Kinetics of Materials - R. Balluff S. Allen W. Carter (Wiley 2005) WW Part 11 pptx

CHAPTER 16 MORPHOLOGICAL EVOLUTION DUE TO CAPILLARY AND APPLIED FORCES: DIFFUSIONAL CREEP AND SlNTERlNG Capillary forces induce morphological evolution of an interface toward uniform d

EXERCISES 385

of the edge and corner grains? Under the same assumptions that apply for

the (N - 6)-rule, find how the growth of a side grain and a corner grain in

a square specimen such as shown in Fig 15.17 depends on the number of neighboring grains, N It is reasonable to assume that the grain boundaries

are maintained perpendicular to the edges of the sample at the locations of their intersections, as shown Local interface-tension equilibration obtains and Young’s equation is satisfied

Corner

Side

grain

grain

Figure 15.17: Two-dimensional grain growth on a square domain

Solution As shown in Fig 15.17, for side grains and corner grains the number of triple junctions is one less than the number of neighboring grains, N For the side grains, the inclination o f the boundary normal changes by 7r from one end t o the other:

Since there is no integer number of neighbors that can produce constant area for a

corner grain, it is impossible t o stabilize grain growth on a rectangular domain

15.4 (a) A cylindrical grain of circular cross section embedded in a large single-

crystalline sheet is shrinking under the influence of its grain-boundary energy Find an expression for the grain radius as a function of time Assume isotropic boundary energy, y, and a constant grain-boundary mobility, MB

(b) Derive a corresponding expression for a shrinking spherical grain embed-

ded in a large single crystal in three dimensions

R2(t) - R2(0) = - 2 M ~ y t (15.58)

386 CHAPTER 15: COARSENING DUE TO CAPILLARY FORCES

(b) Here, the velocity o f the spherical interface normal t o itself is given by Eq 15.28

(15.60) and the spherical grain shrinks twice as fast as the cylindrical grain because of i t s larger curvature

CHAPTER 16

MORPHOLOGICAL EVOLUTION DUE TO CAPILLARY AND APPLIED FORCES:

DIFFUSIONAL CREEP AND SlNTERlNG

Capillary forces induce morphological evolution of an interface toward uniform dif- fusion potential-which is also a condition for constant mean curvature for isotropic free surfaces (Chapter 14) If a microstructure has many internal interfaces, such

as one with fine precipitates or a fine grain size, capillary forces drive mass between

or across interfaces and cause coarsening (Chapter 15) Capillary-driven processes can occur simultaneously in systems containing both free surfaces and internal in- terfaces, such as a porous polycrystal

Applied forces can also induce mass flow between interfaces When tensile forces are applied, atoms from an unloaded free surface will tend to diffuse toward internal interfaces that are normal to the loading direction; this redistribution of mass causes the system to expand in the tensile direction Applied compressive forces can superpose with capillary forces to cause shrinkage In this chapter, we introduce a framework to treat the combined effects of capillary and applied mechanical forces

on mass redistribution between surfaces and internal interfaces

Applications of this framework include diffusional creep in dense polycrystals and sintering of porous polycrystals Diffusional creep and sintering derive from similar kinetic driving forces Diffusional creep is associated with macroscopic shape change when mass is transported between interfaces due to capillary and mechanical driving forces Sintering occurs in response to the same driving forces, but is identified with porous bodies Sintering changes the shape and size of pores; if pores shrink, sintering also produces macroscopic shrinkage (densification)

Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 387

Copyright @ 2005 John Wiley & Sons, Inc

388 CHAPTER 16 MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP, AND SlNTERlNG

Microstructures are generally too complex for exact models In a polycrystalline microstructure, grain-boundary tractions will be distributed with respect to an applied load Microstructures of porous bodies include isolated pores as well as pores attached to grain boundaries and triple junctions Nevertheless, there are several simple representative geometries that illustrate general coupled phenomena and serve as good models for subsets of more complex structures

Both capillarity and stresses contribute to the diffusion potential (Sections 2.2.3 and 3.5.4) When diffusion potential differences exist between interfaces or between internal interfaces and surfaces, an atom flux (and its associated volume flux) will arise These driving forces were introduced in Chapter 3 and illustrated in Fig 3.7 (for the case of capillarity-induced surface evolution) and in Fig 3.10 (for the case

of shape changes due to capillary and applied forces)

For pores within an unstressed body, the diffusion potential at a pore surface will be lower than at nearby grain boundaries if the surface curvature is negative.'

In this case, the material densifies as atoms flow from grain boundaries to the pore surfaces Conversely, macroscopic expansion occurs if the pore surface has average positive curvature

An applied stress, as in Fig 16.1, can reverse the situation by modifying the diffu- sion potential on interfaces if their inclinations are not perpendicular to the loading direction With applied stress and capillary forces, the flux equations for crystal diffusion and surface diffusion are given by Eqs 13.3 and 14.2 For grain-boundary

'The sign of the average pore-surface curvature will generally be negative if the dihedral angles are large and the number of neighboring grains is small In two dimensions-if the pore-surface tension is equal to the grain-boundary surface tension-the average pore-surface curvature will

be positive if there are more than six neighbors, and the pore can grow by absorbing vacancies from its abutting grain boundaries This is equivalent to the ( N -6)-rule (Eq 15.33) If the grain boundaries have variable tensions, pore growth or shrinkage will depend on the particular abutting grain boundary energies However, two-dimensional pores with more than Ncrit = ~ T / ( T - ($)) abutting grains (where ($) is the average dihedral angle (2cos-' r B / ( 2 r S ) ) ) will grow on the average

16.1: MORPHOLOGICAL EVOLUTION FOR SIMPLE GEOMETRIES 389

diffusion, the flux along a boundary under normal stress, u,,, is determined from Eqs 2.21, 3.43, and 3.84,

As in surface diffusion (Eq 14.6), flux accumulation during grain-boundary dif- fusion leads to atom deposition adjacent to the grain boundary The resulting accumulation causes the adjacent crystals to move apart at the rate’

(16.2)

Three conditions are required for a complete solution to the problems illustrated

in Figs 3.10 and 16.1 If the grain boundary remains planar, dL/dt in Eq 16.2 must be spatially uniform-the Laplacian of the normal surface stress under quasi- steady-state conditions must then be constant:

Solving Eqs 16.6 subject to the symmetry condition (dann/drJr,o = 0),

3The justification for the projected interface contribution is presented elsewhere [l-41 The total

force Fapp is that measured by a wetting balance [5]

4By symmetry, there is no angular dependence of unR

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,

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

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 = Solving 0,

2 where the grain-boundary center is located at x = 0 The constant A can be determined from Eq 16.5,

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

Evolution of Bamboo Wire by Bulk Diffusion

(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)

16.1: MORPHOLOGICAL EVOLUTION FOR SIMPLE GEOMETRIES 393

The knR, quantities are the roots of the zeroth-order Bessel function of the first kind,

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]:

B z [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 c y ~ reduces to the exact force

r b a m b o o 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

Because of the proximity effect of surface diffusion, the flux from the regions adjacent to the neck leaves an undercut region in the neck ~ i c i n i t y ~ 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 1 6 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

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 1 6 3 ~ ) are mapped to a corresponding cylindrical surface (indicated by the arrows in Fig 16.3b) The zone extends between z = 1 2 ~ ~ 1 3 The flux equation has the same form as Eq 14.4, so that'

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

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, L B , and Lc in Fig 16.4 and the shear displacements by S A , S B , 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 - d 3 s B

Similarly, along the 2 axis,

Also, the volume must remain constant Therefore,

El1 + E 2 2 = 0

2

t

(16.37) ( 16.38)

(16.39)

Figure 16.4:

subjected to uniaxial applied stress, Two-dimensional polycrystal consisting of identical hexagonal grains g , giving rise to an axial strain rate t From Beer6 [14]

16.2: DIFFUSIONAL CREEP 397

where ~ 1 1 and ~ 2 2 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 ~ 1 1 = dul/dzl, ~ 2 2 =

duz/dzz, and ~ 1 2 = (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,

Substituting Eqs 16.42 into Eq 16.39 yields

Combining Eqs 16.38, 16.37, and 16.43,

To show that boundary sliding must participate in the diffusional creep to main- tain compatibility, suppose that all of the S A , S B , and Sc sliding displacements are zero Equations 16.44 require that the LA, L B , 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 e E l l + sin2 e~~~ + 2 sin e cos e~~~ (16.46)

or, using Eq 16.40-16.44,

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, S A , S B , and Sc, while the normal stresses will promote the diffusion currents responsible for the L A , L B , 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])

with A1 = 32, and for diffusion-controlled Nabarro-Herring creep,

Theoretical shear stress Dislocation glide

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]

400 CHAPTER 16: MORPHOLOGICAL EVOLUTION: DIFFUSIONAL CREEP, AND SlNTERlNG

individual grains to rotate Also, grain-boundary migration and the formation of new grains by recrystallization will affect both microstructure and creep rate.12 Finally, mechanisms besides diffusional transport of mass between internal in- terfaces can contribute to diffusional creep For instance, single crystals containing dislocations exhibit limited creep if the dislocations act as sources and sinks, de- pending on their orientation with respect to an applied stress (see Exercise 16.3)

16.3 SlNTERlNG

Sintering is a kinetic process that converts a compacted particle mass (or powder)

or fragile porous body into one with more structural integrity Increased mechan- ical integrity stems from both neck growth (due to mass transport that increases the particle/particle “necks”) and densification (due to mass transport that reduces porosity) The fundamental sintering driving force-capillarity-derives from re- duction of total surface energy and is often augmented by applied pressure The kinetic transport mechanisms that permit sintering are solid-state processes, and therefore sintering is an important forming process that does not require melt- ing Materials with high melting temperatures, such as ceramics, can be molded into a complex shape from a powder and subsequently sintered into a solid body.I3

16.3.1 Sintering Mechanisms

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 mechanisms include grain-boundary diffusion, volume diffusion, and viscous flow Other mechanisms simply rearrange volume at the pore surfaces and contribute to particle/particle neck growth without reduction

in porosity and shrinkage-t hese mechanisms include surface diffusion and vapor transport Particle compacts and porous bodies have complex geometries, but models for sintering and shrinkage can be developed for simpler geometries such as the one captured in Fig 16.6.14 These models can be used to infer behaviors of

-Time-

Figure 16.6: h’Iodel sint,rririg cxpcriiricrit drrrioristratirig neck growth during sint,ering

by viscous flow of iiiitially splierical 3 imri diameter glass beads at 1000°C over 30 iriiiiutes

Courtesy of Hans-Eckart Exner

I2These phenomena, and their effects on the creep rate, are described in more detail by Sutton

and Balluffi [12], Beer6 [13, 141; and Pilling and Ridley [15]

13General reviews of sintering appear in introductory ceramics texts [21, 221, and a more complete exposition is given in German’s book on sintering [23]

14Further details about such models can be found in Reviews in Powder Metallurgy and Physical

Ceramics or in Physical Metallurgy [24, 251

163 SlNTERlNG 401

complex systems of which these simpler geometries are component parts

Figure 16.7 summarizes the atom-transport paths that can contribute to neck growth and also, in some cases, densification If the particles are crystalline, a grain boundary will generally form at the contact region (the neck) A dihedral angle y will form at the neck/surface junction, and for the isotropic case, conform to Young’s equation, y B = 27’ cos($/2) The seven different transport paths in Fig 16.7 are listed in Table 16.1 with their kinetic mechanisms Atoms generally flow to the neck region, where the surface has a large negative principal curvature and therefore a low diffusion potential compared to neighboring regions Densification will accompany neck growth if the centers of the abutting spheres move toward one another For example with mechanism BS.B, atoms are removed from the boundary region

causing such motion

The dominant mechanism and transport path-or combinations thereof-depend upon material properties such as the diffusivity spectrum, surface tension, temper- ature, chemistry, and atmosphere The dominant mechanism may also change as the microstructure evolves from one sintering stage to another Sintering maps that indicate dominant kinetic mechanisms for different microstructural scales and environmental conditions are discussed in Section 16.3.5

Figure 16.7: (a) Sintering of two abutting single-crystal spherical particles of differing

crystal orientations A grain boundary has formed across the neck region (b) Detail of neck perimeter Seven possible sintering mechanisms for the growth of the neck are illustrated (see the text and Table 16.1)

16.3.2 Sintering Microstructures

Powder compressed into a desired shape at room temperature provides an initial microstructure for a typical sintering process Such a microstructure may be com- posed of equiaxed particles or the particles may vary in size and shape Particle packing may be regular and nearly ‘krystalline,” highly irregular, or mixtures of both Sintering microstructures are generally complex, but some aspects of their mi- crostructural evolution can be understood by investigating primary process models such as those described in Section 16.1 and the simple neck-growth models presented

in Section 16.1.4 However, some microstructural evolution processes are not eas- ily captured by simple models Additional modeling difficulties arise for irregular packings, variability in particle size and shape, and inhomogeneous chemistry

402 CHAPTER 16 MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP, AND SlNTERlNG

Table 16.1: Mass Transport Mechanisms for Sintering

SS.XL Surface Surface Crystal diffusion Nondensifying

Atoms diffuse through the crystal from larger-curvature surface regions to lower- curvature regions

BS.XL Boundary Surface Crystal diffusion Densifying

Atoms diffuse through the crystal from the grain boundary to low-curvature sur- face regions

~~ ~

BS.B Boundary Surface Boundary diffusion Densifying

Atoms diffuse along the boundary to the surface; subsequently, they are trans- ported along the surface by one or more of the SS.XL, SS.S, or SS.V paths

DS.XL Dislocation Surface Crystal diffusion Either

Atoms diffuse through the crystal from climbing dislocations Equivalently, va- cancies diffuse from the surface

ss.s Surface Surface Surface diffusion Nondensifying

Atoms diffuse along the surface from 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 surface regions to lower-curvature surface regions

Atoms are transported by viscous flow by differences in the capillary pressure at nonuniformly curved surfaces

Nevertheless, there are parallel stages in any powder sintering process that can

be used to catalog behavior Each powder sintering process begins with parti- cle/particle neck formation and a porous phase between the weakly attached par- ticles As these necks grow, the particle/pore interface becomes more uniformly curved but remains interconnected throughout the compact Before the porous phase is removed, it becomes disconnected and isolated at pockets where four grain boundaries intersect

Initial, Intermediate, and Final Stages of Powder Sintering Following Coble’s pio- neering work, the microstructural evolution of a densifying compact is separated into an initial stage, an intermediate stage, and a final stage of sintering [26] Fig- ure 16.8 illustrates some of the microstructural features of each stage

The initial stage comprises neck growth along the grain boundary between abut- ting particles The intermediate stage occurs during the period when the necks between the particles are no longer small compared to the particle radii and the porosity is mainly in the form of tubular pores along the three-grain junctions in the compact The geometries of both the initial and intermediate stages therefore have intergranular porosity percolating through the compact

16.3 SlNTERlNG 403

Initial powder Initial stage Intermediate stage Final stage Dense polycrystal- compact of sintering of sintering of sintering line compact

Figure 16.8: Stages of powder sinterin Initial stage involves neck growth Intermediate

state is marked by continuous porosity j o n g three-grain junctions Final stage involves removal of isolated pores at four-grain junctions Figure calculated using Surface Evolver [27] ;

figure concept by Coble [26] Courtesy of Ellen J Siem

The transition from the intermediate to the final stage occurs when the intercon- nected tubular porosity along the grain junctions (edges) breaks up because of the Rayleigh instability (see Section 14.1.2) and leaves isolated pores of equiaxed shape

at the grain corners [7] The final stage occurs when the porosity is isolated and located at multiple-grain junctions Final pore elimination occurs by mass transfer from the grain boundaries to the pores attached to the grain boundaries similar to the transport in the wire-bundle model treated in Section 16.1.2 If grain growth occurs during any stage, the pores may break away from the grain boundaries In such cases, the pores will be isolated from the grain boundaries in the final stage and further densification will be limited by the rate of crystal diffusion of atoms from dislocation sources by the mechanism DS.XL illustrated in Fig 16.7 Failure

to reach full density is often caused by such pore breakaway

16.3.3 Model Sintering Experiments

Experiments have been designed to reveal details of the sintering mechanisms indi- cated by Fig 16.7 and the sintering stages illustrated by Fig 16.8 Such sintering experiments include sphere-sphere model experiments similar to that depicted in Fig 16.6 [28], sintering of rows of spheres [29], sintering of spheres and wires to flat plates [30], and sintering of bundles of wires such as that depicted in Fig 16.9 [31] With their simple geometry, these model experiments reveal fundamental pro- cesses during the various stages of sintering Initial-stage processes are illuminated

by the sphere-sphere experiments, and transitions between the intermediate and final stages are captured in the wire-bundle experiments Figure 16.9d, in par- ticular, demonstrates the important role of grain-boundary attachment for pore removal-essentially all of the grain-boundary segments trapped between the pores have broken free and left the specimen However, one boundary remains and con- tinues to feed atoms to the pores to which it is connected

16.3.4 Scaling Laws for Sintering

Because the surface energy per volume is larger for small particles and because the fundamental driving force for sintering is surface-energy reduction, compacts composed of smaller powders will typically sinter more rapidly Smaller powders are more difficult to produce and handle; therefore, predictions of sintering rate dependence on size are used to make choices of initial particle size Herring’s

404 CHAPTER 16 MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP, AND SINTERING

(c)

Figure 16.9:

at 900°C: (a) 50 11 (b) 100 h, (c) 300 h; and at 1075°C: (d) 408 h

Balluffi [31]

Cross section of bundle of parallel 128 p n diameter Cu wires after sintering

From Alexander and

scaling laws provide a straightforward method to predict sintering rate dependence

In general, a sintering rate is proportional to the mass-transport current, I , due

to sintering driving forces and is inversely proportional to the transported material volume, AV required to produce a given shape change (e.g., the volume associated with neck growth) The current I is the vector product of the atoniic flux :and the area A’ through which the current flows during sintering Therefore, the rates

at which bodies S and B undergo geometrically similar changes will be in the ratio

(16.51)

rateB IB AVs rates AVB IS The current, I , is proportional to the diffusion potential gradient, V@* and to the cross-sectional area A, through which this flux flows Therefore,