Double diffusivity model under stochastic forcing

The “double diffusivity” model was proposed in the late 1970s, and reworked in the early 1980s, as a continuum counterpart to existing discrete models of diffusion corresponding to high diffusivity paths, such as grain boundaries and dislocation lines. It was later rejuvenated in the 1990s to interpret experimental results on diffusion in polycrystalline and nanocrystalline specimens where grain boundaries and triple grain boundary junctions act as high diffusivity paths. Technically, the model pans out as a system of coupled Fick-type diffusion equations to represent “regular” and “high” diffusivity paths with “source terms” accounting for the mass exchange between the two paths. The model remit was extended by analogy to describe flow in porous media with double porosity, as well as to model heat conduction in media with two nonequilibrium local temperature baths, e.g., ion and electron baths. Uncoupling of the two partial differential equations leads to a higher-ordered diffusion equation, solutions of which could be obtained in terms of classical diffusion equation solutions. Similar equations could also be derived within an “internal length” gradient (ILG) mechanics formulation applied to diffusion problems, i.e., by introducing nonlocal effects, together with inertia and viscosity, in a mechanics based formulation of diffusion theory. While being remarkably successful in studies related to various aspects of transport in inhomogeneous media with deterministic microstructures and nanostructures, its implications in the presence of stochasticity have not yet been considered. This issue becomes particularly important in the case of diffusion in nanopolycrystals whose deterministic ILG-based theoretical calculations predict a relaxation time that is only about one-tenth of the actual experimentally verified time scale. This article provides the “missing link” in this estimation by adding a vital element in the ILG structure, that of stochasticity, that takes into account all boundary layer fluctuations. Our stochastic-ILG diffusion calculation confirms rapprochement between theory and experiment, thereby benchmarking a new generation of gradient-based continuum models that conform closer to real-life fluctuating environments.

Figure 1

Variation of the autocorrelation function ${\rho }_1^{\text{rms}}$ or ${\rho }_2^{\text{rms}}$ against the average scaled diffusivity $D$ (assumption $D_1=D_2=D$, scaling factor ${10}^{-10}$). The dots represent the actual data points obtained from a numerical solution of Eq. (15b) for the parameter set ${\kappa }_1=1,{\kappa }_2=0.001$ for identical noise strengths ${\gamma }_1={\gamma }_2=1$ (coefficient values are all scaled dimensionless numbers, based on [11]). Results indicate a monotonic decay with $D$.

Figure 2

Variation of the autocorrelation function ${\rho }_1^{\text{rms}}$ (represented by circles) or ${\rho }_2^{\text{rms}}$ (represented by squares) against the average scaled diffusivity $D$ (assumption $D_1=D_2=D$, scaling factor ${10}^{-10}$) for anisotropic noise case: ${\gamma }_1=1$ and ${\gamma }_2=2$. Results are obtained from a numerical solution of Eqs. (15a) and (15b), respectively, for ${\rho }_1^{\text{rms}}$ and ${\rho }_2^{\text{rms}}$ for ${\kappa }_1=1$ and ${\kappa }_2=0.001$. Coefficient values are all scaled dimensionless numbers, based on [11].

Figure 3

Variation of the spatial correlation function $C_r={\rho }_1^{\text{rms}}\left(r\right)={\rho }_2^{\text{rms}}\left(r\right)$ against displacement $r$ for large $k$ (assumption $D_1=D_2=D={10}^{-9}$). The plot is obtained from a numerical solution of Eqs. (18a) and (18b) for the parameter rescaled version (where ${\alpha }_1=-\frac{{\pi }^2\{5{\gamma }_2{\kappa }_2^2+{\gamma }_1[5{\kappa }_2^2+2{({\kappa }_1+{\kappa }_2)}^8]\}}{8\times 2^{3/4}{({\kappa }_1+{\kappa }_2)}^{14}}$) for identical noise strengths ${\gamma }_1={\gamma }_2=1$ (coefficient values are all scaled dimensionless numbers, based on [11]).

Figure 4

Variation of the spatial correlation function $C_r={\rho }_1^{\text{rms}}\left(r\right)={\rho }_2^{\text{rms}}\left(r\right)$ against displacement $r$ (assumption $D_1=D_2=D={10}^{-9}$). The plot is obtained from a numerical solution of Eqs. (17a) and (17b) for the parameter set ${\kappa }_1=1,{\kappa }_2=0.001$ for identical noise strengths ${\gamma }_1={\gamma }_2=1$ (coefficient values are all scaled dimensionless numbers, based on [11]). The circles represent the real data points (the saturation region is represented by multiple close-lying circles doubling up as a thick solid straight line) while the solid straight line is the extrapolated fit.

Figure 5

Variation of the spatial correlation function $C_r$ for the variables ${\rho }_1^{\text{rms}}\left(r\right)$ (represented by circles) or ${\rho }_2^{\text{rms}}\left(r\right)$ (represented by crosses) against the separation distance $r$ (assumption $D_1=D_2=D={10}^{-9}$) for the anisiotropic noise case: ${\gamma }_1=1$ and ${\gamma }_2=2$. Results are obtained from a numerical solution of Eqs. (17a) and (17b) for ${\kappa }_1=1$ and ${\kappa }_2=0.001$. Coefficient values are all scaled dimensionless numbers, based on [11]. The saturation regimes for both plots are represented by multiple close-lying symbols—circles or crosses, as the case may be—giving them the appearance of a thick line.

Figure 6

Variation of the spatial correlation functions $C_r={\rho }_1^{\text{rms}}\left(r\right)$ against displacement $r$ for the degenerate case ($D_1=D_2={10}^{-9}$; represented by dots) versus the nondegenerate case ($D_1=2D_2,D_2={10}^{-9}$; represented by circles). The plot is obtained from a numerical solution of Eqs. (19a) and (19b) for the parameter set ${\kappa }_1=1,{\kappa }_2=0.001$ for identical noise strengths ${\gamma }_1={\gamma }_2=1$ (coefficient values are all scaled dimensionless numbers, based on [11]).

Figure 7

Variation of the temporal correlation function $C_T$ for the variables ${\rho }_1\left(T\right)$ and ${\rho }_2\left(T\right)$ against the time difference $T$ (assumption $D_1=D_2=D={10}^{-9}$). The circles and crosses, respectively, represent ${\rho }_1^{\text{rms}}\left(T\right)$ and ${\rho }_2^{\text{rms}}\left(T\right)$ for the special case ${\gamma }_1={\gamma }_2=1$ while the corresponding anisotropic noise cases (${\gamma }_1=1,{\gamma }_2=2$) are represented by the solid line and dots, respectively. Results are obtained from a numerical solution of Eqs. (21a) and (21b) for ${\kappa }_1=1$ and ${\kappa }_2=0.001$. Coefficient values are all scaled dimensionless numbers, based on [11, 12].

Figure 8

Variation of the spectral correlation function $C_r$ against the separation distance $r$ (assumption $D_1=D_2=D={10}^{-1},{\gamma }_0=1$). Results are obtained from a numerical solution of Eq. (25a) for ${\lambda }_1={\lambda }_2=1$.

Figure 9

Variation of the spectral correlation function $C_T$ against the time difference $T$ (assumption $D_1=D_2=D={10}^{-1},{\gamma }_0=1$). Results are obtained from a numerical solution of Eq. (25b) for ${\lambda }_1={\lambda }_2=1$.