390 CHAPTER 16 MORPHOLOGICAL EVOLUTION: DIFFUSIONAL CREEP, AND SlNTERlNG Figure 16.2: Force-balance diagram for a body with capillary forces and applied load Fapp. The plane cuts the body normal.to the applied force. There are two contributions from the body itself. One is the projection of the surface capillary force per unit length (rS) onto the normal direction and integrated over the bounding curve. The second is the normal stress onn integrated over the cross-sectional area-in the case of fluids bounded by a surface of uniform curvature K’, onn = ySnS [4]. The constant A is determined from the force balance in Eq. 16.5, (16.8) Using Young‘s force-balance equation (Eq. 14.18), (16.9) at the grain boundary/surface intersection and the elongation rate (Eq. 16.2) be- comes ( 16.10) When the grain boundaries are not spaced too closely, the quantity Tbamboo is generally negative because Rbn % 1 is less than 2J1 - [yB/(2rS)I2 and yB/(2yS) M 1/6 for metals. T, the capillary shrinkage force, arises from a balance between reductions of surface and grain-boundary area. If Fapp is adjusted so that the elongation rate goes to zero, Fapp = -rbamboo, and this provides an experimental method to determine yB/yS, and thus y” if + is measured. This is known as the Udin-Schaler-Wulff zero-creep method [6]. Scaling arguments can be used to estimate elongation behavior. Because K and 1/Rb will scale with Jm and the grain volume, V, is constant, Eq. 16.10 implies that dL - dt cc L2 ( Fapp - ys@) (16.11) 16.1. MORPHOLOGICAL EVOLUTION FOR SIMPLE GEOMETRIES 391 where ?“bamboo M irySRb is replaced by a term that depends on L alone. Elonga- tion proceeds according to5 16.1.2 For the boundary of width 2w in Fig. 16.1, Eq. 16.4 becomes Evolution of a Bundle of Parallel Wires via Grain-Boundary Diffusion ann(Z = *W) = -7 S K (16.13) where K is evaluated at the pore surface/grain boundary intersection. Eq. 16.3 subject to Eq. 16.13 and the symmetry condition (da,,/dz)l,=o = 0, Solving a,,(x) = -(x A2 - w2) - 7% (16.14) 2 where the grain-boundary center is located at x = 0. The constant A can be determined from Eq. 16.5, S - 27 KW - fapp (16.15) The shrinkage rate, Eq. 16.2, becomes dL RAG*D~A 3R.46*DB dt kT 2w3kT (16.16) (fapp + rwires) - - _- - - $ rwires = 2y’(~w - sin -) 2 If surface diffusion or vapor transport is rapid enough, the pores will maintain their quasi-static equilibrium shape, illustrated in Fig. 16.1 in the form of four cylindrical sections of radius R.6 The dihedral angle at the four intersections with grain boundaries, $, will obey Young’s equation. $ is related to 8 by sin($/2) = cos 8. An exact expression can be calculated for the quasi-static capillary force, Ywires, as a function of the time-dependent length L(t). Young’s equation places a geomet- ric constraint among L(t), the cylinder’s radius of curvature R(t), and boundary width w(t); conservation of material volume provides the second necessary equa- tion. With Twire(L) and w(L), Eq. 16.16 can be integrated. This model could be extended to general two-dimensional loads by applying different forces onto the horizontal and vertical grain boundaries in Fig. 16.1. The three-dimensional case, with sections of spheres and a triaxial load, could also be derived exactly. 5An exact quasi-static [e.g., surfaces of uniform curvature (Eq. 14.29)] derivation exists for this model [4]. 6The Rayleigh instability (Section 14.1.2) of the pore channel is neglected. Pores attached to grain boundaries have increased critical Rayleigh instability wavelengths [7]. 392 CHAPTER 16: MORPHOLOGICAL EVOLUTION: DIFFUSIONAL CREEP, AND SlNTERlNG 16.1.3 Morphological evolution and elongation can also occur by mass flux (and its associ- ated volume) from the grain boundary through the bulk to the surface as illustrated in Fig. 3.10~. For elongation of a crystalline material, vacancies could be created at the grain boundary and diffuse through the grain to the surface, where they would be removed. The quasi-steady-state rate of elongation can be determined by solv- ing the boundary-value problem described in Section 3.5.3 involving the solution to Laplace’s equation V2@~ = 0 within each grain of the idealized bamboo structure. For isotropic surfaces and grain boundaries, @A is given by Eqs. 3.76 and 3.84. The expression for bulk mass flux is given by Eq. 13.3, and using the coordinate system shown in Fig. 3.10, symmetry requires that Evolution of Bamboo Wire by Bulk Diffusion (16.17) If the grain boundary remains planar, the flux into the boundary must be uniform, = C = constant (%) z=o Laplace’s equation in cylindrical coordinates is (16.18) (16.19) Assuming that the solution to Eq. 16.19 is the product of functions of z and r and using the separation-of-variables method (Section 5.2.4), @A = [c1 sinh(kz) + cz cosh(kz)][c~J0(kr) + c4Yo(kr)] (16.20) where clrc2,c3,c4, and k are constants to be determined, and Jo and Yo are the zeroth-order Bessel functions of the first and second kinds. Because @A(?- = 0) must be bounded, c4 = 0. Introducing a new variable p(r, z) that will necessarily vanish on the free surface, (16.21) The general solution to Eq. 16.19 is the superposition of the homogeneous solutions, p(r, z) = Jo(k,r) [b, sinh(k,z) + c, cosh(k,z)] (16.22) n The bamboo segment can be approximated as a cylinder of average radius Re, where L nRZL = 1 nR2(z) dz The boundary condition (Eq. 3.76) is then approximated by @A=po+- or, equivalently, p(r = R,, z) = o Re (16.23) (16.24) 16.1: MORPHOLOGICAL EVOLUTION FOR SIMPLE GEOMETRIES 393 The knR, quantities are the roots of the zeroth-order Bessel function of the first kind, Jo (kRc) = 0 (16.25) The symmetry condition Eq. 16.17 is satisfied if b, cosh(k,l/2) + c, sinh(knl/2) = 0, and therefore, ~(r, 2) = C bnJo(knr) sinh(k,z) - coth ( y) cosh(k,r)] (16.26) n The planar grain-boundary condition given by Eq. 16.18 is satisfied if The coefficients, b,kn, of Jo in this Bessel function series can be determined [8]: (16.28) The constant C can be determined by substituting Eqs. 16.26 and 16.28 into the force-balance condition (Eq. 16.5), where (16.29) The total atom current into the boundary is IA = -27rRz JA; therefore, (16.31) coth(k,l/2) Bz [T k2R2 B M 12 for L/Rc M 2 [9]. The elongation-rate expressions for grain-boundary diffusion (Eq. 16.10) and bulk diffusion (Eq. 16.31) for a bamboo wire are similar except for a length scale. The approximate capillary shrinkage force 'Yapprox cy~ reduces to the exact force rbamboo as the segment shapes become cylindrical, Rb % R, % l/&. However, because the grain-boundary diffusion elongation rate is proportional to *DB/R;f, while the bulk diffusion rate is proportional to *DXL/R2, grain-boundary transport will dominate at low temperatures and small wire radii. 394 CHAPTER 16 MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP, AND SlNTERlNG 16.1.4 Figure 16.3 illustrates neck growth between two particles by surface diffusion. Sur- face flux is driven toward the neck region by gradients in curvature. Neck growth (and particle bonding) occurs as a result of mass deposition in that region of small- est curvature. Because no mass is transported from the region between the particle centers, the two spheres maintain their spacing at 2R as the neck grows through rearrangement of surface atoms. This is surface evolution toward a uniform po- tential for which governing equations were derived in Section 14.1.1. However, the small-slope approximation that was used to obtain Eq. 14.10 does not apply for the sphere-sphere geometry. Approximate models, such as those used in the following treatment of Coblenz et al., can be used and verified experimentally [lo]. Neck Growth between Two Spherical Particles via Surface Diffusion Overcut /- volume I Figure 16.3: (a) Model for formation of a neck between two spherical particles due to surface diffusion. (b) Approxiniation in which the surface diffusion zone within the saddle- shaped neck regioii of (a) is mapped onto a riglit circular cylinder of radius 2. t is the distance parameter in the diffusion direction. Arrows parallel to the surface indicate surface-diffusion directions in hoth (a) and (11). From Coblenz et al. [lo]. Because of the proximity effect of surface diffusion, the flux from the regions adjacent to the neck leaves an undercut region in the neck ~icinity.~ Diffusion along the uniformly curved spherical surfaces is small because curvature gradients are small and therefore the undercut neck region fills in slowly. This undercutting is illustrated in Fig. 16.3~. Because mass is conserved, the undercut volume is equal to the overcut volume. Conservation of volume provides an approximate relation between the radius of curvature, p, and the neck radius, x: 1/3 p = 0.26~ (G) (16.32) This surface-diffusion problem can be mapped to a one-dimensional problem by approximating the neck region as a cylinder of radius x as shown in Fig. 16.3b. The fluxes along the surface in the actual specimen (indicated by the arrows in Fig. 16.3~) are mapped to a corresponding cylindrical surface (indicated by the arrows in Fig. 16.3b). The zone extends between z = 12~~13. The flux equation has the same form as Eq. 14.4, so that' JS x *Dsys dr; kT dz (16.33) 7The proxiniity effect is reflected in the strong wavelength dependence of surface smoothing (i.e., l/X4 in Eq. 14.12). sEquation 16.33 ignores the relatively small effect of the increase in energy due to the growing grain boundary. 16 2: DIFFUSIONAL CREEP 395 The curvature has the value 2/R at z = f2rrp/3 and approximately -l/p at z = 0 (neglecting terms of order p/R). The average curvature gradient -3/(2.irp2) can be inserted into Eq. 16.33 for an approximation to the total accumulation at the neck (per neck circumference), 3 6 *Dsys .irkTp2 Is M 26Js M (16.34) The corresponding neck surface area is approximately p (per neck circumference), and therefore the neck growth rate is approximately dx 36*DSySR~ dt .irkTp3 _N N (16.35) Putting Eq. 16.32 into Eq. 16.35 and integrating yields the neck growth law, (16.36) Equation 16.36 predicts that x(t) K t1I5 and that the neck growth rate will therefore fall off rapidly with time. The time to produce a neck size that is a given particle-size fraction is a strong function of initial particle size-it increases as R4. Equation 16.36 agrees with the results of a numerical treatment by Nichols and Mullins [ 111 .’ 16.2 DIFFUSIONAL CREEP Mass diffusion between grain boundaries in a polycrystal can be driven by an ap- plied shear stress. The result of the mass transfer is a high-temperature permanent (plastic) deformation called diffusional creep. If the mass flux between grain bound- aries occurs via the crystalline matrix (as in Section 16.1.3), the process is called Nabarro-Herring creep. If the mass flux is along the grain boundaries themselves via triple and quadjunctions (as in Sections 16.1.1 and 16.1.2), the process is called Coble creep. Grain boundaries serve as both sources and sinks in polycrystalline materials- those grain boundaries with larger normal tensile loads are sinks for atoms trans- ported from grain boundaries under lower tensile loads and from those under com- pressive loads. The diffusional creep in polycrystalline microstructure is geomet- rically complex and difficult to analyze. Again, simple representative models are amenable to rigorous treatment and lead to an approximate treatment of creep in general. 16.2.1 A representative model is a two-dimensional polycrystal composed of equiaxed hexagonal grains. In a dense polycrystal, diffusion is complicated by the necessity Diffusional Creep of Two-Dimensional Polycrystals gDifferent growth-law exponents are obtained for other dominant transport mechanisms. Coblenz et al. present corresponding neck-growth laws for the vapor transport, grain-boundary diffusion, and crystal-diffusion mechanisms [lo]. 396 CHAPTER 16 MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP, AND SlNTERlNG of simultaneous grain-boundary sliding-a thermally activated shearing process by which abutting grains slide past one another-to maintain compatibility between the grains. In the absence of sliding, gaps or pores will develop. Sliding is confined to the grain-boundary region and occurs by complex mechanisms that are not yet completely understood [12]. The need for such sliding can be demonstrated by analyzing the diffusional creep of the idealized polycrystal illustrated in Fig. 16.4 [12-151. The specimen is sub- jected to the applied tensile stress, u, which motivates diffusion currents between the boundaries at differing inclinations and causes the specimen to elongate along the applied stress axis. Figure 16.4 shows the currents associated with Nabarro- Herring creep. Currents along the boundaries can occur simultaneously, and if these dominate the dimensional changes, produce Coble creep. For the equiaxed microstructure in Fig. 16.4, there are only three different boundary inclinations with respect to a general loading direction; these are exhibited by the boundaries between grains A, B, and C indicated in Fig. 16.4. Mass transport between these boundaries will cause displacement of the centers of their adjoining grains. The normal displacements are indicated by LA, LB, and Lc in Fig. 16.4 and the shear displacements by SA, SB, and Sc. These combined grain-center displacements produce an equivalent net shape change of the polycrystal. Compatibility relationships between the displacements must exist if the grain boundaries remain intact. Along the 1 axis, the displacement of grain C relative to grain B must be consistent with the difference between the displacement of grain C with respect to grain A and with that of grain B with respect to grain A. This requirement is met if LA + LB - 2LC = v5sA - d3sB Similarly, along the 2 axis, Also, the volume must remain constant. Therefore, El1 + E22 = 0 2 t (16.37) ( 16.38) (16.39) Figure 16.4: subjected to uniaxial applied stress, g, giving rise to an axial strain rate t. From Beer6 [14]. Two-dimensional polycrystal consisting of identical hexagonal grains 16.2: DIFFUSIONAL CREEP 397 where ~11 and ~22 are the normal strains of the overall network connecting the centers of the grains in the (1,2) coordinate system in Fig. 16.4. These strains are related to the displacements through ~11 = dul/dzl, ~22 = duz/dzz, and ~12 = (1/2)(dul/dz2 $duz/dzl), where the ui are the displacements produced throughout the network of grain centers. For the representative unit cell PQRS in Fig. 16.4, AC1- AB1 d El1 = (16.40) where d is the width of a hexagonal grain, and ABi and ACi are the components of the displacements of the centers of the grains B and C relative to A and are given bY 2AB1= asB - LB 2AB2 = -SB + & LB 2AC2 = SA + &LA (16.41) 2ac1 = -&sA + L~ Therefore. 8 (SB - SA) LA + LB 2d -k 2d El1 = Substituting Eqs. 16.42 into Eq. 16.39 yields Combining Eqs. 16.38, 16.37, and 16.43, and L~ + L~ + L~ = o (16.42) (16.43) (16.44) (16.45) which is equivalent to the constant-volume condition. To show that boundary sliding must participate in the diffusional creep to main- tain compatibility, suppose that all of the SA, SB, and Sc sliding displacements are zero. Equations 16.44 require that the LA, LB, and Lc must also vanish. There- fore, nonzero Si 's (sliding) are required to produce nonzero grain-center normal displacements. 398 CHAPTER 16 MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP AND SlNTERlNG This result can be demonstrated similarly by solving for the strain, E, along the applied tensile stress axis shown in Fig. 16.4 in terms of only the Lz’s or only the 9’s: E = cos 2 eEll + sin2 e~~~ + 2 sin e cos e~~~ (16.46) or, using Eq. 16.40-16.44, 1 LA - LB v5 2 sin 8 cos 0 (L~ + L~)(I - 2cos2e) + d (16.47) ] (16.48) 2 sin 0 cos 0 2sc - SA - SB 1 SA-SB dv5 3 &=-[ (1 - 2 COS~ e) + Equation 16.47 indicates that the creep strain may be regarded as diffusional trans- port accommodated by boundary sliding, and Eq. 16.48 indicates that it may equally well be regarded as boundary sliding accommodated by diffusional trans- port.l0 The creep rate, t, can be obtained by taking time derivatives of E in Eqs. 16.47 and 16.48. The applied tensile stress, 0, shown in Fig. 16.4 will generate stresses throughout the polycrystal, and each boundary segment will, in general, be subjected to a shear stress (parallel to the boundary) and a normal stress (per- pendicular to the boundary). The shear stresses will promote the grain-boundary sliding displacements, SA, SB, and Sc, while the normal stresses will promote the diffusion currents responsible for the LA, LB, and Lc displacements. A de- tailed analysis of the shear and normal stresses at the various boundary segments is available (see also Exercise 16.2) [12-141. 16.2.2 The analysis can be extended to a three-dimensional polycrystal with an equiaxed grain microstructure. As in two-dimensional creep, grain-boundary sliding must accompany the diffusional creep, and because these processes are interdependent, either sliding or diffusion may be rate limiting. In most observed cases, the rate is controlled by the diffusional transport [14, 15, 18, 191. Exact solutions for cor- responding tensile strain rates are unknown, but approximate expressions for the Coble and Nabarro-Herring creep rates under diffusion-controlled conditions where the boundaries act as perfect sources may be obtained from the solutions for the bamboo-structured wire in Section 16.1.1. The equiaxed polycrystal can be approx- imated as an array of bonded bamboo-structured wires with their lengths running parallel to the stress axis and with the lengths of their grains (designated by L in Fig. 3.10) equal to the wire diameter, 2R. This produces a polycrystal with an approximate equiaxed grain size d = L = 2R. The Coble and Nabarro-Herring creep rates of this structure can be approximated by those given for the creep rates of the bamboo-structured wire by Eqs. 16.10 and 16.31 with L = 2R = d and the sintering potential set to zero. In this approximation, the effects of internal normal stresses generated along the vertical boundaries (between the bonded wires) may be neglected because these stresses are zero on average. Using this approximation, for diffusion-controlled Coble creep, Diffusional Creep of Three-Dimensional Polycrystals (16.49) ‘OThis duality has been recognized (e.g., Landau and Lifshitz [16] and Raj and Ashby [17]). 16.2: DIFFUSIONAL CREEP 399 0.0 9 -2.0. * 4.0- 5 3 - -6.0- -8.0 with A1 = 32, and for diffusion-controlled Nabarro-Herring creep, Theoretical shear stress Dislocation glide Dislocation creep . Elastic (16.50) with A2 = 12." Because the Coble creep rate is proportional to *DB/d3 and the Nabarro-Herring rate to *DXL/d2, Coble creep will be favored as the temperature and grain size are reduced. Figure 16.5 shows a deformation map for polycrystalline Ag possessing a grain size of 32 pm strained at a rate of 10-8s-1 [20]. Each region delineated on the map indicates a region of applied stress and temperature where a particular ki- netic mechanism dominates. Experimental data and approximate models are used to produce such deformation maps. The mechanisms include elastic deformation at low temperatures and low stresses, dislocation glide at relatively high stresses, dislocation creep at somewhat lower stresses and high temperatures, and Nabarro- Herring and Coble diffusional creep at high temperatures and low stresses. Coble creep supplants Nabarro-Herring creep as the temperature is reduced. An analysis of diffusional creep when the boundaries do not act as perfect sources and sinks has been given by Arzt et al. [19] and is explored in Exercise 16.1. The creep rate when boundary sliding is rate-limiting has been treated and discussed by Beer6 [13, 141. If a viscous constitutive relation is used for grain- boundary sliding (i.e., the sliding rate is proportional to the shear stress across the boundary), the macroscopic creep rate is proportional to the applied stress, and the bulk polycrystalline specimen behaves as a viscous material. An analysis of the sliding-controlled creep rate of the idealized model in Fig. 16.4 is taken up in Exercise 16.2. Variable boundary behavior complicates the results derived from the uniform equiaxed model presented above. Nonuniform boundary sliding rates may cause cases by factors aslarge as three. See Ashby [20], Burton [18], Arzt et al. [19], and Pilling and Ridley [15]. [...]... rrR~L-*( 2-3 cosB+sin30) 3 and dR -dL 1 =O R: 2R2(2 - 3cosQ sin3 0) + (16.98) (16.99) Substituting Eq 16.99 into Eq 16.97 and employing Eq 16.94, dGI dL -f= = 2nySR:[sin8cos28 - 2 ( 1 -sine)] - _- 27rySR,2 R(2 - 3 cos 0 sin3 0) R in agreement with Eq 16.92 + (16 ,100 ) PART IV PHASE TRANS FORMATI0 NS Phase transformations are of central importance in materials science and engineering An understanding of. .. = x, + 7 g p i = xi = X B - q pg = p; = 1 - xg = 1 - X B - q pA = 1 - xg = 1 - X B p (17.9) + 7) These probabilities must all lie between zero and one; this sets bounds on physical values for the structural order parameter 1 -xg . larger-curvature surface regions to lower- curvature surface regions. ss.v Surface Surface Vapor transport Nondensifying Atoms are transported through the vapor phase from larger-curvature. boundaries serve as both sources and sinks in polycrystalline materials- those grain boundaries with larger normal tensile loads are sinks for atoms trans- ported from grain boundaries under lower. Neck growth can occur by any mass transport mechanism. However, processes that permit shrinkage by pore removal must transport mass from the interior of the particles to the pore surfaces-these