of single crystals and textured polycrystalline materials

Optical interferometry approaches

• Optical interferometry techniques, such as Fabry-Perot interferometry, can be used to measure the thermal expansion of materials with high precision
• In these methods, the sample is placed between two reflective surfaces, forming an optical cavity
• Changes in the sample's thickness due to thermal expansion alter the interference pattern of light passing through the cavity, allowing for the determination of the thermal expansion coefficient

Thermal expansion applications

• Understanding and controlling thermal expansion is crucial for a wide range of applications in engineering, manufacturing, and materials design
• Thermal expansion can be harnessed for specific functions, such as actuation and sensing, or it can be minimized to ensure the dimensional stability of components and devices
• Tailoring the thermal expansion of materials is essential for creating products that can withstand temperature variations and perform reliably in various environments

Thermomechanical actuators and sensors

• Thermal expansion can be used to create thermomechanical actuators and sensors that convert temperature changes into mechanical motion or electrical signals
• Bimetallic strips, which consist of two materials with different thermal expansion coefficients bonded together, can bend or deflect in response to temperature changes, acting as temperature-sensitive switches or actuators
• Thermal expansion-based sensors, such as strain gauges and fiber Bragg gratings, can detect temperature changes by measuring the strain induced in the material due to thermal expansion

Thermal expansion in manufacturing

• Accounting for thermal expansion is essential in manufacturing processes, particularly when dealing with high-precision components or assemblies
• Thermal expansion mismatches between different materials can lead to stresses, deformations, or failures during manufacturing or operation
• Designers must consider the thermal expansion properties of materials when specifying tolerances, selecting materials for different components, and designing interfaces between dissimilar materials

Thermal expansion in materials design

• Tailoring the thermal expansion of materials is a key aspect of materials design for various applications
• In electronic packaging, materials with low or matched thermal expansion coefficients (glass-ceramics, Invar) are used to minimize thermal stresses and ensure the reliability of interconnects
• Composite materials can be designed to have specific thermal expansion properties by controlling the composition, orientation, and distribution of the reinforcing phase (carbon fibers in a polymer matrix)

Thermal expansion related phenomena

• Thermal expansion is closely related to several other phenomena in materials science and engineering
• Understanding the interplay between thermal expansion and these related phenomena is essential for predicting material behavior and designing materials with desired properties
• Some of the key phenomena related to thermal expansion include thermal shock, phase transitions, and thermoelectricity

Thermal shock and thermal stress

• Thermal shock occurs when a material experiences a sudden change in temperature, leading to the development of thermal stresses
• The thermal stresses arise due to the non-uniform expansion or contraction of the material, which can cause cracking, fracture, or delamination if the stresses exceed the material's strength
• Materials with low thermal expansion coefficients and high thermal conductivity are more resistant to thermal shock, as they can accommodate temperature changes more easily and distribute the thermal stresses more uniformly

Thermal expansion and phase transitions

• Phase transitions in materials can be accompanied by significant changes in thermal expansion behavior
• Solid-state phase transitions, such as the transformation from one crystal structure to another (allotropic transformations), can lead to abrupt changes in the thermal expansion coefficient
• The volume changes associated with phase transitions can also induce thermal stresses in materials, which may affect their mechanical properties and stability

Thermal expansion and thermoelectricity

• Thermoelectric materials, which can convert temperature gradients into electrical energy (Seebeck effect) or use electrical energy to create temperature gradients (Peltier effect), are influenced by thermal expansion
• The efficiency of thermoelectric devices depends on the material's thermoelectric figure of merit ($ZT$), which is a function of its electrical conductivity, thermal conductivity, and Seebeck coefficient
• Thermal expansion can affect the performance of thermoelectric materials by altering their electronic and thermal transport properties, as well as by inducing thermal stresses that may degrade the material over time