Anisotropy in Shrinkage During Sintering

While significant progress in modeling of sintering has been accomplished since the original paper by Frenkel ‘’Viscous flow of crystalline bodies under action of surface tension’’, there are still several issues that remain open. One of them is anisotropy during sintering. In this paper we present some recent developments that improve our understanding of sintering anisotropy based on simulations of a two-dimensional array of particles. A number of possible sources of anisotropy are examined and evaluated.


Introduction
Modeling of sintering in the macroscopic scale has become increasingly popular.Improved accuracy in the predictions demands not only proper characterization of the densification of the material but also constitutive models of increased realism that are capable of predicting dimensional changes during sintering.In addition to densification, the final dimensions of a sintering part depend on gravity, non-uniform green density, temperature gradients, interactions with supports, and anisotropy in sintering induced by prior die compaction.With the exception of anisotropy these effects can be modeled.Current modeling approaches [2] [3], however, rely always on the assumption of isotropic shrinkage despite the fact that experimental evidence for anisotropy in shrinkage has existed for quite some time.An extensive bibliography can be found in [4].For example, anisotropic shrinkage during sintering occurs in compacted samples (e.g., [5]), in tape casting [6] and magnetic materials compacted under a magnetic field [7,8].
It is believed that shrinkage anisotropy is related with the directionality of microstructural features developed during preparation of the green part (pressing, tape casting etc.).Although some attempts have been made to identify the origins of such anisotropy, it is only recently that a systematic work in identification of possible explanations and supporting models.In this paper we present a summary of prior study [4] and the latest understanding of anisotropy provided by the work of the authors over the last five years.The relevant work of Cannon [6] and Olevsky [9,10] will also be commented on during this paper where appropriate.
To explore the role of various sources of the anisotropic behavior, we employ numerical simulations of a 2-D periodic arrangement of particles.In this paper we present first the theoretical framework and its numerical implementation, and then we discuss in detail a number of possible origins of anisotropy, the associated model predictions and experimental evidence where possible.

Theoretical framework
The easiest way to understand the possible origins of anisotropy is to consider a simple model that describes sintering in a two dimensional periodic arrangement of particles, see Fig. 1.Grain boundary and surface diffusion can be considered.For the 2D problem, the evolution of the surface of the pore is given by: where v n is the normal velocity of the pore surface, D S = coefficient of surface diffusion, δ S = surface diffusion width, γ = surface energy, Ω = atomic volume, k = Boltzman's constant, T = absolute temperature, K = curvature, s = length coordinate along the pore surface.Normalizing length by the initial particle radius R, time with R 4 /B, the normalized grain boundary diffusion at the tip of the neck is given by [11]: where x is the normalized neck size,

(
) is the grain boundary/surface diffusivity ratio, ψ is the dihedral angle.The average stress on the grain boundary σ (compressive=positive) is normalized by the pore pressure: , where γ is the surface energy and R the initial radius of the undeformed particle.The dimensionless shrinkage rate is given by: ( ) ( ) ( ) A systematic examination of equation (3) indicates that directional differences in sintering rate are related to directional differences in any one or a combination of differences in neck size, neck tip curvature, diffusivity ratio, applied stress or dihedral angle.
The unit problem is solved in two steps when a prior compaction is studied: (a) deformation of the particle during compaction using finite elements, and (b) sintering by considering surface and grain boundary diffusion in the sense of [11,12], in which the differential equations presented above are solved by a finite-difference method in space and integrated by an Euler-forward algorithm.Accuracy and stability are ensured by a tolerance in the time integration procedure.Stability conditions are more intense when the mesh is finer.Fine mesh is only required to resolve the earlier stage of the evolution of the neck which is assumed to begin from a circle of diameter equal 1/100 of the particle size.Such resolution is not needed in the later stages, and an automated mesh point removal was developed that improved stability without loss of accuracy.
The shrinkage anisotropy can defined as the difference in the sintering shrinkage strains between the transverse, W ε and the vertical direction H ε : The symbols W and H used as subscripts will be used henceforth to denote variables in the transverse and vertical directions.The vertical direction will be associated with the prior compaction or the magnetization direction.∆ε is positive when the transverse shrinkage is larger than the one in the compaction direction.Other possible measures of anisotropy can be the ratio , as well as corresponding definitions on the basis of strain rates.Characterization of anisotropy can also be in the spirit of continuum mechanics sintering models.In that case the viscosity in the vertical direction then the ratio of viscosities in the two directions can be expressed as: and the tensile stresses which would stop shrinkage in both directions is shown below.

(
) ( ) During the preparation of this manuscript the work of Olevsky et al. [10] came to our attention when these authors derive similar expressions for the viscosities and stresses (as in equations ( 6) and ( 7)) for an array of ellipsoidal 2D pores in a tetragonal arrangement.

Results and discussion
The effect of prior compaction on sintering anisotropy Die compaction involves uniaxial straining.Although necks do develop in the transverse direction, their size is smaller that the size of the necks in the axial direction.Such difference in neck size will result in substantially different grain boundary fluxes at the tips of the two necks (equation ( 3)) and different shrinkage strains.To quantify this argument we have simulated the compaction and the subsequent sintering process in a 2-D tetragonal arrangement of particles.One quarter of a particle was modeled using the Mises plasticity model of the commercial finite element program ABAQUS.A typical deformed mesh is shown in Fig. 2(a) and the predicted size of the necks is shown in Fig. 2(b).The neck on contacts along the transverse direction is smaller than that along the compaction direction especially in the early stages of the compaction.The difference of neck sizes increases with the amount of compaction for the range of the simulation.If hardening is present, the difference between the two necks at a given compaction level is reduced since material under more heavily deformed contacts hardens and resists further deformation.Hexagonal arrangement shows similar results.Compaction of a hexagonal arrangement of cylinders was also examined in [13].
The results of the finite element simulations provide the initial conditions for the sintering simulations.Fig. 3. shows the evolution of the pore corresponding to a previously compressed particle (10% compaction strain) and its curvature along the pore surface.Fig. 4 shows the evolution of ∆ε with relative density during sintering after a 5% uniaxial compaction for different values of the diffusivity ratio Γ=∆ γ δ γ /(∆ σ δ σ ).
Initially neck growth is driven by the local curvature of the tip of the necks.∆ε is positive and grows quickly while density increases.As the curvatures at the two necks become of the same order with the curvature on the rest of the pore, neck growth slows down especially if the diffusivity ratio is small.In the last stage the length of the pore surface is small compared to the characteristic length for surface diffusion.Thus material that reaches the tip of the grain boundary is practically immediately deposited uniformly over the surface of the pore.The curvature increases as the pore shrinks and pore closure occurs quickly.The numerical results reflect the predictions of equation ( 3) i.e., a strong relationship between neck size and shrinkage strain.Any difference in neck size is amplified in the form of shrinkage strain difference.Therefore anisotropy should be higher for higher compaction strains, that results in a larger difference in the initial neck size in the two directions (see Fig. 2(b)).This was confirmed by the simulation, see Fig. 5. (a).Also, the shrinkage rate is proportional to diffusivity ratio, therefore higher diffusivity ratios should result in larger anisotropy, which agrees with simulation results, see Fig. 5. (a).

Pore gas pressure and anisotropy
Differences in neck sizes due to prior compaction coupled with entrapped gas in the pores will result in differential stresses on the necks along and normal to the prior compression direction, which in turn can contribute to anisotropic shrinkage.There are various situations that can lead to swelling e.g., vapor produced by oxide reduction, entrapped gas, or solubility differences in liquid phase sintering.Swelling is exacerbated when the green density is very high and closed porosity appears early during sintering, or when the particle size is very small and degassing requires long times.Swelling that involves interstitial gas pressure is difficult to model because an objective estimation of the pressure of the trapped gas in the pores is not straightforward but it can be taken into account under certain assumptions.For example, if the mass of the trapped gas is known and remains constant, then the interstitial gas pressure can be obtained as function of the pore dimensions and the evolution of density.Balance of forces in the axial and transverse direction can be invoked to compute the induced tensile stress 0 < σ on the interparticle necks as function of the pore pressure.Simulation results show ( Fig. 6) that sintering anisotropy expressed as the difference in sintering strain ∆ε=ε transverse -ε axial is reduced with normalized pore gas pressure 3 .. 2 = p .The reduction occurs in later stages, because initially the high neck curvature dominates the grain boundary fluxes.The analysis predicts that densification of both vertical and horizontal necks is delayed by pore pressure (as expected) but the effect is stronger on the necks that formed along the transverse direction, which are smaller.In fact the analysis predicts that for certain level of internal pressure the compact will expand in the transverse direction while it will still contract in the axial direction.These results are plausible but they are only qualitative due to the simplified geometry selected.

Anisotropy in the quality of contacts
Compaction of powder particles does not yield perfect grain boundaries.Possible imperfections include: 1. Particle boundary porosity ("interface pores"), that may result from residual roughness geometry or from damage during unloading.2. As surface oxide films (in metals) are broken and clean surfaces are generated, their residuals remain in the contact area.
In both cases the resulting geometry should affect the kinetics of grain boundary diffusion.Under applied pressure surface roughness is only partially flattened.The result is localized porosity in the contact area that is eliminated by local sintering.Recent work with X-ray tomography [14] has shown that similar type of porosity is present in die compacted materials in contacts normal to the direction of compaction and it was argued that this is the result of damage during the unloading of the compact in the die.Kuroki [15] had made the observation that springback and sintering anisotropy were correlated.Until such interface porosity is closed, grain boundary diffusion is delayed locally.In general such porosity sinters fast and the resulting delay in the development of grain boundary diffusion is very small.Preferential presence of such porosity in certain orientation may induce higher shrinkage rates in that direction.When the overall shrinkage is very large then the role of such porosity would be rather small.For alloys that sinter with no significant volume change (such as a number of ferrous alloys) the shrinkage associated with such localized porosity may be important.Rough calculations [14] have shown that in Distaloy AE the shrinkage anisotropy corresponds numerically to the volume of interface pores.
On the other hand, the presence of broken oxide films can play a significant role in anisotropy.Cleaner surfaces under the contacts imply faster grain boundary diffusion.The level of "cleanness" of the contact depends on compaction.It is possible to argue that the quality of contact along the compaction direction can be different than that along the transverse direction.The higher loads on the contacts along the compaction direction may lead to cleaner contacts.If this is true then the effective grain boundary diffusion coefficient on vertical contacts can be higher than the one on the transverse contacts.Such a scenario has been included in our simulations and the corresponding results are shown in Fig. 7.It can be seen that higher grain boundary diffusion on the vertical contacts can reverse the sense of the differential shrinkage, i.e., resulting in higher shrinkage along the compaction direction.In fact it is possible that the sign of the differential shrinkage ∆ε can change during the process.These results can be rationalized by examining equation ( 3) and taking into account the difference in neck size in the two directions.If perfect contacts are assumed (as in the earlier discussion), then the difference in neck induced by compaction results in higher shrinkage in the transverse direction.If the contacts along the compaction direction are of higher quality due to the breaking of the surface oxide, then it is possible that higher shrinkage occurs along the compaction direction.In this case the larger diffusivity ratio on the vertical contacts compensates for the difference in neck size.As the sintering progresses the difference between the neck sizes is reduced and the difference in the diffusivity ratio dominates the anisotropic response causing higher shrinkage rates along the compaction direction.
These results are in direct agreement with experimental results reported in [16].In that paper, a ferrous powder is compacted immediately upon production using static and vibratory compaction as well as after long-term storage that promoted oxidation.The "fresh" iron powder under static compaction exhibited transverse shrinkage larger than the axial one, while oxidized powder showed the opposite behavior.Moreover green specimens produced under low pressure vibratory compaction exhibited low anisotropy in sintering.These results are consistent with our discussion on the role of axial stress in disturbing the presence of the surface oxide layer and enhancing the grain boundary diffusion in particle boundaries normal to the maximum compressive stress.

Elongated particles
Elongated particles (needles, and more commonly platelets) can become oriented as a result of presintering operations (die filling and compaction, tape casting, etc.).In many publications the argument is made (e.g., [17,18]) that the larger number of necks per unit length in the direction of the long axis the sintering rate is higher along that direction.This is not precise.Consider the geometry in Fig. 1.The overall sintering strains rates are related to the local shrinkage rate (i.e. the neck sizes and local curvatures are identical) the larger shrinkage would occur along the short direction of the particle.The true reason for which ellipsoid particles shrink more in the long axis direction is that a large neck forms quickly in the contact along the short axis because less amount of material is required to "fill-in" the initial neck.Given that the shrinkage rate is proportional to the inverse of the square of the neck size, large necks will shrink slower.The difference becomes significant at larger values of the diffusivity ratio (i.e., when grain boundary diffusion dominates).
Nevertheless it is clear that the presence of oriented elongated particles will result in shrinkage anisotropy.The problem of sintering of 2-D elongated particles was addressed in [6] for a periodic array of ellipses with the same starting contact (neck).Ductile equiaxed particles will also achieve a platelet like morphology due to compaction.This problem was address in [9] by considering a 2D periodic arrangement of oriented ellipsoidal voids as well as a more general case by using Monte Carlo simulation.Both analyses showed that the shrinkage strain along the long axis of the ellipse is larger than the one along the short axis in agreement to experiments.
For many particles morphological texture in the compact implies the presence of crystallographic texture because for example the platelet form of the particles has usually a crystallographic origin (e.g., c-axis normal to the platelet plane).Therefore the discussion that follows on the relation between crystallographic texture and sintering anisotropy is relevant.

Crystallographic texture
Ferrites present a practical example of the presence of strong crystallographic texture compact because they are oriented on purpose during compaction in order to achieve maximum magnetic properties.Recent sintering experiments [7,8] on SrFe 12 O 19 have shown that this material shows significant anisotropy in sintering when the compaction is performed under the action of a magnetic field parallel to the compaction axis.The shrinkage is between 1.5 and 2 time faster in the direction of magnetization than in the transverse direction.A reversal in the relative magnitude of shrinkage in the two directions occurs in the very last stage of sintering but is not enough to affect the total shrinkage, which amounts to a longitudinal shrinkage of 22.7% and a transverse shrinkage of 13%.In the absence of a magnetic field during compaction the compact sinters essentially isotropically [7,8].In this material there is both morphological and crystallographic texture.Quantitative analysis of the two indicates that although morphological and crystallographic texture are somehow correlated, the crystallographic texture is stronger, especially in the initial compact, and the morphological texture becomes comparable only at the end of sintering, after significant grain growth has taken place.Therefore it is the crystallographic texture, which unambiguously contributes to shrinkage anisotropy during sintering.
The presence of crystallographic texture implies that systematic differences exist in contacts along different directions.For ferrites, for example, contacts along the magnetization axis will be primarily basal with low misorientation and low energy while boundaries in the perpendicular direction are between prismatic or other planes and will have a higher energy.The differences in grain boundary structure may also induce differences in grain boundary diffusivity but this effect is much more complex to address and there are not so much data in the literature for ceramic systems.Self-diffusion and diffusion of dopants are usually slower for low misorientation/low energy grain boundaries in metallic systems due to the lower defect concentration [19].Indeed a slower diffusion coefficient is also observed for Cr dopant in MgO tilt grain boundary as the tilt angle is decreased [20].In complex oxide structures, however, diffusion of cations may be enhanced along crystallographic planes that contain them.The example of NaO-11Al 2 O 3 , where the diffusion of Na + is essentially confined to the Na-containing planes perpendicular to the c axis of the hexagonal structure was given in [21], and grain boundary diffusion is then expected to be enhanced for these cations in the low energy grain boundaries, perpendicular to the c axis.In the same paper [21] it is stated that the same anisotropic diffusivity may occur in hexaferrites.Grain boundary segregation may also complicate the problem.If impurities or dopants segregate at low energy boundaries, they can enhance the diffusivity in these boundaries, especially if they may locally form a vitreous or liquid phase.Since we do not have data concerning the relation between crystallography and diffusivity in grain boundaries of strontium hexaferrites, we will first examine the effect of anisotropy in grain boundary energies, assuming a constant diffusivity, to explore whether this effect alone may explain the shrinkage anisotropy in hexaferrites.
Using the same particle packing as before we simulate the sintering of round particles (to exclude the morphology effect) and we chose parameters of the simulation to describe the effect of differences in grain boundary energies due to the crystallographic texture.Grain boundary diffusion is probably rapid in this system and the ratio Γ of grain boundary diffusivity to surface diffusivity was taken ≥ 1 in the simulation.For comparison, in case of alumina, Γ is in the range 10-1000 [6].To facilitate the calculations that become very difficult numerically at Γ>10, we present here only simulations with Γ=1, and Γ=10, with the understanding that higher diffusivity ratios should amplify the anisotropy in accordance to equation (3).Values of dihedral angles for alumina, another ceramic with hexagonal structure, will be used as reference for the simulation.Dihedral angles in sintered alumina were measured in the range 60°-160°, with a median value at 106° [22].The highest value (160°) corresponds the lowest grain boundary energy, i.e. probably boundaries between the basal planes of the hexagonal structure.This value will be used for the dihedral angle of boundaries normal to the vertical direction, which corresponds to the magnetization axis and then to the preferential orientation for the c-axis of the platelets, while the average dihedral angle in the other direction will be taken equal to 90°.Assuming constant surface energy, such values would correspond to a ratio of the average grain boundary energy to the minimum grain boundary energy equal to 4. A ratio of 5 was obtained by adjusting results of grain growth simulations to experiments on BaTiO 3 [23], another hexagonal ceramic.To explore the effect of the dihedral angle difference, we will also present results from a simulation with 160° and 120° at contacts normal and transverse to the magnetization direction.Values in the range 120-160° for the dihedral angles where indeed obtained by AFM measurements on thermally etched strontium hexaferrites surfaces.Stage I, when Kx >> sin(ψ/2), the difference between dihedral angles leads to minor anisotropy build up.Increase of the difference in shrinkage strains in the two directions is practically linear.

−
Stage II, when Kx and sin(ψ/2) are of the same order.In this case anisotropy becomes more pronounced with the sintering along Z-direction to be clearly higher.As shrinkage progresses faster in the necks with the high dihedral angle, the corresponding neck size increases faster than that with the lower dihedral angle.This competing events progressively reduce the rate of increase of ∆ε.− Stage III, when the necks normal to the Z-direction have grown sufficiently larger than the ones in the transverse direction, the anisotropy trend as defined by ∆ε is reversed, because the effect of neck size in equation ( 3) is stronger due to the high exponent.It can be clearly seen that higher difference in dihedral angles leads to higher levels of anisotropy.The transition from each stage to the other is not affected significantly by the difference in dihedral angle.The predicted maximum in ∆ε is about 80% higher for a unit cell with dihedral angles of 160 o and 90 o versus one with 160 o and 120 o .The effect of diffusivity ratio is also illustrated in Fig. .It is important to remember that the level of anisotropy will be amplified by higher (and more realistic) diffusivity ratios, as predicted by equation (3) and shown in Fig. .An order of magnitude increase in Γ, increases the maximum in ∆ε by a factor of more than 2. At the same time the transition from each stage to the next is significantly delayed, resulting in a higher final shrinkage strain difference.This behavior occurs because for higher Γ, surface diffusion is not capable of transporting enough material away from the neck and as a result, the rounding of the pore is delayed.
Anisotropy in grain boundary energy can then induce a significant shrinkage anisotropy, especially for systems with a high anisotropy in grain boundary energy and a high diffusivity ratio.In case of strontium hexaferrites where the shrinkage anisotropy is high while the anisotropy in grain boundaries is rather limited, anisotropy in the diffusivities may probably play an additional role.The small reversal in anisotropy of the shrinkage rate observed in the late stage of the process [8] can be explained by the higher neck size normal to the magnetization direction, and is also consistent with the simulations.

Conclusions
We have shown that anisotropy in sintering can originate from a variety of sources including prior compaction, elongated particles, gas pore pressure, interface porosity and crystallographic texture.While the 2-D models provide an interesting theoretical justification for many mechanisms responsible for anisotropic shrinkage, further work is required towards the development of anisotropic constitutive models that can be used for predictive purposes.

Fig. 1 .
Fig. 1.Tetragonal periodic arrangement of 2D particles.Symmetry reduces the problem to the analysis of a quarter of a particle.Ellipsoidal shapes are shown here as an example.

..
Therefore equation (3) can be used to derive the viscosity: account the magnification of the macroscopic stress Σ into the local average stress on the grain boundary If there are no difference in

Fig. 2 .
Fig. 2. (a) Deformed mesh -particle compressed in a die-compaction mode of strain.Thick line indicates the initial undeformed cylindrical shape.(b) Neck sizes in the compaction and transverse directions as a function of the compaction strain, H ε& .

Fig. 3 .
Fig. 3. (a) Evolution of the pore shape, (b) evolution of the curvature along the pore surface during sintering of a previously compressed particle by 10% .
where p is the pore gas pressure normalized in the same way as the stress.Tensile stresses on the grain boundary hinder densification because they reduce the grain boundary flux at the tip of the neck as shown in equation(3).The asymmetry in neck size in the two directions results in a different level of stress on each neck.

Fig. 6 .
Fig. 6.Evolution of anisotropy in sintering (expressed in terms of the difference in sintering strain in the axial and transverse directions), for different values of the normalized pore pressure (pR/γ S ).

Fig. 7 .
Fig.7.Simulation of the effect of contact quality on anisotropic shrinkage, assuming that the diffusivity ratio in vertical and transverse contacts is different.(prior compaction strain =10%).
on the number of necks.Therefore if b a ∆ = ∆

Fig. shows the
Fig. shows the evolution of ∆ε for Γ=1 and dihedral angles of 160°, and 120° respectively.Three stages can be distinguished, with the corresponding shapes of the pore shown in Fig. : −Stage I, when Kx >> sin(ψ/2), the difference between dihedral angles leads to minor anisotropy build up.Increase of the difference in shrinkage strains in the two directions is practically linear.−StageII, when Kx and sin(ψ/2) are of the same order.In this case anisotropy becomes more pronounced with the sintering along Z-direction to be clearly higher.As shrinkage progresses faster in the necks with the high dihedral angle, the corresponding neck size increases faster than that with the lower dihedral angle.This competing events progressively reduce the rate of increase of ∆ε.−Stage III, when the necks normal to the Z-direction have grown sufficiently larger than the ones in the transverse direction, the anisotropy trend as defined by ∆ε is reversed, because the effect of neck size in equation (3) is stronger due to the high exponent.

Fig. 8 .
Fig.8.Difference in shrinkage strain versus relative density between transverse and magnetization directions for a 2-D arrangement of cylinders.Dihedral angle for neck normal and transverse to magnetization directions is take to be 160 o , and 120 o respectively and diffusivity ratio Γ=1.

Fig. offers a
Fig. offers a comparison of the evolution of anisotropy versus density for two levelsof difference in dihedral angles, and two levels of the diffusivity ratio.It can be clearly seen that higher difference in dihedral angles leads to higher levels of anisotropy.The transition from each stage to the other is not affected significantly by the difference in dihedral angle.The predicted maximum in ∆ε is about 80% higher for a unit cell with dihedral angles of 160 o and 90 o versus one with 160 o and 120 o .The effect of diffusivity ratio is also illustrated in Fig..It is important to remember that the level of anisotropy will be amplified by higher (and more realistic) diffusivity ratios, as predicted by equation (3) and shown in Fig..An order of magnitude increase in Γ, increases the maximum in ∆ε by a factor of more than 2. At the same time the transition from each stage to the next is significantly delayed, resulting in a higher final shrinkage strain difference.This behavior occurs because for higher Γ, surface

Fig. 9 .
Fig. 9. Comparison of anisotropy evolution for two levels of difference in dihedral angle in longitudinal and transverse directions.Thicker line corresponds to 160 o , in the longitudinal and 90 o in the transverse direction, which the thinner one corresponds to 160 o and 120 o respectively.