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Direct observation of sintering mechanics of a single grain boundary
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Acta Materialia 60 (2012) 507–516 www.elsevier.com/locate/actamat
Direct observation of sintering mechanics of a single grain boundary F. Wakai a,⇑, H. Fukutome a, N. Kobayashi a, T. Misaki a, Y. Shinoda a, T. Akatsu a, M. Sone b, Y. Higo b a
Secure Materials Center, Materials and Structures Laboratory, Tokyo Institute of Technology, R3-23 4259 Nagatsuta, Midori, Yokohama 226-8503, Japan b Precision and Intelligence Laboratory, Tokyo Institute of Technology, R2-35 4259 Nagatsuta, Midori, Yokohama 226-8503, Japan Received 9 August 2011; received in revised form 16 September 2011; accepted 3 October 2011 Available online 22 November 2011
Abstract A method to analyze the mechanics of sintering of a single grain boundary was developed by using specimens fabricated by focused ion beam micro-machining. The translational motion and rotation of particles in sintering are influenced by the grain boundary diffusion coefficient and grain boundary energy; both are dependent on crystallographic orientation. Three-dimensional computer simulation was used to analyze the shrinkage and rotation as a response to sintering force and torque, respectively. The model experiments on gold were compared with the simulation results to determine the depth of the cusps on the plot of grain boundary energy vs. misorientation angle, the grain boundary diffusion coefficients, and the surface diffusion coefficient. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Sintering; Simulation; Micromechanical modeling
1. Introduction The macroscopic shrinkage in sintering depends on the microscopic structure on the particle scale. Shrinkage is the result of the motion of many particles, which interact with their neighbors. The translational motion and rotation of crystalline particles are influenced by grain boundary diffusion coefficient and grain boundary energy, both of which are dependent on crystallographic orientation [1,2]. The anisotropic grain boundary diffusion coefficient will be one of the origins of anisotropic shrinkage when particles are arranged in a preferred orientation in powder processing. Particle rotation often occurs during sintering, and influences texture formation and microstructural evolution [3]. In the mechanics of sintering, the translational motion, or shrinkage, is driven by the sintering force, and the rotation is driven by torques arising from anisotropic grain boundary energy [4,5] and asymmetric neck shape [6–8]. ⇑ Corresponding author. Tel.: +81 45 924 5361; fax: +81 45 924 5390.
E-mail address: [email protected] (F. Wakai).
The sintering force is originally defined for equilibrium states [9–11], and is measured by mechanical force [12,13] that must be applied to stop the shrinkage. Beere [14] identified the sintering force that arose from the difference between the stress on the neck surface and the average compressive stress on the grain boundary. This definition of sintering force is applicable not only to equilibrium states [15,16], but also to non-equilibrium processes [17]. Shewmon [4] proposed the torque that drives particle rotation by the lowering of the free energy of the tilt boundary for the first time. Exner and Bross [7] analyzed the torque induced by asymmetric neck shape. The particle motion was directly observed experimentally in sintering of row of particles to measure diffusion coefficients [18–20]. The particle rotation in sintering of spheres on a plate was analyzed in order to study the grain boundary energy as a function of misfit angle [5,21–23]. The particle rearrangement occurs due to asymmetric neck geometry [6]. The importance of the crystallographic orientation on the sintering of nano-particles is also demonstrated by in situ observation under transmission electron microscopy [24,25] and molecular dynamics simulations
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.10.003
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⋅ θ
z
γs(n n × t) t n
ez
h
ey
y
w x
Fig. 1. Micro-specimens at pre-existing grain boundaries. A crystal orientation map is superimposed on the scanning electron micrograph.
Fig. 2. Geometry of the micro-specimen in three dimensions. The fixed particle on the right-hand side is translucent to show the grain boundary.
[26–30]. Furthermore, the high-resolution X-ray microtomography made it possible to visualize the real threedimensional (3-D) microstructures in sintering [31–33]. It is desirable to analyze such microstructural evolution from the point of view of micromechanics. Here we developed a method to analyze the mechanics of sintering of a single grain boundary by using specimens fabricated by focused ion beam (FIB) micro-machining, as shown in Fig. 1. The FIB-machined specimen has been used to study mechanical properties of materials in microscopic scales [34]. Zhang et al. [35] have applied the method to study the coarsening of particles in sintering. We combined a computer simulation and the microscopic experiments to investigate forces and torques that lie behind particle motion in sintering. This method is used to study the effect of crystalline orientation on grain boundary energy and grain boundary diffusion coefficient of Au, a typical face-centered-cubic (fcc) metal. The knowledge on the sintering dynamics of each grain boundary will be useful to understand the macroscopic shrinkage from the microscopic structures. 2. Theoretical background 2.1. Linear and angular velocity We consider the sintering of the micro-specimen shown in Fig. 2 assuming coupled grain boundary and surface diffusion. We use a Cartesian coordinate system with the xand z-axes in the plane of boundary and the y-axis perpendicular to this plane. The surface near the neck is the source of vacancy, and the grain boundary is assumed to be a perfect sink/source of vacancies. The motion of the free particle relative to the fixed particle takes place when the vacancies are annihilated at the grain boundary. The diffusive flux jgb along the grain boundary is proportional to the gradient of chemical potential l, which is related to normal stress rn on the boundary [36]:
jgb ¼
dDgb 1 dDgb rs l ¼ rs rn ; kT X kT
ð1Þ
where $s denotes the surface gradient, X is the atomic volume, k is Boltzmann’s constant, T is absolute temperature, and dDgb is the grain boundary diffusion times the grain boundary thickness. The divergence of flux multiplied by X is equal to the normal component of the relative velocity u_ n of two particles: Xrs jgb ¼ u_ n :
ð2Þ
The normal stress distribution is a solution of Poisson’s equation: r2s rn ¼
kT u_ n ; XdDgb
ð3Þ
where r2s denotes the surface Laplacian. The normal velocity u_ n at each point on the grain boundary is related to linear and angular velocity of the free particle: _ u_ n ¼ u_ þ hx;
ð4Þ
where u_ is the linear velocity along y-axis, and h_ is the angular velocity around the z-axis. Here we do not consider rotations around the x-axis and the y-axis for simplicity. The grain boundary intersects with the surface to form a surface triple junction (free surface–boundary–free surface). We solve Eq. (3) with the boundary condition at a triple junction: rn ¼ cs jneck ;
ð5Þ
where jneck is the neck curvature that is dependent on position. The curvature is defined such that j = 2/r (negative) for a spherical particle, where r is the sphere radius. The positive normal stress on the surface csj is tension and the negative one is compression. When an external force is not applied, the integration of the normal stress over the boundary area must balance with the surface tension force along the circumference [19]:
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Z
rn ey dS ¼
Agb
Z
cs ðn tÞdr;
ð6Þ
509
2.2. Surface motion
C
where ey is the unit normal vector to the grain boundary, n is the unit normal vector to the surface, and t is the unit tangent vector along the surface triple junction C as shown in Fig. 2. The moment or torque on the boundary area is given by: Z M¼ xrn dS: ð7Þ Agb
The torque arises from the dependence of grain boundary energy on crystal orientation [4]: @cgb ; ð8Þ M S ¼ Agb @h where Agb is the area of the grain boundary and h is the rotation angle. Mykura [5] has called it the Shewmon torque. As the free particle rotates, the neck shape becomes asymmetrical. The formation of an asymmetric neck induces an additional torque [6–8]. Eq. (3) is solved by the finite element method as will be described in Section 2.4. When the height h of the boundary area is much larger than its width w (h/w 1), the problem can be solved analytically in two dimensions. Exner and Bross [7] and Hsueh and De Jonghe [8] considered asymmetric neck curvature (j1 at x = w/2 and j2 at x = w/2), and derived linear and angular velocity. Their results are expressed as functions of the sintering force and torque. The linear velocity is proportional to the sintering force Fs: 12XdDgb s ð9Þ F ; u_ ¼ kTw3 h neck r ÞAgb ; F s ¼ ðcs j ð10Þ neck ¼ ðj1 þ j2 Þ=2 is the average neck curvature, where j is the average compressive stress, which always acts and r on the grain boundary due to surface tension along the triple junction. It is derived from Eq. (6), and is given as: cs L w ¼ r sin ; ð11Þ Agb 2 where L ffi 2h is the circumference. The equilibrium dihedral angle w is given by the ratio of the grain boundary energy cgb to the surface energy cs: cgb ¼ 2cs cosðw=2Þ ð12Þ The angular velocity is proportional to the torque induced by anisotropic grain boundary energy MS, Eq. (8), and that induced by asymmetric neck shape ME: 720XdDgb h_ ¼ ð13Þ ðM S þ M E Þ kTw5 h w2 h c ðj2 j1 Þ ME ¼ ð14Þ 12 s We call ME an Exner torque. When MS = 0 and j1 > j2, a wedge of material is removed from one side of the grain boundary adjacent to the neck with j1, and material is inserted at the other side. The free particle rotates to the neck with j1 by the Exner torque.
The surface motion can occur by atoms moving along the surface. The diffusive flux js is proportional to the gradients of the curvature: js ¼
cs dDs 1 c dDs rl ¼ s rs j; kT X kT
ð15Þ
where dDs is the surface diffusion times the surface thickness. The normal velocity tn is the rate of accumulation of matter, which is the negative of the surface divergence of the flux by multiplied by X: Xrs js ¼ tn
ð16Þ
then tn ¼
cs XdDs 2 rs j: kT
ð17Þ
This equation corresponds to motion by the negative of the surface Laplacian of the mean curvature [37,38]. 2.3. Coupling between grain boundary diffusion and surface diffusion The relative velocity of two particles is given by linear and angular velocity, Eqs. (9) and (13): _ z r; _ y þ he v ¼ ue
ð18Þ
where r is a position vector from the rotation axis. The total surface motion is the sum of the surface motion, Eq. (17), and the rigid body motion, Eq. (18). The diffusive flux along the grain boundary jgb and that along the surface js must be continuous at the triple junction: ð2Þ jgb ¼ jð1Þ s þ js ;
ð19Þ
where jsð1Þ is the flux along the free particle, and jsð2Þ is the flux along the fixed particle [35]. 2.4. Surface evolver program The sintering by coupled grain boundary and surface diffusion was studied by using Brakke’s Surface Evolver program [39]. The surface of a particle, including both the grain boundary and the surface, is represented as a set of triangular finite elements, or facets. Each facet consists of three edges and three vertices. The surface and the grain boundary have energies proportional to their area. The Surface Evolver program (Ver. 2.30) calculates the velocity of a vertex as the Laplacian of the mean curvature of the surface, Eq. (17), by using the “Laplacian_mean_curvature” routine. The mean curvature, j/2, at each vertex is calculated as a scalar. Then the finite differences are used to calculate the Laplacian of the mean curvature. The actual motion is found by multiplying the velocity by a scale factor. The physical interpretation of the scale factor is the time step. Modeling the dynamics of the evolution requires using a fixed time
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step. The Surface Evolver program has been used to study sintering of two spheres by coupled grain boundary and surface diffusion [17]. We intend to analyze the sintering of the micro-specimen in Fig. 2. Actually the volume of the particle changes during sintering, since the flux jsð1Þ along the small free particle is different from the flux jsð2Þ along the fixed particle in Eq. (19). The smaller particle will shrink after a long time not only by the difference in surface diffusion flux, but also by grain boundary migration [35]. Here we consider a symmetrical model (jsð1Þ ¼ jð2Þ s ) in order to avoid the complication of coarsening and grain growth in our simulation. This symmetrical model is shown in Fig. 3, and is valid as long as we restrict our consideration to the particle motion in the initial stage of sintering. The sintering of the symmetrical model was investigated by assuming isotropic surface energy and surface diffusion coefficient. We considered orientation-dependent grain boundary energy and the grain boundary diffusion coefficient. The linear and angular velocities were determined by solving Eq. (3) under boundary conditions using the finite element method. Fig. 3 shows an example of stress distribution on the rectangular boundary area. The stress in Eq. (11). The stress is compressive at the center due to r csjneck at the triple junction is dependent on position; it is tension at both sides, and compression at the corners. In each time step Dt, vertices on both particle surfaces are shifted by the rigid body motion ± vDt/2. Since this rigid body motion decreases the particle volume, vertices at the surface triple junction are moved to expand the grain boundary area so as to restore the particle volume. The velocity of the triple junction is proportional to jgb (Eq. (1)), and it is schematically shown as arrows in Fig. 3. We intend to satisfy the continuity condition, Eq. (19), through this procedure.
When the specimen rotates, the stress distribution in Fig. 3 and the velocity of the triple junction become asymmetric. The center of the grain boundary and the position of the rotation axis shift from x = 0 to x = xcenter. Then, Eq. (4) is modified as: _ xcenter Þ: u_ n ¼ u_ þ hðx
ð20Þ
In Section 4.2, we examine the effects of the diffusivity ratio dDgb/dDs, the energy ratio cgb/cs, and anisotropic cgb and dDgb on the sintering kinetics. 3. Experimental The experiments were carried out on Au foil (purity 99.99%, Ag 1 ppm, Ca 6 ppm, Cu < 1 ppm, Fe < 1 ppm, Mg < 1 ppm, others < 1 ppm, thickness 10 lm, ribbon width 1.3 mm). The foil was annealed at 1173 K for 1 h in vacuum. The average grain size of the annealed foil was 27 lm, roughly three times larger than the thickness. The grain boundaries were approximately perpendicular to the foil surface. The specimen was placed in a carbon holder. Milling was performed on one side of the foil to reduce its thickness by focused ion beam machining with a Ga+ ion beam. By making use of pre-existing grain boundaries, the microspecimens were cut in the foil as shown in Fig. 1. Each specimen had a rectangular boundary with dimensions of w0 1 lm h0 4 lm between two U-shaped notches as illustrated in Fig. 2. The notch radius was w0/2. The carbon nano-pillars were formed on the Au surface by beaminduced chemical vapor deposition of phenanthrene (C14H10) gas. The nano-pillar shown in Fig. 4 had a hole with a diameter of 50 nm, which could be used as a marker to measure the motion of the free particle. The foil was then annealed at 1023 K in an infrared vacuum furnace. The annealing was periodically interrupted, and the specimen was observed by scanning electron microscopy (S4500, Hitachi Co.). The orientations of grains were determined by electron backscatter diffraction pattern (EBSD). INCA Crystal software (Oxford Instruments) was used for the EBSD analysis. 4. Results 4.1. Shrinkage and rotation in sintering experiments
Fig. 3. Symmetrical simulation model. Stress distribution on the grain boundary is shown on the right-hand side. Positive stress is tension, and negative is compression. The boundary area increases with the shrinkage in sintering. The velocity of the triple junction is shown as arrows in arbitrary units.
Fig. 4 shows that both rotation and shrinkage take place in the evolution of a micro-specimen during annealing. Almost all micro-specimens showed significant rotations and the majority also produced measurable shrinkage. The rotation angle h and the shrinkage u are plotted as functions of time in Figs. 5 and 6. The measurement errors were ±0.5° and ±10 nm for rotation angle and shrinkage, respectively. The particle motion was much faster at the beginning, and both rotation rate and shrinkage rate decreased with time.
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2µm
(a)
(b)
(c)
Fig. 4. Evolution of a micro-specimen of Au annealed at 1023 K in vacuum (specimen 1): annealing time is (a) 0 s, (b) 5000 s, (c) 20,000 s.
Table 1 lists the crystallographic orientation of each specimen before and after the annealing for 20,000 s. The rotations of free particles in specimens 1, 2, and 4 were trapped at specific orientations, indicating the presence of minima on the cgb(h) function at these orientations. The rotation angle and the rotation axis during annealing were calculated for the free particle, and are also listed in Table 1. For example, the h1 0 0i axis of the free particle in specimen 1 was approximately parallel to the z-axis. The particle rotated 13° around the h1 0 0i axis to become 51°h1 1 0i R11 boundary. The specimens 3, 5, and 6 with random grain boundaries still continued to shrink at 20,000 s. A coherent R3 twin boundary (Specimen 7) does not act as a source and sink of vacancy, and does not contribute to rotation and shrinkage in sintering. Fig. 6. Shrinkage of micro-specimens as a function of annealing time.
4.2. Simulation results 4.2.1. Shrinkage without rotation The detailed sintering kinetics can be analyzed by comparing the experimental results with the 3-D simulation of sintering of the symmetrical model. When grain boundary energy is isotropic, the micro-specimen shrinks without
Table 1 Measured axis-angle pairs of grain boundaries in Au before and after the annealing for 20,000 s at 1023 K.
No. 1 2 3 4 5 6 7
Grain boundary orientation
Rotation
Before annealing h hhkli
After annealing h hhkli
h hhkli
Around z-axis h (measured)
50° h8 5 1i 42° h1 1 2i 20° h9 3 1i N.A. 48° h1 2 2i N.A. 60° h1 1 1i R3
51° h1 1 0i 41° h1 1 1i 12° h9 4 2i 45° h1 1 1i 52° h7 6 4i 9° h7 5 5i 60° h1 1 1i
13° h1 0 0i 10° h3 3 1i 9° h9 4 1i N.A. 4° h1 2 3i N.A 0°
13.5° 11.3° 10° 7.4° 4.5° 2° 0°
R11 R7 R19b
R3
rotation. The shrinkage u is plotted as a function of the dimensionless time t* in Fig. 7a: t ¼
Fig. 5. Rotation angle of micro-specimens as a function of annealing time.
cs XdDs t; kTw40
ð21Þ
where w0 is the initial neck width. The slope of the shrinkage curve at cgb/cs = 0 agrees well with the 2-D model (Eq. (9)) that is shown as a broken line at the very early stage of sintering. Fig. 7b shows the neck width w increasing with time. The formation of thermal grooving [37] at cgb/cs > 0
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(a)
(b)
Fig. 8. Linear velocity of free particle in sintering of micro-specimen as a function of the dimensionless time.
ear since the increase of jgb decreases the neck curvature and Fs in Eq. (9). 4.2.2. Shrinkage accompanied with rotation The particle rotates in order to reduce the grain boundary energy when the grain boundary energy is dependent on orientation. Wolf [41] investigated the grain boundary energy of symmetrical tilt grain boundary (STGB) in fcc metals by using embedded-atom-method potential. A part of his result for Au is shown as a function of tilt angle / in Fig. 9. The h0 1 1i STGB shows a deep cusp at the tilt angle of 129.52°. The variation of grain boundary energy in the vicinity of the cusp can be described by the Read– Shockley model [42], shown as a broken line in Fig. 9: Fig. 7. Effect of cgb/cs on sintering of micro-specimens. (a) Shrinkage curves: the broken line shows the slope predicted by the 2-D model (Eq. (9)) at cgb/cs = 0. (b) Width of the boundary area: the evolution of neck shape is also illustrated.
is illustrated in Fig. 7b. It decreases the neck curvature, then, at cgb/cs = 1, the shrinkage is negative at the beginning, as shown in Fig. 7a. _ or shrinkage rate, decreases gradThe linear velocity u, ually with time as shown in Fig. 8, because u_ in Eq. (9) decreases with increasing w. Despite this, the sintering of the micro-specimen can be analyzed more easily than that of the classic model of two spheres, where u_ varies by several orders of magnitude in the initial stage. The evolution of neck shape occurs concurrently with the formation of thermal grooving under the influence of flux jgb along the grain boundary [40], which is proportional to dDgb. The relation between u_ and dDgb/dDs at a given time is non-lin-
cgb ccusp ¼ c0 hðA ln hÞ
ð22Þ
where ccusp is the grain boundary energy at the cusp, h is the misorientation angle from the cusp, and c0 and A are constants. Here, we simulated the particle rotation starting at the misorientation angle ha = 4° and 6° from the cusp in Fig. 9. Grain boundary diffusion is sensitive to the grain boundary structure and chemical composition [1,43,44]. Experiments and theories on the orientation dependence of grain boundary diffusion suggested the existence of minima at specific grain boundaries. Fig. 10 shows three phenomenological models of grain boundary diffusion coefficient as a function of misorientation angle: flat (dDgb is independent of misorientation angle), cusp-zero (dDgb/dDs = 0 at the cusp), cusp-nonzero (dDgb/dDs > 0 at the cusp). The rotation angle h and the shrinkage u are plotted as functions of dimensionless time in Fig. 11a and b, respectively. When rotation stops at the cusp, the shrinkage also stops in the case of the cusp-zero model. On the other
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(a)
Fig. 9. Grain boundary energy cgb of Au h0 1 1i symmetrical tilt grain boundaries [41]. The surface energy cs of Au is assumed to be isotropic, 1.39 J m2. The broken line shows the Read–Shockley model.
(b)
Fig. 11. Simulation results: (a) rotation angle, (b) shrinkage.
Fig. 10. Grain boundary diffusion coefficient dDgb as a function of misorientation angle.
the increase of rotation angle is retarded in specimens 2 and 5 (Fig. 5) due to the presence of a plateau region in the cgb(h) curve. 5. Discussion
hand, the shrinkage continues in the case of the cusp-nonzero model and the flat model. The increase of rotation angle is retarded when the simulation starts from the plateau region (ha = 6°) of the cgb(h) curve in Fig. 9. The simulation also shows that the Exner torque is much smaller than the Shewmon torque in the vicinity of the deep cusp. The experimental results in Figs. 5 and 6 can be classified into the cusp-nonzero type (specimens 1, 3 and 6) and the cusp-zero type (specimens 2 and 4) according to the shrinkage rate after the rotation stops. We suppose that
5.1. Dependence of grain boundary energy on crystal misorientation The elimination of dDgb from Eqs. (9) and (13) gives a simple relation to estimate cgb as a function of the rotation angle: h_ 1 @cgb ; ¼a _ ðu=wÞ cs @h
ð23Þ
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where a is a coefficient that depends on the neck curvature. _ u_ in the simulaThe relationship between o cgb/o h and h= _ u_ is small in tion of the deep cusp is plotted in Fig. 12. h= the plateau region of cgb(h) curve, and it increases near the cusp. The dotted line shows the slope a = 0.0467 that fits to the cusp-zero model. But the linear relation cannot be obtained in the flat model and the cusp-nonzero model _ u. _ As illustrated in Fig. 12, the flux along the at large h= grain boundary makes a bulge at the neck in the flat model and the cusp-nonzero model. The bulge decreases the neck _ u. _ On the other hand, in curvature, then a decreases with h= the cusp-zero model, the neck curvature, and then a, is approximately constant till the orientation reaches the cusp. The experimental results in the cusp-zero type are analyzed by using Eq. (23). The decrease of the grain boundary energy from the initial orientation is plotted as a function of the rotation angle in Fig. 13. Specimen 1 is also included in the plot. Fig. 13 shows that the rotation is trapped at shallow cusps. The depth of the cusp for the R11 boundary is shallower than that shown in Fig. 9. It should be understood since the orientation of the pre-existing grain boundary is different from the symmetrical tilt grain boundary. _ u_ is small in such shallow cusps, Eq. (23) can be Since h= applied to the flat model and the cusp-nonzero model also. Chan and Balluffi [22,23] showed the presence of shallow cusps in the cgb(h) curve of Au from the analysis of the rotation of spheres in sintering. But they did not determine the exact depth of the cusp. Kuhn et al. [45] used the rotation rate to estimate the shape of cusp assuming the flat model on dDgb. Our method using Eq. (23) can analyze the cgb(h) curve near shallow cusps regardless of the type of dDgb models.
Fig. 13. Estimated grain boundary energy near special grain boundaries of Au.
Relatively shallow cusps or inflections in the cgb(h) curve were demonstrated at high temperatures in a systematic study of [1 0 0] twist boundaries in Au [46]. It was suggested that the cusp/inflection in the cgb(h) curve is determined by entropy and enthalpy. Such shallow cusps can be easily detected by observing particle rotation in sintering. It should be noted that the shape of the curve in the vicinity of the energy cusp in Fig. 13 may not be accurate due to the measurement error of shrinkage. But we believe the curves still provide the data to estimate the torque on a particle in real sintering. Of course, as Cahn [47] pointed out, the microscopic analyses on the dislocation motion and defect structures are necessary to understand the rotation correctly in addition to our analysis in particle scales. A grain boundary is specified by five angle parameters: three angles for the relative orientation of two crystals, and two angles for the boundary-plane orientation. When the grain boundary energy is dependent on the boundaryplane orientation, the orientation derivative gives the Herring torque term [36]. Our computer simulation showed that the boundary-plane orientation rotated with the particle rotation as shown in Fig. 11 even though we did not include the Herring torque. The boundary plane rotated simply because the grain boundary was constrained between two U-shaped notches. 5.2. Grain boundary diffusion
_ u_ in the simulation. The Fig. 12. Relationship between o cgb/o h and h= unit of h is radian. The filled circles show the data at the plateau region in the cgb(h) curve.
At a given temperature, dDgb can vary by several orders of magnitude, depending on the grain boundary structure [44]. dDgb of Au at 1023 K are estimated from the initial shrinkage rates in Fig. 6, and are plotted in Fig. 14. There is a huge variation in experimentally obtained dDgb. In order to define the average dDgb, it will be necessary to
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know the distribution function of orientation. Fig. 14 shows a comparison of our results with other measurements. The results are plotted on an Arrhenius diagram using a reduced temperature scale corresponding to the dimensionless parameter TM/T, where TM is the melting temperature of Au. It may be seen that our data would fall on an extrapolation of Gupta’s results [48] on the average dDgb of Au. Ma and Balluffi [49] measured the chemical diffusion of Ag along the grain boundaries of Au as a function of temperature and tilt angle at relatively low temperatures. The variation of dDgb in different grain boundaries in our results is consistent with that of chemical diffusion along grain boundaries. 5.3. Effect of contamination on surface diffusion The surface diffusion coefficient dDs can be directly estimated from the rounding of corners of the micro-specimen [50]. Fig. 15 shows the cross-sectional profile of the Ushaped notch at z = 0, illustrating the rounding at its corners in the 3-D simulation. The relationship between the curvature of the corner and the dimensionless time is plotted in Fig. 15. The curvature is given by: 1
w0 j ¼ 0:55ðt Þ4 :
ð24Þ
Eq. (24) is consistent with Mullins’ theory [37] of microstructural evolution by surface diffusion. Fig. 4 shows that the corner of the micro-specimen is already round after FIB micro-machining. The difference in curvature at the corner cannot be measured exactly after the annealing at 1023 K. This result suggests low dDs < 1020 m3 s1. The surface diffusion coefficients of Au are plotted in Fig. 14 for comparison [51–53]. The large spread in the pre-
Fig. 15. Calculated curvature at the corner of the U-shaped notch as a function of the dimensionless time.
viously reported values was due to the difference in the degree of surface contamination [52]. Experiments have demonstrated that carbon is a very effective impurity to suppress dDs for metals Cu and Au. Carbon is found to segregate on the Au surface [54]. Since carbon is deposited on the micro-specimen as a marker, it is understandable that dDs was much lower than that on the clean surface. The surface may be also contaminated by Ga during FIB micro-machining. The impurities that have lower melting point than the substrate cause a substantial increase in dDs. But our experimental results suggest that the effect of carbon segregation is much stronger than that of Ga. The surface energy of crystalline particles is usually anisotropic. Klinger and Rabkin [55] recently demonstrated that the anisotropy of surface energy leads to particle rotation. But we assumed isotropic surface energy throughout this paper, because it is a valid assumption at temperatures higher than the surface roughening transition. 6. Summary
Fig. 14. Grain boundary and surface diffusion of Au (TM: melting temperature of Au, 1337 K).
Theoretical and experimental approaches are important to obtaining reliable predictions in dealing with the complexity of sintering. We proposed an experimental method to analyze the mechanics of sintering of a single grain boundary by using specimens fabricated by FIB micromachining. A self-consistent theoretical model on the evolution of neck curvature was developed considering rotation, shrinkage and asymmetric neck growth. The microspecimen had a rectangular grain boundary between Ushaped notches. The sintering of the micro-specimen could be analyzed more easily than that of the classic two-sphere model. The linear velocity was driven by the sintering force that was a function of neck curvature. The variation of dDgb in different grain boundaries of Au was determined from the linear velocity. The rotation was driven by torques arising from anisotropic cgb and asymmetric neck shape. A method to estimate cgb(h) was developed to show
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the depth of shallow cusps at specific h. The torque induced by the asymmetric neck shape could be ignored in the vicinity of a deep cusp. dDs was directly estimated from the rounding of corners of the micro-specimen. References [1] Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Clarendon Press; 1995. [2] Kaur I, Mishin Y, Gust W. Fundamentals of grain and interphase boundary diffusion. Chichester: Wiley; 1995. [3] Moldovan D, Wolf D, Phillpot SR. Acta Mater 2001;49:3521. [4] Shewmon PG. Recrystallisation, grain growth and textures. Metals Park, OH: American Society for Metals; 1966. [5] Mykura H. Acta Metall 1979;27:243. [6] Petzow G, Exner HE. Z Metallkde 1976;67:611. [7] Exner HE, Bross P. Acta Metall 1979;27:1007. [8] Hsueh CH, De Jonghe LC. J Am Ceram Soc 1984;67:C215. [9] Kellett BJ, Lange FF. J Am Ceram Soc 1989;72:725. [10] Cannon RM, Carter WC. J Am Ceram Soc 1989;72:1550. [11] Wakai F, Aldinger F. Acta Mater 2003;51:641. [12] Gregg RA, Rhines FN. Metall Trans 1973;4:1365. [13] Ikegami T, Iyi N, Sakaguchi I. Ceram Int 2009;35:3185. [14] Beere W. Acta Metall 1975;23:139. [15] Svoboda J, Riedel H, Zipse H. Acta Metall Mater 1994;42:435. [16] Wakai F, Shinoda Y, Akatsu T. Acta Mater 2004;52:5621. [17] Wakai F, Brakke KA. Acta Mater 2011;59:5379. [18] Kingery WD, Berg M. J Appl Phys 1955;26:1205. [19] Johnson DL. J Appl Phys 1969;40:192. [20] Wilson TL, Shewmon PG. Trans Metal Soc AIME 1966;236:49. [21] Herrmann G, Gleiter H, Ba¨ro G. Acta Metall 1976;24:353. [22] Chan SW, Balluffi RW. Acta Metall 1985;33:1113. [23] Chan SW, Balluffi RW. Acta Metall 1986;34:2191. [24] Theissmann R, Fendrich M, Zinetullin R, Guenther G, Shierning G, Wolf DE. Phys Rev B 2008;78:205413. [25] Asoro MA, Kovar D, Shao-Horn Y, Allard LF, Ferreira PJ. Nanotechnology 2010;21:025701. [26] Zhu H, Averback RS. Phil Mag Lett 1996;73:27. [27] Tsuruta K, Omeltchenko A, Kalia RK, Vashishta P. Europhys Lett 1996;33:441.
[28] Zeng P, Zajac S, Clapp PC, Rifkin JA. Mater Sci Eng A 1998;252:301. [29] Arcidiacono S, Bieri NR, Poulikakos D, Grigoropoulos CP. Int J Multi Flow 2004;30:979. [30] Ding L, Davidchack RL, Pan J. Comp Mater Sci 2009;45:247. [31] Lame O, Bellet D, Michiel MD, Bouvard D. Nucl Inst Methods Phys Res B 2003;200:287. [32] Bernard D, Gendron D, Heintz JM, Borde`re S, Etourneau J. Acta Mater 2005;53:121. [33] Grupp R, No¨the M, Kieback B, Banhart J. Nat Commun 2011;2:298. [34] Maekawa S, Takashima K, Shimojo M, Higo Y. Jpn J Appl Phys 1999;38:7194. [35] Zhang W, Sachenko P, Schneibel JH, Gladwell I. Phil Mag A 2002;82:2995. [36] Herring C. In: Kingston WE, editor. Physics of powder metallurgy. New York: McGraw-Hill; 1951. p. 143. [37] Mullins WW. J Appl Phys 1957;28:333. [38] Cahn JW, Taylor JE. Acta Metall Mater 1994;42:1045. [39] Brakke KA. Exp Math 1992;1:141. [40] Ge`nin FY, Mullins WW, Wynblatt P. Acta Metall Mater 1993;41:3541. [41] Wolf D. Acta Metall Mater 1990;38:781. [42] Read WT, Shockley W. Phys Rev 1950;78:275. [43] Turnbull D, Hoffman RE. Acta Metall 1954;2:419. [44] Mishin Y, Herzig C. Mater Sci Eng A 1999;260:55. [45] Kuhn H, Baero¨ G, Gleiter H. Acta Metall 1979;27:959. [46] Najafabadi R, Srolovitz DJ. J Mater Res 1991;6:999. [47] Cahn JW. Ceram Trans 1988;7:185. [48] Gupta D. J Appl Phys 1973;44:4455. [49] Ma Q, Balluffi RW. Acta Metall Mater 1993;41:133. [50] Kuribayashi H, Hiruta R, Shimizu R, Sudoh K, Iwasaki H. J Vac Sci Technol A 2003;21:1279. [51] Geguzin YE, Kovalev GN, Ovcharenko NN. Sov Phys Solid Stat 1964;5:2627. [52] Olson DL, Patil HR, Blakely JM. Scripta Metall 1972;6:229. [53] Go¨bel H, von Blanckenhagen P. Surface Sci 1995;331–333:885. [54] Gangopadhyay U, Wynblatt P. J Mater Sci 1995;30:94. [55] Klinger L, Rabkin E. Acta Mater 2011;59:6691.
Acta Materialia 60 (2012) 507–516 www.elsevier.com/locate/actamat
Direct observation of sintering mechanics of a single grain boundary F. Wakai a,⇑, H. Fukutome a, N. Kobayashi a, T. Misaki a, Y. Shinoda a, T. Akatsu a, M. Sone b, Y. Higo b a
Secure Materials Center, Materials and Structures Laboratory, Tokyo Institute of Technology, R3-23 4259 Nagatsuta, Midori, Yokohama 226-8503, Japan b Precision and Intelligence Laboratory, Tokyo Institute of Technology, R2-35 4259 Nagatsuta, Midori, Yokohama 226-8503, Japan Received 9 August 2011; received in revised form 16 September 2011; accepted 3 October 2011 Available online 22 November 2011
Abstract A method to analyze the mechanics of sintering of a single grain boundary was developed by using specimens fabricated by focused ion beam micro-machining. The translational motion and rotation of particles in sintering are influenced by the grain boundary diffusion coefficient and grain boundary energy; both are dependent on crystallographic orientation. Three-dimensional computer simulation was used to analyze the shrinkage and rotation as a response to sintering force and torque, respectively. The model experiments on gold were compared with the simulation results to determine the depth of the cusps on the plot of grain boundary energy vs. misorientation angle, the grain boundary diffusion coefficients, and the surface diffusion coefficient. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Sintering; Simulation; Micromechanical modeling
1. Introduction The macroscopic shrinkage in sintering depends on the microscopic structure on the particle scale. Shrinkage is the result of the motion of many particles, which interact with their neighbors. The translational motion and rotation of crystalline particles are influenced by grain boundary diffusion coefficient and grain boundary energy, both of which are dependent on crystallographic orientation [1,2]. The anisotropic grain boundary diffusion coefficient will be one of the origins of anisotropic shrinkage when particles are arranged in a preferred orientation in powder processing. Particle rotation often occurs during sintering, and influences texture formation and microstructural evolution [3]. In the mechanics of sintering, the translational motion, or shrinkage, is driven by the sintering force, and the rotation is driven by torques arising from anisotropic grain boundary energy [4,5] and asymmetric neck shape [6–8]. ⇑ Corresponding author. Tel.: +81 45 924 5361; fax: +81 45 924 5390.
E-mail address: [email protected] (F. Wakai).
The sintering force is originally defined for equilibrium states [9–11], and is measured by mechanical force [12,13] that must be applied to stop the shrinkage. Beere [14] identified the sintering force that arose from the difference between the stress on the neck surface and the average compressive stress on the grain boundary. This definition of sintering force is applicable not only to equilibrium states [15,16], but also to non-equilibrium processes [17]. Shewmon [4] proposed the torque that drives particle rotation by the lowering of the free energy of the tilt boundary for the first time. Exner and Bross [7] analyzed the torque induced by asymmetric neck shape. The particle motion was directly observed experimentally in sintering of row of particles to measure diffusion coefficients [18–20]. The particle rotation in sintering of spheres on a plate was analyzed in order to study the grain boundary energy as a function of misfit angle [5,21–23]. The particle rearrangement occurs due to asymmetric neck geometry [6]. The importance of the crystallographic orientation on the sintering of nano-particles is also demonstrated by in situ observation under transmission electron microscopy [24,25] and molecular dynamics simulations
1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.10.003
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⋅ θ
z
γs(n n × t) t n
ez
h
ey
y
w x
Fig. 1. Micro-specimens at pre-existing grain boundaries. A crystal orientation map is superimposed on the scanning electron micrograph.
Fig. 2. Geometry of the micro-specimen in three dimensions. The fixed particle on the right-hand side is translucent to show the grain boundary.
[26–30]. Furthermore, the high-resolution X-ray microtomography made it possible to visualize the real threedimensional (3-D) microstructures in sintering [31–33]. It is desirable to analyze such microstructural evolution from the point of view of micromechanics. Here we developed a method to analyze the mechanics of sintering of a single grain boundary by using specimens fabricated by focused ion beam (FIB) micro-machining, as shown in Fig. 1. The FIB-machined specimen has been used to study mechanical properties of materials in microscopic scales [34]. Zhang et al. [35] have applied the method to study the coarsening of particles in sintering. We combined a computer simulation and the microscopic experiments to investigate forces and torques that lie behind particle motion in sintering. This method is used to study the effect of crystalline orientation on grain boundary energy and grain boundary diffusion coefficient of Au, a typical face-centered-cubic (fcc) metal. The knowledge on the sintering dynamics of each grain boundary will be useful to understand the macroscopic shrinkage from the microscopic structures. 2. Theoretical background 2.1. Linear and angular velocity We consider the sintering of the micro-specimen shown in Fig. 2 assuming coupled grain boundary and surface diffusion. We use a Cartesian coordinate system with the xand z-axes in the plane of boundary and the y-axis perpendicular to this plane. The surface near the neck is the source of vacancy, and the grain boundary is assumed to be a perfect sink/source of vacancies. The motion of the free particle relative to the fixed particle takes place when the vacancies are annihilated at the grain boundary. The diffusive flux jgb along the grain boundary is proportional to the gradient of chemical potential l, which is related to normal stress rn on the boundary [36]:
jgb ¼
dDgb 1 dDgb rs l ¼ rs rn ; kT X kT
ð1Þ
where $s denotes the surface gradient, X is the atomic volume, k is Boltzmann’s constant, T is absolute temperature, and dDgb is the grain boundary diffusion times the grain boundary thickness. The divergence of flux multiplied by X is equal to the normal component of the relative velocity u_ n of two particles: Xrs jgb ¼ u_ n :
ð2Þ
The normal stress distribution is a solution of Poisson’s equation: r2s rn ¼
kT u_ n ; XdDgb
ð3Þ
where r2s denotes the surface Laplacian. The normal velocity u_ n at each point on the grain boundary is related to linear and angular velocity of the free particle: _ u_ n ¼ u_ þ hx;
ð4Þ
where u_ is the linear velocity along y-axis, and h_ is the angular velocity around the z-axis. Here we do not consider rotations around the x-axis and the y-axis for simplicity. The grain boundary intersects with the surface to form a surface triple junction (free surface–boundary–free surface). We solve Eq. (3) with the boundary condition at a triple junction: rn ¼ cs jneck ;
ð5Þ
where jneck is the neck curvature that is dependent on position. The curvature is defined such that j = 2/r (negative) for a spherical particle, where r is the sphere radius. The positive normal stress on the surface csj is tension and the negative one is compression. When an external force is not applied, the integration of the normal stress over the boundary area must balance with the surface tension force along the circumference [19]:
F. Wakai et al. / Acta Materialia 60 (2012) 507–516
Z
rn ey dS ¼
Agb
Z
cs ðn tÞdr;
ð6Þ
509
2.2. Surface motion
C
where ey is the unit normal vector to the grain boundary, n is the unit normal vector to the surface, and t is the unit tangent vector along the surface triple junction C as shown in Fig. 2. The moment or torque on the boundary area is given by: Z M¼ xrn dS: ð7Þ Agb
The torque arises from the dependence of grain boundary energy on crystal orientation [4]: @cgb ; ð8Þ M S ¼ Agb @h where Agb is the area of the grain boundary and h is the rotation angle. Mykura [5] has called it the Shewmon torque. As the free particle rotates, the neck shape becomes asymmetrical. The formation of an asymmetric neck induces an additional torque [6–8]. Eq. (3) is solved by the finite element method as will be described in Section 2.4. When the height h of the boundary area is much larger than its width w (h/w 1), the problem can be solved analytically in two dimensions. Exner and Bross [7] and Hsueh and De Jonghe [8] considered asymmetric neck curvature (j1 at x = w/2 and j2 at x = w/2), and derived linear and angular velocity. Their results are expressed as functions of the sintering force and torque. The linear velocity is proportional to the sintering force Fs: 12XdDgb s ð9Þ F ; u_ ¼ kTw3 h neck r ÞAgb ; F s ¼ ðcs j ð10Þ neck ¼ ðj1 þ j2 Þ=2 is the average neck curvature, where j is the average compressive stress, which always acts and r on the grain boundary due to surface tension along the triple junction. It is derived from Eq. (6), and is given as: cs L w ¼ r sin ; ð11Þ Agb 2 where L ffi 2h is the circumference. The equilibrium dihedral angle w is given by the ratio of the grain boundary energy cgb to the surface energy cs: cgb ¼ 2cs cosðw=2Þ ð12Þ The angular velocity is proportional to the torque induced by anisotropic grain boundary energy MS, Eq. (8), and that induced by asymmetric neck shape ME: 720XdDgb h_ ¼ ð13Þ ðM S þ M E Þ kTw5 h w2 h c ðj2 j1 Þ ME ¼ ð14Þ 12 s We call ME an Exner torque. When MS = 0 and j1 > j2, a wedge of material is removed from one side of the grain boundary adjacent to the neck with j1, and material is inserted at the other side. The free particle rotates to the neck with j1 by the Exner torque.
The surface motion can occur by atoms moving along the surface. The diffusive flux js is proportional to the gradients of the curvature: js ¼
cs dDs 1 c dDs rl ¼ s rs j; kT X kT
ð15Þ
where dDs is the surface diffusion times the surface thickness. The normal velocity tn is the rate of accumulation of matter, which is the negative of the surface divergence of the flux by multiplied by X: Xrs js ¼ tn
ð16Þ
then tn ¼
cs XdDs 2 rs j: kT
ð17Þ
This equation corresponds to motion by the negative of the surface Laplacian of the mean curvature [37,38]. 2.3. Coupling between grain boundary diffusion and surface diffusion The relative velocity of two particles is given by linear and angular velocity, Eqs. (9) and (13): _ z r; _ y þ he v ¼ ue
ð18Þ
where r is a position vector from the rotation axis. The total surface motion is the sum of the surface motion, Eq. (17), and the rigid body motion, Eq. (18). The diffusive flux along the grain boundary jgb and that along the surface js must be continuous at the triple junction: ð2Þ jgb ¼ jð1Þ s þ js ;
ð19Þ
where jsð1Þ is the flux along the free particle, and jsð2Þ is the flux along the fixed particle [35]. 2.4. Surface evolver program The sintering by coupled grain boundary and surface diffusion was studied by using Brakke’s Surface Evolver program [39]. The surface of a particle, including both the grain boundary and the surface, is represented as a set of triangular finite elements, or facets. Each facet consists of three edges and three vertices. The surface and the grain boundary have energies proportional to their area. The Surface Evolver program (Ver. 2.30) calculates the velocity of a vertex as the Laplacian of the mean curvature of the surface, Eq. (17), by using the “Laplacian_mean_curvature” routine. The mean curvature, j/2, at each vertex is calculated as a scalar. Then the finite differences are used to calculate the Laplacian of the mean curvature. The actual motion is found by multiplying the velocity by a scale factor. The physical interpretation of the scale factor is the time step. Modeling the dynamics of the evolution requires using a fixed time
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step. The Surface Evolver program has been used to study sintering of two spheres by coupled grain boundary and surface diffusion [17]. We intend to analyze the sintering of the micro-specimen in Fig. 2. Actually the volume of the particle changes during sintering, since the flux jsð1Þ along the small free particle is different from the flux jsð2Þ along the fixed particle in Eq. (19). The smaller particle will shrink after a long time not only by the difference in surface diffusion flux, but also by grain boundary migration [35]. Here we consider a symmetrical model (jsð1Þ ¼ jð2Þ s ) in order to avoid the complication of coarsening and grain growth in our simulation. This symmetrical model is shown in Fig. 3, and is valid as long as we restrict our consideration to the particle motion in the initial stage of sintering. The sintering of the symmetrical model was investigated by assuming isotropic surface energy and surface diffusion coefficient. We considered orientation-dependent grain boundary energy and the grain boundary diffusion coefficient. The linear and angular velocities were determined by solving Eq. (3) under boundary conditions using the finite element method. Fig. 3 shows an example of stress distribution on the rectangular boundary area. The stress in Eq. (11). The stress is compressive at the center due to r csjneck at the triple junction is dependent on position; it is tension at both sides, and compression at the corners. In each time step Dt, vertices on both particle surfaces are shifted by the rigid body motion ± vDt/2. Since this rigid body motion decreases the particle volume, vertices at the surface triple junction are moved to expand the grain boundary area so as to restore the particle volume. The velocity of the triple junction is proportional to jgb (Eq. (1)), and it is schematically shown as arrows in Fig. 3. We intend to satisfy the continuity condition, Eq. (19), through this procedure.
When the specimen rotates, the stress distribution in Fig. 3 and the velocity of the triple junction become asymmetric. The center of the grain boundary and the position of the rotation axis shift from x = 0 to x = xcenter. Then, Eq. (4) is modified as: _ xcenter Þ: u_ n ¼ u_ þ hðx
ð20Þ
In Section 4.2, we examine the effects of the diffusivity ratio dDgb/dDs, the energy ratio cgb/cs, and anisotropic cgb and dDgb on the sintering kinetics. 3. Experimental The experiments were carried out on Au foil (purity 99.99%, Ag 1 ppm, Ca 6 ppm, Cu < 1 ppm, Fe < 1 ppm, Mg < 1 ppm, others < 1 ppm, thickness 10 lm, ribbon width 1.3 mm). The foil was annealed at 1173 K for 1 h in vacuum. The average grain size of the annealed foil was 27 lm, roughly three times larger than the thickness. The grain boundaries were approximately perpendicular to the foil surface. The specimen was placed in a carbon holder. Milling was performed on one side of the foil to reduce its thickness by focused ion beam machining with a Ga+ ion beam. By making use of pre-existing grain boundaries, the microspecimens were cut in the foil as shown in Fig. 1. Each specimen had a rectangular boundary with dimensions of w0 1 lm h0 4 lm between two U-shaped notches as illustrated in Fig. 2. The notch radius was w0/2. The carbon nano-pillars were formed on the Au surface by beaminduced chemical vapor deposition of phenanthrene (C14H10) gas. The nano-pillar shown in Fig. 4 had a hole with a diameter of 50 nm, which could be used as a marker to measure the motion of the free particle. The foil was then annealed at 1023 K in an infrared vacuum furnace. The annealing was periodically interrupted, and the specimen was observed by scanning electron microscopy (S4500, Hitachi Co.). The orientations of grains were determined by electron backscatter diffraction pattern (EBSD). INCA Crystal software (Oxford Instruments) was used for the EBSD analysis. 4. Results 4.1. Shrinkage and rotation in sintering experiments
Fig. 3. Symmetrical simulation model. Stress distribution on the grain boundary is shown on the right-hand side. Positive stress is tension, and negative is compression. The boundary area increases with the shrinkage in sintering. The velocity of the triple junction is shown as arrows in arbitrary units.
Fig. 4 shows that both rotation and shrinkage take place in the evolution of a micro-specimen during annealing. Almost all micro-specimens showed significant rotations and the majority also produced measurable shrinkage. The rotation angle h and the shrinkage u are plotted as functions of time in Figs. 5 and 6. The measurement errors were ±0.5° and ±10 nm for rotation angle and shrinkage, respectively. The particle motion was much faster at the beginning, and both rotation rate and shrinkage rate decreased with time.
F. Wakai et al. / Acta Materialia 60 (2012) 507–516
511
2µm
(a)
(b)
(c)
Fig. 4. Evolution of a micro-specimen of Au annealed at 1023 K in vacuum (specimen 1): annealing time is (a) 0 s, (b) 5000 s, (c) 20,000 s.
Table 1 lists the crystallographic orientation of each specimen before and after the annealing for 20,000 s. The rotations of free particles in specimens 1, 2, and 4 were trapped at specific orientations, indicating the presence of minima on the cgb(h) function at these orientations. The rotation angle and the rotation axis during annealing were calculated for the free particle, and are also listed in Table 1. For example, the h1 0 0i axis of the free particle in specimen 1 was approximately parallel to the z-axis. The particle rotated 13° around the h1 0 0i axis to become 51°h1 1 0i R11 boundary. The specimens 3, 5, and 6 with random grain boundaries still continued to shrink at 20,000 s. A coherent R3 twin boundary (Specimen 7) does not act as a source and sink of vacancy, and does not contribute to rotation and shrinkage in sintering. Fig. 6. Shrinkage of micro-specimens as a function of annealing time.
4.2. Simulation results 4.2.1. Shrinkage without rotation The detailed sintering kinetics can be analyzed by comparing the experimental results with the 3-D simulation of sintering of the symmetrical model. When grain boundary energy is isotropic, the micro-specimen shrinks without
Table 1 Measured axis-angle pairs of grain boundaries in Au before and after the annealing for 20,000 s at 1023 K.
No. 1 2 3 4 5 6 7
Grain boundary orientation
Rotation
Before annealing h hhkli
After annealing h hhkli
h hhkli
Around z-axis h (measured)
50° h8 5 1i 42° h1 1 2i 20° h9 3 1i N.A. 48° h1 2 2i N.A. 60° h1 1 1i R3
51° h1 1 0i 41° h1 1 1i 12° h9 4 2i 45° h1 1 1i 52° h7 6 4i 9° h7 5 5i 60° h1 1 1i
13° h1 0 0i 10° h3 3 1i 9° h9 4 1i N.A. 4° h1 2 3i N.A 0°
13.5° 11.3° 10° 7.4° 4.5° 2° 0°
R11 R7 R19b
R3
rotation. The shrinkage u is plotted as a function of the dimensionless time t* in Fig. 7a: t ¼
Fig. 5. Rotation angle of micro-specimens as a function of annealing time.
cs XdDs t; kTw40
ð21Þ
where w0 is the initial neck width. The slope of the shrinkage curve at cgb/cs = 0 agrees well with the 2-D model (Eq. (9)) that is shown as a broken line at the very early stage of sintering. Fig. 7b shows the neck width w increasing with time. The formation of thermal grooving [37] at cgb/cs > 0
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(a)
(b)
Fig. 8. Linear velocity of free particle in sintering of micro-specimen as a function of the dimensionless time.
ear since the increase of jgb decreases the neck curvature and Fs in Eq. (9). 4.2.2. Shrinkage accompanied with rotation The particle rotates in order to reduce the grain boundary energy when the grain boundary energy is dependent on orientation. Wolf [41] investigated the grain boundary energy of symmetrical tilt grain boundary (STGB) in fcc metals by using embedded-atom-method potential. A part of his result for Au is shown as a function of tilt angle / in Fig. 9. The h0 1 1i STGB shows a deep cusp at the tilt angle of 129.52°. The variation of grain boundary energy in the vicinity of the cusp can be described by the Read– Shockley model [42], shown as a broken line in Fig. 9: Fig. 7. Effect of cgb/cs on sintering of micro-specimens. (a) Shrinkage curves: the broken line shows the slope predicted by the 2-D model (Eq. (9)) at cgb/cs = 0. (b) Width of the boundary area: the evolution of neck shape is also illustrated.
is illustrated in Fig. 7b. It decreases the neck curvature, then, at cgb/cs = 1, the shrinkage is negative at the beginning, as shown in Fig. 7a. _ or shrinkage rate, decreases gradThe linear velocity u, ually with time as shown in Fig. 8, because u_ in Eq. (9) decreases with increasing w. Despite this, the sintering of the micro-specimen can be analyzed more easily than that of the classic model of two spheres, where u_ varies by several orders of magnitude in the initial stage. The evolution of neck shape occurs concurrently with the formation of thermal grooving under the influence of flux jgb along the grain boundary [40], which is proportional to dDgb. The relation between u_ and dDgb/dDs at a given time is non-lin-
cgb ccusp ¼ c0 hðA ln hÞ
ð22Þ
where ccusp is the grain boundary energy at the cusp, h is the misorientation angle from the cusp, and c0 and A are constants. Here, we simulated the particle rotation starting at the misorientation angle ha = 4° and 6° from the cusp in Fig. 9. Grain boundary diffusion is sensitive to the grain boundary structure and chemical composition [1,43,44]. Experiments and theories on the orientation dependence of grain boundary diffusion suggested the existence of minima at specific grain boundaries. Fig. 10 shows three phenomenological models of grain boundary diffusion coefficient as a function of misorientation angle: flat (dDgb is independent of misorientation angle), cusp-zero (dDgb/dDs = 0 at the cusp), cusp-nonzero (dDgb/dDs > 0 at the cusp). The rotation angle h and the shrinkage u are plotted as functions of dimensionless time in Fig. 11a and b, respectively. When rotation stops at the cusp, the shrinkage also stops in the case of the cusp-zero model. On the other
F. Wakai et al. / Acta Materialia 60 (2012) 507–516
513
(a)
Fig. 9. Grain boundary energy cgb of Au h0 1 1i symmetrical tilt grain boundaries [41]. The surface energy cs of Au is assumed to be isotropic, 1.39 J m2. The broken line shows the Read–Shockley model.
(b)
Fig. 11. Simulation results: (a) rotation angle, (b) shrinkage.
Fig. 10. Grain boundary diffusion coefficient dDgb as a function of misorientation angle.
the increase of rotation angle is retarded in specimens 2 and 5 (Fig. 5) due to the presence of a plateau region in the cgb(h) curve. 5. Discussion
hand, the shrinkage continues in the case of the cusp-nonzero model and the flat model. The increase of rotation angle is retarded when the simulation starts from the plateau region (ha = 6°) of the cgb(h) curve in Fig. 9. The simulation also shows that the Exner torque is much smaller than the Shewmon torque in the vicinity of the deep cusp. The experimental results in Figs. 5 and 6 can be classified into the cusp-nonzero type (specimens 1, 3 and 6) and the cusp-zero type (specimens 2 and 4) according to the shrinkage rate after the rotation stops. We suppose that
5.1. Dependence of grain boundary energy on crystal misorientation The elimination of dDgb from Eqs. (9) and (13) gives a simple relation to estimate cgb as a function of the rotation angle: h_ 1 @cgb ; ¼a _ ðu=wÞ cs @h
ð23Þ
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where a is a coefficient that depends on the neck curvature. _ u_ in the simulaThe relationship between o cgb/o h and h= _ u_ is small in tion of the deep cusp is plotted in Fig. 12. h= the plateau region of cgb(h) curve, and it increases near the cusp. The dotted line shows the slope a = 0.0467 that fits to the cusp-zero model. But the linear relation cannot be obtained in the flat model and the cusp-nonzero model _ u. _ As illustrated in Fig. 12, the flux along the at large h= grain boundary makes a bulge at the neck in the flat model and the cusp-nonzero model. The bulge decreases the neck _ u. _ On the other hand, in curvature, then a decreases with h= the cusp-zero model, the neck curvature, and then a, is approximately constant till the orientation reaches the cusp. The experimental results in the cusp-zero type are analyzed by using Eq. (23). The decrease of the grain boundary energy from the initial orientation is plotted as a function of the rotation angle in Fig. 13. Specimen 1 is also included in the plot. Fig. 13 shows that the rotation is trapped at shallow cusps. The depth of the cusp for the R11 boundary is shallower than that shown in Fig. 9. It should be understood since the orientation of the pre-existing grain boundary is different from the symmetrical tilt grain boundary. _ u_ is small in such shallow cusps, Eq. (23) can be Since h= applied to the flat model and the cusp-nonzero model also. Chan and Balluffi [22,23] showed the presence of shallow cusps in the cgb(h) curve of Au from the analysis of the rotation of spheres in sintering. But they did not determine the exact depth of the cusp. Kuhn et al. [45] used the rotation rate to estimate the shape of cusp assuming the flat model on dDgb. Our method using Eq. (23) can analyze the cgb(h) curve near shallow cusps regardless of the type of dDgb models.
Fig. 13. Estimated grain boundary energy near special grain boundaries of Au.
Relatively shallow cusps or inflections in the cgb(h) curve were demonstrated at high temperatures in a systematic study of [1 0 0] twist boundaries in Au [46]. It was suggested that the cusp/inflection in the cgb(h) curve is determined by entropy and enthalpy. Such shallow cusps can be easily detected by observing particle rotation in sintering. It should be noted that the shape of the curve in the vicinity of the energy cusp in Fig. 13 may not be accurate due to the measurement error of shrinkage. But we believe the curves still provide the data to estimate the torque on a particle in real sintering. Of course, as Cahn [47] pointed out, the microscopic analyses on the dislocation motion and defect structures are necessary to understand the rotation correctly in addition to our analysis in particle scales. A grain boundary is specified by five angle parameters: three angles for the relative orientation of two crystals, and two angles for the boundary-plane orientation. When the grain boundary energy is dependent on the boundaryplane orientation, the orientation derivative gives the Herring torque term [36]. Our computer simulation showed that the boundary-plane orientation rotated with the particle rotation as shown in Fig. 11 even though we did not include the Herring torque. The boundary plane rotated simply because the grain boundary was constrained between two U-shaped notches. 5.2. Grain boundary diffusion
_ u_ in the simulation. The Fig. 12. Relationship between o cgb/o h and h= unit of h is radian. The filled circles show the data at the plateau region in the cgb(h) curve.
At a given temperature, dDgb can vary by several orders of magnitude, depending on the grain boundary structure [44]. dDgb of Au at 1023 K are estimated from the initial shrinkage rates in Fig. 6, and are plotted in Fig. 14. There is a huge variation in experimentally obtained dDgb. In order to define the average dDgb, it will be necessary to
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know the distribution function of orientation. Fig. 14 shows a comparison of our results with other measurements. The results are plotted on an Arrhenius diagram using a reduced temperature scale corresponding to the dimensionless parameter TM/T, where TM is the melting temperature of Au. It may be seen that our data would fall on an extrapolation of Gupta’s results [48] on the average dDgb of Au. Ma and Balluffi [49] measured the chemical diffusion of Ag along the grain boundaries of Au as a function of temperature and tilt angle at relatively low temperatures. The variation of dDgb in different grain boundaries in our results is consistent with that of chemical diffusion along grain boundaries. 5.3. Effect of contamination on surface diffusion The surface diffusion coefficient dDs can be directly estimated from the rounding of corners of the micro-specimen [50]. Fig. 15 shows the cross-sectional profile of the Ushaped notch at z = 0, illustrating the rounding at its corners in the 3-D simulation. The relationship between the curvature of the corner and the dimensionless time is plotted in Fig. 15. The curvature is given by: 1
w0 j ¼ 0:55ðt Þ4 :
ð24Þ
Eq. (24) is consistent with Mullins’ theory [37] of microstructural evolution by surface diffusion. Fig. 4 shows that the corner of the micro-specimen is already round after FIB micro-machining. The difference in curvature at the corner cannot be measured exactly after the annealing at 1023 K. This result suggests low dDs < 1020 m3 s1. The surface diffusion coefficients of Au are plotted in Fig. 14 for comparison [51–53]. The large spread in the pre-
Fig. 15. Calculated curvature at the corner of the U-shaped notch as a function of the dimensionless time.
viously reported values was due to the difference in the degree of surface contamination [52]. Experiments have demonstrated that carbon is a very effective impurity to suppress dDs for metals Cu and Au. Carbon is found to segregate on the Au surface [54]. Since carbon is deposited on the micro-specimen as a marker, it is understandable that dDs was much lower than that on the clean surface. The surface may be also contaminated by Ga during FIB micro-machining. The impurities that have lower melting point than the substrate cause a substantial increase in dDs. But our experimental results suggest that the effect of carbon segregation is much stronger than that of Ga. The surface energy of crystalline particles is usually anisotropic. Klinger and Rabkin [55] recently demonstrated that the anisotropy of surface energy leads to particle rotation. But we assumed isotropic surface energy throughout this paper, because it is a valid assumption at temperatures higher than the surface roughening transition. 6. Summary
Fig. 14. Grain boundary and surface diffusion of Au (TM: melting temperature of Au, 1337 K).
Theoretical and experimental approaches are important to obtaining reliable predictions in dealing with the complexity of sintering. We proposed an experimental method to analyze the mechanics of sintering of a single grain boundary by using specimens fabricated by FIB micromachining. A self-consistent theoretical model on the evolution of neck curvature was developed considering rotation, shrinkage and asymmetric neck growth. The microspecimen had a rectangular grain boundary between Ushaped notches. The sintering of the micro-specimen could be analyzed more easily than that of the classic two-sphere model. The linear velocity was driven by the sintering force that was a function of neck curvature. The variation of dDgb in different grain boundaries of Au was determined from the linear velocity. The rotation was driven by torques arising from anisotropic cgb and asymmetric neck shape. A method to estimate cgb(h) was developed to show
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