2.2. Diffusion
2.2.4. Diffusion in polycrystals
The diffusion of hydrogen in the samples studied in this work are strongly influenced by grain boundaries. However, since they are polycrystals they are no isolated grain boundary. Therefore, the classical Fisher model does not apply. Diffusion in polycrystalls was first discussed by L.G. Harrison [64]. He introduced three regimes (A, B and C), which have since been called Harrison regimes. In the following they will be introduced briefly and their difference will be explained. Later on, the Harrison regimes have been divided further. Here, mainly the division of regime B into regime B1 to B4 will be considered, as introduced by Mishin and Razumovskii [172]. Finally it will be discussed how the grain size influences which regime will be reached under which conditions.
A graphical overview of all three Harrison regimes is given in figure 2.10. The three Harrison regimes can be separated by timet, grain size d and the ratio of the diffusion coefficients of grain boundary and volume ∆ = DGB/DV. In most cases this ratio ∆ is unknown. In a given measurement all regimes can be reached by changing several parameters such as time, temperature and grain size. For example at a fixed positionz in the sample, regime C is valid for short times, and the profile transforms via regime B into regime A for longer times [51, 66].
Regime A In regime A the GB diffusion and volume diffusion are similarly fast so that the diffusing species forms an isotropic diffusion front in the material (see figure 2.10). This leads to an effective diffusion coefficient Def f [51]:
Def f = sφGBDGB + (1−φGB)DV
1−φGB+sφGB . (2.24)
d δ
y
Regime C Regime B Regime A
z
z∼(Defft) 1/2 z∼(DGBt) 1/2
z∼t 1/4
Figure 2.10: Visualiza-tion of the three Harrison regimes. Regime A shows an isotropic front which depends on the effective diffusion coefficient Def f. Regime B can be described by the solutions for iso-lated grain boundaries (see chapter 2.2.3). Regime C is completely dominated by the grain boundary diffusion and can be de-scribed by an isotropic front which depends on the grain boundary diffusion coefficientDGB.
The effective diffusion coefficient is the geometric average of the grain boundary and grain diffusion coefficients. φGB is the volume fraction of the GB, whiles is the segregation factor discussed above. Equation 2.24 is valid for columnar grains. For cubic grains, Belova and Murch found [173]:
Def f = sDGB(ψ(1−ψ)sDGB + (1−ψ+ψ2)DV)
(1−φGB +sφGB) (ψDV + (1−ψ)sDGB). (2.25) Here, ψ =ψy =ψz is the grain boundary area fraction iny- andz-direction. It can be shown that Def f from equation 2.25 is always smaller than from equation 2.24.
This originates from grain boundary diffusion perpendicular to the main diffusion in the z-direction.
Regime B In regime B, the diffusion is dominated by the grain boundary with outwards diffusion into the grain. The spacing of the grain boundaries is large in comparison with the volume reached by diffusion in the measurement time window.
Hence, the boundaries can be treated as isolated. Therefore, Fisher’s model (dis-cussed in section 2.2.3) gives the analytic solution to this problem (see equation
2.23).
Regime C For regime C the grain diffusion is so low that all transport occurs in the grain boundaries. The result is again an isotropic front, however only within the grain boundaries. Therefore, the grain boundary diffusion coefficient DGB is also the diffusion coefficient of the whole system. This means that the measurement directly gives the grain boundary diffusion coefficient. Furthermore, this is the only regime where grain boundary segregation does not need to be considered, since the actual size of the grain boundary region does not influence the diffusion process. The diffusion process takes place in the grain boundary so more grain boundaries lead to more transport, but not faster or slower transport. Mishin and Razmunovskii studied the diffusion in moving grain boundaries with a constant velocity v [174].
They found that the movement does not change the diffusion speed in regime C.
This is not true for Regime B, where the penetration depth is lower for a moving grain boundary [174]. Deng et al. studied the diffusion in Regime C for more than one type of grain boundary [175]. They studied how grain boundaries with fast and slow diffusion lead to an effective diffusion coefficient of the overall grain boundary network (the effective diffusion coefficient should not be confused with Def f in Regime A). No dependency on the grain size was found. They determined that the effective diffusion depends on the percolation of the fast diffusion grain boundaries. Before the fast diffusion grain boundaries form a closed network, the overall diffusion is governed by the slow diffusion grain boundaries. After a closed network is formed the effective diffusion of the overall system becomes similar to the discussion above of Harrison regimes. Two regions with different diffusion coefficients interact and the overall diffusion depends on the volume fractions of the regions and the ratio of the diffusion coefficients of the regions.
Regime B1, Regime B2, Regime B3 and Regime B4 Mishin and Razmumovskii further divided Harrison regimes [172]. They based this on the two dimensionless coordinates β = (δDGB)/(2D3/2t1/2) (see section 2.2.3) andα =δ/(2(Dt)1/2). β, as said above, describes the relationship of the y- andz-direction of the bulk diffusion [61]. For example, if β >> 1 bulk diffusion occurs mainly in the y-direction and no relevant transport is found in the z-direction. This means that transport in the z-direction can only happen by the grain boundary. α gives information about the relationship between diffusion along the grain boundary and leakage into the bulk. For α >> 1 leakage can be ignored, while for α << 1 the grain boundary diffusion becomes quasi steady and leakage dominant. This ignores grain boundary segregation, which can be included by exchanging the geometric width δ by sδ as discussed above.
The two coordinates are not independent:
α
β = DV
DGB <<1. (2.26)
The inequality comes from the fact that DGB >> DV is assumed. For the inequality to be true neither α≈1and β ≈1norα >>1andβ <<1can be true. This leaves five different cases which correspond to regime C and four additional cases, which were labeled regimes B1 to B4 by Mishin and Razmumovskii. Harrison regime A is missing, because an isolated grain boundary is classified by these different regimes.
However, all six regimes can be discussed in dependence on the grain size d.
Regime C is defined by no relevant leakage from the grain boundary and dominant grain boundary diffusion. This can be described in terms of α >> 1 and β >> 1. For the regime B1, α≈1and leakage occurs into the bulk from the grain boundary.
This increases in Regime B2 as α << 1. This means that the grain boundary diffusion starts to become quasi-steady. However, asβ >>1still stands, the volume diffusion still applies predominantly in y-direction. This changes for regime B3 and β ≈ 1. Finally, in regime B4, β << 1 and the bulk diffusion in the z-direction becomes dominant. This change from regime C through regimes B1 to B4 happens with increasing time at a fixed temperature or vice versa. However, because the time dependence scales with the square root of time, in real experiments, it is in generally impossible to reach all regimes at a fixed temperature.
The regimes in dependence on the grain size Not all the regimes discussed above will be reached for a given sample. The dependence on the regimes on the grain size will therefore be discussed in the following. To simplify the problem the regimes B1 and B3 will not be further discussed as they are transition states between regime C, B2and B4. Additional information can be found in "Fundamentals of grain and interphase boundary diffusion" by Kaur et al. [51].
A polycrystal with grain size d can be classified by its grain size in relation to two characteristic diffusion lengths. The two lengths are characteristic of diffusion in the bulk L = (DVt)1/2 and in the grain boundary LCGB = (DGBt)1/2. However, LCGB is only characteristic for the diffusion in regime C, since it ignores the leakage from the grain boundary into the grain. For regime B2 one can define LBGB = ((δDGB)1/2t1/4)/(4DV)1/4, which takes the leakage into account. LCGB is always much larger than L. However, the same is not true for LBGB because it grows slower in time than the other two length scales. Therefore LBGB intersectsLCGB andLat times t0 = (s2δ2)/(4DV) and t00 = (sδDGB)2/(4DV3) respectively. Two corresponding sizes can be defined:
L0 = sδ 2
DGB DV
1/2
(2.27)
and
L00= sδ 2
DGB DV
. (2.28)
Using these sizes polycrystals can be divided into three classes: coarse grained (L0 << L00 << d), fine grained (L0 << d << L00) and ultra-fine grained (d <<
L0 << L00). The advantage of this separation is that it can be defined for all times (and temperatures), which is not the case if e.g. d is compared with L and LGB. The three classes and their sequence of regimes are collected in table 2.3.
class of polycrystals relationship of length scales sequence of regimes coarse grained L0 << L00<< d C→ B2→ B4→ A
fine grained L0 << d << L00 C→ B2→B2’→ A’
ultra-fine grained d << L0 << L00 C→ C’→B2’→ A’
Table 2.3.: Classes of polycrystals as divided by Kaur et al. [51]. The sequence of the regimes follows increasing time and does not include regime B1 and B3
For coarse grained polycrystals, one starts in regime C and goes to the different regimes B with increasing time. The only difference from the previous discussed is thatL grows large enough that it reaches d. This marks the transition from regime B4 to regime A, and the initially isolated grain boundaries start to interact. As said before the bulk diffusion becomes dominant in regime B4. Because the grain boundaries are still isolated in regime B4 they have no relevant influence on the overall diffusion. Therefore the diffusion is fully described by the grain diffusion coefficient DV and not by an average as in Harrisons regime A. No information about the grain boundary diffusion can be extracted in regime B4.
Fine grained polycrystals start in regime C and go into regime B2 for longer times (or higher temperature). However, the grain boundary length scale LBGB becomes larger than the grain sizedbefore regime B4 would be reached. Therefore the grain boundaries are no longer isolated. However, this is not equivalent to regime A, as the bulk diffusion lengthL(<< LBGB)is still smaller than the grain size. This transition regime is labeled regime B’2. For calculation purposes regime B’2 can be treated like regime B2 (which means Harrison regime B as presented above). However, the boundary width δ has to be exchanged with an effective width δef f = φGB/V. φGB is the volume fraction of the grain boundary of volume V. Depending on the geometry, the difference between δ and δef f is only a factor on the order of unity. After regime B’2 regime A’ is reached. Regime A’ is identical to regime A from the diffusion process point of view, however, it can be shown that d << L00 inducesDef f ≈qsδDGB/d (for further information see [51]). q is a geometric factor depending on the grain shape.
Ultra-fine grained polycrystals go from regime C into a new regime C’. In regime C’ the diffusant penetrates deep into the grain boundary without any considerable
leakage. Similar to regime B’2 the process can be mathematically treated like regime C. The only difference is that the surface fraction of the grain boundaries has to be exchanged by the grain boundary volume fraction. From regime C’ the diffusion process goes into regime B’2 and finally into regime A’.
Figure 2.11 shows for which grain size and∆ =DGB/DV which of the three different classes of polycrystals are reached. The lines dividing the classes are calculated from equations 2.27 and 2.28. sδis assumed to be1 nmas typical sizes of grain boundaries [176, 177].
1 nm 1 102 104 106 108 1010 1012 1014 1016
grain size d
∆
class: ultrafine grained
class: fine grained
class: coarse grained
10 nm 100 nm 1µm 10µm 100µm 1 mm
Figure 2.11.: Overview of the classes of polycrystals in dependence on the grain size d and ∆ = DGB/DV. The dividing lines are calculated from L0 and L00, assuming a grain boundary width of 1 nm. The red box is the region that was analyzed here for the Mg-H system by finite-element simulations using COMSOL Multiphysics (see chapter 3.6).