Chemical Expansion and Thermochemical Strain in Functional Oxides
Functional oxides often change their dimensions when temperature, oxygen partial pressure, or defect concentration changes. These dimensional changes are not always caused by ordinary thermal expansion. In non-stoichiometric oxides, oxygen incorporation or release changes the population and electronic state of lattice defects, which can alter average cation–oxygen bond lengths, lattice parameters, and sometimes crystal symmetry.
This phenomenon is commonly described as chemical expansion. When chemical and thermal effects occur together, the resulting dimensional response is often called thermochemical strain. Understanding it is essential for interpreting dilatometry, designing solid oxide cells, evaluating oxygen-permeation membranes, and preventing cracking or delamination during redox and thermal cycling.
1. Thermal expansion versus chemical expansion
Thermal expansion is the dimensional response of a material to a temperature change. For an isotropic material over a limited temperature interval:
\[ \varepsilon_{th}=\frac{\Delta L}{L_0} \approx \alpha_{th}\Delta T \]
where \(\alpha_{th}\) is the linear thermal-expansion coefficient.
Chemical expansion is a dimensional response to a change in composition or defect concentration. In an oxygen-deficient oxide written as \(ABO_{3-\delta}\), the oxygen deficiency is represented by \(\delta\). A local linear approximation can be written as:
\[ \varepsilon_{chem} \approx \alpha_{chem}^{\delta}\Delta\delta \]
where \(\alpha_{chem}^{\delta}\) is a chemical-expansion coefficient defined with respect to oxygen deficiency.
The superscript is useful because the literature also uses chemical-expansion coefficients defined against oxygen content, oxygen partial pressure, or another composition variable. The definition and sign convention must therefore be stated explicitly.
For small changes, the total free strain may be approximated as:
\[ \varepsilon_{free} \approx \varepsilon_{th}+\varepsilon_{chem} \]
This additive expression is a first-order approximation. It neglects higher-order coupling, anisotropy, phase transformations, and composition-dependent elastic properties.
2. Why oxygen non-stoichiometry changes lattice dimensions
Oxygen removal from an oxide can be represented by the reduction reaction:
\[ O_O^x \rightleftharpoons V_O^{\bullet\bullet} +\frac{1}{2}O_2(g)+2e' \]
The reaction produces oxygen vacancies and electrons. Depending on the electronic structure, electrons may occupy extended states or localize on transition-metal cations, changing their formal oxidation states and effective ionic sizes.
For example, reduction of a transition-metal cation can increase its effective ionic radius in many oxide coordination environments. This can lengthen metal–oxygen bonds and contribute to lattice expansion. However, the direction and magnitude of the dimensional response are not universal: vacancy relaxation, changes in covalency, octahedral tilting, defect association, and structural transitions can modify or even reverse the simple radius-based expectation.
Chemical expansion is therefore a coupled response involving:
- Oxygen vacancy or interstitial concentration.
- Changes in transition-metal oxidation state.
- Local coordination and bond-length relaxation.
- Defect association and ordering.
- Octahedral tilting or other structural distortions.
- Elastic interactions between defects and the surrounding lattice.
A simple ionic-radius argument can provide intuition, but it is not sufficient to predict a quantitative lattice parameter.
3. Defining chemical-expansion coefficients
For an isotropic material, the volumetric chemical-expansion coefficient with respect to oxygen deficiency can be defined locally as:
\[ \beta_{chem}^{\delta} = \frac{1}{V} \left(\frac{\partial V}{\partial\delta}\right)_{T,p} \]
The corresponding linear coefficient is:
\[ \alpha_{chem}^{\delta} = \frac{1}{L} \left(\frac{\partial L}{\partial\delta}\right)_{T,p} \]
For small isotropic strains:
\[ \beta_{chem}^{\delta} \approx 3\alpha_{chem}^{\delta} \]
This factor of three is not generally valid for anisotropic crystals or when the measured strain is not isotropic.
For a cubic material with lattice parameter \(a\) and unit-cell volume \(V=a^3\), the volumetric relation becomes:
\[ \frac{\Delta V}{V} \approx 3\frac{\Delta a}{a} \]
for small changes. Thus, a lattice-parameter measurement can provide an estimate of the linear chemical strain in a cubic phase, provided that the phase remains unchanged and the lattice response is representative of the bulk specimen.
3.1 Finite composition intervals
If the response is nonlinear, a single coefficient may be misleading. Instead, report the measured relationship between lattice parameter or strain and oxygen deficiency:
\[ a=a(\delta,T) \]
A local coefficient can then be evaluated over a stated composition interval. For a finite interval, an average linear coefficient is:
\[ \overline{\alpha}_{chem}^{\delta} = \frac{\Delta L/L_0}{\Delta\delta} \]
The reference state, temperature, oxygen partial pressure, and composition interval must accompany the reported value.
4. Chemical strain is not always isotropic
In cubic materials with a statistically isotropic defect distribution, chemical expansion is often approximated as isotropic. In lower-symmetry crystals, however, changes in oxygen content can produce different responses along different crystallographic directions.
The strain should then be treated as a tensor:
\[ \boldsymbol{\varepsilon}_{chem} = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz}\\ \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz}\\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \end{bmatrix} \]
For an orthorhombic or tetragonal crystal, the principal lattice directions may respond differently to oxygen-content changes. A single scalar coefficient can conceal this anisotropy.
In epitaxial films, textured ceramics, and single crystals, the measured strain also depends on crystallographic orientation and mechanical constraint. It is therefore important to distinguish the intrinsic stress-free chemical strain from the strain measured in a constrained specimen.
5. Thermochemical strain during operation
A functional oxide operating at elevated temperature may experience simultaneous changes in temperature and oxygen chemical potential. A useful first-order decomposition is:
\[ \varepsilon_{free}(T,\delta) = \int_{T_0}^{T}\alpha_{th}(T',\delta)\,dT' + \int_{\delta_0}^{\delta} \alpha_{chem}^{\delta}(T,\delta')\,d\delta' \]
This expression assumes that thermal and chemical contributions can be separated along the selected path. In real materials, both coefficients can depend on temperature and composition, and the final strain can depend on the thermodynamic path.
The oxygen deficiency itself is not an independent constant during operation. It is set by temperature, oxygen chemical potential, composition, and defect equilibria:
\[ \delta=\delta(T,p_{O_2},\text{composition}) \]
Consequently, changing temperature at a fixed gas composition can change both ordinary thermal expansion and oxygen non-stoichiometry. A measured dilatometry curve in a reactive atmosphere may therefore contain both contributions.
6. From free strain to mechanical stress
Chemical expansion does not necessarily generate stress in a freely expanding, homogeneous specimen. Stress develops when the chemical strain is constrained or when different regions attempt to expand by different amounts.
Examples include:
- A dense oxide layer bonded to an electrolyte.
- A porous electrode attached to a dense support.
- A composition gradient across a membrane.
- A redox front moving through a ceramic.
- Different phases with dissimilar chemical-expansion coefficients.
- Thermal and chemical gradients during start-up or shut-down.
For a simple isotropic, linearly elastic material, the stress–strain relationship can be expressed as:
\[ \boldsymbol{\sigma} = \mathsf{C}: \left( \boldsymbol{\varepsilon}_{total} - \boldsymbol{\varepsilon}_{free} \right) \]
where \(\mathsf{C}\) is the elastic stiffness tensor and the colon denotes tensor contraction. The free strain includes the relevant thermal and chemical eigenstrains.
For an ideal thin film that is fully constrained in-plane by a much thicker substrate, an approximate equibiaxial stress under plane-stress conditions is:
\[ \sigma_{\parallel} \approx -\frac{E}{1-\nu}\,\varepsilon_{free,\parallel} \]
Here, \(E\) is Young’s modulus, \(\nu\) is Poisson’s ratio, and \(\varepsilon_{free,\parallel}\) is the in-plane strain the film would undergo if unconstrained. This relation assumes an isotropic elastic film, uniform strain, perfect bonding, and no stress relaxation. The sign convention is tensile-positive; a positive free expansion constrained by the substrate produces compressive in-plane stress.
This equation should not be applied directly to porous electrodes, thick multilayers, cracked specimens, or materials undergoing creep without a more appropriate mechanical model.
7. Chemical-expansion mismatch in multilayer devices
Solid oxide cells and oxygen-separation membranes contain layers that differ in composition, microstructure, and transport function. If adjacent layers have different chemical-expansion responses, a change in oxygen partial pressure can create interfacial stress.
Potential consequences include:
- Cracking within an electrode or electrolyte.
- Delamination at an electrode/electrolyte interface.
- Loss of contact between particles in a porous electrode.
- Warping of asymmetric membranes.
- Changes in porosity and gas-transport pathways.
- Accelerated degradation during repeated redox cycling.
A mechanically compatible design should consider more than the thermal-expansion coefficients measured in air. The oxygen chemical potential under operating and transient conditions may produce additional strain.
The relevant comparison is the difference in free strain between the adjoining materials under the same temperature and gas conditions:
\[ \Delta\varepsilon_{mismatch} = \varepsilon_{free,1} - \varepsilon_{free,2} \]
This mismatch is a useful design indicator, but the actual stress distribution depends on layer thicknesses, elastic moduli, geometry, porosity, interfaces, and time-dependent relaxation.
8. How to measure chemical expansion
8.1 Dilatometry under controlled atmosphere
Dilatometry measures dimensional change as a function of temperature or time. To separate thermal expansion from chemical expansion, measurements should be performed under controlled and documented oxygen partial pressures.
A useful strategy is:
- Establish a stable reference atmosphere and temperature.
- Record the dimensional response after equilibration.
- Change oxygen partial pressure at constant temperature.
- Record the transient dimensional response.
- Repeat at additional temperatures and oxygen partial pressures.
- Check reversibility by returning to the initial atmosphere.
- Compare the result with an independent measure of oxygen non-stoichiometry.
A gas switch can itself produce a transient at the instrument or specimen. The response time of the gas-delivery system should be characterized.
8.2 Thermogravimetric analysis (TGA)
TGA measures mass changes during oxygen uptake or release. For a material with a known composition and a well-defined reaction, mass change can be converted into a change in oxygen non-stoichiometry.
For a sample of initial mass \(m_0\), a measured mass change \(\Delta m\) can be related to oxygen exchange if the exchanged species and reaction stoichiometry are known. The calculation must account for the formula mass, sample composition, and the number of oxygen atoms per formula unit.
TGA does not measure strain directly. Its value is greatest when paired with dilatometry, allowing dimensional change to be correlated with oxygen-content change.
8.3 In-situ or operando X-ray diffraction
XRD can track lattice parameters and phase evolution as temperature or oxygen partial pressure changes. For a cubic phase:
\[ V_{cell}=a^3 \]
For a tetragonal phase:
\[ V_{cell}=a^2c \]
For an orthorhombic phase:
\[ V_{cell}=abc \]
The lattice parameters should be refined consistently across the measurement series. Instrument calibration, sample displacement, thermal gradients, peak overlap, and phase coexistence can all affect the apparent lattice response.
A change in lattice parameter is not automatically proof of chemical expansion. Thermal expansion, phase transformation, residual stress, and compositional changes must be considered.
8.4 Combining methods
A strong experimental study combines complementary measurements:
| Method | Main information | Important limitation |
|---|---|---|
| Dilatometry | Macroscopic dimensional change | Does not independently identify the cause |
| TGA | Oxygen uptake or release | Does not directly measure strain |
| In-situ XRD | Lattice parameters and phase evolution | May not represent macroscopic strain in a textured or constrained specimen |
| ECR | Chemical oxygen-transport response | Requires conductivity to be sufficiently sensitive to oxygen content |
| EIS | Frequency-dependent electrochemical response | Does not directly determine chemical expansion |
The combination of TGA and dilatometry is particularly useful for estimating chemical-expansion coefficients, while in-situ XRD helps determine whether the response is associated with lattice changes or phase evolution.
9. A practical calculation workflow
Consider an isotropic cubic oxide measured at a fixed temperature under two oxygen partial pressures. Suppose the refined lattice parameter changes from \(a_0\) to \(a_1\), while an independent composition measurement determines the corresponding change in oxygen deficiency from \(\delta_0\) to \(\delta_1\).
The linear strain is:
\[ \varepsilon_{chem} = \frac{a_1-a_0}{a_0} \]
The average chemical-expansion coefficient is:
\[ \overline{\alpha}_{chem}^{\delta} = \frac{(a_1-a_0)/a_0}{\delta_1-\delta_0} \]
For small isotropic changes, the volumetric strain is approximately:
\[ \frac{\Delta V}{V} \approx 3\varepsilon_{chem} \]
Before interpreting the result, verify that:
- Both measurements refer to the same temperature.
- The material remains in the same crystallographic phase.
- The oxygen deficiency is determined independently or through a validated calibration.
- The lattice parameters are refined using a consistent structural model.
- The composition interval is sufficiently small for an average coefficient to be meaningful.
- The specimen is not subject to changing external mechanical constraint.
If the sample is anisotropic, use direction-dependent lattice strains rather than the isotropic factor of three.
10. Common interpretation errors
Confusing thermal and chemical expansion. A length increase during heating in air is not automatically evidence of oxygen loss. The oxygen content must be evaluated under the actual atmosphere.
Assuming all reduction causes expansion. Reduction-induced cation-size changes often promote expansion, but local structural relaxation and phase changes can produce more complicated behaviour.
Using a coefficient without defining its variable. A coefficient with respect to oxygen deficiency is not numerically interchangeable with one defined against oxygen content or oxygen partial pressure.
Inferring oxygen deficiency from XRD alone. Lattice parameters can respond to several variables. Independent chemical or thermogravimetric evidence is needed for a defensible composition assignment.
Applying isotropic equations to anisotropic crystals. Lower-symmetry structures may exhibit direction-dependent chemical strain.
Equating free strain with operating stress. Stress depends on mechanical constraint, geometry, elastic properties, interfaces, and time-dependent relaxation.
11. Design implications for functional oxides
Chemical expansion should be treated as a functional property, not merely a secondary structural effect.
For oxygen electrodes, repeated changes in oxygen content can generate local strain and contribute to microstructural degradation. For dense membranes, gradients in oxygen chemical potential can create nonuniform expansion through the thickness. In multilayer cells, mismatch between adjoining layers can concentrate stress near interfaces.
A practical design strategy is to:
- Measure oxygen non-stoichiometry over the intended operating window.
- Determine thermal and chemical strain under representative gas conditions.
- Evaluate anisotropy and possible phase transitions.
- Compare free-strain responses of adjoining layers.
- Assess elastic constraint and stress relaxation.
- Test dimensional and phase reversibility during repeated redox cycles.
Materials selection should balance transport properties, surface kinetics, chemical stability, and mechanical compatibility. Maximizing oxygen vacancy concentration or conductivity alone is not a sufficient design criterion.
12. Key takeaways
- Chemical expansion is a composition- and defect-driven dimensional response, distinct from ordinary thermal expansion.
- Oxygen non-stoichiometry can alter cation valence, local bonding, lattice parameters, and crystal symmetry.
- Chemical-expansion coefficients must be defined with respect to a specific composition variable and reference state.
- Chemical strain can be anisotropic and nonlinear.
- Mechanical stress arises when free chemical or thermal strain is constrained or spatially nonuniform.
- Dilatometry, TGA, and in-situ XRD provide complementary information and are strongest when used together.
- Thermochemical compatibility is important for the durability of solid oxide cells and oxygen-transport membranes.
References
- D. S. Tsvetkov, V. V. Sereda, D. A. Malyshkin, I. L. Ivanov, and A. Yu. Zuev, “Chemical lattice strain in nonstoichiometric oxides: an overview,” Journal of Materials Chemistry A 10 (2022), 6351–6375. https://doi.org/10.1039/D1TA08407K
- S. R. Bishop, D. Marrocchelli, C. Chatzichristodoulou, N. H. Perry, M. B. Mogensen, H. L. Tuller, and E. D. Wachsman, “Chemical Expansion: Implications for Electrochemical Energy Storage and Conversion Devices,” Annual Review of Materials Research 44 (2014), 205–239. https://doi.org/10.1146/annurev-matsci-070813-113329
- D. Marrocchelli, C. Chatzichristodoulou, and S. R. Bishop, “Defining chemical expansion: the choice of units for the stoichiometric expansion coefficient,” Physical Chemistry Chemical Physics 16 (2014), 9229–9232. https://doi.org/10.1039/C4CP01096E
- D. Marrocchelli, S. R. Bishop, H. L. Tuller, and B. Yildiz, “Understanding Chemical Expansion in Non-Stoichiometric Oxides: Ceria and Zirconia Case Studies,” Advanced Functional Materials 22 (2012), 1958–1965. https://doi.org/10.1002/adfm.201102648
- J. G. Swallow, J. J. Kim, J. M. Maloney, D. Chen, J. F. Smith, S. R. Bishop, H. L. Tuller, and K. J. Van Vliet, “Dynamic chemical expansion of thin-film non-stoichiometric oxides at extreme temperatures,” Nature Materials 16 (2017), 749–754. https://doi.org/10.1038/nmat4898