Grain boundary diffusion: recent progress and future research

Materials Science and Engineering A260 (1999) 55 – 71

Grain boundary diffusion: recent progress and future research Y. Mishin a,*, Chr. Herzig b a

Department of Materials Science and Engineering, Virginia Polytechnic Institute and State Uni6ersity, Blacksburg, VA 24061, USA b Institut fu¨r Metallforschung, Uni6ersita¨t Mu¨nster, Wilhelm-Klemm-Str. 10, D-48149, Mu¨nster, Germany

Abstract Grain boundary (GB) diffusion often controls the evolution of structure and properties of engineering materials at elevated temperatures. A knowledge of diffusion characteristics of GBs and deep fundamental understanding of this phenomenon are critical to many materials applications. In this paper we give an overview of boundary diffusion theory with emphasis on the interpretation of concentration profiles measured in diffusion experiments. We consider the most important situations encountered in boundary diffusion experiments, such as diffusion in the B and C regimes and diffusion in the presence of segregation. We also discuss the recent progress in the atomistic interpretation of GB diffusion. We conclude with an outlook for future research in this area. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Grain boundary diffusion; Grain boundary segregation; Radiotracer method; Monte Carlo simulations

1. Introduction Grain boundary (GB) diffusion plays a key role in many processes occurring in engineering materials at elevated temperatures, such as Coble creep, sintering, diffusion-induced GB migration (DIGM), different discontinuous reactions, recrystallization and grain growth [1]. On the other hand, GB diffusion is a phenomenon of great fundamental interest. Atomic migration in a GB should be treated as a correlated walk of atoms in a periodic quasi-2D system with multiple jump frequencies [2,3]. It should be remembered that many fundamental properties of random walkers, such as the return provability pr depend on the dimensionality of the system (e.g. pr =1 in 1D and 2D systems and pr B 1 in 3D systems [4]). GB diffusion is sensitive to the GB structure and chemical composition, and on the other hand it can be studied by modern radiotracer methods without disturbing the GB state. Due to these unique features, GB diffusion measurements can be used as a tool to study the structure and physical properties of GBs. Although the fact that GBs provide high diffusivity (‘short circuit’) paths in metals has been known since the 1920–1930s, the first direct proof of GB diffusion * Corresponding author. Tel.: +1-540-231-9019; fax: + 1-540-2318919; e-mail: [email protected].

was obtained in the early 1950s using autoradiography [5]. The autoradiographic observations were followed by two important events, the appearance of the famous Fisher [6] model of GB diffusion, and the development and extensive use of the radiotracer serial sectioning technique. It was largely due to these events that GB diffusion studies were put on quantitative grounds and GB diffusion measurements became the subject of many investigations. During the four decades that followed, the experimental techniques for GB diffusion measurements have been drastically improved and have been extended to a wide temperature range and a broad spectrum of metallic, semiconductor and ceramic materials. On the other hand, the Fisher model, being still the foundation of GB diffusion theory, has been subject to careful mathematical analysis and extended to many new situations encountered in either diffusion experiments or various metallurgical processes. The fundamentals and recent achievements in the area of GB diffusion have been recently summarized by Kaur, Mishin and Gust [1]. A complete collection of experimental data obtained up to the end of the 1980s can be found in the handbook of Kaur, Gust and Kozma [7]. In this section we review the most recent progress in this area and formulate some recommendations for future research. We begin by presenting a brief overview of the fundamentals of GB diffusion in order to establish a general background for the remainder of

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the section. We then discuss the recent developments in three research directions which have relevance to fracture at GBs, namely, GB diffusion – segregation effects, diffusion studies on oriented bicrystals, and atomistic modeling of GB diffusion. We conclude with a short summary and outlook. We restrict our attention to metals and metallic systems.

concentration of the diffusing atoms in the volume and cb(y, t) is their concentration in the boundary. The second term in the right-hand side of Eq. (2) takes into account the leakage of the atoms from the GB to the volume. The solution of Eq. (1) and Eq. (2) should meet a certain surface condition (see below) and natural initial and boundary conditions at x and y“ . Fisher [6], considered a constant source condition at the surface:

2. Fundamentals of grain boundary diffusion

c(x, 0, t)=c0 = const.

2.1. Basic equations of grain boundary diffusion Most mathematical treatments of GB diffusion are based on Fisher’s model [6], which considers diffusion along a single GB. According to Fisher’s model, the GB is represented by a high-diffusivity, uniform, and isotropic slab embedded in a low-diffusivity isotropic crystal perpendicular to its surface (Fig. 1). The GB is described by two physical parameters: the GB width, d, and the GB diffusion coefficient Db. It is assumed that Db D, where D is the volume diffusion coefficient. One more physical parameter, the GB segregation factor, will be introduced shortly. In a diffusion experiment, a layer of foreign atoms or tracer atoms of the same material is created at the surface and the specimen is annealed at a constant temperature T for a time t. During the anneal the atoms diffuse from the surface into the specimen in two ways, directly into the grains and, much faster, along the GB. In turn, the atoms which diffuse along the GB eventually leave it and continue their diffusion in the lattice regions adjacent to the boundary, thus giving rise to a volume diffusion zone around the boundary. Mathematically, this diffusion process is described by a set of two coupled equations:



  

( c ( c (c + , =D (x 2 (y 2 (t 2

2

( 2c 2D (c (cb = Db 2b + (t (y d (x

where

x \d/2,

(1) (2)

x = d/2

These equations represent diffusion in the volume, and along the GB, respectively. Here c(x, y, t) is the

This condition can be established by depositing a layer of the diffusant with thickness h such that h(Dt)1/2. The constant source condition also applies when diffusion occurs from a gas phase. Suzouka [8,9], introduced a so called instantaneous source, or thin layer condition: c(x, y, 0)= Md(y)

Here M is the amount of diffusant deposited per unit area of the surface, and d is Dirac’s delta-function. This surface condition suggests that the initial layer of the diffusant is completely consumed by the specimen during the diffusion experiment, i.e. h (Dt)1/2. In modern diffusion experiments the thin layer condition is established more often than the constant source condition. This tendency is explained by the desire of the experimentalists to work with extremely thin radiotracer layers which would not disturb the structural or chemical state of the GBs during the experiment. Of course, this imposes more strict requirements on the specific radioactivity of the diffusant and the sensitivity of the detection technique.

2.2. The role of grain boundary segregation The coupling condition between functions c(x, y, t) and cb(y, t) essentially depends on whether we study self-diffusion or impurity diffusion, and whether the matrix material is a pure metal or an alloy. For tracer self-diffusion in a pure metal the coupling condition simply expresses the continuity of the concentration across the boundary: cb(y, t)= cv(y, t)

(3)

where cv(y, t) c(9 d/2, y, t) has the meaning of the volume concentration next to the GB. For impurity diffusion in a pure metal, the coupling condition involves the equilibrium segregation factor s and reads: cb(y, t)= scv(y, t)

Fig. 1. Schematic geometry in the Fisher model of GB diffusion.

(dc/dy)y = 0 = 0

(4)

where s does not depend on y or t. This equation rests on two assumptions: 1. The impurity atoms in the GB keep in local thermodynamic equilibrium with the atoms in the lattice adjacent to the boundary. In other words, the GB segregation is locally at equilibrium at any depth.

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2. The GB segregation follows the so-called Henry isotherm, which reads cb =scv, s being a function of temperature only. This isotherm only applies when both cb and cv are small enough, in which case the thermodynamic activities of the impurity in the GB and in the lattice are proportional to the respective concentrations (Henry’s law). Another interesting situation is GB self-diffusion in concentrated alloys [10,11]. Consider, for example, GB diffusion of tracer B* in a binary alloy A – B. In this case Eq. (4) again applies, but the underlying assumptions are now as follows: 1. An equilibrium GB segregation of component B forms in the alloy before, and persists during, the diffusion anneal. (This can be reached by a long pre-diffusion anneal of the sample at the intended diffusion temperature.) 2. The amount of tracer B* diffused into the sample is very small and does not affect the distribution of component B in the sample. 3. Isotope equilibrium with respect to atoms B is maintained between the GB and the adjacent lattice at any depth. Condition 3 is very important, it implies that the tracer concentrations involved in Eq. (4) are proportional to the respective net concentrations of B atoms. Then, the factor s in Eq. (4) equals the ratio cb/cv of the corresponding net concentrations of B. This ratio characterizes the equilibrium GB segregation in the alloy. Note that the net concentrations of B do not have to be small, and the GB segregation can be either in the linear (Henry-type) regime or in the saturation regime.

n= 6/5 [15,16]. Moreover, the linear part of the profile scales with the reduced depth: v=

 

4D y (sdDb )1/2 t

Most GB diffusion measurements are carried out using radiotracers and the serial sectioning technique [12,13]. After the tracer deposition and the diffusion anneal, thin layers of the material parallel to the source surface are removed from the specimen (either mechanically or by ion-beam sputtering) and the radioactivity of each section is determined using a crystalline g-detector or a liquid scintillation counter. The quantity measured in such experiments is the average layered concentration of the diffusant, c¯, as a function of the penetration depth y. This function, called a concentration (or penetration, or diffusion) profile, bears the information about the GB diffusion parameters. It is therefore subject to a mathematical treatment designed to extract that information. Much work has been devoted to the development of approximate analytical solutions of the model and simple recipes for profile processing. It was found that, if the profile calculated using the exact analytical solution [14], of the model is plotted as log c¯ versus y n with different powers n, it becomes almost a straight line if

1/4

(5)

with a constant slope of about 0.78, i.e. − ( ln c¯ /(v 6/5 : 0.78. It immediately follows that, knowing the slope of the linear part of the profile, one can calculate the product sdDb as: sdDb = 1.322(D/t)1/2(− ( ln c¯ /(y 6/5) − 5/3

(6)

(constant source). The volume diffusion coefficient D is supposed to be known from independent measurements. Eq. (6) only applies if the following conditions are met: (a) The so-called LeClaire [16] parameter b, defined as: b=

sdDb 2D(Dt)1/2

(7)

must be large enough, in practical conditions b\10. (b) The parameter, a=

sd 2(Dt)1/2

(8)

must be small enough, in practical conditions aB0.1. Similarly, for diffusion from an instantaneous source, it was found that −( ln c¯ /(v 6/5 : 0.775, provided that a is small enough and b\104 [8,9]. Then, the product sdDb can be determined from the linear part of the plot log c¯ versus y 6/5 as: sdDb = 1.308(D/t)1/2(− ( ln c¯ /(y 6/5) − 5/3

(9)

If bB 10 , the numerical constants in Eq. (9) should be slightly modified [1]. The above relations only consider one part of the profile, which is dominated by GB diffusion. In reality the overall penetration profile will consist of two parts as shown schematically in Fig. 2: “ A near-surface part due to the direct volume diffusion from the surface. The concentration in this region follows a Gaussian function or an error function, depending on the surface condition. If measured accurately, this part can be used to determine the volume diffusion coefficient. “ A long-penetration tail due to the simultaneous GB diffusion and lateral volume diffusion from the boundary to the adjacent grains. It is this tail that should become a straight line when plotted in the log c¯ versus y 6/5 format. The slope of this line is used to determine the GB diffusivity, as indicated above. Accurately measured experimental profiles do follow the y 6/5-rule in a wide concentration range; examples of such profiles are shown in Fig. 3. In practice, the linearity of a log c¯ versus y 6/5 plot is used as a proof of predominant GB diffusion, on one hand, and as a criterion of the profile quality, on the other hand. 4

2.3. Methods of profile processing

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Fig. 4. Arrhenius lines for self-diffusion in fcc metals in the lattice (D) [21], along GBs (Db) [22], on the surface (Ds) [20] and in the liquid phase (Dl) [20] Fig. 2. Schematic shape of a typical penetration profile in the presence of GB diffusion. If a 1 and b  1 (Eqs. (7) and (8)), the tail of the profile is a straight line in the coordinates log c¯ versus y 6/5.

Eqs. (6) and (9) are the basic equations that are used in everyday practice of GB diffusion measurements. We note that they only give the triple product sdDb, while the individual values of s, d and Db thus remain un-

known. Even for self-diffusion in a pure metal (s=1), we still need to know d in order to determine the GB diffusion coefficient Db separately. The traditional assumption that d=0.5 nm, introduced by Fisher [6], seems to be a good approximation. This value of d is consistent with evaluations of the GB width by highresolution transmission electron microscopy and other techniques [1,17]. Furthermore, recent combined B regime and C regime measurements (see definition of these regimes below), of GB self-diffusion in Ag indicate that d =0.5 nm is a very reasonable estimate of d in metals [18,19]. The assumption d=0.5 nm is commonly used in the interpretation of GB diffusion measurements.

2.4. Empirical rules Like lattice diffusion, GB diffusion usually follows an Arrhenius-type temperature-dependence: Db = Db0exp(− Eb/RT)

Fig. 3. Typical penetration profiles for self-diffusion in polycrystalline silver, after Sommer and Herzig [18]. This diagram demonstrates that the GB related tail of an accurately measured profile becomes a straight line when plotted as log c¯ versus y 6/5.

(10)

Eb being the activation energy of GB diffusion, Db0, the pre-exponential factor, and R, the gas constant. For GB self-diffusion in metals, Eb is a factor of two smaller than the activation energy Ev of lattice diffusion; more exactly, for most metals Eb/Ev :0.4–0.6. Typically, GB self-diffusion in metals is several (four to six) orders of magnitude faster than volume diffusion, depending on the temperature (Fig. 4). This great difference in the diffusion coefficients is mainly due to the difference in the activation energies, while the pre-exponential factors are not much different. For all metals, Db values approach a common value of about 10 − 9 – 3× 10 − 9 at the melting temperature Tm [20]. Even near

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the melting point Db remains significantly larger than D and approaches the diffusion coefficient in the liquid phase (Fig. 4). In Table 1, we summarize the empirical correlations between GB diffusion data and Tm for three classes of metals (fcc, bcc and hcp) derived by Brown and Ashby [21] and Gust et al. [22]. The physical justification of such correlations is rather obscure; there is no obvious link, at the atomic level, between a diffusion act and the process of melting. Nevertheless, such correlations do work and are therefore very useful as a guide for the systematics of the available GB diffusion data and the evaluation of new data.

2.5. Classification of diffusion kinetics A GB diffusion experiment is a complicated process that involves several elementary processes, such as direct volume diffusion from the surface, diffusion along the GBs, partial leakage from the GBs to the volume, and the subsequent volume diffusion near the GBs. In a fine-grained polycrystalline sample, diffusion transport between individual GBs can also play an important role. Depending on the relative importance of such elementary processes, one can observe essentially different situations, or regimes of kinetics. In each regime there are one or two elementary processes that essentially control the overall kinetics. Each regime prevails in a certain domain of diffusion-anneal temperatures, times, grain sizes and/or other relevant parameters. The knowledge of all possible regimes is extremely important for both designing diffusion experiments and interpreting their results. This is because the diffusion characteristics that can be extracted from the penetration profile depend on the kinetic regime and should therefore be identified a priori. We shall consider Harrison’s [23] classification of diffusion kinetics, which introduces three regimes called type A, B and C (Fig. 5). This is the first, and still the most widely used, classification for a polycrystal. Other classifications are discussed in [1]

Table 1 Empirical correlation between GB self-diffusion characteristics and the melting temperature for three structural classes of metals Brown and Ashby [21]

Gust et al. [22]

Structure

dDb0 (m s−1)

dDb0 (m3 s−1)

Eb (J mol−1)

fcc bcc hcp

9.44×10−15 83.0Tm 3.35×10−13 97.6Tm 2.74×10−14 89.8Tm

9.7×10−15 9.2×10−15 1.5×10−14

75.4Tm 86.7Tm 85.4Tm

Eb (J mol−1)

Fig. 5. Schematic illustration of type A, B and C diffusion kinetics according to Harrison’s classification.

2.5.1. Type A kinetics The A regime is observed in the limiting case of high temperatures, and/or very long anneals, and/or small grain sizes. In such conditions the volume diffusion length, (Dt)1/2, is greater than the spacing d between the GBs and the volume diffusion fields around neighbouring GBs overlap each other extensively. Thus, the condition of type A kinetics is: (Dt)1/2  d (11) Under this condition an average tracer atom visits many grains and GBs during the anneal time t. On a macroscopic scale the whole system appears to obey Fick’s law for a homogeneous medium with some effective diffusion coefficient Deff. According to Hart [24], Deff = fDb + (1− f)D. Here f is the volume fraction of GBs in the polycrystal, i.e. f= qd/d, q being a numerical factor depending on the grain shape (q=1 for parallel GBs). If there is GB segregation of the diffusing atoms, then f should be multiplied by s, resulting in f= qsd/d [1]. The diffusion profile measured in the A regime follows a Gaussian function (instantaneous source) or an error function (constant source), and the only quantity determined from the profile is Deff. Since Db  D, Deff is generally larger than D, which explains why the diffusion coefficient measured on polycrystals is often larger than the true value of D. If d is very small, then Deff is dominated by the first term, and Deff : qsdDb/d. Then, knowing Deff and the grain size, one can determine the usual product sdDb.

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Y. Mishin, Chr. Herzig / Materials Science and Engineering A260 (1999) 55–71

2.5.2. Type B kinetics If the temperature is lower, and/or the diffusion anneal is shorter, and/or the grain size is larger than in the previous case, then diffusion is dominated by the so called B regime. The condition of this regime is: sd (Dt)1/2 d

(12)

Here again, GB diffusion takes place simultaneously with volume diffusion around the GBs, but in contrast to the A regime the volume diffusion fields of neighbouring GBs do not overlap each other (Fig. 5). Individual GBs are thus isolated, and the solutions worked out for an isolated GB are also valid for the polycrystal. The above condition Eq. (12) implies that a  1. It is additionally assumed for the B regime that b 1, which means that the tracer penetrates along the GBs much deeper than in the volume [16]. Then, the penetration profile has a two-step shape shown schematically in Fig. 2. The GB-related tail of the profile depends on the dimensionless variable v only (see Eq. (5)), the physical reason being the quasi-steady character of GB diffusion in this case [1]. The triple product sdDb is the only quantity that can be determined in the B regime. It is in this regime that Eq. (6) and Eq. (9) can be applied for the profile processing. The conditions of the B regime are the most commonly encountered in GB diffusion measurements. For reasonable anneal times, the B regime comprises the widest and the most convenient temperature range.

2.5.3. Type C kinetics If, starting from the B regime, we go to lower temperatures and/or shorter anneal times, we eventually arrive at the situation when volume diffusion is almost ‘frozen out’, and diffusion takes place only along the GBs, without any essential leakage to the volume (Fig. 5). In this regime, called type C, we have: (Dt)1/2  sd. This relation shows that the criterion for the C regime is a 1. In practical conditions the regime is identified as C if a \ 10. The concentration profile in the C regime is either a Gaussian function (instantaneous source) or an error function (constant source) with the diffusion coefficient Db. If the profile is measured experimentally (which is extremely difficult to implement, because the amount of tracer diffused into the sample is very small), one can determine Db separately from s and d. If the profile is measured accurately in a wide concentration range, one can distinguish between the B and C regimes from the shape of the profile, and not only from the a-value. This is illustrated in Fig. 6, where the C regime profile for GB self-diffusion in Ag was measured using the carrier-free radiotracer 105Ag [18].

3. Grain boundary diffusion and segregation Since the discovery of GB diffusion, this topic has been the subject of special interest, owing to the prominent role of GB segregation in the mechanical behaviour of materials. In this section we shall discuss the progress achieved in this area in recent years.

3.1. Determination of the segregation factor from grain boundary diffusion As was pointed out above, GB diffusion experiments are normally performed in the B regime, and the penetration profiles are treated in terms of the y 6/5-method (Eqs. (6) and (9)). For impurity diffusion this method gives the triple product sdDb, which is the only quantity that can be determined in type B conditions. While the GB width, d, can be treated as a known constant (d=0.5 nm), the GB segregation factor s and the GB diffusion coefficient Db are still to be determined. Because both quantities are essentially temperature-dependent and may vary by orders of magnitude, knowing only their product means knowing almost nothing about each of them individually.

Fig. 6. A typical penetration profile of GB self-diffusion in polycrystalline Ag measured in the C regime (a = 17) [18]. In order to measure this profile in a wide concentration range, a carrier-free 105Ag radiotracer layer was implanted at the ISOLDE/CERN facility. After a microtome sectioning, the radioactivities of the sections were determined using a well-type intrinsic Ge g-detector with a high sensitivity and extremely low background. The tail of the profile shows a downward curvature when plotted as log c¯ versus y 6/5 (B regime format, lower scale) but becomes a straight line when plotted as log c¯ versus y 2 (C regime format, upper scale).

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In some systems the impurity segregation factor can be determined by independent direct measurements. In such measurements, a sample of the segregated material is fractured in situ along GBs and the chemical composition of the fracture surface is analyzed using Auger electron spectroscopy (AES) or some other technique. Then, knowing the segregation factor from such measurements and the product sdDb from GB diffusion measurements, one can calculate Db. Unfortunately, direct measurements of s are only possible for intrinsically brittle materials (such as ceramics or some intermetallic compounds) and are practically impossible for most pure metals. Of course, even an intrinsically ductile material can assume intergranular brittleness as a result of strong segregation of certain impurities. However, the impurity concentrations required for such embrittlement are usually higher than those involved in diffusion measurements. In accurate radiotracer experiments the average tracer concentration in the sample is typically below 1 ppm. Because of this difference in the impurity level, the combination of the segregation factors obtained in fracture experiments with sdDb values measured by the radiotracer technique can be inadequate. Another possibility to separate s and Db is to perform GB diffusion measurements in a wide temperature range including both B and C regimes. As explained above, from the measurements in the C regime one can directly determine Db. Then, by combining the obtained Db values with sdDb values extrapolated from B regime measurements at higher temperatures one can determine sd =sdDb/Db. It has been experimentally established that the product sdDb for impurity diffusion normally follows the Arrhenius law: sdDb = (sdDb)0 exp( −Qb/RT)

(14)

where Qb is the apparent activation energy. On the other hand, according to Henry’s isotherm of segregation the GB segregation factor is a function of temperature only, namely: s =s0 exp(− Es/RT)

(15)

Es being the segregation energy. Since Db also obeys the Arrhenius law, Eq. (10), it follows that: Qb =Eb + Es

(16)

If Qb and Eb are known from diffusion measurements in the B, and C regimes, respectively, one can determine the segregation energy Es This approach is very attractive, because it offers a key to solving two important problems at the same time. On one hand, it opens a possibility to determine the GB diffusion coefficient separately. By making systematic measurements of volume and GB diffusion in different matrix-impurity systems, one can get better

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insights into the mechanisms of the GB diffusion–segregation effects. On the other hand, as a by-product of diffusion measurements one can determine the GB segregation factor and the segregation energy. This possibility is especially valuable for non-brittle materials, where direct segregation measurements are not possible. A major problem associated with this approach is that measurements in the C regime are extremely difficult. Strong segregation favours such measurements by extending the temperature range dominated by the C regime towards higher temperatures. Indeed, since a= sd/2(Dt)1/2, it is clear that larger s values allow one to meet the C regime condition a1 at higher temperatures. Nevertheless, due to the enormous experimental difficulties, C regime measurements have practically not been performed until very recently. It is only a few years since reliable and systematic C regime measurements became possible, due mainly to the use of carrierfree radioisotope layers and extremely sensitive detectors with a large counting efficiency and low background. To date, combined B and C regime measurements have been performed in a few systems: Te in Ag [25], Au in Cu [26], Se in Cu [27], Se in Ag [28] and Ni in Ag [28]. We shall first discuss the results for two systems, Te in Ag [25] and Au in Cu [26], which represent the extreme cases of very strong, and very weak segregation, respectively. GB diffusion of Te in Ag was studied in the temperature range 378–970 K using the radiotracers 123Te (deposited by vacuum evaporation) and 121Te (carrierfree, implanted at the ISOLDE/CERN facility). Each penetration profile was tentatively treated according to both B and C regimes. The actual regime that dominated the experiment was established from (i) the profile shape (whether it became linear in the respective coordinates), and (ii) the comparison of the obtained a and b values with those required for the B and C regimes. It was found that the B regime prevailed in the temperature range \ 600 K (Fig. 7). At those temperatures the sdDb values determined using the log c¯ versus y 6/5 plots and Eq. (9), followed the Arrhenius relation with: (sdDb)0 = 2.34× 10 − 15 m3 s − 1 and Qb = 47.47 kJ mol − 1 It was also established that the measurements below 500 K were dominated by the C regime. At such temperatures the GB diffusion coefficients were determined by fitting the profiles by a Gaussian function and were found to follow the Arrhenius relation with: Db0 = 1.01× 10 − 4 m2 s − 1 and Eb = 86.75 kJ mol − 1 The transition temperature range 500–600 K, was thus dominated by an intermediate regime between B

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Fig. 8. Temperature-dependence of the GB segregation factor s of Te in Ag [25] and Au in Cu [26] determined from combined B and C regime GB diffusion measurements.

Db0 = 4.87× 10 − 6 m2 s − 1

and

Eb = 91.03 kJ mol − 1, Fig. 7. The Arrhenius plot of sdDb(circles) and dDb (squares) (d =0.5 nm) values for Te impurity diffusion along GBs in Ag [25]. The B and C regimes prevail above 600 K, and below 500 K, respectively. The range 500 – 600 K is dominated by the transient B2 regime where the apparent values of both sdDb and dDb are significantly underestimated compared with the true ones. The difference between the two Arrhenius lines is due to the GB segregation of Te.

respectively. The values of s deduced from the GB diffusion data are shown in Fig. 8. As expected, the absolute values of s, 8–11, and the GB segregation energy of Au in Cu, Es = − 9.7 kJ mol − 1, are relatively small. The segregation energy has a reasonably lower

and C. In this range, the sdDb and Db values obtained by processing the profiles according to the B, and C regimes, respectively, showed significant downward deviations from the respective Arrhenius lines (Fig. 7). This behaviour is perfectly consistent with the theoretical analysis [1,29], and is indicative of the transient regime. In Fig. 7, the difference between dDb values measured in the C regime and sdDb values extrapolated from the B regime measurements gives the segregation factor s. The high absolute values of s (103 –105) are consistent with the very small solubility of Te in Ag. The segregation energy determined from the Arrhenius plot of s (Fig. 8), equals Es = − 43.3 kJ mol − 1. In contrast, Au in Cu is a system with complete mutual solubility of the components, so that the GB segregation was expected to be rather weak. The results of combined B and C regime measurements for this system [26], using the carrier-free radiotracer 195Au, are shown in Fig. 9. The Arrhenius parameters obtained in these regimes are: (sdDb)0 =2.11×10 − 15 m3 s − 1 Qb =81.24 kJ mol and

−1

and

Fig. 9. The Arrhenius plot of sdDb (circles) and dDb (squares) (d= 0.5 nm) values for Au impurity diffusion along GBs in Cu [26]. The B and C regimes prevail above 618, and below 526 K, respectively. The difference between the two Arrhenius lines is due to the GB segregation of Au.

Y. Mishin, Chr. Herzig / Materials Science and Engineering A260 (1999) 55–71

Fig. 10. Arrhenius diagram of lattice self-diffusion [34] and impurity diffusion of Te [31], Se [32] and Ni [33] in Ag.

absolute value than the Es = −13.0 kJ mol − 1 reported for Au surface segregation in Cu – 7.5 at.% Au alloy [30]. Results of combined B and C regime measurements are now available for Te [25], Se [28] and Ni [28] GB diffusion, as well as GB self-diffusion [18], in the same high-purity Ag material. These data are already sufficient to draw some conclusions about the systematics. Figs. 10–12 illustrate the Arrhenius diagrams of volume diffusion [31 – 34], GB diffusion [18,25,28] and GB segregation [25,28] in these systems. The observed diffusion–segregation behaviour can be rationalized in terms of the competition between two trends: 1. Fast (or slow) diffusers in the lattice tend to remain

Fig. 11. Arrhenius diagram of GB self-diffusion [18] and impurity GB diffusion of Te [25], Se [28] and Ni [28] in Ag. The impurity diffusion coefficients were measured in the C regime, the self-diffusion coefficients were calculated from dDb measurements in the B regime assuming d =0.5 nm.

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Fig. 12. Arrhenius diagram of GB segregation factors of Te [25], Se [28] and Ni [28] in Ag. The segregation factors were determined from combined B and C regime measurements.

fast (or slow) diffusers in the GBs. 2. Strong GB segregation tends to reduce the GB diffusion rate. The stronger the segregation tendency, the greater is the retardation effect. For Te and Se, the two trends act in opposite directions. On one hand, both elements are fast diffusers in the Ag lattice (Fig. 10), which is probably due to the significant vacancy–impurity binding. On the other hand, Se and especially Te show very strong segregation tendency (Fig. 12), which in turn implies strong binding of the impurity atoms to certain favoured positions in the GB structure. It was even suggested [10,11,25,28], that at least at low temperatures, the impurity atoms tend to form local configurations reminiscent of ‘embryos’ of ordered compounds that are present in the phase diagrams Ag–Te and Ag–Se [35]. In any case, the strong binding to selected GB sites produces the well-known trapping effect, which results in decreased diffusion mobility, especially at relatively low temperatures. The competition between the two trends mentioned above leads to a mixed behaviour of GB diffusion, with the impurity diffusion being faster than self-diffusion at high temperatures and slower than that at low temperatures (Fig. 11). The transition temperature is about 600 K for Te and only 400 K for Se, which again reflects the stronger segregation tendency of Te. Ni, on the other hand, is a slow diffuser (Fig. 10), and at the same time a very strong segregant in Ag (Fig. 12). In this case the two trends act in the same direction and result in dramatically slow Ni diffusion in Ag GBs (Fig. 11). It should be pointed out that Ni is expected to demonstrate a greatly different diffusion– segregation behaviour at the atomic level than Te and Se, which in fact was the reason why Ni was chosen for the experiments. Ni forms no compounds with Ag [35]

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and has a larger surface tension than Ag, and yet it shows a very strong segregation tendency with s values as large as 103 –104. These and other interesting aspects of Ni behaviour in Ag are discussed in [28] The examples above clearly demonstrate that separate measurements of s and Db are now possible for both strongly and weakly segregating impurities and lead to very reasonable results. At present, such measurements offer the only possibility to study GB segregation in materials where direct segregation measurements are hampered by their ductility or tendency to transgranular fracture.

3.2. Beyond the linear segregation Up to this point we have assumed that the GB segregation followed the Henry isotherm, i.e. that factor s in Eq. (4) was a function of temperature only. This approximation is only valid when both the volume concentration cv(y, t) and the GB concentration cb(y, t) are small. In systems with strong segregation tendencies, especially at low temperatures, the GBs can show a tendency to become saturated with the impurity and thus to show a non-linear relation between cb and cv. Such non-linear dependence means that the ratio cb/cv in the coupling condition Eq. (4), is no longer constant; instead, it depends on the volume concentration cv, which in turn changes with depth y. The depth dependence of cb/cv may affect the shape of the profile and should therefore be taken into consideration in the profile interpretation. This problem was first analyzed by Martin and Perraillon [36] and more recently in [37,38]. These studies took into account the non-linearity of the GB segregation by using McLean’s isotherm instead of the Henry isotherm. McLean’s isotherm has the form: cb =

scv 1+(s −1)cv

(17)

where cb and cv are expressed in mole fractions and s depends on the temperature according to Eq. (15). Using similar approximations to those introduced by Fisher [6], one can solve the basic equations of the model analytically. Typical penetration profiles log c¯ /c¯0 versus v, calculated using such solutions are shown in Fig. 13, c¯0 being the surface value of the average concentration c¯. The profiles consist of two parts: 1. The GB saturation region (v B1) where c¯ decreases rapidly and the profile has a strong upward curvature. In this region the GB concentration remains almost constant, cb :1, while the volume concentration cv rapidly drops. 2. The linear-segregation region (v \ 1) where the profile is consistent with Fisher’s exponential solution [6]. In this region both cv and cb are small and the Henry isotherm is a good approximation. It is

this part that can be used for the determination of sdDb using the standard methods of profile processing. The profile behaviour illustrated in Fig. 13 is rather general and could be predicted using more exact solutions of the model as well. Examples of experimental profiles containing two steps (Fig. 13) can be easily found in the literature. However, the explanation of all such profiles by the GB saturation effect would be premature. The near surface region is affected by many other factors, such as direct volume diffusion from the surface, diffusion along dislocations, GB motion, the presence of a spectrum of GBs with different diffusivities, etc. A reliable interpretation of the first part of the profile as caused by the saturation effect requires specially designed experiments in each individual case. Experiments of this type have recently been carried out by Bernardini et al. [39]. Bernardini et al. [39] studied tracer diffusion of 110Ag in polycrystalline samples of pure Cu and Cu–0.091 at.% Ag alloy. All measurements were performed in the B regime. The profiles measured on pure Cu samples showed a two-step shape (Fig. 14). The explanation of this shape by the saturation effect seemed plausible in view of the very strong segregation tendency in this system. In the alloys samples, on the other hand, the ratio cb/cv was pre-determined by the equilibrium segregation in the alloy. Since this ratio was essentially constant along the GBs, the diffusion profiles were expected to have only one step. The remarkable result of the measurements was that the profiles obtained on the alloy samples indeed followed a linear log c¯ versus y 6/5 dependence until very close to the surface (Fig. 14). Moreover, using the ratios (sdDb)pure/(sdDb)alloy at different temperatures and assuming that (Db)pure : (Db)alloy, Bernardini et al. evaluated the segregation factors and the segregation energy of Ag in Cu. The obtained value Es = − 299 13 kJ mol − 1 was in good

Fig. 13. Typical GB penetration profiles calculated with McLean’sisotherm of GB segregation [38].

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Now that separate determination of Db and s values is possible, it would be very interesting and timely to revisit some of the above systems and study the bulk composition dependence of both diffusion and segregation characteristics of A and B over a wide range of temperatures and compositions. In such systems the solute segregation isotherm in different temperature ranges can be reconstructed from the diffusion data. It should be remembered that, for such measurements to be successful, the GB segregation in the alloy does not have to be Henry type. Thus, there is a good chance that such effects as GB saturation, discontinuities in the segregation isotherm, and similar effects can be studied without breaking the samples.

4. Diffusion in oriented bicrystals

Fig. 14. Penetration plots of Ag GB diffusion in pure Cu and Cu(Ag) alloy at 617 K [39]. The arrow indicates the first step associated with the GB saturation effect.

agreement with the experimental data for dilute Cu(Ag) alloys obtained by AES (Es ] − 40 kJ mol − 1) [40]. The results of this study indicate that the GB saturation, (i) can indeed influence the shape of the penetration profiles quite noticeably, and (ii) can be studied separately from other effects if the experiments are properly designed.

3.3. Grain boundary diffusion in solid solutions GB diffusion and segregation in solid solutions is a very attractive topic of future research. For a binary solution A–B, a full consideration involves the tracer diffusion coefficients of both components in the volume and in the GBs, as well as the respective GB segregation factors. In contrast to the case of impurity diffusion, all these quantities generally depend on the bulk composition. There are binary systems where both components have suitable radiotracers, and their GB diffusion characteristics can thus be established as functions of the bulk composition. However, there is only one system, Fe–Sn [41], where, along with GB diffusion coefficients, the solute segregation factors were also determined by independent measurements. In all other systems (e.g. Ag– Sn [42], Ag – Ni [43]), only the products (sdDb)A and (sdDb)B could be determined, and not the segregation factors, which made the interpretation of the results extremely difficult.

Diffusion measurements on polycrystalline samples are influenced by the whole ensemble of GBs that are present and thus yield diffusion characteristics of some ‘average’, or ‘typical’ GB. Measurements on individual GBs with a known misorientation between the grains are much more informative, although more difficult to implement. Such measurements are performed on oriented bicrystals produced by either bonding or directional crystallization. The production of oriented bicrystals is an extremely difficult problem by itself. It is not an easy task to reach the desired misorientation uniformly along the GB with high accuracy, and it is next to impossible to avoid fluctuations in the local orientation of the GB plane around its average orientation. Nowadays the production of top quality oriented bicrystals looks more like art than technology, and only a few groups have a good enough command of that art. Diffusion measurements themselves are also extremely demanding, because with only one GB in the sample the radioactivity of the sections is very small. However, such measurements deliver invaluable information on the anisotropy and misorientation dependence of the GB diffusivity, which are in turn intimately connected with GB structure. The first measurements of the anisotropy of GB diffusion were performed by Hoffman [44], for self-diffusion along [001] tilt GBs in Ag. For small-angle misorientations the anisotropy was especially strong, with diffusion parallel to the tilt axis (D b) being a factor of 15 faster than perpendicular to the tilt axis (D Þ b ). These observations were explained in terms of the dislocation model of low-angle GBs. However, some anisotropy (with D b/D Þ b : 2) was still present even when the tilt angle u was as large as 45°. Since the dislocation model could not be directly applied at such misorientations, the measurements of Hoffman indicated that even high-angle GBs were not amorphous

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but instead had a well ordered anisotropic structure. In Fig. 15 we show more recent results of diffusion anisotropy measurements, again in [001] tilt symmetric GBs in Ag, in the range of high-angle misorientations [45]. Similar results were obtained on other metallic systems [1]. The orientation dependence of GB diffusion is another interesting topic. If the misorientation between two grains changes continuously and GB diffusion is measured along a fixed direction in the boundary plane (say, parallel to the tilt axis), will the diffusion characteristics change monotonically, or will they show maxima and/or minima at some ‘special’ orientations? The first attempt to answer this question was made in the work of Turnbull and Hoffman [46], once again on [001] tilt symmetric GBs in Ag. Turnbull and Hoffman studied small-angle GBs and found a continuous behaviour of the diffusivity in that case. As far as highangle GBs are concerned, there is experimental evidence that the diffusivity shows minima at misorientations with low values of S (the inverse density of coincidence sites in the GB). For example, Budke et al. [47] have recently studied self-diffusion and Au-impurity diffusion in a series of well-characterized Cu GBs in a narrow range (Du = 6°) around the S = 5,u = 36.9° (310)[001] tilt misorientation. For both impurity diffusion and self-diffusion, the diffusivity showed a welldefined minimum and the activation energy a maximum very close to the perfect u =36.9° misorientation (Fig. 16). At first sight, these observations stand in contrast with the continuous orientation dependencies found earlier on [001] tilt symmetric GBs in Ni [48] and Ag [45,49] (cf. Fig. 15). However, if the minima of Db are confined to narrow ranges around some special misorientations, as was observed in [47], they could have been overlooked in the previous studies [45,48,49], where the measurements were only taken in every 5 – 7°. Since the existence of the minima is a topic of great fundamental

Fig. 15. Orientation dependence of self-diffusion in [001] tilt symmetric GBs in silver at T =771 K, after Sommer et al. [45].

Fig. 16. Orientation dependence of 195Au diffusion in Cu [001] tilt symmetric GBs measured on a series of well characterized bicrystals [47]. The GB diffusivity P =sdDb is plotted as a function of the tilt angle u. The results are independent of the purity of the used Cu material (Cu1 and Cu2).

importance, further measurements on oriented bicrystals should be done in the future.

5. Atomistic theory of grain boundary diffusion Most theoretical studies of GB diffusion have been focused on phenomenological models, while the atomistic aspects have received much less attention and are still poorly understood. The recent years have seen an increased interest in the atomistic interpretation of GB diffusion. Because of the lack of experimental methods capable of providing atomic-level information on GB diffusion, atomistic computer simulations are the main tool for getting insights into this phenomenon. The recent progress in this area is associated not so much with new results for some specific systems as with the development of new concepts, models and methods of simulation. The recognition of the key role of correlation effects in GB diffusion is probably one of the most remarkable achievements of the recent years. Below we shall discuss some new simulation methods along with their advantages and limitations. We shall also discuss some recent simulation results.

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5.1. Methods of atomistic simulation Atomistic simulations of GB diffusion employ either molecular dynamics (MD) or molecular statics combined with the kinetic Monte Carlo method. An advantage of MD is that one can identify diffusion mechanisms that could not be anticipated a priori [50–52], which is very important when dealing with GB diffusion. In MD, one can calculate the GB diffusion coefficient in a given direction x as: % (DXi )2 i

, 2Nt where DXi is the displacement of atom i in the x direction and t is the diffusion time. The summation runs over the trajectories of N atoms within the GB core. (For simplicity, we consider tracer self-diffusion in a pure metal.) The drawback of MD is the difficulty in the accumulation of good statistics of atomic jumping. Because it takes many MD-steps before one of the atoms in the system makes a jump, the generation of sufficiently long trajectories requires enormous computational resources. This is the main reason why diffusion calculations by MD are essentially limited to relatively high temperatures. The static approach in its modern form has only emerged in the past 2 or 3 years. The simulation procedure includes the following steps [3]: 1. Calculation of the underlying GB structure at 0 K. Such calculations are performed by standard molecular statics with many-body potentials either of the embedded-atom method (EAM) [53] or of FinnisSinclair type [54]. 2. Calculation of point defect concentrations (occupation probabilities) at different sites in the GB core. These are determined from the equilibrium concentrations of the point defects in the lattice and their binding energies to the GB sites. The binding energies are calculated by molecular statics. 3. Calculation of the defect migration energies between different sites in the GB core. In such calculations, the atom that is making a jump is moved from its initial position towards the final position by small steps, and the simulation block is relaxed after each step. The total energy attains a maximum between the initial and final positions, which is identified as the saddle-point energy of the jump. The difference between the saddle-point energy and the initial energy of the block equals the defect-migration energy. For the vacancy mechanism, the vacancy jumps to the neighbouring positions are simulated by moving the respective atoms towards the vacancy. For the interstitial and interstitialcy mechanisms, the interstitial atom is moved towards a neighbouring vacant site, or a site occupied by some other atom, respectively.

Db =

67

4. Calculation of Db by means of kinetic Monte Carlo simulations. This step will be discussed below. So far, this scheme or some parts of it have been implemented only for GB self-diffusion in pure metals [2,55–57], and very recently in NiAl [58]. Because molecular statics yields the GB structure and the defect energetics at 0 K, the extrapolation towards higher temperatures inevitably involves some approximations. The full relaxation around the jumping atom is one such approximation. Also, the atoms are not free to choose the migration mechanism, such a mechanism has to be postulated a priori. The latter feature, on the other hand, can be used as an advantage. One can postulate several mechanisms, study each of them individually, and quantify their relative importance. Another advantage of molecular statics is that, when combined with Monte Carlo simulations, it gives more accurate values of Db. Knowing the point-defect concentrations and migration energies involved in the given mechanism, one can calculate the frequencies of individual atomic jumps. For example, for self-diffusion by the vacancy mechanism the frequency of atomic jumps from site i to site j equals Gij = cvjyji, where cvj is the vacancy occupation probability at site j and yji is the vacancy jump frequency from j to i. The latter can be evaluated as y0 exp(− oji /kT), where y0 is the attempt frequency (atomic vibration frequency in the jump direction), and oji is the vacancy migration energy. However, the question that is still to be answered is, how to determine the GB diffusion coefficient in a given direction x, knowing all individual jump frequencies Gij and the jump length projections xij ? The determination of Db reduces to the calculation of the jump correlation factor. While for lattice diffusion the correlation effects have been studied extensively, for GB diffusion such studies have not even been attempted until very recently [2,55,59,60]. It is now established [2,3,60], that the relatively low symmetry of the GB structure, its quasi-2D confinement, and the jump multiplicity lead to much stronger correlation effects in comparison with diffusion in a regular lattice. The degree of jump correlations essentially depends on the distribution of ‘easy’ and ‘difficult’ jumps in the GB structure. While the correlation factor of lattice self-diffusion is simply a numerical constant, the correlation factors of GB diffusion can be very small, strongly temperature-dependent, and strongly anisotropic. These features are illustrated in Fig. 17 for GB self-diffusion in NiAl (see [2,3] for other examples). The neglect of the jump correlation effects results in a significantly underestimated activation energy and an overestimated GB diffusion rate, especially at low temperatures. The jump-correlation factor is a complicated function of all quantities Gij and xij, as well as the temperature. Two methods are now available for its calculation:

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5.1.1. Kinetic Monte Carlo method The kinetic Monte Carlo method has been successfully used in simulations of lattice diffusion in metals, alloys, and compounds [61]. It can be adopted, almost without changes, for GB diffusion [3]. In this method one creates a slab-like simulation block of thickness d consisting of many repeat units of the boundary structure. Assuming the vacancy mechanism, one creates a vacancy at some arbitrary site and allows it to walk through the system. The direction of every jump of the vacancy depends on the local jump frequencies and is decided by generating a random number. After a sufficiently long walk of the vacancy, one calculates the total correlation factor fb of atoms in a given direction x: N

% (DXj ) fb =

2

j=1 m

(18)

% x 2i

i=1

Here DXj is the x-projection of the net displacement of atom j along its trajectory induced by the vacancy, xi is the x-projection of the i-th jump of the vacancy and m the total number of vacancy jumps. Knowing the correlation factor, one determines the GB diffusion coefficient in the x-direction: Db random =fbDb random

(19)

where 1 Db random = % na Gax 2a 2a

(20)

is the so-called random diffusion coefficient calculated by ignoring the correlation effects. In Eq. (20) na is the geometric probability of jump a = (i “j ), Ga =cjyji is the basic frequency of jump a, and xa is the x-projec-

Fig. 17. Correlation factors of Al diffusion perpendicular ( fbÞ) and parallel ( fb ) to the tilt axis in a stoichiometric S5 (210)[001] symmetrical tilt GB in NiAl [58]. This plot illustrates that the GB correlation factor can be very small, temperature-dependent and strongly anisotropic.

tion of jump a. This simulation scheme is more economical than MD, because every jump attempt is successful.

5.1.2. Combined matrix-Monte Carlo method This method [2], is even more economical. The starting point is the decomposition of the diffusion coefficient in terms of local jump frequencies and the associated partial correlation factors fax [2,59]: 1 Db = % na Gax 2afax (21) 2 a In turn, the partial correlation factors can be determined by the matrix method using the standard equation [62]: fx = I+2XT(E− T) − 1d

(22)

Here fx is an n-component column vector of partial correlation factors (n being the number of distinct jumps), d is a column vector with n components xa , X 1 is a diagonal n× n matrix with diagonal elements x − a , E is an n× n unit matrix and I is an n-component unit column vector. For the vacancy mechanism, the elements of the jump matrix T can be expressed in terms of the next jump probabilities P*ab (the probability that jump a is followed by jump b by exchanging with the same vacancy) and escape probabilities Ea (the probabilities for the vacancy to escape after jumps a). The escape probabilities are involved in the calculation, because they are related to the frequencies of arrival of ‘fresh’ vacancies at different neighbouring sites of the tracer atom. In lattice diffusion, the ‘fresh’ vacancies have no effect on the jump correlations: in a centrosymmetrical lattice they induce jumps by vectors r and − r with equal probability and are thus averaged out in the calculation of the correlation factor. In a GB, however, the local inversion symmetry is generally broken and the effect of ‘fresh’ vacancies is essential [2]. The quantities P*ab and Ea can be calculated by Monte Carlo simulations of individual vacancy-tracer encounters. In this case we simulate multiple vacancy walks starting from the same configuration arising after a given atomic jump a. Each walk is terminated when the vacancy induces a next tracer jump, b. If this does not happen after a large number of vacancy jumps, the walk is terminated anyway, and the vacancy is identified as randomized. This procedure is repeated for all other distinct vacancy-tracer configurations. In this simulation scheme the defect walks are, on average, shorter than in the standard Monte Carlo method. Also, only the coordinates of the tracer atom and the defect are essential for the algorithm; there is no need to store the coordinates of a large amount of particles in the computer memory. More important, the knowledge of the partial correlation factors is very useful for the analysis of the correlation effects and the GB structure-diffusion relationship.

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Fig. 18. Arrhenius diagram of calculated self-diffusion coefficients parallel (open symbols) and perpendicular (filled symbols) to the tilt axis in a S5 (210)[001] GB in NiAl [58]. Squares: stoichiometric boundary, circles: Ni-rich boundary, triangles: Al-rich boundary.

5.2. Simulation results Ma et al. [55] calculated the point-defect formation and migration energies in a few [001] symmetrical tilt GBs in Ag using molecular statics with an EAM potential. For the calculation of Db, Ma et al. used its decomposition Eq. (21), in terms of partial correlation factors. They did not use the matrix equation Eq. (22); instead, they calculated f‘ax’s directly by Monte Carlo simulations. Their Monte Carlo scheme, however, had some drawbacks, which were discussed in [2]. The obtained total correlation factors fb were relatively large and temperature-independent. Along with the vacancy mechanism, Ma et al. considered interstitial-related mechanisms as well, and came to the conclusion that diffusion along the tilt axis could be dominated by such mechanisms. This conclusion is consistent with more recent MD simulations of GB self-diffusion in Ag performed using the same EAM potential [52]. It should be mentioned, however, that EAM potentials typically underestimate the repulsion between atoms at short distances [63] and thus the interstitial formation and migration energies. Nevertheless, the study of Ma et al. [55,59], was of great importance, because it called attention to the role of correlation effects in GB diffusion. In [2], the point-defect energetics established by Ma et al. [55], were used for a more detailed study of diffusion and jump correlations in the S = 5 (310)[001] GB in Ag. In that study, Db values were calculated using the combined matrix-Monte Carlo method. It was found that the correlation effects were strongly temperature-dependent and anisotropic. For diffusion perpendicular to the tilt axis, the correlation factor ( fbÞ) was smaller than that for diffusion along the tilt axis ( fb ). The difference between fbÞ and fb changed from rather small at high temperatures ( fbÞ :fb :0.7) by almost a decade at 400 K ( fbÞ : 0.01, fb :0.1).

The same tendency to strong and temperature-dependent correlations was observed in a more recent study of self-diffusion in a S= 5 (210)[001] GB in the ordered B2 compound NiAl [58]. The EAM potential used in that study was fitted to reproduce the pointdefect energetics in NiAl, as well as the vacancy formation and migration energies in pure Ni and Al. Both stoichiometric and off-stoichiometric structures of the boundary were simulated. It was assumed that the GB structure remained perfectly ordered at any temperature and that the atoms diffused in the GB core by vacancy jumps between like sites only. This assumption is consistent with the high ordering energy of NiAl, as well as with the next-nearest-neighbour jump mechanism of diffusion in the lattice of NiAl [64]. The GB diffusion coefficients obtained for both Ni and Al followed the Arrhenius law with reasonable accuracy in a wide temperature range (Fig. 18). For a fixed diffusion direction, diffusion of a given species was fastest in the GB boundary enriched in that species and slowest in the GB boundary depleted in that species. Ni interstitials could easily form at certain sites in the GB core, but were totally immobile.

6. Summary and future research Diffusion along GBs is a phenomenon of both practical significance and great fundamental interest. Modern GB diffusion measurements employ novel experimental techniques for precise radiotracer measurements combined with elaborate mathematical treatments of the experimental profiles. The great volume of experimental data accumulated to date follows clear systematics and provides a reasonably good understanding of this process at a phenomenological level.

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One of the most impressive achievements in recent years is the implementation of GB diffusion measurements at relatively low temperatures in the C regime. Such measurements, combined with traditional measurements in the B regime, open the long awaited possibility of separate determination of the GB diffusion coefficient Db and the impurity segregation factor s. Combined B and C regime measurements have been carried out for several systems, including systems with very strong and very weak segregation tendencies. Of particular interest are the data for Te, Se and Ni in Ag, which can be rationalized in terms of the trapping-induced retardation of GB diffusion in the presence of strong segregation. At present, the separate measurements of Db and s offer the only way to study GB segregation of impurities in materials where cleavage along GBs is not feasible. The next step in this direction would be to perform combined B and C regime measurements in solid-solution systems over a range of alloy compositions. Such measurements will lead to a better understanding of the interrelation between GB diffusion and GB segregation, particularly in conditions of solute saturation. Investigations of diffusion along individual well-characterized GBs in metallic bicrystals are of extreme importance to the understanding of the GB structure– diffusion relationship. The traditional scheme is to perform diffusion measurements on a series of symmetrical tilt orientations around a chosen low-S orientation. The accuracy and reliability of such measurements have drastically improved in the recent years due to the use of carrier-free radiotracers and highly sensitive detection techniques, on one hand, and the improved technology of growth and characterization of oriented bicrystals, on the other hand. Yet, the interpretation of the experimental results is at times ambiguous due to the non-uniformity of the GB structure, the presence of second tilt and twist components, fluctuations in the orientation of the GB plane, and other factors which lie beyond the diffusion measurements. Nevertheless, the recent measurements on Cu bicrystals clearly indicate a sharp minimum of the diffusivity very close to the S =5 (310)[001] tilt orientation. Future studies of this type would greatly benefit from careful structural examinations of the GBs in the diffusion samples by highresolution electron microscopy. Simultaneous atomistic modeling of GB diffusion in the same misorientation range would be a very useful complimentary study. A real challenge would be to perform combined B and C regime measurements of impurity diffusion in oriented bicrystals. By such measurements one could study the orientation dependence of not only diffusion but also segregation in the same set of GBs. With only one GB in the sample, the signal to be measured in the C regime lies below the sensitivity of conventional detectors. However, owing to the recent progress in radiotracer

techniques, such measurements may become possible in the near future. Atomistic theory of GB diffusion has recently emerged as a new, rapidly growing area of research. Although specific results are still scarce, there is progress in the development of new atomistic simulation methods and identification of their advantages and limitations. Probably the most significant achievement in this area is the recognition of the key role of jump correlations in GB diffusion. Although methods for the computation of the jump correlation factors are already available, further analysis of correlation effects is needed. At the same time, the new methods and ideas will now be applied for GB diffusion simulations in specific systems. After GB self-diffusion in pure metals, GB impurity diffusion will be the next candidate as a subject of such studies. The existing correlation theory and simulation methods are not immediately transferable to impurity diffusion and will have to be reworked accordingly. GB diffusion in ordered intermetallic compounds is another attractive topic for the near future. In conclusion, it should be emphasized again that GB diffusion measurements can be effectively used as a tool to study GB structure and properties averaged over a 10 − 4 m scale (typical diffusion length along GBs). GBs are buried interfaces, as opposed to an open surface, which makes their study more difficult. Cleavage of a material along GBs may disturb their initial state and in addition is not feasible for many materials. High resolution transmission electron microscopy is probably the most effective technique for GBs, but it also involves some problems. In this situation GB diffusion measurements can serve as a useful complimentary technique. Since GB diffusion is very sensitive to the GB structure and composition, and because radiotracer experiments practically do not disturb the initial state of the GBs, GB diffusion measurements are capable of providing very useful information on the structural and chemical state of GBs.

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