ISSN 0031918X, The Physics of Metals and Metallography, 2012, Vol. 113, No. 12, pp. 1107–1113. © Pleiades Publishing, Ltd., 2012. Original Russian Text © A.G. Kesarev, V.V. Kondrat’ev, I.L. Lomaev, 2012, published in Fizika Metallov i Metallovedenie, 2012, Vol. 113, No. 12, pp. 1173–1179.

THEORY OF METALS

Special Features of GrainBoundary Diffusion in Nanostructural and Submicrocrystalline Materials Caused by Structural Heterogeneity of Grain Boundaries A. G. Kesarev, V. V. Kondrat’ev, and I. L. Lomaev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi, 18, Ekaterinburg, 620990, Russia email: [email protected] Received April 05, 2012, in final form May 29, 2012

Abstract—Mechanisms of grainboundary diffusion are analyzed based on models of structurally heteroge neous grainboundaries and multipath diffusion in a system of structural elements responsible for accelerated mass transfer. The effective (averaged) coefficients of grainboundary diffusion are obtained in models of continuous and discrete distribution of partial activation energies. The results of originally processed diffu sion data in some submicrocrystalline materials are presented. The conclusion is drawn that, at low temper atures, the mass transfer in these materials is achieved through a small portion of active diffusion paths, pre sumably via grain triple junctions. Keywords: nanostuctural and submicrostructural crystalline materials, grainboundaries, diffusion DOI: 10.1134/S0031918X12120046

INTRODUCTION It is known [1] that grainboundary diffusion (GBD) in nanostuctural and submicrostructural crys talline materials (SMCs) obtained by severe plastic deformation (SPD) proceeds faster than in conven tional polycrystals. Correspondingly, GBD activation energy is two times or more lower than in coarse grained materials [2] and its value is near the activation energy of surface diffusion. The decrease in GBD in SMC materials after heat treatment is also typical, which, taking into account the unchanged grain sizes, indicates the relaxation of nonequilibrium and, in general, heterogeneous structural state of GBs. There is another interesting feature in these materi als that is worth noting, i.e., the low preexponential fac tor in the GBD coefficient Dg (several orders below usual value) [3]. In this case, diffusion data are usually processed according to the standard Fisher model [4], in which the grain boundary (GB) is considered to be a uniform isotropic layer with the width δ at the interpla nar scale, and is characterized by the diffusion coeffi cient Dg. In most cases, the diffusion experiments are conducted at lowered temperatures (in C regime) when, in the absence of relaxation processes in GBs and under the condition of frozen bulk diffusion, the coeffi cient Dg is found individually from the concentration profile. Therefore, this significant decrease in the coef ficient cannot be associated with a decrease in the GB thickness. This note is equally related to the Вtype dif fusion regime, in which the product Dgδ is found. In this

case, the use of the standard Fisher model leads to abnormally small δ (e.g., in [5] δ  0.05 nm). As will be shown in this work, these special features can be explained by the sensitivity of diffusion to the inner structure of GBs at low temperatures (in the known models [6], there are amorphous islands, core areas of dislocation, misfits of lattice sites, etc.), and by the possible contribution from diffusion paths to mass transfer (high angle GBs, triple junctions, GB dislocations, and disclinations) characterized by the spectrum of activation energies. For example, the dependence of GB energy and, as a result, the depen dence of the GBD activation energy on the misorien tation angle in bicrystals has been found [4]. In nanoc rystalline and SMC materials, diffusion proceeds pre dominantly in GB segments, where a change in the diffusion characteristics may be caused by excess energy or excess free volume [7]. In the case of com pacted materials, a significant contribution in mass transfer can be made by diffusion via micropore sur faces. Nano and SMC materials produced by SPD have high internal stresses localized near GBs [8]. According to [9], these stresses can form GB zones of accelerated mass transfer. Thus, during the interpretation of diffusion data, problems arise concerning the physical sense of diffu sion characteristics Dg and δ found from profile mea surements and the possibility of calculating partial dif fusion coefficients in a GB. In this case, the choice of a diffusive annealing regime within the adequate GBD

1107

1108

KESAREV et al.

model is the most important. The solution of these problems is beyond the scope of the standard Fisher model and requires the development of new ideas on GBD based on the model of structurally heteroge neous GBs and multipath diffusion in a system of structural elements responsible for accelerated mass transfer, which is the purpose of this work. The results of the original treatment of the existing literature data on GBD in some SMC materials are presented as practical application. MODELS OF HETEROGENEOUS GRAIN BOUNDARIES In diffusion experiments, the density of layer activ ity or the proportional characteristic, the average layer concentration C (x, t) depending on the distance x to free surface of a sample (a source of diffusing agent) and time t are measured. The solution of the diffusion problem within the simple Fisher model [4] allows one to find the GB characteristics, i.e., Dg and δ, from the concentration curves. In a practically important regime for nanocrystals С fulfilled under the condition DV t Ⰶ δ, where DV is the bulk diffusion coefficient, the concentration profiles C (x, t) are described by an error function (steady source) or the Gauss function (instant source of diffusion agent) 2 ⎞ ⎛ ⎞ ⎛ C (x, t ) ~ erfc ⎜ x ⎟ ; C (x, t ) ~ 1 exp ⎜ − x ⎟ , (1) Dgt ⎝ 4D g t ⎠ ⎝ 2 Dgt ⎠ where from GBD coefficient Dg is found. Under the condition 100δ < DV t < L/20 [4], where L is the aver age grain size, the Btype diffusion regime is realized. For this regime, the Whipple general solution is known [4] for the case of a steady source, which enables one to calculate the triple product from profile measure ments of C (x, t) as follows:

−5 3

D (2) sD g δ = 1.322 V ⎛⎜ − ∂ ln6 C5 ⎞⎟ , t ⎝ ∂x ⎠ where s is GB segregation coefficient (s = 1 for self dif fusion, diffusion of isotopes). It can be assumed that the coefficient DV can be found from independent dif fusion experiments. Furthermore, generalized models of GB and diffusion regimes may be of interest, and their solutions can be expressed by formulas (1) or (2), where Dg and δ are structurally dependent effective characteristics of GBs; their weak time dependence is also possible [10]. There are two different approaches to describing structurally heterogeneous GBs and diffusion within them. The first approach can be considered micro scopic because it assumes the detailed modeling of GBs. In [10] a model of a GB saturated zone with a width of Δ ≥ δ is considered, which is in equilibrium with the GB characterized by Dg and δ and serves as a

sink for the diffusion agent (diffusion along the GB area is slow). The appearance of this zone of quasi bulk diffusion is caused by the localization of internal stresses near nonequilibrium GBs [9]. There is almost no bulk diffusion beyond the saturated zone. This regime corresponds to the conditions D t Ⰷ Δ, DV Ⰶ D Ⰶ Dg, where D is the averaged diffusion coef ficient in the GB area. As a result, an analog of the equation of GBD in the С regime is obtained, but with the increased effective GB width δef = (δ + 2Δ) and decreased effective GBD coefficient

D gef =

(δ +δ2Δ) D .

(3)

g

Thus, the effective coefficient of GBD under the conditions of the saturation of the GB zone is several times lower than the initial coefficient, which corre sponds to short annealing times, i.e., D t Ⰶ Δ. A similar model was proposed earlier in [11] for the interpretation of Mössbauer data obtained on poly crystalline samples after low temperature annealing. In this case, the generalization of the Fisher model involves the introduction of a GB atomic layer charac terized by equilibrium impurity concentration. Another approach to the GBD problem is essen tially macroscopic because it uses he continual description of the diffusion of atoms in media with microstructure [12]. It is assumed that a physically small volume contains a family of mutually penetrat ing n diffusion paths with different diffusion activation energies and concentrations of diffusing agent. Only the paths for which the percolation criterion is fulfilled are taken into account. Then, for partial concentra tions of impurity C i*, the following interconnected system of diffusion equations can be written:

∂Ci* = Di ΔCi* + ∂t

n

∑ν C* ij

j

(i = 1,2,…, n).

(4)

j =1

Here, C i* = Cipi is the volume fraction of the diffusing agent concentration in the i path, which occupies the rel ative volume fraction pi; Сi is the usual volume concentra tion in the i path; and the normalization condition n

∑p

i

n

= 1,

i =1

∑ C * = C, i

i =1

is fulfilled, where C is the entire volume concentration of a diffusing agent in paths of accelerated diffusion; νij are frequencies of transitions between i and j paths, Di = Di0exp(–Qi/kT) are the partial diffusion coeffi cients, and Di0 and Qi are the corresponding preexpo nential factors and activation energies. In [13], this approach was used to describe GBD in the В regime for GBs that consist of two structural components with

THE PHYSICS OF METALS AND METALLOGRAPHY

Vol. 113

No. 12

2012

SPECIAL FEATURES OF GRAINBOUNDARY DIFFUSION

considerably different diffusion coefficients D1 Ⰷ D2 and generally unequal effective widths δ1 ≠ δ2. The values Qi ∈ [Qm, QM], where Qm, QM are mini mum and maximum of activation energies, respec tively. Since partial coefficients Di most strongly depend on activation energies Qi, we shall hereafter assume that a difference between preexponential fac tors for different paths can be neglected, i.e.: Di0 ≡ D0. We also suppose that all diffusion paths of accelerated mass transfer form intergranular space with entire effective width ΔS. Equation (4) allows for a simple analysis in a Сtype regime (bulk diffusion is frozen) for two prac tically interesting extreme cases that correspond to short and relatively long annealing times. In the first case, diffusion only proceeds along one path of accel erated diffusion, and the percolation of a diffusion agent in other paths can be neglected. Then, the effec tive diffusion coefficient will be of maximum magni tude and heterogeneity of the boundary and will only have an effect on the resulting average concentration. In the second case, there is diffusive mixing between the paths such that the concentrations of the diffusing agent at some distance from a source become equal for all paths (Ci = C). Then, according to (4), GBD is characterized by average weighed diffusion coefficient as follows: n

D gef

=

∑pD. i

i

where DM = D0exp(–QM/kT) corresponds to the max imum activation energy QM. For QM, the activation energy for structureless GBs in coarse grain materials Qg can be taken. Under the far stronger condition

l M > L,

We shall consider in detail the second case, i.e., when entire diffusion mixing (concentration equilib rium) between paths is realized, in detail. The feature that structurally heterogeneous GBs are characterized by homogenous concentrations is essential for this stage. In the model of parallel diffusion paths (e.g., the boundary–near boundary zone [10]), mixing proceeds between different paths when the minimum diffusion length becomes compatible with an effective thickness of GBs, which corresponds to the condition

l M = DM t > Δ S ,

(6)

THE PHYSICS OF METALS AND METALLOGRAPHY

∫ exp ( − kT ) F (Q ) dQ,

QM

D = D0

Q

(8)

Qm

where F(Q) is the distribution function of diffusion paths normalized per unit by activation energies Q on the segment [Qm, QM]. We shall introduce Q , the activation energy Q aver aged with F(Q), and abmodality ΔQ = Q – Q. The expansion of the exponential function by Tailor series gives an average diffusion coefficient expressed via the central moments of activation energies ∞

( )

n

Q 1 − 1 Δ Q n. D = D0 exp ⎛⎜ − ⎞⎟ ( ) ⎝ kT ⎠ n=1 n ! kT

∑

i =1

MUTIPATHS APPROACH TO GBD THEORY

(7)

when diffusion length becomes compatible with the grain size, GBD becomes onedimensional and the concentration profiles depend only on the distance from the source. The logical generalization for expression (5) for the continuous activation energy distribution assumes that, in this case, the role of the effective coefficient D gef will be played by the average coefficient

(5)

An important conclusion follows from (5), i.e., if ine qualities p1D1 Ⰷ p2D2 Ⰷ …, are fulfilled then effective fac tor of GBD D gef  p1D1 is determined only by the diffu sion path with a maximum diffusion coefficient D1, and effective preexponential factor p1D10 in D gef can take very small values under condition p1 Ⰶ 1. The intermediate case, i.e., when a diffusion agent is transferred between various paths at a finite velocity is complicated for analy sis and for twopaths diffusion is characterized by three parameters, including two partial diffusion coefficients D1, D2 and frequency ν = ν21 [13].

1109

(9)

At rather small differences between the diffusion activation energies for different paths, i.e., when

QM − Qm (10) Ⰶ 1, kT one can confine consideration to two nonzero terms in (9) as follows: 2 Q ⎡ ΔQ ⎤ D  D0 exp ⎛⎜ − ⎞⎟ ⎢1 + ⎥. ⎝ kT ⎠ ⎣ 2(kT )2 ⎦

(11)

Here, Δ Q 2 is the dispersion of the activation energy. It follows from this that, under condition (10), the effective diffusion coefficient is determined by the average activation energy. As a rule, diffusion experi ments in nanostructural materials are conducted at lowered temperatures, when the case opposite to (10) is the most interesting. The common expansion (9) is valid, but becomes inefficient because it contains an infinite number of unknown parameters. We shall sep arately consider the cases of continuous and discrete spectra of activation energies. Let us assume that the ensemble of accelerated dif fusion paths is characterized by the continuous distri Vol. 113

No. 12

2012

1110

KESAREV et al.

bution function F(Q) = F(Qmq) = f(q). Then, integral (8) can be presented as follows

∫

exp ( −µq ) f ( q ) dq,

(12)

i =2

1

Qm Ⰷ 1. In order to estimate integral (12) at kT μ → ∞, we shall use Laplace method [15]. It is known that, for two infinitely differentiated functions h(t)and ϕ(t), at μ → ∞, under the condition that the function h(t) reaches its maximum at point a, and its derivative at this point is h'(a) < 0, the following evaluation is performed: and μ =

b

∫ ϕ (t )e

µh( t )

dt = −e

µh(a)

a

⎡ N 1 ϕ k −1 ( a ) ⎛ 1 ⎞⎤ , (13) + O ⎢ ⎜ N +1 ⎟⎥ k ⎝ λ ⎠⎥⎦ ⎢⎣ k =1 λ h' ( a )

∑

ϕ (t ) where ϕ0(t) = ϕ(t), ϕk + 1(t) = − d ⎡⎢ k ⎤⎥ . In our case, dt ⎣ h'(t ) ⎦ k t = q, h(t) = –q, ϕ0(t) = f(q), and ϕk(t) = d k f(q). As a dq result, the following expression is derived:

( )∑

N n −1 ⎤ Qm ⎡ nd F D  D0 exp − ⎢ ( kT ) ⎥. kT ⎣⎢ n=1 dQ n−1 Q =Qm ⎦⎥

(14)

Under the condition (kT)⎛⎜ 1 dF ⎞⎟ Ⰶ 1, one can ⎝ F dQ ⎠ Q =Qm confine himself to the first expansion term in [14] and use the simple expression

( kTQ ) .

D  D0[kTF (Q m )]exp −

m

(15)

It can be seen that expression (15) differs princi pally from (11). In the case of (15), the temperature enters explicitly into the preexponential factor and the effective diffusion coefficient is determined by mini mum activation energy. We shall now turn to the case (5), when the activa tion energy along diffusion paths takes a discrete series of values Qi and is characterized by probabilities pi. Then, the averaged diffusion coefficient is

∑ p exp ( − kT ) n

D = D0

Qi

i

i =1

(

) ( )

(16)

n ⎡ (Q − Q1) ⎤ pi Q exp − i = D0 p1 ⎢1 + ⎥ exp − 1 p kT kT ⎢⎣ ⎥⎦ i =2 1 (we assume that Qi ≤ Qi + 1, Q1 ≡ Qm). If the interval of energies ΔQ ~ kT contains a large number of paths of accelerated diffusion, their distribution can be consid ered quasicontinuous, and the abovementioned

∑

∑ p exp ( − n

QM Qm

D = D0Qm

model can be considered. In the opposite case, when |Q1 – Qi| Ⰷ kT, and the following equation is fulfilled:

pi

1

)

Qi − Q1 Ⰶ 1, kT

(17)

furthermore, one can confine himself to the first expansion term in (16) and assume

( kTQ ) .

D  D0 p1 exp −

1

(18)

Expression (18) describes diffusion only via active paths with activation energies Q1, the fraction of which is p1. A necessary condition for the fulfillment of ine quality (17) is the fulfillment of all (n – 1) inequalities Qi – Q1 Ⰷ kTln(p1/pi), (i ≥ 2). If Qi ≥ Q2, only the suf ⎛1 − p1 ⎞ ficient condition Q2 – Q1 Ⰷ kTln⎜ ⎟ can be taken ⎝ p1 ⎠ into account. According to the accepted approximations (taking into account only the first expansion terms in (14) and (16)) the models of the continuous and discrete activa tion energy distribution occur in the same way, and p1 in (18) corresponds to pm = kTF(Qm) in (15). As is easy to see, the physical meaning of the value pm is that the activation energies of diffusion paths responsible for mass transfer during GBD are concentrated in the kT band adjusted to Qm. The choice of the first or second model for a given system can be made after more detailed measurements are performed to find the tem perature dependence of the preexponential factor because, in the case of the continuous distribution of activation energy Q, there is a linear dependence of pm = kTF(Qm) on temperature, while at a discrete dis tribution of Q, there is no T dependence of this mag nitude. TREATMENT OF DIFFUSION DATA We shall apply the considered models of multipaths diffusion for processing of literature diffusion data [2, 3, 16] using the temperature dependencies of GBD coefficients in SMC materials (see Table 1). Analysis of the temperature dependence of GBD coefficient for coarsegrained materials is usually per formed using a standard method of linear regression of the logarithm of the diffusion coefficient on the inverse temperature as follows: (19) ln(D) = a + b 1 , T where under conditions close to the С regime and according to the model of structureless GBs, when Dg = D0exp(–Qg/RT) (hereafter activation energy is measured in J/mol), the preexponential factor and activation energy are found by the formulas

D0 = exp ( a ) , Q g = −bR.

THE PHYSICS OF METALS AND METALLOGRAPHY

Vol. 113

(20) No. 12

2012

SPECIAL FEATURES OF GRAINBOUNDARY DIFFUSION

1111

Table 1. Literature data on diffusion Diffusion agent in material Cu in Ni

Dg, m2/s, coarsegrained

Temperature, K 398 423 443 773 823 873 973 1073 1123 1273 1373 423 448 473

Ni in Mo

Co in Ti

Dg, m2/s, SMC

Reference

5.06 × 10–15 9.6 × 10–15 2.2 × 10–14

[3]

1.0 × 10–13 4.4 × 10–13 7.8 × 10–13

[2]

8.4 × 10–15 1.7 × 10–14 4.1 × 10–14

[16]

4.0 × 10–13 1.26 × 10–12 3.9 × 10–12

8.7 × 10–15 1.9 × 10–13 1.0 × 10–12 9.0 × 10–17 3.6 × 10–16 2.0 × 10–15

Table 2. Model of discrete distribution of diffusion paths D0, m2/s

Qg, kJ/mol

Q1, kJ/mol

p1, J–1

Cu in Ni

1.65 × 10–4

127.6

47.1

4.4 × 10–5

Ni in Mo Co in Ti

1.83 × 10–3 4.19 × 10–4

243.3 102.8

125.1 52.5

2.9 × 10–4 6.0 × 10–5

Diffusion agent in material

This method is also suitable for a discrete spectrum. When GBD is governed by one diffusion path with a minimum activation energy (18), we obtain

p1 = 1 exp ( a ) , Qm ≡ Q1 = −bR. D0

(21)

Here, factor D0 is considered to be approximately the same as in the case of the GBs of coarsegrained materials (20). This approach is justified because the predicted differences between D0 factors for coarse grained and SMC materials can be dozens of percents (they are related to the difference between the Debye frequencies in conventional and nonequilibrium GBs), and D0 and effective factor D0p1 (18) differ by several orders of magnitude. In the case of the continuous distribution of the activation energy, when expression (15) is applied for

the effective coefficient of GBD, it is necessary to con duct linear regression for a difference between loga rithms of the GBD coefficient and temperature (22) ln(D) − ln(T ) = a1 + b1 1 . T Then, the minimum value of the activation energy and the energy distribution function for this value are found from the equations (23) Qm = −b1R, F ( Qm ) = 1 exp(a1), RD0 where the D0 factor also corresponds to the coarse grained state of a material. The results of treatment of the experimental data in the frames of the models of discrete and continuous distribution of diffusion paths by energies are pre sented in Tables 2 and 3. There, for comparison, the

Table 3. Model of continual distribution of diffusion paths Diffusion agent in material Cu in Ni Ni in Mo Co in Ti

D0, m2/s

Qg, kJ/mol

Qm, kJ/mol

F(Qm), J–1

pm

1.65 × 10–4 1.83 × 10–3 4.19 × 10–4

127.6 243.3 102.8

43.61 116.4 48.84

4.7 × 10–9 1.2 × 10–8 5.9 × 10–9

1.5 × 10–5–1.7 × 10–5 1.1 × 10–4–1.4 × 10–4 2.3 × 10–5–2.1 × 10–5

THE PHYSICS OF METALS AND METALLOGRAPHY

Vol. 113

No. 12

2012

1112

KESAREV et al.

Table 4. Diffusion GBD lengths (minimum lM, maximum lm) Diffusion agent in material Cu in Ni Ni in Mo Co in Ti

L, nm

lM, nm

lm, nm

300 450 320

7–50 2 × 103–1 × 104 1 × 103–6 × 103

1 × 104–2 × 104 4 × 104–1 × 105 1 × 104–3 × 104

activation energies Qg for GBs in coarsegrained mate rials are shown. It can be seen that paths for acceler ated diffusion have far lower activation energies than the activation energy Qg obtained in the standard Fisher model. It is approximately onehalf of Qg for Ni in Mo and Co in Ti systems and about onethird for Cu in the Ni system. As follows from Table 2, the frac tion of the active diffusion path along which a diffu sion agent is transferred at great distances, is ~10–4. Thus, a portion of the accelerated diffusion paths is very small and varies from 10–4 to 10–5 for all systems. Our consideration can be applied at the stage when diffusive mixing proceeds between the accelerated dif fusion paths. In the case when diffusive properties in a sample are equal in all directions (no texture) and mix ing proceeds via transverse diffusion paths, at a scale ~L of the grain size, this condition takes the form (7). Taking annealing time 5 h as in [2, 16], we shall estimate the diffusion length lM for maximum activa tion energies, which will be taken as Qg (see Table 2) and we shall test the fulfillment of condition (6). The calculated values of these diffusion lengths and the corresponding values lm = Dmt for active diffusive paths, where Dm = D0exp(–Qm/RT), are shown in Table 4. It can be seen that lm is several orders of mag nitude higher than lM, and the condition of mixing is appropriately met for the Ni in Mo and Co in Ti systems, while for Cu in an Ni system, this condition is violated. Taking into account the finite velocity of the outflow of a diffusing agent from the active path, this can be consid ered a possible reason for the effective activation energy being lowered. Most likely, the latter case requires the use of more complicated models, which take into account the finitude of the mixing rate [13]. A small volume fraction of active diffusive paths shows that they may be triple junctions. According to [17], specific length of the triple junctions per unit vol ume is ~λ/L3, where L is average grain size, the coef ficient λ ≈ 3 (depends on grain configuration). If it is assumed that the thickness of a triple junction is equal to the unit cell parameter a and the entire GB width is ΔS > a (in dislocation model of nonequilibrium GBs [9], it has been shown that the efficient diffusion length of the GB can reach several unit cell parame ters); then, the volume fraction of GBs p1(tj ) that make

triple junctions can be approximately calculated as follows: (24) p1(tj) ≈ 1 (a Δ S )(a L). 3 The factor of 1/3 in (24) is added for the sake of geom etry and corresponds to a system of cubic grains in which just onethird of the triple junctions participates in diffusion from a source to the bulk of a sample. For grain sizes from 270 [16] to 450 [2] nm and extremely high ratios ΔS/a ≈ (3–5), the fraction of triple junc tions p1(tj ) is from 4.4 × 10–5 to 1.2 × 10–4, which coin cides in order of magnitude with the values of volume fraction of active diffusion paths p1 shown in Table 2. CONCLUSIONS These special features of GBD in nanostructural and SMC materials produced by SPD technique, such as the low activation energy and small preexponential factor, are explained by the heterogeneity of GBs when mass transfer proceeds along individual (active) diffu sion paths with lowered activation energies, the vol ume fraction of which is small, and the diffusing agent actively outflows to other regions in GBs. In this case, depending on the time of diffusion annealing, specific concentrations of the diffusing agent in different diffu sion paths can differ or coincide (diffusion mixing between the paths is achieved). When there is complete mixing between diffusion paths, a heterogeneous GB can be characterized by the efficient (averaged) diffusion coefficient, which is determined by processing the diffusion data. If there is no such mixing, the description of GBD by the stan dard Fisher model is incorrect. In the last case, more detailed experiments are required, as well as the use of the theory of structurally heterogeneous GBs, such as, e.g., in [13]. In the nanostructural materials considered in this work, mass transfer along triple junctions is most probable at lowered temperatures. ACKNOWLEDGEMENTS This work was carried out under the program of Russian Academy of Science, key number “Structure No. g.r. 01.02.006.13392” and partially financially supported by the Ural Branch of the Russian Academy of Sciences, project No. 12U21004.

THE PHYSICS OF METALS AND METALLOGRAPHY

Vol. 113

No. 12

2012

SPECIAL FEATURES OF GRAINBOUNDARY DIFFUSION

REFERENCES 1. Yu. A. Kolobov and R. Z. Valiev, GrainBoundary Diffu sion and Properties of Nanostructural Materials (Nauka, Novosibirsk, 2001) [in Russian]. 2. G. P. Grabovetskaya, I. P. Mishin, I. V. Ratochka, S. G. Psakhie, and Yu. R. Kolobov, “GrainBoundary Diffusion of Nickel in Submicrocrystalline Molybde num Processed by Severe Plastic Deformation,” Tech. Phys. Lett. 34, 136–138 (2008). 3. G. P. Grabovetskaya, I. V. Ratochka, Yu. R. Kolobov, and M. N. Puchkareva, “A Comparative Study of GrainBoundary Diffusion of Copper in Ultrafine Grained and CoarseGrained Nickel,” Phys. Met. Metallogr. 83, 310–313 (1997). 4. I. Kaur and W. Gust, Fundamentals of Grain and Inter phase Diffusion, 2nd ed. (Ziegler, Stuttgart, 1989). 5. V. B. Vykhodets, E. V. Vykhodets, B. A. Gizhevskii, et al., “Grain Boundary SelfDiffusion of Tracer 18O Atoms in Nanocrystalline Oxide LaMnO3 + δ),” JETP Lett. 87, 115–119 (2008). 6. O. A. Kaibyshev and R. Z. Valiev, Grain Boundaries and Properties of Metals (Metallurgiya, Moscow, 1987) [in Russian]. 7. V. N. Chuvil’diev, Nonequilibrium Grain Boundaries in Metals. Theory and Applications (Fizmatlit, Moscow, 2004) [in Russian]. 8. R. Z. Valiev and I. V. Aleksandrov, Nanostructural Materials Obtained by Severe Plastic Deformation (Logos, Moscow, 2000) [in Russian].

THE PHYSICS OF METALS AND METALLOGRAPHY

1113

9. A. G. Kesarev and V. V. Kondrat’ev, “Some Problems of Diffusion Theory in Nanostructured Materials,” Inorg. Mater.: Appl. Res. 2, 65–69 (2011). 10. A. G. Kesarev, V. V. Kondrat’ev, and I. L. Lomaev, “On the Theory of GrainBoundary Diffusion in Nanostructured Materials under Conditions of Satura tion of the Subboundary Region by the Diffusant,” Phys. Met. Metallogr. 112, 44–52 (2011). 11. V. V. Popov, “Analysis of Possibilities of Fisher Model Development,” Solid State Phenom. 138, 133–144 (2008). 12. E. C. Aifannis, “A New Interpretation of Diffusion in HighDiffusivity Paths,” Acta Metall. 27, 683–691 (1979). 13. V. V. Kondrat’ev and I. Sh. Trakhtenberg, “Grain Boundary Diffusion of Atoms in Model of Structurally Heterogeneous Boundaries,” Fiz. Met. Metalloved. 62), 434–441 (1986). 14. V. A. Sevast’yanov, Course of Probability and Mathemat ical Statistics (Fizmatlit, Moscow, 1982) [in Russian]. 15. A. M. Il’in and A. R. Danilin, Asymptotical Methods in Analysis (Fizmatlit, Moscow, 2009) [in Russian]. 16. Yu. R. Kolobov, A. G. Lipnitskii, I. V. Nelasov, and G. P. Grabovetskaya, “Investigations and Computer Simulations of the Intergrain Diffusion in Submicro and Nanocrystalline Metals,” Russ. Phys. J. 51, 385– 399 (2008). 17. A. G. Lipnitskii, “Grain and Triple Joint Boundary Energy in Nanocrystalline Materials,” Materialovede nie, No. 2, 2–10 (2009).

Vol. 113

No. 12

2012