#### Abstract

Grain boundary (GB) diffusion in engineering materials at elevated temperatures often determines the evolution of microstructure, phase transformations, and certain regimes of plastic deformation and fracture. Interpreting experimental data with the use of the classical Fisher model sometimes encounters contradictions that can be related to violation of Fick’s law. Here, we generalize the Fisher model to the case of non-Fickian (anomalous) diffusion ubiquitous in disordered materials. The process is formulated in terms of the subdiffusion equations with time-fractional derivatives of order and for grain volume and GB, respectively. It is shown that propagation along GB for the case of a localized instantaneous source and weak localization in GB () is approximately described by distributed-order subdiffusion with exponents and . The mean square displacement is calculated with the use of the alternating renewal process model. The tail of the impurity concentration profiles along the axis is approximately described by the dependence for all , as in the case of normal GB diffusion, so the 6/5-law itself can serve as an identifier of a more general phenomenon, namely, anomalous GB diffusion.

#### 1. Introduction

Grain boundary (GB) diffusion can control many processes in materials and determine the evolution of microstructure, phase transformations, and some regimes of plastic deformation and fracture [1]. Usually, atoms diffuse along grain boundaries (GB) by orders of magnitude faster than in crystalline grains. In recent decades considerable progress has been made in the accuracy and reliability of measurements of GB diffusion parameters [1–3]. However, many fundamental aspects including mechanisms at the atomic level remain unclear. One of the reasons for this discrepancy is that experimental data on GB diffusion is usually interpreted with the Fisher model [4] and its modifications [1–3] based on the Fick law, which can be violated. The goal of the present paper is to generalize the classic model to the case of non-Fickian (anomalous) diffusion and compare predictions of the new model with the classic solutions.

The Fisher model considers a system with a single isolated GB of thickness placed in a semi-infinite medium. Usually, the diffusion coefficient along GB is assumed to be significantly larger than in grain volume. The impurity atoms initially located on the surface of a semi-infinite medium diffuse along GB and in volume, at a constant annealing temperature during the time [5]. The propagation in such a system is usually described by a set of diffusion equations. The exact solutions for the Fisher model with constant and instantaneous source were found by Wipple and Suzuoka [6, 7]. Levin and McCallum [8] have shown that the dependence of the average concentration on is linear for sufficiently large [1]. Such dependence often serves as an identifier of the GB diffusion phenomenon. It is natural to expect the anomalous GB diffusion to be characterized by index different from 6/5.

Several papers discuss slowing down the diffusion of light elements due to GBs. In [9], various aspects of the interaction of hydrogen with grain boundaries in nickel (Ni) were investigated by first-principle calculations by means of the density functional theory. Localization and migration of hydrogen strongly depend on type of GB. GBs can act as a two-dimensional obstacle (wall) for H migration. In [10] molecular dynamics simulation of dissolution and diffusion of hydrogen in GB of tungsten has shown that high local concentration of H in GB (for example, 30% at 900 K) leads to an unordered structure of GB that can provide trapping of H. Autoradiography [11] applied to polycrystals with different grain sizes showed that the increased GB area leads to the suppression of migration of C isotope in -iron. The opposite effect is observed for Co isotope. The authors of [11] conclude that GBs in -iron act as traps for interstitial atoms and slow down their diffusion.

Besides that, a number of experimental facts indicate anomalous diffusion (mainly, subdiffusion) of impurities and defects in disordered materials (see, e.g., [12–16]). Usually the phenomenon is characterized by the expansion law for the diffusion packet . In the case of , the diffusion packet width increases with time more slowly (subdiffusion), and in the case faster (superdiffusion) than in the normal case. Diffusion equations with fractional-order derivatives are an effective tool to describe anomalous diffusion [15, 17]. GB diffusion also demonstrates anomalous diffusive behavior [1, 18]. In [19], the fractional calculus approach for GB diffusion is justified within the classic Fisher model.

In the present paper, we consider a model of anomalous diffusion along the grain boundaries accounting for localization in grains and GB. The process is described by the anomalous diffusion equations containing time derivatives of fractional orders and for grains and boundary, respectively. The mean square displacement along GB and the penetration depth distribution are calculated for the case of a localized instantaneous source and weak localization (). The propagation along the boundary is approximately described by subdiffusion with a mixture of dispersion parameters and . The solutions are examined by the Monte Carlo simulation. Numerically, we also consider the cases of constant and instantaneous source on the surface. The impurity density and the mean square displacement are calculated with the use of the alternating renewal process model. It is remarkable that the tail of the impurity concentration profiles along GB ( axis) is approximately described by the same dependence for all , as in the case of normal GB diffusion.

#### 2. Anoamlous Diffusion and Fractional Stable Distributions

Despite of different specific mechanisms generating anomalous diffusion processes in different physical phenomena its main features can be obtained from the continuous time random walk (CTRW) scheme as time [20, 21].

The basic idea of the simplest (one-dimensional decoupled) version of CTRW model is that different jump lengths , as well as waiting times between two successive jumps, are random variables independent of each other and among themselves. One supposes that jump lengths and waiting times have the following distributions: and both directions for a test particle leaving a trap are equal in probabilities. If and we observe normal diffusion; all other values of and lead to anomalous diffusion with characteristic exponents . The anomalous diffusion packet for and . When we have subdiffusion regime; when we observe superdiffusion regime. The asymptotic term of the probability density function (propagator) obeys the fractional diffusion equation (see, e.g., [22]):Here is the Riemann-Liouville fractional derivative and is a fractional power of the Laplace operator (see for details [23]). This equation has a solution in the self-similar form: where the functionis the so-called fractional stable density [24, 25] expressed through the one-sided stable density and symmetric stable density . The stable densities are usually defined by their characteristic functions. In the one-dimensional case Here, is an asymmetry parameter. The asymmetric fractional stable density is simply defined by

The characteristic function of a fractional stable lawis expressed through the Mittag-Leffler function

Further, we consider only subdiffusive case for transport in GB and bulk, because this type of anomalous diffusion is observed more frequently in disordered solids. General representation (7) will be helpful for calculation of MSD with the use of an alternating renewal process.

#### 3. Fractional Generalization of the Fisher Model

In the Fisher model, GB is modeled as a high-diffusivity uniform slab of width perpendicular to the surface; this slab is embedded into a low-diffusivity isotropic medium [1, 4]. Diffusive motions in GB and surrounded medium are characterized by coefficients and such that . In a typical experiment, a layer of impurity atoms or tracer atoms is created at the surface and then the sample is annealed at a constant temperature during time [2].

Consider the modified Fisher model, accounting for the generalized diffusion in the grains and in GB . The width of GB . Equation for subdiffusion in grains isThe same equation with instead of and instead of is used for diffusion in GB ().

For small GB thicknesses, assuming slight variation in diffusant concentration across GB, we arrive at a single boundary condition for the modified Fisher model

Using the Fourier transformation on and Laplace transformation on of equation (10) and boundary condition (11) in the case of instantaneous source with initial condition , we obtain This transform solution generalizes Suzuoka’s solution [7]. For a particular case when parameters and concentration of traps in grains and GB coincide (), the expression can be inverted and represented in the formwhere is the Suzuoka solution [7] with anomalous diffusion coefficient instead of the ordinary one and is the one-sided -stable density [26]. The integral in (13) can be calculated by simple Monte Carlo algorithm by averaging over one-sided -stable variables . The expression similar to (13) is applicable for the case of a constant source and the Whipple solution [6].

#### 4. Fractional Alternating Renewal Process and Grain Boundary Diffusion

It is useful to consider the process in terms of a single-particle random walk. The displacement of a particle along depends on the distribution of times that the particle spends in GB and grains [19]. Let us simplify the situation by considering an instantaneous source at GB at (the case of a localized source). The time scale is divided into intervals of motion in grains and GB. To determine the distribution of total time spent in GB we consider the alternating renewal process [27]. Renewal events are associated with the transitions from the grains to GB and vice versa. For definiteness, we assume that at time the particle enters GB. In [27], we derived an expression for the double Laplace transform of PDF of total residence time in one of the states (e.g., in GB):where and are the Laplace images of PDF’s of waiting times and in GB and, in the grain volume,are images of the corresponding distribution functions. The elementary times and are independent random variables.

We consider the subordinated Brownian motion directed by fractional Poisson processes with different order and rate for grains and GB (see Appendix A). The density and distribution of waiting time in GB are characterized by the following asymptotic Laplace transforms:and in grain volume:where small shift from the boundary is introduced to overcome the problem of return times for the Brownian motion (then we assume ). Here, and are orders of fractional Poisson process for grains and GB, and are corresponding rates, and and can be considered as average times between subsequent localization events. Substituting these expressions into (15) goes down and we came to the following transform: where The inverse double Laplace transform leads to an expression (see details in [27])containing one-sided fractional-stable densities

If , that is valid for in the case (weak localization in GB), we have

Knowing distribution of the displacement along GB, due to the motion in grains and due to motion in GB one can write PDF of the total displacement along GB as a convolution, As a result, we have

The mean square displacement averaged over an ensemble of tracers is Taking the following property of the stable subordinator into account we obtain the subdiffusive behavior with two fractional exponents:

Averaging over time with the use of (23) (see Appendix B) leads to

Density (23) is an asymptotic expression for the case . It leads to MSD consisting of two power laws with exponents and . This behavior is well confirmed by simulation for and . But for , Monte Carlo computation indicates an additional component in proportional to : This term is clearly interpreted, when the process of propagation along GB is considered a distributed-order subdiffusion. When , diffusion along GB is accompanied by trapping into sites distributed in GB and delay due to sojourn in grains characterized by time distribution (18). In this case, the process is approximately described by the following distributed-order fractional equation: where denotes the fractional Caputo derivative. So, the MSD evolution is characterized by three power laws with exponents , , and .

If , we have the following scaling:which is well confirmed by Monte Carlo simulation (see Figure 1).

#### 5. Monte Carlo Verification

The process of diffusion in grains and in GB was simulated as the subordinated Brownian motion. The motion during the operational time (the delocalization time) is modeled according to the Langevin equation where is a two-dimensional white noise and . The diffusion coefficients along the -th direction depend on the position of the particle (GB or grain). Diffusion in grains is isotropic with coefficient for the parent process. In GB, the parent Brownian motion along GB is characterized by , and across GB the diffusion coefficient . The displacement for a small interval of the operational time is determined by the relation , where are standard normal random variables with zero mathematical expectation and unit variance. After each displacement, the particle can be trapped with probability in GB and probability in grain volume.

Figures 1 and 2 demonstrate the Monte Carlo simulation results for the cases and , respectively. In Figure 1, parameters and (i.e., trapping in GB is absent), , , , , and different values of subdiffusion exponent in grains. The figure shows that the dependence (34) is confirmed.

In Figure 2 the parameters are the same, except with , and and different indicated in figure. The influence of traps in GB is amplified for increased and the intermediate asymptotic behavior of the mean squared displacement can be described by the law . GB subdiffusion along can be approximately considered as subdiffusion with a mixture of localization time distributions characterized by two fractional orders and . Suchbehavior can be described by fractional diffusion equations with multifractal memory kernelsconsidered in details in [28].

Figure 3 shows calculated PDF of the penetration depth along the grain boundary for the case of subdiffusive motion in grains and a localized instantaneous source. It is worthy to note that the tails of these distributions are also successfully approximated by the dependence , well known from the theory of normal diffusion along the grain boundaries [1–3]. The importance of this dependence is that it serves as an indicator of GB diffusion that distinguishes the phenomenon from ordinary (bulk) diffusion. Moreover, this fact is confirmed by the corresponding tails of the symmetric fractional stable density , which is an approximate solution of the subordinated diffusion equation. These tails are also straightened out in the coordinates vs (Figure 4).

In Figure 5, the results of Monte Carlo simulation for the density distribution of the tracer penetration depth along the grain boundary for subdiffusion in grains in the case of a constant distributed source on the surface for different values of are shown. Other parameters ; , , , , , , .

#### 6. Conclusion

Interpreting experimental data with the use of the classical Fisher model sometimes encounters contradictions that can be related to violation of Fick’s law. Here, we generalize the Fisher model to the case of non-Fickian (anomalous) diffusion ubiquitous in disordered materials. The process is formulated in terms of the subdiffusion equations with time-fractional derivatives of order and for grain volume and GB, respectively. We found that, for the case of a localized instantaneous source and weak localization in GB (), the GB diffusion along is approximately described by distributed-order subdiffusion with exponents and . It is remarkable that tails of the tracer concentration profiles along the axis are approximately described by dependence for all , as in the case of normal GB diffusion, so the 6/5-law itself can serve as an identifier of a more general phenomenon, namely, anomalous GB diffusion.

#### Appendix

#### A. Subordinated Brownian Motion Directed by Fractional Poisson Process

We model subdiffusion in grains and GB as a subordinated Brownian motion directed by fractional Poisson process. Below, main definitions and relations used in our calculations are given.

Let be a Markovian process with one-dimensional density and a process with nonnegative increments. The following terminology is used: is called the parent process, is the directing process, and is said to be subordinated to the parent process using the operational time . The one-dimensional probability density for the subordinated process is expressed through the density of the directing process by

We consider subordinated Brownian motion with zero mean, so the parent process obeys the simple stochastic differential equation where is a standard Brownian motion with and is a diffusion coefficient. has the following characteristic function: Characteristic function of the subordinated process is where

To model subdiffusion, we take the fractional Poisson process [29, 30] as directing process defined by the Laplace transform of waiting time density :The original function is the so-called ‘fractional exponential density’ expressed via the Mittag-Leffler function: The density is expressed through the one-sided Lévy stable density the corresponding Laplace transform is where can be interpreted as an average time between two subsequent localization events.

If operational time is given, the probability density function of the total time is also expressed through the stable density:

The distribution density of first passage time or exit time for the subordinated process is related to the corresponding density for the Brownian motion by expression: In terms of the Laplace transformation:

The density of the subordinated Brownian motion directed by fractional Poisson process of order satisfies the subdiffusion equation containing the fractional Riemann-Liouville derivative of the same order:

#### B. To Calculation of the Mean Square Displacement

To obtain (31), we average (30) over total time in GB according to using approximate PDF (23). For asymptotic large times , . From the first term in (30) we derive the following: From the second term in (30), we obtain

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that there are no conflicts of interest.

#### Acknowledgments

The author thanks the Ministry of Education and Science of the Russian Federation (state program 3.2111.2017/4.6) and the Russian Foundation for Basic Research (projects 16-42-732113 and 18-51-53018) for financial support.