Transient Diffusion Creep

• Transient diffusion creep is a viscoelastic deformation mechanism where grain interiors store strain energy elastically while grain-boundary sliding is accommodated by diffusion.
• The mechanism transitions from Andrade-like behavior at high frequencies to steady-state Coble creep at low frequencies, captured by an extended Burgers model.
• Finite element and thermodynamic formulations link diffusion, vacancy transport, and grain-boundary kinematics to practical insights on transient and dislocation-controlled creep.

Transient diffusion creep is a viscoelastic deformation mechanism in polycrystals in which grain interiors store strain energy elastically, grain boundaries slide freely in shear, and diffusion accommodates the incompatibilities created by that sliding. In the simple reference formulation of diffusionally accommodated/-assisted grain-boundary sliding, the long-time response tends to steady-state diffusion creep, specifically Coble creep, whereas the short-time or high-frequency response is Andrade-like rather than a single-relaxation Debye or ordinary Burgers peak. The resulting rheology is well described by a parameterised extended Burgers model that behaves as a Maxwell model at low frequencies and as an Andrade model at high frequencies (Rudge, 11 Jul 2025).

1. Definition and scope

Transient diffusion creep differs from steady-state diffusion creep by its explicit time- and frequency-dependence. At low frequency, diffusion has time to equilibrate, so the response tends to steady-state diffusion creep and the material behaves as a Newtonian viscous fluid. At high frequency, diffusion can act only in small neighborhoods near stress concentrations, so the response develops a broad anelastic regime with a power-law attenuation spectrum. In the reference finite-element treatment, this transient regime is not an auxiliary correction to steady flow; it is the effective viscoelastic rheology generated by diffusionally accommodated grain-boundary sliding itself (Rudge, 11 Jul 2025).

In a broader thermodynamic formulation, transient diffusion creep is also the irreversible deformation that arises when vacancy diffusion is coupled to lattice-site creation and annihilation. In that setting, the solid is treated as a non-hydrostatically stressed multicomponent crystalline medium with non-conserved lattice sites, and creep is transient when vacancy concentration gradients are not yet equilibrated or when grain-boundary sink/source activity evolves in time (Mishin et al., 2013).

The term should be distinguished from diffusion-controlled dislocation creep. In coupled dislocation–vacancy models, vacancy diffusion controls dislocation climb, climb unlocks glide, and the creep strain is mainly produced by dislocation glide at a rate controlled by climb. That process is diffusion-controlled, but it is not classical steady-state diffusion creep in the Nabarro–Herring or Coble sense (Keralavarma et al., 2017).

2. Grain-scale mechanism of diffusionally accommodated grain-boundary sliding

The reference microphysical picture assumes linearly elastic grains with intrinsic shear modulus $\mu$ and Poisson’s ratio $\nu$, grain boundaries that are shear-free, and accommodation by grain-boundary diffusion. The finite-element calculations use regular hexagons in $2$D and tetrakaidecahedra in $3$D as reference geometries. These geometries are not merely convenient tessellations: the angles at which grains meet at triple junctions control the stress singularities, and those singularities control the Andrade exponent (Rudge, 11 Jul 2025).

The grain-scale elastic problem is posed through

$\nabla \cdot \sigma = \mathbf{0},$

with

$$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$

On grain boundaries, shear stress is zero, and diffusion accommodates the normal displacement discontinuity. The upscaling is performed by periodic homogenisation on a single grain with periodic boundary conditions, using equivalent energetics to match stored and dissipated energy between micro- and macro-scale descriptions. The implementation uses P2 vector elements for the displacement field, P2 scalar elements for the boundary pressure, and iterative solution with MINRES and GAMG preconditioning (Rudge, 11 Jul 2025).

In the thermodynamic vacancy-based description, the same physics appears in a different language. The site density balance is

$$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$

and the kinematic bridge between vacancy-mediated site production and permanent deformation is

$$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$

For a one-dimensional bicrystal, the grain boundary acts as a localized vacancy sink/source, so vacancy diffusion, grain-boundary migration, and relative rigid translation of grains appear as a coupled relaxation process rather than an imposed constitutive law (Mishin et al., 2013).

3. Constitutive representation and asymptotic regimes

The low-frequency limit of the reference model is steady-state Coble creep. For hexagonal grains, the steady-state viscosity is

$$\eta = \frac{1}{144}\frac{k T d^3}{\delta D_{gb} \Omega}.$$

The corresponding Maxwell time is

$\tau_M=\frac{\eta}{G_0},$

where $\nu$0 is the unrelaxed modulus. The compliance is represented by an extended Burgers form,

$\nu$1

with time-domain counterpart

$\nu$2

For the chosen power-law spectrum, the model reduces to a Maxwell material at long periods and to an Andrade material at short periods (Rudge, 11 Jul 2025).

The Andrade asymptotic form is

$\nu$3

which implies

$\nu$4

Here $\nu$5 is the relaxation strength, $\nu$6 is the Andrade exponent, $\nu$7 is the Andrade time, and $\nu$8 is the long-period cutoff of the extended Burgers spectrum. In the reference geometries, $\nu$9, $2$0, and $2$1 vary with Poisson’s ratio, while $2$2 is fixed by triple-junction geometry (Rudge, 11 Jul 2025).

Geometry	Triple-junction angles	Parameters at $2$3
Hexagonal grains	$2$4	$2$5, $2$6, $2$7
Tetrakaidecahedral grains	$2$8	$2$9, $3$0, $3$1

For these same fits, the effective low-frequency cutoff is $3$2 for hexagons and $3$3 for tetrakaidecahedra. The geometric control of $3$4 follows from the purely elastic stress singularity

$3$5

a diffusion boundary-layer length

$3$6

and the scaling relation

$3$7

For the $3$8 junction, $3$9, giving $\nabla \cdot \sigma = \mathbf{0},$0; for the tetrakaidecahedral junction geometry, $\nabla \cdot \sigma = \mathbf{0},$1, giving $\nabla \cdot \sigma = \mathbf{0},$2 (Rudge, 11 Jul 2025).

4. Thermodynamic and generalized continuum formulations

A general irreversible-thermodynamic treatment of creep by vacancy diffusion and lattice-site generation identifies the entropy production rate and the conjugate driving forces for diffusion, phase-field evolution, and creep deformation. The internal energy density is written as

$\nabla \cdot \sigma = \mathbf{0},$3

with nonclassical chemical potentials

$\nabla \cdot \sigma = \mathbf{0},$4

In the slow-creep limit, diffusion is driven by gradients of these nonclassical diffusion potentials, and the bicrystal example shows explicitly that vacancy concentration evolution, grain-boundary migration, local lattice velocity, and permanent shape change form a coupled transient relaxation problem (Mishin et al., 2013).

A more general homogenised continuum description identifies diffusion creep with a micropolar, or Cosserat, fluid rather than a purely Cauchy viscous fluid. The independent kinematic measures are

$\nabla \cdot \sigma = \mathbf{0},$5

so grain rotation becomes an explicit degree of freedom. Homogenisation of discrete rigid grains yields constitutive tensors for force-stress, couple-stress, and translation–rotation coupling. For irregular hexagonal grains, the theory illustrates a potential coupling between rotational and translational degrees of freedom. If only plating out or removal of material at grain boundaries is included, the constitutive laws are degenerate: modes of deformation that involve pure tangential motion at the grain boundaries are not resisted. That degeneracy can be removed by including resistance to grain-boundary sliding or by imposing additional constraints on the deformation (Rudge, 2021).

These generalized formulations imply that transient diffusion creep need not be represented solely by a scalar Newtonian viscosity. A plausible implication is that transient behavior depends on how vacancy transport, grain-boundary kinematics, and rotational compatibility are coarse-grained into continuum variables.

5. Diffusion-controlled transients in atomistic and dislocation-resolved models

Atomistic Phase Field Crystal simulations provide a direct route from diffusion-mediated defect kinetics to macroscopic creep laws. In nanopolycrystalline systems, the measured stress and grain-size exponents are $\nabla \cdot \sigma = \mathbf{0},$6 and $\nabla \cdot \sigma = \mathbf{0},$7, closely matching idealized Nabarro–Herring creep. These exponents are observed in the presence of significant stress-assisted diffusive grain-boundary migration, indicating that Nabarro–Herring creep and stress-assisted boundary migration contribute in the same manner to the macroscopic constitutive relation. Near melting, the same simulations show a crossover to $\nabla \cdot \sigma = \mathbf{0},$8 and $\nabla \cdot \sigma = \mathbf{0},$9, interpreted as re-entrant Coble creep caused by grain-boundary premelting (Berry et al., 2015).

Discrete dislocation dynamics with explicit vacancy diffusion resolves a different diffusion-controlled transient. The model couples glide, climb, and vacancy transport through

$$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$0

In quasi-equilibrium or jammed states, glide is largely exhausted and slower diffusion-mediated climb becomes decisive. At low applied stress, the stress exponent approaches $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$1, consistent with diffusion creep scaling, and the pure diffusion contribution to strain rate also scales with exponent $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$2. At higher stresses, explicit coupling of glide and climb yields $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$3 to $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$4, with an activation energy of about $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$5, close to the self-diffusion energy of aluminum, about $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$6 in the parameter set used (Keralavarma et al., 2017).

Taken together, these results delimit two distinct diffusion-controlled transients. One is classical transient diffusion creep in which diffusion accommodates grain-boundary sliding. The other is diffusion-controlled dislocation creep in which vacancy transport enables climb and thus sustained glide. The overlap in terminology is real, but the underlying kinematics differ.

6. Comparison with laboratory observations and attenuation data

The simple reference model provides a lower bound on attenuation in laboratory experiments. The strongest agreement reported is with borneol, where the experimental $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$7 and $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$8 are close to the model values and the attenuation curves are very similar, especially for the tetrakaidecahedral geometry. By contrast, some experiments on synthetic olivine, synthetic dunite, and MgO produce lower $$\sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij}, \qquad e_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).$$9, different relaxation strengths, and often stronger attenuation at high homologous temperatures. In many such comparisons the modeled attenuation lies below the laboratory data (Rudge, 11 Jul 2025).

For mantle-like parameters, the modeled attenuation is

$$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$0

at seismic frequencies, whereas observations suggest closer to

$$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$1

The discrepancy is too large to be explained by transient diffusion creep alone in the simple model. The interpretation advanced for this mismatch is that real materials include additional dissipative processes not present in the reference geometries, including elastically accommodated grain-boundary sliding, premelting-related effects, impurities, dislocations, or other temperature-dependent grain-boundary/triple-line processes. The simple model also emphasizes the theoretically expected grain-size dependence $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$2 for grain-boundary diffusion; more recent experiments reporting $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$3 are closer to that prediction and therefore reduce extrapolated attenuation (Rudge, 11 Jul 2025).

An important methodological caution follows from the wider transient-creep literature. Andrade-like or logarithmic decay is not uniquely diagnostic of transient diffusion creep. Separate studies attribute $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$4 creep to thermal activation plus elastic stress redistribution, to evolving event statistics and Omori-like decay, or to transient yield-stress scaling near arrest and fluidisation. Those mechanisms are formulated for damage models, elastoplastic models, paper experiments, and amorphous solids rather than for grain-boundary diffusion, but they show that similar $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$5 transients can emerge from different microscopic physics [(Weiss et al., 2022); (Castellanos et al., 2019); (Laurson et al., 2011); (Popović et al., 2021)].

7. Distinction from dislocation-controlled transient creep

A recurrent misconception is that transient creep in high-temperature polycrystals is generically explained by diffusion. Work on olivine aggregates argues against that generalization. In olivine deforming by dislocation creep, aggregate behavior is much closer to the isostress endmember than the isostrain endmember, aggregate strain rates are about $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$6 faster than isostrain predictions at laboratory conditions, and the resulting steady-state viscosity difference grows to $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$7–$$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$8 under upper-mantle conditions. Under that interpretation, transient creep is controlled primarily by intragranular dislocation interactions and evolving back stress, not by intergranular grain-to-grain stress transfer (Wallis et al., 2024).

Independent microstructural evidence reinforces that conclusion. In torsion experiments on olivine aggregates at $$\frac{\partial n_s}{\partial t}+\nabla\cdot(n_s\mathbf v_L)=r_s,$$9, $$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$0, and constant shear strain rate $$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$1, both coarse- and fine-grained materials strain harden during the transient. In coarse-grained samples, dislocation density increases from $$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$2 to $$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$3 with strain; intragranular stress heterogeneity increases from a distribution width of about $$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$4 to about $$\nabla\cdot\mathbf v_L-\nabla\cdot\tilde{\mathbf v}_L=\frac{r_s}{n_s}.$$5; and elevated stresses are associated with regions of high geometrically necessary dislocation density. These observations indicate that dislocation interactions are the primary cause of strain hardening during transient creep and do not support a diffusion-creep-dominated explanation for these transients (Wiesman et al., 2024).

The boundary between transient diffusion creep and transient dislocation creep is therefore substantive rather than semantic. Transient diffusion creep concerns viscoelastic relaxation by diffusionally accommodated grain-boundary sliding or by vacancy-mediated site creation and annihilation. Dislocation-dominated transients, even when diffusion controls climb or recovery, belong to a different rheological class. A plausible implication is that identifying the relevant transient mechanism requires direct microstructural evidence—such as dislocation-density evolution, stress heterogeneity, grain-size sensitivity, and attenuation spectra—rather than reliance on a macroscopic power-law transient alone.