Diffusion-reaction model of positron annihilation for complex defect pattern

The increasing structural complexity in modern material science is often associated with grain sizes in the µm- and the sub-µm-regime. Therefore, when positron annihilation is applied for studying free-volume type defects in such materials, positron trapping at grain boundaries (GBs) cannot be neglected, even when other defect types are in the primary focus. For this purpose, the available diffusion-reaction model for positron trapping and annihilation at GBs is extended to competitive trapping at two different types of intragranular defects. Closed-form expressions for the mean positron lifetime and the relative intensities of the defect-specific positron lifetime components are given. The model is presented for cylindrical-shaped crystallites, but is valid in the general sense for spherical symmetry as well with appropriate replacements. The model yields the basis for properly determining defect concentrations, even for the inconvenient but common case that one intragranular defect type exhibits a lifetime component similar to that in GBs. It turns out, that positron trapping at GBs matters even for µm-sized crystallites and should not be neglected for precise studies of intragranular defects.


Introduction
Positron annihilation represents a well-established and sensitive probe technique for the study of atomic-scale sized free volumes in condensed matter [1][2][3].In the classical former field of point-defect studies in coarse-grained metals, positron (e + ) trapping at grain boundaries (GBs) could be safely set aside.The increasing structural complexity in modern material science is, however, often associated with grain sizes in the µm-and the sub-µm-regime.In these application fields of positron annihilation, e + -trapping at GBs can no longer be neglected, even when other defect types, rather than GBs, are in the primary focus of study.
For this purpose, a model for e + -trapping and annihilation both in GBs and two different types of intragranular defects is presented in this paper.Such complex defect scenarious may prevail, e.g. in metals after strong plastic deformation [4][5][6][7] or else in corroded [8] or hydrogen-loaded fine-grained metals [9,10], where in addition to GBs, dislocations and voids may act as positron trap.Another example are porous metals with a high-fraction of interfaces, lattice vacancies and voids [11][12][13].Although three different defect-associated positron lifetimes may in practice hardly be distinguishable, the model is of high relevance in the inconvenient but common case that one intragranular defect type (e.g.vacancy or dislocation) exhibits a lifetime component similar to that in GBs.The error in determining the concentration of this type of intragranular defect is critically assessed when in such a case, positron trapping at GBs is not taken into account.
As outlined and worked out earlier, e + trapping at GBs has to be treated in the framework of diffusion-reaction models in order to take the diffusion-limitation in the trapping process properly into consideration [6,[14][15][16][17][18][19][20].The mathematical approach [15], which is used here, as well as by another group [16], has the attractive feature that it yields closed-form expressions of the major e + annihilation parameters.This enables direct insight in the physical details of e + annihilation characteristics as well as a convenient application for the analysis of experimental data.
A quantitative analysis of the present model with characteristic data sets leads to the conclusion that positron trapping at GBs matters even for µm-sized crystallites and should not be neglected for precise studies of intragranular defects.From this point of view, the present work is of general relevance for positron annihilation studies of polycrystalline materials.

The model
The model describes the positron annihilation characteristics in polycrystalline materials where positrons are trapped and annihilated both in defects inside the grains and freevolume type defects in GBs.Extending the available models for spherical-shaped grains with a single type of point defect inside the grains [6,16,17], the present model considers two different types of intragranular defects with concentrations C i (i = 1, 2) and crystallites with cylindrical shape of infinite length.Trapping at the intergranular defects is handled by standard rate theory, whereas for e + trapping at GBs both the e + diffusion and the transition reaction at the grain boundary is taken into account (so called diffusion-reaction controlled trapping process).Although for the sake of conciseness, the calculation is done for cylindrical symmetry, with appropriate replacements the model results are valid for spherical symmetry as well (see section 2.5).The model results for both geometries will be compared with each other (see section 3) The behavior of the positrons is described by their bulk (free) lifetime τ f , by their characteristic lifetimes (τ i , i = 1, 2) in the two types of intragranular defects, by their lifetime (τ b ) in the GBs, and by their bulk diffusivity D. Trapping at the intragranular defects and the grain boundary are characterized by the specific e + trapping rates σ i (i = 1, 2) and α, respectively.
The temporal and spatial evolution of the density ρ g of free positrons within the grains is given by: where C i denotes the concentrations of intragranular defects.The e + trapped in GBs of cylindrical-shaped crystallites (radius r 0 ) are described by the area density, ρ b , i.e. the number of e + per GB unit area, for which the rate equation holds.The temporal evolution of the number of e + trapped in the two types of intragranular defects is given by where the number N f of positrons in the free state follows from integration of ρ g over the crystallite volume Note that detrapping of e + from the intragranular defects or from the GB trapped state is not considered.The continuity of the e + flux at the grain boundary defines the boundary condition: As initial condition, it is as usually assumed that positrons are exclusively in the free state, homogeneously distributed in the grains with initial density ρ g = ρ g (0).This is well justified since the GB volume can be neglected with respect to the crystallite volume.The grain boundary is therefore considered as strictly two-dimensional in the present model.Following the earlier works [15][16][17], the time dependence is handled by the Laplace transforms which leads to the basic equations for this diffusion-reaction model with cylindrical symmetry with and The solution of the differential equation (7a) satisfying equation (7e) can be written as with I 0 (γr 0 ) and I 1 (γr 0 ) as the modified Bessel functions.For analysis of positron annihilation experiments of relevance is the total probability n(t) that a e + implanted at t = 0 has not yet been annihilated at time t.n(t) is given by the fraction of e + per unit length of cylinder at time t: Taking into account the solutions of Ñi (equation (7d)), the Laplace transform of n(t) reads Solving the integral ( ´r0 0 rI 0 (γr)dr = (r 0 /γ)I 1 (γr 0 )) for the solution of the differential equation (8) and inserting ρb (p) (equation (7c)) yields: with For a single type of intragranular defect, i.e.C 2 = 0, ñ(p) simplifies to The Laplace transform ñ(p) (equation ( 12)) represents the entire solution of the present diffusion and trapping model from which both the mean positron lifetime and the positron lifetime spectrum can be deduced.In the following, the solutions for the general case (section 2.1), for exclusive grain boundary trapping (section 2.2), for the limiting case of reaction-controlled GB trapping (section 2.3) as well as approximate solutions for minor diffusion limitation (section 2.4) will be presented.In addition, the corresponding solutions for spherical symmetry will be shortly addressed (section 2.5).

General solution
The mean positron lifetime τ is obtained by taking the Laplace transform at p = 0: with The positron lifetime spectrum follows from ñ(p) (equation ( 12)) by means of Laplace inversion.The single poles p of ñ(p) yield the characteristic annihilation rates, i.e. the inverse of the positron lifetime components; their relative intensities is given by the residues of ñ(p).
The intensity of the grain-boundary trapped state with the characteristic annihilation rate τ −1 b (pole p = −τ −1 b ) reads or with The characteristic annihilation rates τ −1 i (poles p = −τ −1 i , i = 1, 2) of the intragranular-trapped states are associated with the intensities with The poles of ñ(p) (equation ( 12)) defined by α + γD Θ(γr 0 ) = 0 are given by the real positive solutions of (J 1 (x), J 0 (x): Bessel functions of first kind) with x = γ ⋆ j,0 r 0 , where γ ⋆ j,0 is associated with the characteristic annihilation rates λ 0,j as follows: where the index j enumerates the roots of equation (21).Equation ( 21) is based on the relation γI 1 (γr 0 )/I 0 (γr This transcendental equation has the typical form for this type of reactioncontrolled boundary condition with cylindrical symmetry [21].
From the residues the associated intensities are obtained.
The second-order pole of ñ(p) (equation ( 12) 2 ) yields no contribution.Closer inspection by means of series expansion of Θ(z) shows that the intensity associated with this pole cancels.Therefore, in summary the e + lifetime spectrum reads The sequence of annihilation rates λ 0,j > τ −1 f characterizes the rates by which positrons are removed from the free state.In practice, only one or a few components of the sequence are of relevance (see section 3).
The analogous extension of the above equations to more than two types of intragranular defects is straightforward, but of less relevance due to the limited number of e + lifetime components, that can in practice be resolved.

Special case of exclusive grain boundary trapping
For negligible trapping inside the grains, i.e.

Limiting case of reaction-controlled GB trapping
For the sake of completeness, it is shown in the following that the present diffusion-reaction model contains as special case of high e + diffusivity and/or small grain size the standard reaction model.For γr 0 ≪ 1, the function Θ (equation ( 13)) can be expanded.Restriction to first order (Θ(z for which the Laplace transform (equation ( 12)) becomes independent of the diffusivity D: By means of Laplace inversion of equation ( 29) the wellknown solution of the simple trapping model for three types of e + traps [22,23] is recovered.From the poles of equation ( 29), the e + annihilation rates in the intragranular defect trapped states (τ −1 i , i = 1, 2) and grain-boundary trapped state (τ −1 b ), and, in addition, the rate constant follows.The rate τ −1 0 , by which e + are removed from the free state, is given by the sum of the free e + annihilation rate τ −1 f and the trapping rates 2αr −1 0 and σ i C i (i = 1, 2) of GBs and intragranular defects, respectively.Calculation of the residues yields the corresponding relative intensities With ñ(p) (equation ( 29)) for p = 0 the mean positron lifetime reads

Discussion
We start the discussion with considering how the key parameters, i.e. the mean positron lifetime τ and the relative intensities I b , I i (i = 1, 2) of the defect-specific e + -lifetime components depend on the crystallite size.Figure 1 shows the intensities I b and I i (i = 1, 2) according to the general solution (equations ( 19) and (17b)) as a function of crystallite radius r 0 ; figure 2(among others) the corresponding r 0 -variation of the mean positron lifetime τ (equation ( 15)).The figures pertain to the practical relevant case that the e + -lifetimes in the GB-trapped state and in the defect-trapped state of type 1 are the same, i.e., τ b = τ 1 , and of vacancy-type.Parameters τ f , τ b , τ 1 , D, α, and σ i typical for metals are used (see caption of figures 1 and 2, D = 0.5 × 10 −4 m 2 s −1 [24]; lattice vacancies σ 1 = 4 × 10 14 s −1 [25]).For the second intragranular component, a lifetime τ 2 = 300 ps as typical for small vacancy agglomerates is assumed.For the sake of illustration, concentrations C 1 = C 2 = 10 −5 of the two intragranular defect types are used which correspond to the center of the sensibility regime of the technique of positron annihilation.
As shown in figure 2, the mean positron lifetime τ decreases with decreasing r 0 for the selected parameter set, since without or minor GB trapping τ is enhanced to values above τ b due to the proportionate trapping at the defect type 2 with longer component τ 2 = 300 ps.For the purely reaction-controlled limit, the τ -curve is shifted to higher r 0 -values, because in that case the limitation of trapping due to diffusion is neglected, i.e. the trapping at GBs is stronger.The approximation for minor diffusion-limitation describes the τ − r 0 -variation very well in the regime of small r 0 -values.
For comparison, also the case of exclusive grain boundary trapping (i.e.C 1 = C 2 = 0) is shown in figure 2, for diffusionreaction limited trapping, for entirely reaction-controlled trapping, as well as for minor diffusion limitation.The mean e +lifetime shows the typical sigmoidal increase from τ f to τ b with decreasing crystallite size.For a given r 0 -value, τ is lower for diffusion-reaction limited trapping compared to purely reaction-controlled trapping because of diffusion limitation.19), i = 1, dotted line), I 2 (equation ( 19), i = 2, dashed line), as a function of radius r 0 of cylindrical-shaped grains.For selected r 0 -values also the intensity I 0,j =1 of first component of the series I 0,j (equation ( 23  Again, the solution for minor diffusion-limitation is a good approximation in the regime of small r 0 -values.
The figures 1 and 2 pertain to cylindrical-shaped crystallites.Figure 3 shows the comparison between cylindrical-and spherical-shaped crystallites for the intensity of the GB component, I b , and the mean positron lifetime, τ , for the same parameter sets used in figures 1 and 2. Since the GB area related to the crystallite volume is higher for spherical-shaped crystallites than for cylindrical-shaped ones, meaning that the mean diffusion length to reach the GB is shorter, e + -trapping at GBs for spherical is stronger than for cylindrical symmetry for the same crystallite size.Correspondingly, the variation of I b and τ with crystallite radius r 0 for spherical-shaped crystallites is shifted to higher r 0 -values compared to cylindricalshaped ones.The same conclusion has been drawn earlier by Dryzek and coworker [26], however, for a diffusion-reaction model where trapping at intergranular defects was not taken into account.
In the following we address the important issue that even for µm-sized crystallites e + trapping at GBs matters for correctly determining of concentrations of intragranular defects.From figure 1, already, it becomes clear that trapping at intragranular defects would be overestimated if e + trapping and annihilation at GBs is not taken into consideration, meaning that the intensity I b due to GBs is added to that of the intragranular trap component (I 1 ).To visualize this quantitatively, figure 4 shows the supposed excess concentration ∆C 1 that arises when the sum I b + I 1 of GB intensity and intragranular trap type 1 intensity is interpreted as arising from trap type 1, exclusively.It is evident, that in this way the trap concentration C 1 would be severely overestimated, with increasing trend for decreasing crystallite size.For spherical-shaped crystallites, this effect is even stronger than for cylindrical-shaped ones owing to the stronger GB trapping in the former case (compare red and black curves in figure 4).
The extent of overestimation depends on the characteristic parameters for GB trapping, i.e. on the e + -diffusivity D and the specific trapping rate α.The figures shown above refer to typical parameter sets for D and α.Specific e + trapping rates α of GBs range from 200 ms −1 for Zn alloys to 3 × 10 3 ms −1 for Al alloys [14].A substantially lower specific e + trapping rate α = 70 ms −1 is reported for interfaces between matrix and semi-coherent precipitates [27].To quantify the influence of α and D, table 1 shows the intensity of the GB component I b (equation (17b)) for crystallites with a diameter (2r 0 ) of 10 −6 m and various α-values, i.e. the maximum value of with exact solution of I b (equation (17b)) and I 1 (equation ( 19)) for concentration C 1 .The GB trapping is still significant when competitive trapping at intragranular defects occurs as quoted in table 1 for a defect concentration C 1 of 10 −5 , when the e + -lifetimes in both trapped states are identical (τ 1 = τ b ).If in such a case one would not consider the e + -trapping at GBs, quantitatively meaning that the GB component I b is added to the intensity I 1 of the intragranular defect component (see caption figure 4), the concentration C 1 of this defect would be overestimated strongly, e.g. by ∆C 1 = 0.55 × 10 −5 or 0.44 × 10 −5 for the examples I b = 0.25 (D = 10 −4 m 2 s −1 ) or I b = 0.20 (D = 0.5 × 10 −4 m 2 s −1 ) of cylindrical-shaped crystallites given in table 1, i.e. the determined concentration would be 55% or 44% too high, respectively.For spherical-shaped crystallites, the overestimation would even be 87% or 69% for the same parameter set for D = 10 −4 m 2 s −1 or 0.5 × 10 −4 m 2 s −1 , respectively.
In conclusion, e + -trapping at GBs matters even for µmsized crystallites and should not be neglected for precise studies of intragranular defects.

Figure 2 .
Figure 2. Mean positron lifetime τ as a function of radius r 0 of cylindrical-shaped grains for exact solution (equation (15), black solid line), for limiting case of entirely reaction-controlled GB trapping (equation (32) black dashed line), and for approximate solution for minor diffusion limitation (equation (35), black dotted line) for the same parameter set as in figure 1.In addition both trapping models and the approximate solution for minor diffusion limitation are shown for exclusive grain boundary trapping (equation (26); equations (32) and (35) with C i = 0, blue lines).

Figure 3 .
Figure 3. Intensity of GB component, I b (dotted lines), and mean positron lifetime τ (solid and dashed lines) as a function of radius r 0 of cylindrical-shaped grains (black) in comparison to spherical-shaped grains (red).The curves for the cylindrical symmetry are the same as shown in figure 1 (I b ) and in figure 2 (τ ) for the exact solution.For the spherical symmetry the same parameter sets as quoted in the caption of figure 1 is used.The lower τ -curves (dashed lines) correspond to exclusive GB trapping (compare figure 2).

Table 1 . 3 ×
Intensity I b (equation (17b)) of GB component for exclusive GB trapping (C 1 = 0) at cylindrical shaped crystallites with radius r 0 = 0.5 µm and for the case of competitive trapping at intragranular defects with concentration C 1 and identical e + lifetime (τ 1 = τ b ) for selected values of e + diffusion coefficient D and specific GB trapping rate α.Parameters: τ f = 120 ps, τ b = τ 1 = 180 ps, σ 1 = 4 × 10 14 s −1 , C 2 = 0. 10 3 ms −1 for GBs, the value of 70 ms −1 for semicoherent interfaces, and the value 10 3 ms −1 used in the figures considered characteristic for GBs.In addition to the D-value taken for the figures, the I b -intensities are quoted for a value twice (D = 10 −4 m 2 s −1 ) as high.The high I b values show that substantial GB trapping occurs although the crystallite diameter of 1 µm by far exceeds the characteristic e + diffusion length Dτ f ≃ 0.11 µm (for D = 10 −4 m 2 s −1 and τ f = 120 ps).Even for semicoherent interfaces with reduced α, GB trapping may not be neglected.