10. Sintering & Grain Growth

- 12주차 -

▪ Based on the foregoing discussion, a powder compact can reduce its energy by following various paths, some of which can lead to coarsening, others to densification.

This brings up the central and critical question in sintering:

What governs whether a collection of particles will densify or coarsen?

To answer the question, models for each of the paths considered before must be developed and compared, with the fastest path determining the behavior of the compact.

For instance, a compact in which surface diffusion is much faster than bulk diffusivity would tend to coarsen rather than densify.

In practice, the question is much more difficult to answer, however, because the kinetics of sintering are dependent on many variables, including particle size and packing, sintering atmosphere, degree of agglomeration, temperature, and presence of impurities.

10.3 Sintering Kinetics

The difficulty of the problem is best illustrated by comparing the densification kinetics of an “as-received” yttria-stabilized zirconia powder compact to that of the same powder that was rid of agglomerates before sintering (Fig. 10.10).

The marked reduction in the temperature, by about 300℃, needed to fully densify the compact which was prepared from an agglomerate-free compact is obvious.

In this section, some of the many sintering models proposed over years to model this complex process are outlined.

Because of the complex geometry of the problem, analytic solutions are only possible by making considerable geometric and diffusion flow field approximations, which ae rarely realized in practice.

Consequently, the models discussed next have limited validity and should be used with extreme care when one is trying to predict the sintering behavior of real powders.

Much of the usefulness of these sintering models thus lies more in appreciating the general trends that are to be expected and identifying the critical parameters than in their predictive capabilities.

Sintering stages

▪ Coble described a sintering stage as an “interval of geometric change in which pore shape is totally defined (such as rounding of necks during the initial stage of sintering) or an interval of time during which the pore remains constant in shape while decreasing in size”.

During the initial stage, the interparticle contact area increases by neck growth (Fig. 10.11b) from 0 to ≈0.2, and the relative density increases from about 60 to 65 percent.

The intermediate stage is characterized by continuous pore channels that are coincident with three-grain edges (Fig. 10.11c). During this stage, the relative density increases from 65 to 90 percent by having matter diffuse toward, and vacancies away from the long cylindrical channels.

The final stage begins when the pore phase is eventually pinched off and is characterized by the absence of a continuous pore channel (Fig. 10.11d).

Individual pores are either of lenticular shape, if they reside on the grain boundaries, or rounded, if they reside within a grain.

An important characteristic of this stage is the increase in pore and grain boundary mobilities, which have to be controlled if the theoretical density is to be achieved.

Clearly, the sintering kinetics will be different during each of the aforementioned stages. To further complicate matters, in addition to having to treat each stage separately, the kinetics will depend on the specific atomic mechanisms operative. Despite these complications, most, if not all, sintering models share the following common philosophy:

1. A representative particle shape is assumed.2. The surface curvature is calculated as a function of geometry.3. A flux equation that depends on the rate-limiting step is adopted.4. The flux equation is integrated to predict the rate of geometry change.

In the following subsections, this approach is used to predict the rates of various processes occurring during various stages.

▪ Initial-Stage Sintering

▪ Given the multiplicity of paths available to a powder compact during this stage, it is impossible to address them all in detail.

Instead, the following approach has been adopted:

The rate of neck growth by evaporation condensation (path 1 in Fig. 10.9a) is worked out in detail;

The final results for the other mechanisms, namely, surface, grain boundary, lattice diffusion, and viscous sintering, are given without proof.

Evaporation-condensation model (path 1 in Fig. 10.9a)

▪ In this mechanism, the pressure differential between the surface of the particle and neck area results in a net matter transport, via the gas phase from the surface to the neck.

The evaporation rate (in molecules of MX per square meter per second), is given by the Langmuir expression

𝑗𝑗 =𝛼𝛼∆𝑃𝑃

2𝜋𝜋𝑚𝑚𝑀𝑀𝑀𝑀𝑘𝑘𝑘𝑘= 𝐾𝐾𝑟𝑟∆𝑃𝑃

where 𝛼𝛼 and 𝑚𝑚𝑀𝑀𝑀𝑀 are, respectively, the evaporation coefficient and the mass of the evaporating gas molecules; ∆𝑃𝑃 is the pressure differential between the surface and neck areas.

By applying Eq. (4), it can be easily shown that the pressure differential between these two regions is

∆𝑃𝑃 =Ω𝑀𝑀𝑀𝑀𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝛾𝛾𝑠𝑠𝑠𝑠

𝑘𝑘𝑘𝑘1𝜌𝜌

+1𝑟𝑟≈Ω𝑀𝑀𝑀𝑀𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝛾𝛾𝑠𝑠𝑠𝑠

𝜌𝜌𝑘𝑘𝑘𝑘

(7)

(8)

where 𝜌𝜌 and 𝑟𝑟, defined in Fig 10.12a, are, respectively, the radius of curvature of the neck area and the sphere radius.

Furthermore, according to this figure,

(𝑟𝑟 + 𝜌𝜌)2= (𝑥𝑥 + 𝜌𝜌)2+𝑟𝑟2

where 𝑥𝑥 is the neck radius.

For 𝑥𝑥 ≪ 𝑟𝑟, this equation simplifies to

𝜌𝜌 =𝑥𝑥2

2(𝑟𝑟 − 𝑥𝑥)≈𝑥𝑥2

2𝑟𝑟

Multiplying the flux of material arriving to the neck area by Ω𝑀𝑀𝑀𝑀 yields the rate at which the neck will grow, or

𝑑𝑑𝑥𝑥𝑑𝑑𝑑𝑑

= 𝑗𝑗Ω𝑀𝑀𝑀𝑀

(9)

(10)

Combining Eqs. (7) to (10) and integrating yields

𝑥𝑥𝑟𝑟

3=

6𝛼𝛼𝛾𝛾𝑠𝑠𝑠𝑠Ω𝑀𝑀𝑀𝑀2 𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓2𝜋𝜋𝑚𝑚𝑀𝑀𝑀𝑀𝑘𝑘𝑘𝑘[𝑘𝑘𝑘𝑘]𝑟𝑟2

𝑑𝑑

This equation predicts that the rate of growth of the neck region (1) is initially quite rapid but then flattens out, (2) is a strong function of initial particle size, and (3) is a function of the partial pressure 𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 of the compound, which in turn depends exponentially on temperature.

▪ To recap: Equation (11) was derived by assuming a representative shape (Fig. 10.12a), from which the surface curvature was calculated as a function of geometry [Eq. (9)].

A flux equation [Eq. (7)] was then assumed and integrated to yield the final result.

By using essentially the same procedure, the following results for other models are obtained.

(11)

▪ Densification Kinetics

Intermediate sintering model

▪ Most of the densification of a powder compact occurs during the intermediate state.

Unfortunately, this stage is the most difficult to tackle because it depends strongly on the details of particle packing − a variable that is quite difficult to model.

To render the problem tractable, Coble made the following assumptions:

1. The powder compact is composed of ideally packed tetrakaidecahedra of length 𝑎𝑎𝑝𝑝 separated from each other by long porous channels of radii 𝑟𝑟𝑐𝑐(Fig. 10.14a).

2. Densification occurs by the bulk diffusion of vacancies away from the cylindrical pore channels toward the grain boundaries (curved arrows in Fig. 10.14b).

3. A linear, steady-state profile of the vacancy concentration is established between the source and the sink.

4. The vacancies are annihilated at the grain boundaries; the grain boundaries act as vacancy sinks.

Making these assumptions, one can show (App. 10C) that during the intermediate-stage sintering, the fractional porosity 𝑃𝑃𝑐𝑐 should decrease linearly with time according to

𝑃𝑃𝑐𝑐 ≈ (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑑𝑑)𝐷𝐷𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑑𝑑3

𝛾𝛾𝑠𝑠𝑠𝑠Ω𝑀𝑀𝑀𝑀𝑘𝑘𝑘𝑘

(𝑑𝑑𝑓𝑓 − 𝑑𝑑)

where 𝑑𝑑𝑓𝑓 is the time at which the cylindrical channels vanish and 𝑑𝑑 is the average diameter of the sintering particles, which is assumed to scale with 𝑎𝑎𝑝𝑝.

What is interesting about this process is that it is self-accelerating, since as the cylinder gets smaller in diameter, its curvature increases and the vacancy concentration gradient also increases.

This process cannot and does not go on indefinitely; as the cylindrical pores get longer and thinner, at some point they become unstable and break up into smaller spherical pores along the grain boundary and/or at the triple points between grains (see Fig. 10.11d).

It is at this point that the intermediate sintering stage gives way to the final-stage sintering, where both the annihilation of the last remnants of porosity and the simultaneous coarsening, i.e., grain growth, of the microstructure occur.

Pore elimination (during the final stage of sintering)

▪ When atoms diffuse toward the pores and vacancies are transported away from the pores to a sink such as grain boundaries, dislocations, or external surfaces of the crystal, the pores will be eliminated.

Grain boundary diffusion mechanism

The elimination of the last remnants of vacancies usually occurs only if they remain attached to and are eliminated at the grain boundary.

The appropriate geometry is shown schematically in Fig. 10.15b, and by following a derivation similar to the one carried out in Appl. 10D, it can be shown that

𝜌𝜌𝑝𝑝4 − 𝜌𝜌𝑝𝑝,04 = −

8𝛿𝛿𝑔𝑔𝑎𝑎𝛾𝛾𝑠𝑠𝑠𝑠Ω𝑀𝑀𝑀𝑀𝑘𝑘𝑘𝑘

1

log𝑅𝑅𝑔𝑔𝑎𝑎𝜌𝜌𝑝𝑝

𝑑𝑑

where 𝑅𝑅𝑔𝑔𝑎𝑎 is defined in Fig. 10.15b.

In other words, if grain boundary diffusion is the operative mechanism, the average pore size should shrink with 𝑑𝑑1/4.

▪ Coarsening and Grain Growth Kinetics▪ Any collection of particles will coarsen with time, kinetics permitting, where coarsening implies an increase in the ensemble’s average particle size with time.

Comparing Fig. 10.2a and b shows the clear similarity between coarsening and grain growth.

This section deals with the kinetics of the microstructural evolution of a collection of particles during sintering (Fig. 10.2b) and the grain growth kinetics associated with the final stages of sintering (Fig. 10.2a)

Coarsening

▪ To model coarsening, consider a powder compact consisting of a distribution of particles, with an average particle radius 𝑟𝑟𝑓𝑓𝑠𝑠.

Assuming all particle to be sphere, the average partial pressure over the ensemble is given by Eq. (5), or

𝑃𝑃𝑓𝑓 = 𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 1 +2Ω𝑀𝑀𝑀𝑀𝛾𝛾𝑠𝑠𝑠𝑠𝑟𝑟𝑓𝑓𝑠𝑠𝑘𝑘𝑘𝑘

Similarly, the partial pressure 𝑃𝑃𝑟𝑟 over any particle of radius 𝑟𝑟 ≠ 𝑟𝑟𝑓𝑓𝑠𝑠 is also given by Eq. (5), using the appropriate radius.

(12)

Consequently, grains that are smaller than the average will “evaporate” away, while those that are larger will grow with time (Fig. 10.17a).

If the interface kinetics are rate-limiting, it follows that the velocity 𝑣𝑣 of the solid/gas interface is linearly dependent on the driving force, and

𝑣𝑣 =𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑

= 𝐾𝐾𝑟𝑟(𝑃𝑃𝑓𝑓𝑠𝑠 − 𝑃𝑃𝑟𝑟)

where 𝐾𝐾𝑟𝑟 is a proportionality constant related to the mobility of the interface.

Comparing Eqs. (5), (12), (13), one obtains

𝑣𝑣 =𝑑𝑑𝑟𝑟𝑑𝑑𝑑𝑑

= 𝐾𝐾𝑟𝑟 𝑃𝑃𝑓𝑓𝑠𝑠 − 𝑃𝑃𝑟𝑟 =2Ω𝑀𝑀𝑀𝑀𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝐾𝐾𝑟𝑟𝛾𝛾𝑠𝑠𝑠𝑠

𝑘𝑘𝑘𝑘1𝑟𝑟𝑓𝑓𝑠𝑠

−1𝑟𝑟

A plot of this equation is shown in Fig. 10.17b for two different average particle sizes.

In addition to demonstrating that small particles shrink and larger ones grow, the figure also shows that smaller grains (𝑟𝑟 < 𝑟𝑟𝑓𝑓𝑠𝑠) disappear much faster than larger grains grow.

(13)

(14)

Furthermore, as time progresses and the average grain size increases, the growth rate for all particles is reduced and eventually goes to zero.

The model can be taken further by making the simplifying assumption that the rate of increase of the average particle size is identical to that of particles that are twice the average size.

In other words, by assuming

𝑑𝑑𝑟𝑟𝑎𝑎𝑎𝑎𝑑𝑑𝑓𝑓

= 𝑑𝑑𝑟𝑟𝑑𝑑𝑓𝑓

at 𝑟𝑟 = 2𝑟𝑟𝑓𝑓𝑠𝑠

Eq. (14) can be integrated to yield the final result

𝑟𝑟𝑓𝑓𝑠𝑠2 − 𝑟𝑟0,𝑓𝑓𝑠𝑠2 =

2𝛾𝛾𝑠𝑠𝑠𝑠Ω𝑀𝑀𝑀𝑀𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝐾𝐾𝑟𝑟𝑘𝑘𝑘𝑘

𝑑𝑑

where 𝑟𝑟0,𝑓𝑓𝑠𝑠 is the average particle size at 𝑑𝑑 = 0.

(15)

Equation (15) predicts a parabolic increase in the average grain size with time.

It also predicts that the coarsening kinetics are enhanced for solids with high vapor/surface interface energy (𝛾𝛾𝑠𝑠𝑠𝑠 ↑) and high vapor pressure (𝑃𝑃𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ↑), both predictions in fair agreement with experimental observations.

For example, it is now well established that covalently bonded solids such as Si3N4, SiC, and Si coarsen, rather than densify, because they have relatively high vapor pressure.

Grain growth

▪ During the final stages of sintering, in addition to the elimination of pores, a general coarsening of the microstructure by grain growth occurs.

During this process, the average grain size increases with time as the smaller grains are consumed by larger grains as shown in Fig. 10.18.

Controlling and understanding the processes that lead to grain growth are important for two reasons.

The first, discussed in greater detail in subsequent chapters, is related to the fact that grain size is a major factor determining many of the electrical, magnetic, optical, and mechanical properties of ceramics.

The second is related to suppressing what is known as abnormal grain growth, which is the process whereby a small number of grains grow very rapidly to sizes that are more than an order of magnitude larger than average in the population (Fig. 10.21b).

In addition to the detrimental effect that the larger grains have on the mechanical properties (see Chap. 11), the walls of these large grains can pull away from porosities, leaving them trapped within them, which in turn limits the possibility of obtaining theoretical densities in reasonable times.

Before one proceeds with the model, it is important to appreciate that the origin of the driving force responsible for coarsening.

Consider the schematic microstructure composed of cylindrical grains of varying curvatures, shown in Fig. 10.19a.

Since in this structure the dihedral equilibrium angle (∅) has to be 120°, it follows that grains with more than six sides will tend to grow, while those with less than six sides will tend to shrink (i.e., the critical value 𝑐𝑐𝑐𝑐 = 360

180−∅).

This occurs by migration of grain boundaries in the direction of the arrows (Fig. 10.19a).

To appreciate the origin of the driving force, consider the atomic-scale schematic of such a boundary (Fig. 10.19b).

At this level, it should be obvious why an atom on the convex of the boundary would rather be on the concave side − it would, on average, be more tightly bound, i.e., have lower potential energy.

Consequently, the atoms will jump from right to left, which means that the grain boundary will move from left to right, as shown in Fig. 10.19a.

Looking at the problem at that level, one can easily see why straight (i.e., no curvature) grain boundary would be stable and would not move.

The final result for grain growth is

𝑑𝑑𝑓𝑓𝑠𝑠2 − 𝑑𝑑𝑓𝑓𝑠𝑠,02 =

4𝑀𝑀𝛾𝛾𝑔𝑔𝑎𝑎Ω𝑀𝑀𝑀𝑀𝛽𝛽

𝑑𝑑

where 𝑑𝑑𝑓𝑓𝑠𝑠,0 is the average grain size at 𝑑𝑑 = 0, and 𝛽𝛽 is a geometric factor that depends on the curvature of the boundary.

For example, for a solid that is made up entirely of straight, noncurved grain boundaries, no grain growth would occur and 𝛽𝛽 would be infinite.

The process just described is sometimes referred to as Ostwald ripening and is characterized by a parabolic dependence of grain size on time.

(16)

Effect of microstructure and grain boundary chemistry on boundary mobility

▪ In deriving Eq. (16), the implicit assumption that the grain boundaries were pore-, inclusion-, and essentially solute-free − a very rare occurrence, indeed − was made, and as such Eq. (16) predicts the so-called intrinsic grain growth kinetics.

Needless to say, the presence of “secondary phases” or solute at the boundaries can have a dramatic effect on their mobility, and from a practical point of view it is usually the mobility of these phases that is rate-limiting.

Extrinsic or solute drag: If the diffusion of the solute segregated at the grain boundaries is slower than the intrinsic grain boundary mobility, it becomes rate-limiting. In other words, if the moving grain boundary must drag the solute along, that tends to slow it down.

Pores: What is true of second phases is also true of pores. Pores cannot enhance boundary mobility: they only leave it unaffected or reduce it.During the final stages of sintering as the pores shrink, the mobility of the boundaries will increase.

As the grains get larger and the pores fewer, the grain mobility increases accordingly.

In some cases, at a combination of grain size and density, the mobility of the grain boundaries becomes large enough that the pores can no longer keep up with them; the boundaries simply move too fast for the pores to follow and consequently unpin themselves.

This region is depicted on the grain size versus density trajectory in the upper right-hang corner of Fig. 10.20a.

If theoretical density is to be achieved, it is important that the grain boundary versus density trajectory not intersect this separation region.

The importance of having the pores near grain boundaries is illustrated in Fig. 10.20b; the migration of the boundary downward has swept and eliminated all the pores in its wake.

Pores that are trapped within the grains will remain there because the diffusion distances between sources and sinks become too large.

There are essentially two strategies that can be employed to prevent pore breakaway, namely, reduce grain boundary mobility and/or enhance pore mobility.

An example of how slowing grain boundary mobility enhances the final density is shown in Fig. 10.6g, where the grain size versus density trajectories for two aluminas, one pure and the other doped with 250 ppmMgO, are compared.

It is obvious from the results that the doped alumina achieves higher density− the reason is believed to be the result of impurity drag on the boundary by the MgO.