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Volume 59, Issue 9, May 2011, Pages 3510-3518

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Volume 59, Issue 9, May 2011, Pages 3510-3518 e a,⇑ RW of form e 2 und gr n g res equ was the analysis of a groove formed at the intersection of a mine the triple junction line tension [5,6]. The method is based on the equilibrium of four line tensions at their point of intersection—the grain bou...

Volume 59, Issue 9, May 2011, Pages 3510-3518
e a,⇑ RW of form e 2 und gr n g res equ was the analysis of a groove formed at the intersection of a mine the triple junction line tension [5,6]. The method is based on the equilibrium of four line tensions at their point of intersection—the grain boundary triple line and three and to design the grain microstructure of fine-grained 2. Measurement of triple line tension The fundamental principle underlying the measurement of triple line tension is the establishment of thermodynamic equilibrium at the intersection of four triple junctions, namely three groove root junctions (triple line between ⇑ Corresponding author. E-mail address: gottstein@imm.rwth-aachen.de (G. Gottstein). Available online at www.sciencedirect.com Acta Materialia 59 (2011) 3510–35 grain boundary with a surface [1]. The fact that grain boundary motion can be influenced by a thermal groove [2] is established as textbook knowledge. High-resolution transmission electron microscopy observations have shown that the intersection of a grain boundary with a free surface leads to a reconstruction [3]. This proves the existence of a driving force, which is likely to be related to the existence of a grain boundary–free surface line tension [4]. We have recently designed a method to correctly deter- and nanocrystalline materials through their effect on recov- ery, recrystallization and grain growth. Knowledge of the magnitude of the grain boundary line tension can provide a quantitative estimate of the contribution of grain bound- ary triple junctions to the driving force for grain growth. In this study we will show that the Zener force and the Gibbs– Thompson relation and related phenomena are also affected by the triple line energy. 1. Introduction The term “triple line” identifies the intersection of three interfaces, either external interfaces or internal interfaces of bulk materials. Triple lines are differentiated based on the interfaces that are intersecting, e.g. three phase boundaries, three grain boundaries, or one grain boundary and two phase boundaries. One of the earliest studies of triple junctions in materials triple lines at the bottom of the thermal grooves of the merging boundaries. The line tension of the grain bound- ary–free surface triple lines is determined by comparing the dihedral angle at the root of a flat and a curved grain boundary groove. Grain boundary triple junctions have been recognized recently to constitute another structural element of poly- crystals, which can strongly impact microstructural evolu- tion [7–9]. They open up new opportunities to control Triple junction B. Zhao a, G. Gottstein a Institut fu¨r Metallkunde und Metallphysik, b Institute of Solid State Physics, Russian Academy Received 22 December 2010; received in revised Available onlin Abstract The grain boundary–free surface triple line tension and grain bo introduced novel approach. The effect of triple line tension on grain results showed that the triple line tension has a considerable effect o cially for nanocrystalline materials. � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights Keywords: Grain boundary triple junction; Zener drag; Gibbs–Thompson 1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All doi:10.1016/j.actamat.2011.02.024 ffects in solids , L.S. Shvindlerman a,b TH Aachen University, Aachen, Germany Sciences, Chernogolovka, Moscow District, Russia 10 February 2011; accepted 14 February 2011 1 March 2011 ary triple line tension were investigated in copper using a recently owth, Zener drag and Gibbs–Thompson relation was studied. The rain growth, particle–boundary interactions and void shape, espe- erved. ation; Triple line energy www.elsevier.com/locate/actamat 18 rights reserved. (b eria Groove root 1-2 Gro ove roo t 2- 3 z y u2-3(r) r t (a) B. Zhao et al. / Acta Mat two crystal surface and a grain boundary) and one triple junction formed by the intersection of three grain bound- aries (Fig. 1). The equilibrium in the z-direction yields clTP ¼ clS1�2 sin f1�2 þ clS1�3 sin f1�3 þ clS2�3 sin f2�3; ð1Þ where clTP and c lS i�j are the grain boundary triple line tension and the line tension of the triple lines at the bottom of each thermal groove, respectively. fi�j are the angles at each Groove root 1-3 x Triple line r Fig. 1. (a) Schematic 3-D view of the line geometry at a triple junction. (b) Fig. 2. (a) AFM topography measurement on a triple junction of a Cu tricr γB γSγS θ γB γSγS θ (a) (b Fig. 3. (a) Grain boundary groove formed at a straight, non-curved grain bou grain boundary with a curved groove root [6]. Grain3Grain2 u2-3(r) lS 21−γ lS 31−γ lS 32−γ Grain3Grain2 u2-3(r) lS 21−γ lS 31−γ lS 32−γ ) lia 59 (2011) 3510–3518 3511 groove root of the corresponding grain boundary at the center of the triple junction (Fig. 2). Eq. (1) requires the determination of the triple line tensions of the groove roots clSi�j. This information can be retrieved from the curvature of the groove root triple lines. Fig. 3a depicts a typical thermal groove formed at a tilt grain boundary which extends perpendicular to the free surface. If the orientation of the surface on both sides of Grain1 Triple line lγ Grain1 Triple line lγ AFM 3-D view of the line tension equilibrium at a triple junction [6]. ystal after annealing at 980 �C for 2 h. (b) Profiles along the lines in (a). ξξ ) ndary with no variation in height and (b) grain boundary formed at a flat eria 3512 B. Zhao et al. / Acta Mat the grain boundary is the same and the root of the groove is straight, the specific grain boundary energy is given by: cB ¼ 2cS cos h 2 ; ð2Þ where h is the dihedral angle at the groove root under the assumption that the grain boundary groove is symmetric, and cB, cS are the grain boundary tension and the free sur- face tension, respectively. Fig. 4. Top view of AFM topography measurement of a grain boun Fig. 5. (a) Top view of AFM topography measurement on a Cu wire after ann and across the thin Cu wire. lia 59 (2011) 3510–3518 The curved groove root in Fig. 3b gives: cB � clS R ¼ 2cS cos n 2 ; ð3Þ where R is the radius of curvature at a given point of the groove root. The dihedral angle n has the same meaning as h but may be different in magnitude owing to the curva- ture. Combining Eqs. (1) and (2) yields the grain bound- ary–free surface line tension clS: dary groove in a Cu bicrystal after annealing at 980 �C for 2 h. ealing at 300 �C for 2 h. AFM image step size 7 nm. (b) Profiles parallel to eria clS ¼ 2cS cos h 2 � cos n 2 � � � R: ð4Þ The measurement of the angle h was performed on Cu bicrystals with perfect surface quality [6] Fig. 4. The dihedral angle h was determined from an atomic force microsopy (AFM) scan perpendicular to the grain boundary groove, and was found to be h ¼ 161:0� � 2:3�. The frequency distribution of the measured dihedral angle was Gaussian. Measurement of the angle n (Eq. (3)) is more difficult. For that, we determined the dihedral angle of the grain boundary groove on very thin Cu wires, which were grown by strain control within thin-film cracks [10] Fig. 5. Copper wire grids were deposited on a silicon substrate, and annealed at 300 �C in a vacuum furnace for 2 h. To obtain maximum accuracy, the measurements were performed with high-aspect-ratio tips (Olympus AC11160BN-A2) in an atomic force microscope in non- contact mode. The resolution of these tips was calibrated with a tip-check sample, and the geometry of the tip was derived by blind tip reconstruction. According to this cali- bration, the tip radius was about 10 nm with a slope of 72�. Therefore the step size of the AFM image of Cu wires was also chosen to be610 nm for fast and accuratemeasurement. The radius of the wire was in the range 80–150 nm. The dihedral angles of the grooves were in the range 150–157� depending on the groove curvature. This yielded on average a line tension of the grain boundary groove root (triple line grain boundary–free surfaces) clS = (2.5 ± 1.1) � 10�8 J/m. Based on the assumptions that 1. the free surface energy is independent of orientation, cS = 1.75 J/m 2, 2. the grain boundary–free surface line tension is constant along the groove root, and 3. all grain boundary–free surface line tensions are the same, the grain boundary triple line tension in Cu was determined to be clTP ¼ ð6:0� 3:0Þ � 10�9 J=m. The sign of the grain boundary triple line tension depends on the grain boundary–free surface line tension, which is positive under equilibrium conditions. It was dem- onstrated in Ref. [6] that a negative grain boundary–free surface line tension would cause the grain boundary–free surface interface to form a convex bulge which would be higher than the surface. Therefore a negative grain bound- ary–free surface line tension and the normal (classical) shape of a thermal groove system cannot be in equilibrium. Consequently, the grain boundary triple line tension must be positive. This is different from the line tension for a spe- cific system which can undergo a first-order wetting transi- tion, where the line tension will change sign from negative to positive values with increasing temperature [11–13]. Pompe et al. [14] found that the line tension of a three- B. Zhao et al. / Acta Mat phase system (solid–liquid–vapor) is of the order of �2� 10�10 to þ8� 10�11 J/m. The experiments in Ref. [14] were carried out on hexaethylene glycol, aqueous CaCl2 solution and water with surface tensions in the range of (4.5–7.2) � 10–2 J/m2. The measured grain boundary– free surface line tension in our experiments on Cu amounted to clS ¼ ð2:5� 1:1Þ � 10�8 J=m and the respec- tive grain boundary triple line tension was determined to be clTP ¼ ð6:0� 3:0Þ � 10�9 J=m with cS ¼ 1:75 J=m2 obvi- ously in reasonable agreement with Ref. [14]. 3. Impact of triple line tension on grain growth 3.1. Effect of triple junctions on the driving force for grain growth It was shown in Refs. [15,16] that the influence of grain boundary junctions (triple line and quadruple junction in 3-D polycrystals) on grain growth kinetics can be described in terms of a parameter K. The rate of 3-D grain growth can be expressed as: dhDi dt ¼ mBP 1þ 1KQP þ 1KTP � � ; ð5Þ where D is the grain size, and KQP and KTP are the quadru- ple point and triple junction parameters, respectively [17]. We note that 1K is a measure of the drag effect of a given junction on grain growth. However, junctions do not only exert a drag on grain boundary motion, but also contribute to the driving force for grain growth due to the junction en- ergy. The classical equation for the driving force P for con- tinuous grain growth reads P ¼ 2cBR . Here, 2/R is the grain boundary curvature. If, for simplicity, we assume R � hDi, the total driving force from grain boundaries and triple junctions reads: P ¼ 2cBhDi þ 36clTP phD2i : ð6Þ The ratio PTPPB ¼ 18clTP pcB ¼ a defines the grain size for which the grain boundary and triple junction contributions to the driving force are equal. Assuming the measured value of clTP � 6� 10�9 J=m2 for Cu, a constant grain boundary surface tension of Cu cB ¼ 0:6 J=m2 and a constant ratio clTP=cB, we arrive at a � 55 nm. In other words, up to a mean grain size of about 55 nm the driving force stemming from triple junctions is larger than that of the grain bound- aries. As a consequence, a correct examination of grain growth in nanocrystalline materials at least up to mean a grain size a cannot be performed if the driving force of tri- ple junctions is not taken into account [17]. 3.2. Effect of triple line tension on grain boundary–particle interaction In 1948 Zener proposed a concept of how the interaction lia 59 (2011) 3510–3518 3513 between a grain boundary and a particle can be estimated quantitatively [18]. u < arccos 1g, the grain boundary would intersect the parti- cle, and f* becomes a retarding force for grain boundary motion. In Cu, cB ¼ 0:6 J=m2, and if we assume cBP � clS ¼ ð2:5� 1:1Þ � 10�8 J=m, the critical size R* is about 40 nm; if we take the value cBP � clTP ¼ ð6:0� 3:0Þ � 10�9 J=m, the critical size R* is about 10 nm. According to Fig 7, DG0I > 0 for R < R * and increases with growing area of intersection. The minimum value of DG0 is obtained at ticle 0.0 0.5 1.0 1.5 2.0 -6 -5 -4 10nm 20nm 30nm GB h/R Fig. 7. Dependency of DG0I on particle radius in Cu for c BP ¼ clTP ; h is the penetration depth of the boundary in the particle as defined in Fig. 6. eria When a spherical solid particle intersects a grain bound- ary, it replaces part of the grain boundary area, and by doing so reduces the free energy of the system: DG ¼ �pr2 � cB; ð7Þ where G is the Gibbs free energy, cB is the boundary energy, and r is the radius of the intersected circular area (Fig. 6). However, this area has to be regenerated when the grain boundary detaches from the particle. The respective retard- ing force, i.e. the “Zener force”, is given by: f ¼ dDG dr ¼ �2pr � cB; ð8Þ which retards the motion of the grain boundary. For the past 60 years all considerations of grain bound- ary motion and grain growth in solids with both immobile and mobile particles have been based on this concept, i.e. the triple line that forms at the intersection of sphere and boundary was simply disregarded. In Section 2 we showed that a triple line is a defect in its own right with specific thermodynamic properties, and that the line tension of a triple junction can be measured. Hence, such measure- ments can be utilized to study its effect on particle bound- ary interaction [19]. Then, Eq. (7) has to be rewritten as: DG0I ¼ �pr2 � cB þ 2pr � cBP ; ð9Þ where cBP is the line tension of the triple junction. Triple line “grain boundary-particle” Fig. 6. Schematic view of a par 3514 B. Zhao et al. / Acta Mat Let us define the coefficient g ¼ cB�RcBP . The derivative of Eq. (9) reads: dDG0I du ¼ 2pRcBP sinu � ðg cosu� 1Þ: ð10Þ For g < 1, DG0I attains a maximum value at u ¼ 0. When g > 1, the maximum shifts to cosu ¼ 1g, and DG0I assumes a minimum at u ¼ 0. The sign of DG0I depends on the par- ticle radius, as shown in Fig. 7. From Eq. (9), we obtain the interaction force between the grain boundary and the particle: F � ¼ �2pr � cB þ 2pcBP ð11Þ At the critical particle size R� ¼ cBPcB , f* changes sign. Hence, for R < R* the particle will not attach to the grain boundary. When the particle radius is larger than R*, and ϕr R Grain boundary Particle h r R r R r R h intersecting a grain boundary. -3 -2 -1 0 1 2 3 -60° 90°60°0°-90° ΔG 0 I(1 0- 16 J) GB lia 59 (2011) 3510–3518 I u ¼ �90�, therefore the particle will not attach to the grain boundary at all. When R > R*, a further minimum of DG0I ðuÞ appears at u ¼ 0�. On the other hand, the impact of the triple junction on particle–boundary interaction brings about the formation of an energy barrier. This bar- rier can only be overcome with the help of an external driv- ing force, as follows: DGI ¼ �pr2 � cB þ 2pr � cBP � PV : ð12Þ The external driving force P will expend a work propor- tional to the swept volume V. The energy barrier still exists, but the grain boundary will become curved under the driv- ing force. If we assume that the radius of the curved grain boundary is equal to the radius of the particle, the change in the system energy DGC due to the curved grain boundary reads: eria DGC ¼ �pr2 � cB þ 2pRh � cB � PV : ð13Þ Fig. 8 gives an example of the dependencies DGI(u) and DGC(u) for R = 30 nm, P = 25 MPa. Accordingly, the grain boundary will first circumvent the particle (energy curve DGC, broken line in Fig. 8) until the angle u� is reached. Then it will intersect the particle (energy curve DGI, thick solid profile in Fig. 8). The angle u� and the crit- ical driving force for the process are: u� ¼ arcsin 1� g 2 1þ g2 ð14Þ Pcrit ¼ ð1þ g 2ÞcB g2R ð15Þ A detailed derivation is given in the Appendix A. When the driving force P > Pcrit, the grain boundary can pass the particle, and the interaction force is given by Eq. (11). The maximum interaction force f �max ¼ f �ðRÞ does 0.0 0.1 0.2 0.3 0.4 0.5 -6 -4 -2 0 2 Δ G (1 0- 16 J) h/R ΔGI ΔGI0 ΔGC -90° -60°* Δ Fig. 8. The dependencies of DG0I , DGI and DGC on h/R for R = 30 nm, cBP ¼ clTP ¼ ð6:0� 3:0Þ � 10�9 J=m. B. Zhao et al. / Acta Mat not depend on the driving force. In other words, a particle cannot cross a grain boundary spontaneously, but a certain driving force needs to be applied to the grain boundary to overcome the barrier. In the course of the process, the grain boundary first becomes curved and eventually will pass the particle. So far we have assumed that the particle is immobile. In the following we will dwell on the interaction between mov- ing grain boundary and mobile particles [15,20–23]. Since the mobility of a second-phase particle strongly depends on its size, the problem becomes particularly important for fine-grained and nanocrystalline materials. In Ref. [23] this problem was considered in the framework of the Zener approach for steady-state motion of the boundary– particle system. The latter assumes that the shape of a par- ticle which moves together with a grain boundary changes with the velocity of their joint motion. It is of interest to evaluate the size of such a particle. For joint motion, the velocity of a particle and grain boundary have to be equal: vB ¼ vP ; ð16Þ where vB ¼ 2mBcBhDi=2 , andmB, cB and hDi are the reduced bound- arymobility, grain boundary surface tension andmean grain size, respectively. The particle velocity vP ¼ mPpr ¼ 110 � DsbXakT �R4 pRcB [15], where Ds, b, Xa and R are the interface diffusion coefficient, lattice constant, atomic volume and particle ra- dius, respectively. For a Cu polycrystal with hDi � 10 lm ¼ 10�5 m at 937 K, cB ¼ 0:6 J=m2, Ds � 10�13 m2=s, b ¼ 3 � 10�10 m and Xa ¼ 10�5 m3=mol, the radius R of a particle moving together with a grain boundary is in the range 1–2 nm. For R � 2 nm, g ¼ cB�RcBP < 1 and the critical driving force P for the grain boundary motion which is nec- essary to permit a second-phase particle to intersect the grain boundary is given by Eq. (15), with parameters cB ¼ 0:6 J=m2, cBP � clTP ¼ 6� 10�9 J=m, R � 2� 10�9 m and Pcrit ¼ 7:7� 109 J=m3. This value is four orders of mag- nitude larger than the capillary driving force! Even for a mean grain size < D >� 20 nm the critical driving force Pcrit is larger than the driving force for grain growth. In other words, themotion of a 2–3 nm particle adsorbed to themov- ing grain boundary according to the Zener model under a capillary driving force is impossible [23]. 3.3. Effect of triple line tension on the Gibbs–Thompson relation Let us consider the shape of a void at a grain boundary in a polycrystal. For homogeneous materials the contact angle h is constant, and the equilibrium surface will constitute a sur- face of rotation [18,19] (Fig. 9).Owing to the symmetry of the problem, the free energy of the system can be expressed as: DG ¼ Z y1 y0 4px ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ x02 p� � dy � cS � px20cB þ 2px0clS ; ð17Þ where cS is the free surface energy, cB is the grain boundary energy and clS is the grain boundary–free surface triple line θ γS γB R r x y (x0,y0) (x1,y1) θ R Fig. 9. Schematic view of a void intersecting a grain boundary. lia 59 (2011) 3510–3518 3515 tension. Since the volume of the void V is constant: V ¼ 2 Z y1 y0 px2dy ¼ const; ð18Þ the problem can be reduced to the isoperimetric problem of calculus of variations: J ¼ Z y1 y0 4pxcS ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ x02 p þ 2kpx2 � � dy � �px20cB þ 2px0clS : ð19Þ The extrema of the function J correspond to the extrema of the function xðyÞ, i.e. the shape of the void, in accor- dance with the Euler equation: U� x0Ux0 ¼ C1 ð20Þ and the transverse conditions: �Ux0 þ @U @x0 ���� ���� y¼y0 ¼ 0; ð21Þ jU�
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