Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.issp.ac.ru/libm/papers/266_eng.pdf Äàòà èçìåíåíèÿ: Tue Mar 5 21:43:22 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 02:14:21 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: http www.badastronomy.com bad tv foxapollo.html

ISSN 0021 3640, JETP Letters, 2012, Vol. 96, No. 9, pp. 582­587. © Pleiades Publishing, Inc., 2012. Original Russian Text © B.B. Straumal, P.V. Protsenko, A.B. Straumal, A.O. Rodin, Yu.O. Kucheev, A.M. Gusak, V.A. Murashov, 2012, published in Pis'ma v Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 2012, Vol. 96, No. 9, pp. 651­656.

Contribution of Tilt Boundaries to the Total Energy Spectrum of Grain Boundaries in Polycrystals
B. B. Straumala, b, *, P. V. Protsenkoc, A. B. Straumalb, A. O. Rodinb, Yu. O. Kucheeva, b, A. M. Gusakd, and V. A. Murashovc
a Institute of Solid State Physics, Russian Academy of Sciences, ul. Akademika Ossipyana, Chernogolovka, Moscow region, 142432 Russia * e mail: straumal@issp.ac.ru b National University of Science and Technology MISiS, Leninskii pr. 4, Moscow, 119049 Russia Department of Colloidal Chemistry, Faculty of Chemistry, Moscow State University, Moscow, 119899 Russia d Department of Theoretical Physics, Faculty of Physics, Cherkassy National University, Cherkassy, 18027 Ukraine

c

Received September 25, 2012

By measuring temperatures Tw for the transition from the incomplete to complete wetting of grain boundaries in poly and bicrystals, the width of the spectrum of tilt grain boundaries and their contribution to the total energy spectrum of grain boundaries in polycrystals have been experimentally estimated. It has been shown that the tilt grain boundaries correspond to a rather narrow (only 5­10%) portion in the total energy spec trum of grain boundaries in polycrystals. In metals with a low stacking fault energy (copper, tin, zinc), the tilt grain boundaries belong to 10­20% of the grain boundaries with the highest transition temperatures Tw (hence, with low energies). In a metal with a high stacking fault energy (aluminum), the values of Tw for the tilt grain boundaries lie nearly in the middle between the minimum (Tw ,min) and maximum (Tw ,max) transition temperatures from the incomplete to complete wetting of grain boundaries. This means that grain boundaries with the structure corresponding to a lower energy than that of the symmetric twin boundaries (or stacking faults) can exist in aluminum. DOI: 10.1134/S0021364012210114

As early as the turn of the 1940s and 1950s, it became clear that the grain boundaries in metals are not at all amorphous spacers between crystallites; on the contrary, they have a definite crystallographic structure. The so called tilt grain boundaries turned out to be the simplest ones for both theoretical and experimental analysis. They are the boundaries between crystallites, which are rotated with respect to each other about a fixed common crystallographic axis hk l (usually with low values of the Miller indi ces) by a certain misorientation angle ; the rotation axis hk l lies in the plane of the boundary. It is clear that the {hkl}1 planes for one grain coincide (some times with a small translation along the rotation axis) with the {hkl}2 planes for another grain. Even in the earlier works dealing with individual boundaries, it was shown that the grain boundary energy GB gradually grows from zero with , attains the maximum value at a certain max value, and then again decreases down to zero [1­5]. The value of max is determined by the symmetry of the rotation axis hk l . At small angles, the tilt grain boundary consists of an array of lattice

dislocations and the GB() function is given by the Read­Shockley expression
GB

= Gb ln ( e / 2 ) / [ 4 ( 1 ­ ) ] ,

(1)

where b is the Burgers vector of lattice dislocations, G is the shear modulus, is the Poisson ratio, and is a numerical constant, which is about unity for crystals with the face centered cubic (fcc) lattice and ranges from one to four in crystals with the body centered cubic (bcc) lattice [7]. Later, it was found that, at certain values of the mis orientation angle , there arise so called coincidence site lattices (CSLs). They are superlattices common to two lattices of the rotated grains (grains 1 and 2). Parameter determines how many sites belonging to each lattice, 1 and 2, is the share of one CSL site. The minimum possible value = 3 for the fcc lattice corre sponds to a twin boundary (or stacking fault). It turned out that the GB() curve in the vicinity of exhibits deep dips [8­10]. Their shape is also described by Read­Shockley relation (1) but with the difference ­ instead of the angle and with the Burgers vector b = b/ of grain boundary dislocations, which is the lattice vector for the displacement shift lattice

582


CONTRIBUTION OF TILT BOUNDARIES TO THE TOTAL ENERGY SPECTRUM

583

reciprocal to CSL [11, 12] instead of the Burgers vec tor b of lattice dislocations. A similar dependence with the dips also appears when the plane of the tilt grain boundary is rotated about the hk l axis, whereas the misorientation angle remains fixed [13, 14]. Gradually (by default), it was commonly accepted that the energy spectrum of the tilt boundaries begin ning from zero energy or from GB( =3) to GB(max) is just the energy spectrum of grain boundaries in a polycrystal as a whole (or, at least, its main part). This is because any simple experimental technique for comparing energies of tilt grain boundaries with differ ent misorientation axes and the total energy of tilt boundaries with that of a general type without the coinciding lattice planes was absent until recently. In [1­10, 13, 14], the energy of grain boundaries was measured by the technique of thermal etching groove. This method always gives the relative value expressed in units of the energy of the outer surface, to which the grain boundary comes out in the course of measure ments. The other methods (for example, zero creep measurements [15­18]) allow one to determine the average value of GB for polycrystals without giving information on the full amplitude of GB. An experimental method for comparison of ener gies GB for quite different kinds of grain boundaries has been developed recently. It is based on measuring temperatures of the transition from the incomplete to complete wetting of grain boundaries by the liquid phase (melt) [19­22]. If the two or multicomponent solid solution is in an equilibrium contact with the melt, two situations are possible. 1. The melt partially wets internal interfaces between crystallites (grain boundaries) in the polycrys tal. This means that the grain boundary energy GB is lower than the energy 2SL of two interfaces between the solid phase and the melt. In this case, the contact angle at the place where the grain boundary comes out to the interface between the solid and liquid phases is determined by the relation
GB



GB, max



GB, max low



GB, min

T

w, max

T

w2

T

w1

T

w, min

= 2 SL cos ( / 2 ) .

(2)

2. The melt completely wets the grain boundaries. Then, GB 2SL and the contact angle vanishes ( = 0). In this case, the melt layer should substitute for the grain boundary, separating the crystallites from each other. In 1977, it was demonstrated for the first time that the transition from the incomplete to complete wet ting is a phase transition, which can occur at variations of the temperature or pressure [23, 24]. It is possible if the GB(T) and 2SL(T) curves cross at some tempera ture Tw below the melting temperature Tm. The value of Tw depends on the grain boundary energy: the higher GB, the lower Tw. The wetting transition in polycrystals starts at Tw, min, when the complete wetting occurs at grain boundaries with the maximum energy
JETP LETTERS Vol. 96 No. 9 2012

Fig. 1. (a) Schematic curve illustrating the dependence of the grain boundary energy on the misorientation angle. The dips correspond to special types of grain boundaries close to the coincidence misorientations. (b) Part of the phase diagram for Cu­In alloys [26]. Solid lines corre spond to the phase transitions in the bulk, thin lines denote tie lines for the wetting transitions at grain boundaries, and points indicate the annealing temperatures and the com positions of the alloys under study.

GB, max. It terminates at Tw, max, when the complete wetting occurs at grain boundaries with the minimum energy GB ,min (Fig. 1a). The comparison of Tw values measured at specially grown bicrystals [25] with indi vidual tilt boundaries [19­22] having the minimum and maximum energies with Tw, min and Tw, max mea sured at polycrystals with a large number of grain boundaries just provides a tool to determine the width of the spectrum of tilt grain boundaries and their con


584

STRAUMAL et al.

Fig. 2. (a) Microstructure of Cu­2 wt % Zn alloy after annealing at 220°C in the two phase region of the phase diagram. (b, c) Microstructure of Cu­4 wt % In alloy after annealing in the two phase region of the phase diagram at 772 and 972°C, respectively. The arrows indicate completely (A) and incompletely (B) wetted grain boundaries.

tribution to the total energy spectrum of grain bound aries in polycrystals. For the studies using induction melting in vacuum, we prepared alloys of copper with 4, 8, 12, 16, and 22 wt % of indium. We used copper with a purity of 99.998 wt % and indium with a purity of 99.999 wt %. We also prepared an alloy of tin with 2 wt % of zinc (using both tin and zinc with a purity of 99.999 wt %). The prepared ingots 10 mm in diameter were cut into disks 2 mm thick, which were sealed into evacuated quartz ampoules. The ampoules with the samples were annealed in a SUOL resistance furnace for 2 h at tem peratures ranging from 715 to 1000°C (copper­ indium alloys) and for 7 h at 201, 207, 212, 215, 217, 222, and 227°C (tin­zinc alloy). The points corre sponding to the values of annealing temperature and to

the content of copper­indium alloys are shown in Fig. 1b. After annealing, the samples were quenched in water (the ampoules were broken) and then were mechanically ground and polished using a diamond paste with grit down to 1 m. The obtained slices were studied by optical spectroscopy using a Neophot 32 microscope equipped with a 10 Mpix Canon Digital Rebel XT digital camera, as well as by scanning elec tron microscopy and by electron probe microanalysis (EPMA) using a Tescan Vega TS51030 MM device equipped with a LINK energy dispersion spectrome ter (Oxford Instruments). A grain boundary on the copper based (Cu) or tin based (Sn) solid solution was treated as completely wetted if the melt layer com pletely covered the boundary from one triple junction to another (boundaries A in Fig. 2). The boundary was
JETP LETTERS Vol. 96 No. 9 2012


CONTRIBUTION OF TILT BOUNDARIES TO THE TOTAL ENERGY SPECTRUM

585
w, max

considered as partially wetted if the melt layer was ter minated (boundaries B in Fig. 2). Figure 2b illustrates the microstructure of Cu­ 12 wt % In alloy after annealing at 772°C. It includes two structural components, dark and light. The dark structural component (host) is the copper based solid solution with the indium content corresponding to the solidus line at the annealing temperature (i.e., about 7 wt % of In). The light component, located at grain boundaries and triple junction, was formed in the course of quenching from the indium rich melt with the composition corresponding to the liquidus line. In the structure shown in Fig. 2b, there are a very small number of grain boundaries completely wetted by the melt. The melt forms isolated although very flat drop lets at grain boundaries. The first grain boundaries completely wetted by the melt appear at Tw, min = 715°C (see Fig. 1b). For Tw, min = 715°C, on the phase diagram presented in Fig. 1b, we show a tie line for the onset of the wetting phase transition for high angle grain boundaries. At this temperature, the grain boundaries with the highest energy become com pletely wetted (see the schematic curve in Fig. 1a). Figure 2c illustrates the microstructure of Cu­4 wt % In alloy after annealing at 972°C. In this structure, nearly all high angle boundaries are completely wet ted. The last large angle grain boundaries partially wetted by the melt disappear at Tw, max = 986°C (see Fig. 1.b). For Tw, max = 986°C, on the phase diagram presented in Fig. 1b, we show a tie line for the offset of the wetting phase transition for high angle grain boundaries. At this temperature, the grain boundaries with the lowest energy become completely wetted (see Fig. 1a). Figure 3a shows the temperature dependence of the percentage of the wetted grain boundaries in Cu. It grows from zero at Tw, min = 715°C to 100% at Tw, max = 986°C. The measurements of Tw were reported in [22] for two tilt grain boundaries in copper with the misorien tation axis 110 and with the misorientation angles = 77° and 141°. These values are Tw2 = 960°C and Tw1 = 930°C, respectively. The misorientation angle = 77° is close to = 70.5° characteristic of the sym metric twin boundaries and the corresponding energy GB2 is close to the minimum value possible for the tilt grain boundaries in copper. The energy GB1 for the boundary with = 141° is higher by 40%. It is close to the maximum energy possible for the tilt grain bound aries with 110 . Both tie lines, at Tw2 = 960°C and Tw1 = 930°C, are shown in Fig. 1b. We see that the dif ference Tw2 ­ Tw1 = 30°C is very small (about 11%) in comparison to the difference Tw, max ­ Tw, min = 271°C (see Figs. 1b and 3a). At the same time, both temper atures, Tw2 = 960°C and Tw1 = 930°C, are very close to Tw, max = 986°C.
JETP LETTERS Vol. 96 No. 9 2012

T

w, min

Tw1 Tw2 T

w, min)

Sn­Zn Cu­In

Zn­Sn

w, min)(Tw, max

­T

Zn­In

Al­Zn

In Sn­2 wt % Zn alloy (see Fig. 2a), the complete wetting of all grain boundaries occurs at Tw, max = 220°C. However, even at 201°C (i.e., at a temperature slightly exceeding the eutectic temperature of 198.5°C [26]), 90% of grain boundaries in tin are already wet ted by the melt. This means that Tw, min is lower than the eutectic temperature. It can be formally deter mined by the extrapolation to zero of the temperature dependence of the percentage of the wetted grain

(T ­ T
Fig. 3. (a) Temperature dependence of the percentage of the wetted grain boundaries in copper­indium alloys. We show the onset (Tw, min = 715°C) and offset (Tw, max = 986°C) temperatures of the wetting transition in polycrys tals, as well as the wetting transition temperatures at indi vidual tilt grain boundaries with misorientation angles = 77° and 141° (Tw2 = 960°C and Tw1 = 930°C). (b) (Open circles) Maximum and (closed circles) minimum wetting transition temperatures Tw for the tilt grain boundaries in metals with different stacking fault energies. The values of Tw are normalized to the difference between the maximum (Tw, max) and minimum (Tw, min) values of the wetting transition temperature in polycrystals. The crosses indicate the temperatures at which half of the grain boundaries in polycrystals are wetted.


586

STRAUMAL et al.

boundaries. The measurements of Tw were reported in [27] for two tilt grain boundaries in tin with the misori entation axis 100 and with the misorientation angles close to those characteristic of the coincidence misorientations 5 and 17. The values of Tw1 and Tw2 almost coincided and were equal to 217°C. This tem perature is also very close to Tw,max, as in copper­ indium alloys. Figure 3b shows temperatures for the transition from the incomplete to complete wetting as functions of the stacking fault energies. On the vertical axis, zero and unity correspond to Tw, min and Tw, max, respectively. In copper, the stacking fault energy according to dif ferent sources ranges from 30 to 45 mJ/m2 [28], and for tin, we have 30 mJ/m2 [29]. In addition to the data obtained in the present work for copper­indium and tin­zinc alloys, Fig. 3b also shows the points corre sponding to other systems studied by us earlier. In [30], for example, the values Tw, min = 440°C and Tw, max = 620°C were reported for Al­Zn polycrystals, and the temperatures Tw1 = 530°C and Tw2 = 525°C were measured in [31] for 110 tilt boundaries in alumi num corresponding to misorientation angles equal to 15° and 35°. According to the data reported in [8], the energy of tilt grain boundaries at these two misorienta tion angles is close to the maximum and minimum values, respectively. The stacking fault energy in alu minum is 140 mJ/m2 [32]. In zinc­indium alloys (Teutct = 143.5°C and Tmelf(Zn) = 419.58°C [26]), already 18% of grain boundaries are wetted at 210°C, whereas at 360°C, the percentage of wetted boundaries is 63% [33]. This means that both Tw, min and Tw, max can be found only by extrapolation (as a result, we have 150 ± 20°C and 550 ± 20°C, respectively). The temperatures of the transition from the incomplete to complete wetting were measured for the [ 10 1 0 ] tilt grain boundaries in zinc with the misorientation angles 1 = 19° and 2 = 66° and for the tilt grain boundary [1, 1, ­2, 0] with 3 = 79° [33]. Angles 2 and 3 are close to those cor responding to the coincidence misorientations, whereas 1 characterizes a grain boundary of a general type [34]. The transition temperatures are Tw1 = 362°C, Tw2 = 376°C, and Tw3 = 375°C, respectively. The stacking fault energy in zinc according to different sources ranges from 160 to 200 mJ/m2 [29]. In zinc­tin alloys (Teutct = 198.5°C [26]), already 19% of grain boundaries are wetted at 260°C, whereas at 395°C, the percentage of wetted boundaries is 68% [33]. Therefore, both Tw, min and Tw, max can be found only by extrapolation (as a result, we have 100 ± 20°C and 450 ± 20°C, respectively). The temperatures of the transition from the incomplete to complete wetting were determined for the [1, 0, ­1, 0] tilt grain bound aries in zinc with the misorientation angles 1 = 16°

and 2 = 60° [19]. Angle 2 is close to that correspond ing to the coincidence misorientations, whereas 1 characterizes a grain boundary of a general type [34]. The transition temperatures are Tw1 = 386.5°C and Tw2 = 381°C, respectively. To summarize, the width of the spectrum of tilt grain boundaries and their contribution to the total energy spectrum of grain boundaries in polycrystals have been estimated for the first time by measuring temperatures Tw for the transition from the incom plete to complete wetting of grain boundaries in poly and bicrystals. It turned out that all tilt grain bound aries (from those with coincidence misorientations with the lowest grain boundary energy to the grain boundaries of the general type with the highest energy) correspond to a rather narrow (only 5­10%) portion in the total energy spectrum of grain boundaries in polycrystals. In metals with a low stacking fault energy (copper, tin, zinc), the tilt grain boundaries belong to 10­20% of the grain boundaries with the highest tran sition temperatures Tw (hence with low energies). In the metal with a high stacking fault energy (alumi num), the values of Tw for the tilt grain boundaries lie nearly in the middle between the minimum (Tw, min) and maximum (Tw, max) transition temperatures from the incomplete to complete wetting of grain bound aries. This means that grain boundaries with the struc ture corresponding to a lower energy than that of the symmetric twin boundaries (or stacking faults) can exist in aluminum. We are grateful to A.N. Nekrasov for his help in performing experiments and to Profs. B.S. Bokshtein and A.L. Petelin for helpful discussions. This work was supported by the Russian Foundation for Basic Research (project nos. 11 03 01198 and 11 08 90439), by the State Foundation for Basic Research of Ukraine (project no. F407040), and by the National University of Science and Technology MISiS (Devel opment Program). REFERENCES
1. K. T. Aust and B. Chalmers, Proc. R. Soc. A 201, 210 (1950); Proc. R. Soc. A 204, 359 (1950). 2. C. G. Dunn, F. W. Daniels, and M. J. Bolton, J. Metals 2, 1245 (1950). 3. A. P. Greenough and R. King, J. Inst. Metals 79, 415 (1951). 4. C. G. Dunn and F. Lionetti, J. Metals 1, 125 (1949). 5. R. C. Pond and D. A. Smith, Int. Met. Rev. 21, 61 (1976). 6. J. P. Hirth and J. Lothe, Theory of Dislocations (McGraw Hill, New York, 1968; Atomizdat, Moscow, 1972). 7. E. Rabkin and I. Snapiro, Acta Mater. 48, 4463 (2000). 8. G. Hasson, J. Y. Boos, and I. Herbeuval, Surf. Sci. 31, 115 (1972).
JETP LETTERS Vol. 96 No. 9 2012


CONTRIBUTION OF TILT BOUNDARIES TO THE TOTAL ENERGY SPECTRUM 9. T. Muschik, W. Laub, U. Wolf, et al., Acta Metall. Mater. 41, 2163 (1993). 10. A. Barg, E. Rabkin, and W. Gust, Acta Metall. Mater. 43, 4067 (1995). 11. A. N. Orlov, V. N. Perevezentsev, and V. V. Rybin, Grain Boundaries in Metals (Metallurgiya, Moscow, 1980) [in Russian]. 12. W. Bollmann, Crystal Defects and Crystalline Interfaces (Springer, Berlin, 1970). 13. A. Otsuki and M. Mitsuno, Trans. Jpn. Inst. Met. (Suppl.) 27, 789 (1986). 14. A. Otsuki, Interface Sci. 9, 293 (2001). 15. I. Sawai and M. Nishida, J. Anorg. Chem. 190, 375 (1930). 16. G. Tammann and W. Boehme, Ann. Phys. 12, 820 (1932). 17. H. Udin, A. J. Shaler, and J. Wulff, J. Metals 1, 1936 (1949). 18. B. S. Bokshtein, D. V. Vaganov, and S. N. Zhevnenko, Phys. Met. Metallogr. 104, 564 (2007). 19. B. B. Straumal, W. Gust, and T. Watanabe, Mater. Sci. Forum 294, 411 (1999). 20. B. B. Straumal, W. Gust, and D. A. Molodov, Interf. Sci. 9, 127 (1995). 21. B. Straumal and W. Gust, Mater. Sci. Forum 207, 59 (1996). 22. B. Straumal, T. Muschik, W. Gust, et al., Acta Metall. Mater. 40, 939 (1992).

587

23. J. W. Cahn, J. Chem. Phys. 66, 3667 (1977). 24. C. Ebner and W. F. Saam, Phys. Rev. Lett. 38, 1486 (1977). 25. V. N. Semenov, B. B. Straumal, V. G. Glebovsky, et al., J. Cryst. Growth 151, 180 (1995). 26. Binary Alloy Phase Diagrams, Ed. by B. Massalski et al. (ASM International, Materials Park, 1993). 27. V. A. Murashov, B. B. Straumal, and P. V. Protsenko, Bull. Russ. Ac. Sci.: Phys. 74, 1551 (2010). 28. C. B. Carter and I. L. F. Ray, Philos. Mag. 35, 189 (1977). 29. L. E. Murr, Interfacial Phenomena in Metals and Alloys (Addison Wesley, Boston, 1975). 30. B. Straumal, G. Lopez, W. Gust, et al., in Nanomateri als by Severe Plastic Deformation. Fundamentals­Pro cessing­Applications, Ed. by M. J. Zehetbauer and R. Z. Valiev (Wiley VCH, Weinheim, 2004), p. 642. 31. B. B. Straumal, A. S. Gornakova, O. A. Kogtenkova, et al., Phys. Rev. B 78, 054202 (2008). 32. J. Cai, F. Wang, C. Lu, et al., Phys. Rev. B 69, 224104 (2004). 33. A. S. Gornakova, B. B. Straumal, S. Tsurekawa, et al., Rev. Adv. Mater. Sci. 21, 18 (2009). 34. G. Gottstein and L. Shvindlerman, Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Appli cations (Oxford Univ. Press, Oxford, 2011).

Translated by K. Kugel

JETP LETTERS

Vol. 96

No. 9

2012