Grain boundary grooving at the singular surfaces

Acta mater. 48 (2000) 1533±1540 www.elsevier.com/locate/actamat

GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES E. RABKIN 1{, L. KLINGER 1 and V. SEMENOV 2 1

Department of Materials Engineering, TECHNION-Israel Institute of Technology, 32000, Haifa, Israel and 2Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, 142432, Russia (Received 1 July 1999; received in revised form 16 November 1999; accepted 16 November 1999) AbstractÐThe unusual topographies of the grain boundary thermal grooves in Ni-rich NiAl were observed after annealing at 14008C. One of the surfaces forming the grain boundary groove exhibited no curvature measurable in the atomic force microscope, thus indicating its singular character. The theory of grain boundary grooving at singular surfaces was developed in the small-slope approximation and under assumption of negligible di€usivity on these surfaces. The calculated groove shapes are in good agreement with the experimental data and di€er considerably from the shapes predicted by the classical Mullins grooving theory for isotropic surfaces. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Atomic force microscopy (AFM); Intermetallic compounds; Surface di€usion; Grain boundaries

1. INTRODUCTION

Thermal grooves formed at the sites of grain boundary (GB) intersections with the free surface provide important information about the surface and GB energies and surface self-di€usivity. Atomic force microscopy (AFM) o€ers the possibility to measure the groove topography and dihedral angle at the root of the groove with very high accuracy which was unattainable using earlier methods [1]. Quantitative analysis of the GB groove shape is based on the classical model of W.W. Mullins, developed more than 40 years ago [2]. In this model, the GB groove grows by the mechanism of surface di€usion, the driving force for surface di€usion being the gradient of surface curvature. Later, it was proved experimentally that for the GB grooves of below approx. 10 mm, surface di€usion is the dominating mechanism of groove growth [3]. The most serious simpli®cation made in the Mullins model is the assumption of full isotropy of the surface energy, gs. Obviously, this assumption justi®es the use of the continuum approach, with the macroscopic curvature as the only driving force for surface di€usion. However, the importance of the surface energy anisotropy in determining the dihedral angle of the groove and the groove shape was recognized soon after Mullins' original work [4]. In

{ To whom all correspondence should be addressed.

the case of the coherent twin boundary, the surface anisotropy may even cause the formation of a ridge instead of a groove [4]. In the continuous approach, the role of surface anisotropy in curvature-driven surface di€usion is merely reduced to the renormalization of gs to gs+@2gs/@y 2, where y is the inclination of the surface [5]. Recently, Bonzel and Mullins considered the evolution of the pre-perturbed surface topography of the vicinal surface, which is essentially anisotropic [6]. It is known that the vicinal surface consists of ¯at terraces corresponding to the singular inclination (i.e. the inclination corresponding to the cusp on the gs(y ) plot) separated by steps. It was found that in the smallslope approximation, the ¯ux of surface atoms is again proportional to the gradient of surface curvature de®ned in the proper frame of reference, but gs should be substituted by the complex expression which depends on the energy of an isolated step, the energy of interaction between the steps and the direction of the perturbation. Nevertheless, the gradient of curvature is not a necessary pre-requisite for the evolution of surface topography by the surface di€usion mechanism. Mullins developed a theory of growth of the individual singular facet, in which he assumed that the surface di€usion ¯ux along this facet is constant [7]. This ¯ux is induced by the di€erence of chemical potentials at the contact lines between the singular facet and non-singular surfaces. The constant di€u-

1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 4 3 2 - 2

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RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

sion ¯ux along the surface implies a linear variation of the chemical potential along it. More recently, Carter et al. formulated a theory of shape evolution of the fully anisotropic crystal [8]. The driving force for surface di€usion arises in this case due to the deviation of the crystal shape from the equilibrium Wul€ shape. It was shown that the chemical potential at the singular surface without any curvature is a parabolic function of the distance along this surface. From the above, it is clear that, though there are no thermodynamic limitations on the growth of ani-

sotropic GB grooves, the regularities of their growth should be di€erent from those for isotropic grooves. In the present work, we observed the GB grooves with the ¯at singular walls which do not exhibit any curvature. The mechanism of their formation is discussed and the quantitative theory for the growth kinetics is suggested. 2. EXPERIMENTAL

The rod of a NiAl intermetallic compound of a nominal composition 48 at.% Ni+52 at.% Al was

Fig. 1. The AFM image of the GB groove with the ¯at wall (a) and the corresponding line pro®le taken perpendicular to the triple line (b).

RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

produced from Ni of 99.95 at.% purity and Al of 99.99 at.% purity by casting in vacuum. The as-cast alloy was then remelted and puri®ed in a vacuum electron-beam ¯oating zone melting apparatus. Two passes of the the rod through the melted zone were made with a velocity of 6 mm/min. A coarsegrained (averaged grain size 2 mm) polycrystal was obtained as a result of remelting. A disc of 3 mm thickness was then cut from the rod by spark ero-

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sion and annealed in a vacuum of 10ÿ5 Pa at a temperature of 14008C for 1 h in order to relax the internal stresses and to equilibrate the microstructure. After the ®rst annealing, the surface of the disc was ground and polished to mirror quality using SiC papers and diamond pastes down to 0.3 mm particle size. A second annealing at 14008C for 1 h was then conducted. The polished surface was studied after annealing using light microscopy

Fig. 2. The AFM image of the GB groove with the ¯at (on the left) and faceted (on the right) walls (a) and the corresponding line pro®le taken perpendicular to the triple line (b). The macroscopic curvature of the faceted vicinal surface can be seen clearly.

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RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

(LM) and AFM. The AFM measurements were made by an Autoprobe CP AFM (Park Scienti®c) operated in the contact mode. Si Ultralevers with a nominal tip radius of 10 nm were used. The scanning direction was chosen always approximately perpendicular to the GB in order to reduce the artifacts connected with the abrupt change of surface slope at the GB groove root. Both the topography signal and the feedback loop error signal were collected. The chemical composition of the specimen after the second annealing (59 at.% Ni and 41 at.% Al) was determined by energy dispersive X-ray analysis in a scanning electron microscope (SEM). The decrease of Al content indicates that part of it evaporated during the remelting process. 3. RESULTS

In this paper only the GB grooves with singular walls will be considered. Detailed analysis of all observed groove morphologies will be given elsewhere [9]. The typical GB groove of interest is shown in Fig. 1(a). In Fig. 1(b), the corresponding line height pro®le taken perpendicular to the GB is presented. This groove exhibits several features di€erent from the classical Mullins grooves [1, 2]: . the height levels of two adjacent grains are di€erent, this is associated with the phenomenon of near-GB lattice rotation, which is considered in detail elsewhere [9, 10]; . the right part of the groove pro®le exhibits considerable curvature and the overall shape is qualitatively consistent with the Mullins theory [2]; however, the amplitude of the hump is higher than predicted by the Mullins theory; . a large portion of the left part of the pro®le does not exhibit any curvature at all, the height pro®le of the groove wall can be ®tted by the straight line; . the amplitude of the characteristic di€usional hump in the left part of the pro®le is extremely small. The ¯at morphology of the left wall of the groove implies that the corresponding surface is trapped in the local cusp of the gs(y ) plot, i.e. that this surface is singular. In Fig. 2, another groove is shown using higher magni®cation. Again, the right wall exhibits a signi®cant macroscopic curvature, while the left wall is absolutely ¯at. In Fig. 2(a) the vicinal character of the right wall can be clearly seen. On the microscopic level, the macroscopic curvature of the vicinal surface manifests itself in the variations of step density along the surface. The steps on the right wall of the groove of Fig. 2(a) can be seen better on the feedback loop error signal image presented in Fig. 3. This signal is generated if the AFM tip hits a sharp surface feature during scanning. In this imaging mode, the AFM can resolve individual atomic steps on the surface of the

scanned areas of several micrometers in size [11]. The steps are clearly seen on the vicinal surface, while no steps can be observed on the other surface, thus con®rming its singular character. Figure 1 clearly demonstrates that the GB groove extends to the left for approximately the same distance as it extends to the right. At the same time, it is clear that the atomic transport conditions are very di€erent at the singular and non-singular surfaces. In the next section, the growth mechanism of these asymmetric grooves will be considered. 4. THE MODEL

The small amplitude of the di€usional hump at the singular wall of the GB groove (see Fig. 1) clearly indicates that transport by surface di€usion along this singular surface plays a minor role in the development of the groove. The morphology of the majority of anisotropic GB grooves observed in this work was of the type shown in Fig. 1. Consideration of the physical reasons for the di€usional inactivity of the singular surface in the process of GB groove growth is beyond the scope of the present paper. We will only note that both the linear [7] and the parabolic [8] variation of the chemical potential with distance along the singular surface assumed in previous works were based on the purely steady-state arguments, without consideration of the physical mechanisms of surface di€usion. In our opinion, the elastic stress ®eld from the

Fig. 3. The feedback loop error signal image of the same GB groove as presented in Fig. 2. Steps on the vicinal surface (below, dark) can be clearly seen, while no steps on the singular surface are visible (above, light). In this imaging mode, the AFM can detect and visualize the atomic steps of one lattice parameter in height. On the right part of the GB, the irrelevant surface contamination appears.

RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

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brium, assuming that the inclination of the nonsingular surface at the triple line, y0, is not changing with time. For y0<<1 (our observations show that this is indeed the case, see Figs 1 and 2) the smallslope approximation [2] can be used, which leads to the Mullins equation for the non-singular surface: @y @ 4y ˆ ÿB 4 , @t @x

x
…2†

with the boundary conditions

Fig. 4. Illustration of the model of GB grooving at the singular surface. The role of this surface is reduced merely to determining of the angle y1 of the triple line glide in the direction of non-singular surface. This glide causes the GB shift from the original grain boundary (OGB) position.

line of torque moment applied at the GB groove root is the most important reason for chemical potential variation along the singular surface. The elastic energy of this stress ®eld varies as (1/r )4, where r is the distance from the GB groove root [12]. This variation is inconsistent with both the linear and parabolic laws [7, 8]. In this case, the migration of singular surface can only proceed by the nucleation and propagation of steps. However, no steps were observed in Fig. 2, which again favors the assumption about the di€usional inactivity of the singular surface. We will consider the following model (see Fig. 4): the singular surface is inclined on the angle y1 to the specimen surface and forms one of the two walls of the GB groove. The other wall is formed by the non-singular surface, which is curved and allows some surface di€usion to occur along it. The main assumption of the model is that all the material from the GB groove is transported to the outer surface along this non-singular wall. The singular surface does not participate in material transport and the triple point glides along the line y ˆ ÿm1 x

…1†

as the groove deepens. In this equation m1 ˆ tgy1 : The triple line glide should inevitably cause some GB migration and the corresponding change of the dihedral angles. Finding the equilibrium values of dihedral angles at the triple junction at which at least one interface is trapped in the cusp of gs(y ) plot is a delicate issue. Recently, A.H. King has shown that in this case the mechanical equilibrium can be degenerated and the intervals of dihedral angles for the non-singular interfaces at the triple junction are allowed [13]. We will avoid the problems associated with the degeneration of equili-

y…x 4 1† ˆ 0

…3a†

@ y ˆm @ x xˆx

…3b†

@ 3 y ˆ0 @ x 3 xˆx

…3c†

where m ˆ tgy0 and B is the Mullins coecient de®ned according to B ˆ dDs

gs O kT

…4†

where d, Ds and O are the thickness of surface di€usion layer, surface di€usion coecient and atomic volume, respectively. kT has its usual meaning. Equation (3c) re¯ects the fact that no surface di€usion along the singular surface (0 < x < x ) occurs. Strictly speaking, the system of equations (2)±(4) has been derived by Mullins for the single-component system, while in the present case we are dealing with the binary intermetallic compound. Recently, it was demonstrated that if the point at the interface at which the normal capillarityinduced stress at the interface vanishes can be found, equations (2)±(4) also can be applied for the binary system, assuming that Ds is appropriately expressed through the partial surface di€usion coef®cients of the components and their molar volumes [14]. In our case x=1 represents such a point, so the use of equations (2)±(4) is justi®ed. The system of equations (2)±(4) represents the generalization of the original Mullins problem, which for y1=p/2 (m1 4 1) can be reduced to the case considered in Ref. [2]. Following the method of Ref. [2], we are looking for the solution of equations (2)±(4) giving constant groove shape. For the surface di€usion mechanism, this means that all linear dimensions change with time, t, according to the t 1/4 law. We will, therefore, introduce the new dimensionless coordinates u, g and x0 according to x ˆ u…Bt†1=4

…5a†

y ˆ m…Bt†1=4 g…u†

…5b†

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RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

x ˆ x0 …Bt†1=4

…5c†

from the boundary conditions (6c) and (6d):

In the dimensionless coordinates, the system of equations (1) and (2)±(4) is transformed to 1 g 0000 ˆ ÿ … g ÿ ug 0 † 4

…6a†

g…u 4 1† ˆ 0

…6b†

0

…6c†

g 000 …u ˆ x0 † ˆ 0

…6d†

x0 a

…6e†

where a ˆ m=m1 and the prime, double prime, etc. denote the derivatives d=du, d 2 =du2 , etc., respectively. In this formulation, a is the only new parameter which makes the problem (6) di€erent from the formulation of the classical Mullins problem [2]. For a 4 0, the condition x0 4 0 should be ful®lled because the function g…u† is everywhere ®nite, and problem (6) is reduced to the Mullins problem [2]. As was demonstrated in Ref. [2], the general solution of equation (6a) with the boundary condition (6b) can be represented in the following form: g…u† ˆ a1 g1 …u† ‡ a2 g2 …u†

…7†

where a1 and a2 are the u-independent constants and the functions g1 and g2 are two linearly independent solutions of equation (6a) with the boundary condition (6b). It is convenient to choose these functions in such a way that they satisfy the following boundary conditions in the point u = 0: 1 g1 …0† ˆ ÿ p 1 ÿ 0:78012; 2G…5=4† p p g2 …0† ˆ p 11:38273 2G…5=4† g10 …0† ˆ 1

g20 …0† ˆ 0

1 g100 …0† ˆ ÿ p 1 ÿ 0:61372; 2G…3=4† p p 1 ÿ 1:02277 g200 …0† ˆ ÿ p 2G…3=4† g1000 …0† ˆ 0

g2000 …0† ˆ 1

g2000 …x0 † g10 …x0 †g2000 …x0 † ÿ g20 …x0 †g1000 …x0 †

a2 ˆ ÿ

g …u ˆ x0 † ˆ 1

g…u ˆ x0 † ˆ ÿ

a1 ˆ

g1000 …x0 † 0 000 g1 …x0 †g2 …x0 † ÿ g20 …x0 †g1000 …x0 †

…9a†

…9b†

Substituting equations (9) and (7) into the boundary condition (6e), we arrive at the following transcendental equation for determining x0: x0

g20 …x0 †g1000 …x0 † ÿ g10 …x0 †g2000 …x0 †  G…x0 † ˆ a g1 …x0 †g2000 …x0 † ÿ g2 …x0 †g1000 …x0 †

…10†

Function G(x0) is shown in Fig. 5, together with the functions g1 and g2. Finally, the shape g…u† of the GB groove for any given a can be determined according to the following algorithm: from equation (10), the value of x0 is determined for a given a; then the constants a1 and a2 are calculated according to equations (9), and ®nally the shape of the non-singular part of the GB groove is determined according to equation (7). Let us ®rst consider the situation a<<1. Using the boundary conditions for the functions g1 and g2 [see equations (8)], we can simplify equation (10) to the form a ˆ G…x0 †1 ÿ

x0 g1 …0†

…11†

Substituting the expression (11) to equation (6e), we ®nd the coordinate of the triple line:

…8a†

…8b†

…8c†

…8d†

where G is the Euler's gamma-function. These functions have been calculated by Mullins and represented as the in®nite power series [2]. They are shown in Fig. 5. Once the functions g1 and g2 are known, the constants a1 and a2 can be determined

Fig. 5. The dependence of the Mullins functions g1, g2 and of the new function G de®ned according to equation (9) on the dimensionless length u.

RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

g…x0 † ˆ ÿ

x0 ˆ g1 …0†1 ÿ 0:78012 a

1539

…12†

The groove depth given by equation (12) coincides exactly with the depth of the classical Mullins groove. This is reasonable, because according to the de®nition of a, the situation a<<1 corresponds to the singular surface perpendicular to the original specimen surface, and this geometry coincides with the geometry of the classical Mullins problem [2]. For large a, the function G…u† demonstrates upward deviations from the linear behaviour (see Fig. 5). In this case, the depth of the GB groove will be smaller than the value given by equation (12). The calculated GB groove pro®les g…u† for several values of parameter a are presented in Fig. 6. The calculated shapes are in good qualitative agreement with the experimentally observed ones (see Figs 1 and 2). The di€erence in shape between the singular GB groove and the Mullins groove is quanti®ed in Fig. 7, in which the dependencies of the groove width, W, and the groove depth, D, both reduced to the dimensionless values according to equations (5a) and (5b), on parameter a are shown. The values of W and D are de®ned in Fig. 4. For a=0, the depth of singular groove coincides with that of the Mullins groove, but its width is only half of that for the Mullins groove. This is because, for the singular groove under assumptions of the model, only one surface is available for atomic transport, which decreases the amount of material transported to the outer surface by a factor of two if compared with the Mullins groove. Figure 7 demonstrates

Fig. 7. The dependence of the singular groove dimensionless width, W=…Bt†1=4 , and depth, D=m…Bt†1=4 , on parameter a, compared with the parameters of the classical Mullins groove [2].

clearly that W is the increasing function of a, which means that the singular groove becomes very wide for small y1 (inclination of the singular surface to the specimen surface). However, because the rate of material transport along the non-singular surface stays ®nite, the widening of the groove is associated with the decrease of its dimensionless depth.

5. CONCLUSIONS

Fig. 6. Calculated according to the model GB groove shapes for di€erent a.

From the results of the present study, the following conclusions can be drawn. Firstly, unusual GB groove morphologies were observed by the AFM on the surface of Ni±41 at.% Al alloy after annealing at 14008C. One of the two groove walls did not exhibit any measurable curvature in spite of the fact that the distances from the groove root to the unperturbed external surfaces were approximately the same in both directions from the root. Evidence was present that the ¯at groove wall corresponds to the singular surface orientation. A low amplitude of the di€usional hump at this singular surface indicated that surface di€usion along the singular surface plays a minor role in the growth of GB grooves. Further, the model of GB grooving for the situation in which one of the groove walls coincides with the singular surface and exhibits a negligible surface di€usivity was developed. The essential idea of the model is that the groove develops and deepens due to material transport along the non-singular surface only, while the triple line glides along the singular surface. The GB groove shapes calcu-

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RABKIN et al.: GRAIN BOUNDARY GROOVING AT THE SINGULAR SURFACES

lated in small-slope approximation are in a good agreement with the experimental observations. AcknowledgementsÐThis research was supported by the German Federal State Niedersachsen and by the INTAS grant No. 97-0118. E.R. also wishes to thank the support by Technion V.P.R. FundÐAlexander Goldberg Memorial Research. Helpful discussions with Professor W. Gust from the University of Stuttgart and with Professor W.C. Carter from M.I.T. are heartily appreciated. REFERENCES 1. SchoÈllhammer, J., Chang, L.-S., Rabkin, E., Baretzky, B., Gust, W. and Mittemeijer, E. J., Z. Metallkd., 1999, 90, 687. 2. Mullins, W. W., J. appl. Phys., 1957, 28, 333. 3. Mullins, W. W. and Shewmon, P. G., Acta metall., 1959, 7, 163. 4. Mykura, H., Acta metall., 1961, 9, 570.

5. Sutton, A. P. and Ballu, R. W., Interfaces in Crystalline Materials. Clarendon Press, Oxford, 1995. 6. Bonzel, H. P. and Mullins, W. W., Surf. Sci., 1996, 350, 285. 7. Mullins, W. W., Phil. Mag., 1961, 6, 1313. 8. Carter, W. C., Rosen, A. R., Cahn, J. W. and Taylor, J. E., Acta metall. mater., 1995, 43, 4309. 9. Rabkin, E., Klinger, L., Semenov, V., Berner, A. and Izyumova, T., Acta mater., in preparation. 10. Rabkin, E., Semenov, V. and Bischo€, E., Z. Metallkd., in press. 11. Baretzky, B., Reinsch, B., Ta€ner, U., Schneider, G. and RuÈhle, M., Z. Metallkd., 1996, 87, 332. 12. Landau, L. D. and Lifshitz, E. M., Theory of Elasticity. Nauka, Moscow, 1965. 13. King, A. H., in Computational and Mathematical Models of Microstructural Evolution, ed. J. Bullard et al. MRS Symposium Proceedings Series, Vol. 529, 1998. 14. Rabkin, E., Estrin, Y. and Gust, W., Mater. Sci. Engng A, 1998, 249, 190.