Nabarro–Herring creep

The Nabarro–Herring creep rate, ${\displaystyle {\dot {\epsilon }}_{NH}}$, can be derived by considering an individual rectangular grain (in a single or polycrystal).[1] Two opposing sides have a compressive stress applied and the other two have a tensile stress applied. The atomic volume is decreased by compression and increased by tension. Under this change, the activation energy to form a vacancy is altered by ${\displaystyle \pm \sigma \Omega }$. The atomic volume is ${\displaystyle \Omega }$ and the stress is ${\displaystyle \sigma }$. The plus and minus indication is an increase or decrease in the activation energy due to the tensile and compressive stresses, respectively. The fraction of vacancy concentrations in the compressive (${\displaystyle N_{\nu }^{C}}$) and tensile (${\displaystyle N_{\nu }^{T}}$) regions are given as:

${\displaystyle N_{\nu }^{C}\approx \exp \left(-{\frac {Q_{f}}{kT}}\right)\exp \left(-{\frac {\sigma \Omega }{kT}}\right)}$,

${\displaystyle N_{\nu }^{T}\approx \exp \left(-{\frac {Q_{f}}{kT}}\right)\exp \left({\frac {\sigma \Omega }{kT}}\right)}$

In these equations ${\displaystyle Q_{f}}$ is the vacancy formation energy, ${\displaystyle k}$ is the Boltzmann constant, and ${\displaystyle T}$ is the absolute temperature. These vacancy concentrations are maintained at the lateral and horizontal surfaces in the grain. These net concentrations drive vacancies to the compressive regions from the tensile ones which causes grain elongation in one dimension and grain compression in the other. This is creep deformation caused by a flux of vacancy motion.

The vacancy flux, ${\displaystyle J_{\nu }}$, associated with this motion is given by:

${\displaystyle J_{\nu }=-D_{\nu }\left({\frac {\delta N_{\nu }}{\delta x}}\right)}$

where ${\displaystyle D_{\nu }}$ is the vacancy diffusivity. This is given as:

${\displaystyle D_{\nu }=D_{0\nu }\exp \left({\frac {-Q_{m}}{kT}}\right)}$

where ${\displaystyle D_{0\nu }}$ is the diffusivity when 0 vacancies are present and ${\displaystyle Q_{m}}$ is the vacancy motion energy. The term ${\displaystyle {\frac {\delta N_{\nu }}{\delta x}}}$ is the vacancy concentration gradient. The term ${\displaystyle \delta x}$ is proportional to the grain size ${\displaystyle d}$ and ${\displaystyle \delta N_{\nu }=N_{\nu }^{T}-N_{\nu }^{C}}$. If we multiply ${\displaystyle J_{\nu }}$ by ${\displaystyle d^{2}}$ we obtain:

${\displaystyle {\frac {\delta V}{\delta t}}\approx D_{0\nu }d\exp \left[-{\frac {Q_{f}+Q_{m}}{kT}}\right]\left[\exp \left({\frac {\sigma \Omega }{kT}}\right)-\exp \left(-{\frac {\sigma \Omega }{kT}}\right)\right]}$

where ${\displaystyle {\frac {\delta V}{\delta t}}}$ is the volume changed per unit time during creep deformation. The change in volume can be related to the change in length along the tensile axis as ${\displaystyle \delta V\approx d^{2}\delta d}$. Using the relationship between ${\displaystyle \delta V}$ and ${\displaystyle \delta d}$ the NH creep rate is given by:

${\displaystyle {\dot {\epsilon }}_{NH}={\frac {1}{d}}{\frac {\delta d}{\delta t}}}$

${\displaystyle {\dot {\epsilon }}_{NH}=\left({\frac {D_{0\nu }}{d^{2}}}\right)\exp \left[-{\frac {Q_{f}+Q_{m}}{kT}}\right]\left[\exp \left({\frac {\sigma \Omega }{kT}}\right)-\exp \left(-{\frac {\sigma \Omega }{kT}}\right)\right]}$

This equation can be greatly simplified. The lattice self-diffusion coefficient is given by:

${\displaystyle D_{L}=D_{0\nu }\exp \left[-{\frac {Q_{f}+Q_{m}}{kT}}\right]}$

As previously stated, NH creep occurs at low stresses and high temperatures. In this range ${\displaystyle \sigma \Omega <<kT}$. For small ${\displaystyle x}$, ${\displaystyle \exp(\pm x)\approx 1\pm x}$. Thus we can re-write ${\displaystyle {\dot {\epsilon }}_{NH}}$ as:

${\displaystyle {\dot {\epsilon }}_{NH}=A_{NH}\left({\frac {D_{L}}{d^{2}}}\right)\left({\frac {\sigma \Omega }{kT}}\right)}$

where ${\displaystyle A_{NH}}$ is a constant that absorbs the approximations in the derivation.

Alternatively, this can be derived in a different method where the constant ${\displaystyle A_{n}}$ has different dimensions. In this case, the NH creep rate ${\displaystyle {\dot {\epsilon }}}$ is given by:[5]

${\displaystyle {\dot {\epsilon }}={A_{n}}{\frac {Db^{3}\sigma }{d^{2}kT}}}$

Coble creep is closely related to Nabarro–Herring creep and is controlled by diffusion as well. Unlike Nabarro–Herring creep, mass transport occurs by diffusion along the surface of single crystals or the grain boundaries in a polycrystal.[1] For a general expression of creep rate, the comparison between Nabarro–Herring and Coble creep can be presented as follows:[6]

${\displaystyle {\dot {\epsilon }}={\frac {ADGb}{kT}}{\left({\frac {\sigma }{G}}\right)}^{n}{\left({\frac {b}{d}}\right)}^{p}}$

G is the shear modulus. The diffusivity is obtained form the tracer diffusivity, ${\displaystyle D^{*}}$. The dimensionless constant ${\displaystyle A_{n}}$ depends intensively on the geometry of grains. The parameters ${\displaystyle A}$, ${\displaystyle n}$ and ${\displaystyle p}$ are dependent on creep mechanisms. Nabbaro–Herring creep does not involve the motion of dislocations. It predominates over high-temperature dislocation-dependent mechanisms only at low stresses, and then only for fine-grained materials. Nabarro–Herring creep is characterized by creep rates that increase linearly with the stress and inversely with the square of grain diameter.

In contrast, in Coble creep atoms diffuse along grain boundaries and the creep rate varies inversely with the cube of the grain size.[2] Lower temperatures favor Coble creep and higher temperatures favor Nabbaro–Herring creep because the activation energy for vacancy diffusion within the lattice is typically larger than that along the grain boundaries, thus lattice diffusion slows down relative to grain boundary diffusion with decreasing temperature.[2]

• Creep in dense, polycrystalline magnesium oxide[7] and iron-doped polycrystalline magnesia[8]
• Compressive creep in polycrystalline beryllium oxide[9]
• Creep in polycrystalline Al₂O₃ that has been doped with Cr, Fe, or Ti[10]
• Creep in dry synthetic dunite[11] which results in trace melt and some grain growth
• Reproduced for nanopolycrystalline systems in Phase Field Crystal simulations (theory matched in terms of creep stress and grain size exponents)[12]