u tor i a I

1 Grain Growth Stagnation by Inclusions or Pores C.H. Womer and P.M. Hazzledine

The mechanism of grain growth stagna­tion by second-phase particles or pores, also known as Zener pinning, has been analyzed through a variety of theoretical approaches. These range from Zener's original analysis through recent advances made by computer modeling. The Zener phenomenon is rou­tinely utilized in metals and ceramics processing to prevent grain growth and to take full advantage of the small grain size in the mechanical properties of finished poly­crystalline aggregates. On the other hand, in the processing of electrical steels the final product must have a coarse grain size, and an appropriate amount of precipitates helps to achieve this goa/.

INTRODUCTION

The framework for the study of the influence of inclusions or pores on the inhibition of grain growth in polycrys­talline materials was put in its con­temporary form by C. Zener, as quoted by Smith 1 in his famous paper on micro­structure. Consequently, in the scientific literature, this phenomenon is known as Zener pinning or Zener drag, and it is recognized as one Zener's several im­portant contributions to materials sci­ence.2,3 Of course, there are earlier ref­erences to this phenomenon in scientific writings going back as far as 1917, as noted by Hillert,4 but it was not until Smith's publication thatitbecamea well­posed problem, both from the theoretical and experimental sides.

The term "pinning," as used to in­dicate the action of a point defect (in a macroscopic sense) anchoring an ex­tended one, is not restricted to the Zener phenomenon. In fact, other types of pin­ning appear in materials science and solid-state physics, such as the pinning of dislocations by precipitates or other one-dimensional pinning centers5 (an operative mechanism in the hardening of crystalline materials); the pinning of domains walls by nonmagnetic precipi­tates or pores6,7 (a phenomenon respon­sible for the hardening of magnetic materials); and the pinning of flux lines in superconducting materials (a well­known fact that recently has received considerable attention due to the interest in high-temperature superconducting ceramics). Flux pinning seems to be the mechanism controlling the current den­sity capacity in these high-temperature superconductors.8,9

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Textbook writers also have discussed the Zener pinning effect, usually giving an account of Zener's original approach, as revealed by a rapid scan of the schol­arly references. IO-!3 In this way, the phe­nomenon has found a place in the "ma­terials culture" of materials scientists and engineers.

Following any of the above-mentioned textbooks, let us pose the problem of the Zener drag phenomenon. Consider a polycrystalline material that undergoes recrystallization. After the initial peri­ods of recovery and recrystallization, the grain-growth period follows. During grain growth, if precipitates (of the ap­propriate size, number, distribution, etc.) are present, it is possible that they can inhibit the growth of the whole set of grains, resulting in an intermediate or finished product with a small grain size. Of course, the same phenomenon can occur in an amorphous material under­going recrystallization and grain growth. A qualitative understanding of the situ­ation can be obtained if we consider Figure 1. The growing boundary is pinned by the array of inclusions, which produce a restraining stress on the grain boundary and thus inhibit its growth. Figure 2 shows this phenomenon in an Fe-3Si alloy; the boundary is pinned by a MnS precipitate.14

Zener's original analYSis considered that the boundary has a surface tension

•

• • growing direction

• •

• •

Figure 1 . A curved boundary migrates across a distribution of pinning centers.

(or energy per unit area) denoted by y, and so the motion of the deformed boundary (Figure 3) is restrained by a force sometimes called Zener's force (F):

F = 2ltry sinS cosS (1)

where r is the radius of the precipitate, and S is the angle shown in the figure. Although not explicitly stated, Zener assumed that the precipitate is spherical and incoherent with the matrix grains. Using Equation 1, he estimated the force per particle as the maximum F value, namely ltry. After that, he calculated the pressure due to the whole set of second­phase particles by multiplying it by the number of particles per unit area of grain boundaries. This yields the so-called Zener pressure (P z):

Pz(OJ= 3fy/(4r) (2)

where f is the volume fraction of the precipitates. A further development is to relate this expression to the final grain size. In this "back of the envelope" calcu­lation, it is possible to assume that the growing force is yIp, where p is the net radius of curvature of the growing boundary. In this way, in the original paper, the estimate for p is

p=4r/(3f) (3)

(Hillert4 notes that the numerical factor 4/3 was misprinted in Smith's original paper as 3/4.)

These results raise several questions about Zener's hypothesis, and this ac­counts for many improved treatments appearing in the literature since that time. However, the essentials of Zener's rea­soning are present in the actual theoreti­cal treatments. Basically, they try first to establish the force per particle; second, they tackle the statistical problem of ob­taining a net or average expression for the Zener restraining pressure. Last but not least, there is the question of the relationship between the grain's radius of curvature and the grain size, which is a nontrivial question. The simple as­sumption that p is approXimately equal to the grain size allows Equation 3 to be used to estimate the final grain size. This last approach is widely used in the met­allurgical and ceramics literature.

A last comment concerning this semi­nal contribution to the understanding of this phenomenon is the statement that we are dealing (at least in the incoher-

JOM • September 1992

ence case) with a geometrical approach as opposed to a physico-chemical treat­ment. In such an analysis, pores act in the same way as incoherent precipitates.

ANALYTICAL THEORIES

The development of our under­standing of this phenomenon, mainly in a theoretical context, has been re­viewed.IS-I? However, it is interesting to reanalyze the theoretical approaches.

Let us first consider the problem of a spherical incoherent precipitate de­forming the grain boundary in the man­ner shown in Figure 3.

Gladman18,19 assumes that the shape of the deformed boundary is a hyperbo­loid of revolution; hence, he calculates the increase in area due to this deforma­tion. This increase, times the surface energy, gives the energy increase (per unit area); its derivative is the pinning pressure. In the same papers, Gladman proposes a mechanism for normal grain growth, so, in fact, he analyzes both problems simultaneously. The assump­tion of a shape for the deformed bound­ary is essential to the conclusions of the model, and it seems that there are no reasons to assume that it is a hyper­boloid, Further, Gladman assumes that the growing boundary is flat, in conflict with the idea of curvature-driven growth, In Reference 20, a discussion on the shape of a deformed boundary can be found. In spite of the above-men­tioned objections, Gladman's model points in the correct direction: a treat­ment of the interaction energy is, poten­tially, an alternative to the direct force approach.

An alternative route to the pinning problem was proposed by Hellman and Hillert,21 with the so-called "dimple modeL" This approach follows the semi­nal work of Hillert on grain growth,22 In essence, they proposed that the deformed boundary adopts the shape of a catenoid of revolution, and by applying the ap­propriate boundary conditions (namely, that the deformed boundary must be perpendicular to the inclusion and that it meets smoothly with the growing grain with radius of curvature p), they were able to characterize this surface properly, In this way, it is possible to obtain the relationship between the pinning force and the interaction distance between the boundary and the inclusion. The propo­sition to establish the shape of the de­formed boundary as a catenoid has been independently deduced by Hazzledine etal .23 and by Neset aL15Theshapes found by these authors, although written in an apparently different manner, are equivalent, as shown in Reference 20. The essential result of Hellman and Hillert's treatment is to introduce a cor­rection factor (~) to the original Zener expression, which depends on the value of the pi r parameter. Thus, it is possible

1992 September • JOM

•

Figure 2. Grain boundaries ~inned in an Fe-3Si alloy.

to write, for the Zener pressure, the fol­lowing equation:

Pz{1} = ~13fy/ (4r) J (4)

The ~ parameter ranges between 0.4 and 2 for values ofp / rbetween 1 and 105(Le., it shows a very weak smooth depen­dence on p/r). Also, in this model, it is suggested on theoretical grounds that p = 6Ro (where 2Ra is the grain size of the matrix grains) . In this way, the Zener pinning problem is solved. A contro­versy arose on the limits of the interac­tion between the particle and the bound­ary,20.24 but it was promptly resolved.25

A further development of the same model has been published,26 giving a physical argument for the establishment of the shape of the deformed boundary.

A minimum energy principle would give a condition for the shape in the form of an Euler-Lagrange variational equation, the solution of which gives the catenoid of revolution, confirming Hellman and Hillert's findings . Further, it was shown that, from the calculated area increase, it is possible to obtain Zener force (Equa­tion 1). The ~-correction factor was cal­culated as an average over the 8-angles and over the distances, using the above­mentioned25 determination for the point of detachment between the particle and the boundary. In the calculation of the Zener pressure, a hypothesis proposed by Louat2?,28 was used. Louat noted that particles lie both behind and ahead of the moving boundary (positive and nega­tive x in Figure 3) and these two sets of

x

Figure 3. The dimple model. A spherical incoherent precipitate disturbs a growing boundary,

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2.0

1.0 r--~:"""'''''---~:'''------.---I

0.0 L-_---'-__ --'-_---"'--_-'-_----.l 1 3

log (P/r)

Figure 4. The correction factor (~) for Zener pinning pressure in Equation 4. (a) Zener.' (b) Hellman and Hillert.2' (c) Hazzledine et al.23

(d) Louat.28 (e and f) Worner and Cabo.25

particles exert opposite forces on the boundary. To a large extent, they cancel, but there are more particles dragging the boundary than there are pulling it forward. This is because dragging par­ticles remain attached to the boundary when their centers are more than r from the boundary, whereas particles ahead of the boundary do not interact until their centers lie within r of the boundary. By canceling the forward and backward forces, and considering the drag from those particles with centers more than r behind the boundary, the Zener pres­sure may be calculated.

The results for the correction factor~, following the different reviewed treat-

A recent paper by Hazzledine53 compared the direct hardening caused by particle-dislocation interaction with the indirect hardening caused by grain boUndary­precipitate interaction. He eoncluded that the indirect (Hall-Petch) hardening at low temperatures and low strains is alWays dominant. The Zener pinning effect is an appropriate means to achieve a.small grain size and, hence, a high yield stress. For this and other reasons, the most desirable materials propenies at low tempera­tures are often achieved with Zener -pinned grains. At high temperatures, on the other hand, m~terials with smaH pinned grains may deform by a diffusive creep mechanism and show superplasticity. This is another useful property conferred by Zener pinning.

What follows is a scan of the literature, highHghting divetsemateriaisapplicationsoftheZenerphenomenon. This review is far from complete, but It provides a falr picture of the present situation on this subject. Inevita­bly f there is some ai'bitrariness in the order.of presenta­tion 01 the topics.

In steels, Gladmanta19 studied the influence of ni· trides and carbides on grain growth in austenite, and Hellman and Hilieft?' worked on. ferrite containing ce­mentite. Anand and Gur1an~descrjbedthegraln growth behavior in quenched and tempered steels containing cementife. There are recent references to this phenom­enon in papers by Ali BepariM (aluminum nitrides and vanadium carbides, nitrides, or carbonillides in austen­~e), L'Ecuyer and L'Esperancess (modeling recrystalli­zation in a molybdenum-bearing high-strength, low­alloy steel), Wan!f6 (t~anium nitride in. austenite), Palma et aL 57 (the sintering behavior of high-speed steels, carbides in austenite), and Sinha et al.5& (carbides and! or nillides in maraging steel). Also, Alkselsen et al.51

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ments, are displayed in Figure 4. The dependence of the Zener pinning

pressure on the volume fraction of pre­cipitates-a question posed as a result of Monte Carlo-type simulation of the phe­nomenon, as will be discussed in the next section of this paper-has been the subject of further study in the frame­work of the dimple model. From Equa­tion 3, and using the reasonable hypoth­esis that the radius of curvature is pro­portional to the grain size, it follows that the final stagnated grain size has a 1/ f dependence, in agreement with Zener's original equation. However, Hillert29 shows that the f-dependence must be £-'l.93. Also, W6rner30 and Hunderi et al.16

performed the same calculation, obtain­ing-by different routes-the same de­pendence, with only slightly different numerical coefficients (Le., £-'l.B7).

The question of pinning by coherent precipitates has been treated in an au­thoritative manner by Ashby et apt They proposed the basic equations that must replace Equation I, as it is necessary to consider now three grain-boundary sur­face tensions, so losing the boundary condition of the perpendicularity be­tween the boundary and the inclusion. In this sense, we are now treating a phys­ical rather than a purely geometrical problem, and it is possible to say that the boundary "wets" the precipitate, even envelopes it. Ashby et al. assumed that the growing boundary is macroscopi­cally plane, and they determined the

APPLICATIONS analyzed the grain growth in the heat-affected zone of Iow-carbon steels.

Work on thin films could give a clue for determining the f-dependence of the final stagnated grain sizEt, but it scarcely appears in the literature, with the notable exception of the papers of Thompson60 and Longworth and Thompson.61 In the latter paper, the authors noted that in aluminum films for electronic applications, it is necessary to obtain a coarse grain size in order to decrease electromigration. This is one of the cases in which a small grain size is not tne goal in the thermomechanical treabnent of the materials.

The other materials requiring a eoarse-grained str\Jc­ture are siDeon-iron alloys (electrical steels). It seems that the necessary condition. to achieve this goal is to obtain first a pinned structure of primary recrystallized gralns, Work in this field has been published recently by Yashiki and KanekD62-and by Hou et al. 63,S( The work of AbbruzzeseS( on computer modeling aimed to repro­duce the Fe-3Si sheet structure,

There are numerous papers on Zener pinning ap­plied to nonferrous materials. The seminal paper of Ashby et al.3' presented empiri~1 evidence for AJ-Cu and AJ-CO alloys. Tweed eral. 65.66 studied aluminum with alumina particfes. Chan and HumphreysB7 experimented with aluminum-silicon alloys. Randle and RalpflE8 worked with a Nimonic alloy. Raman,£9 by using the pinning of boundaries by pores, estimated the driving force for boundary migration in AI-Mg. Other studied systems inclOded Cu-Si02,1~ ~copper-based Shape-memory alIoYS,7! copper-ooron systems,12' and two-phase tita· nium alloys.13,7. Patterson et al.?5.76 propose to deter· minethedegreeofpioning using a slereological method. Recent papers on nickel sinteringp tensile properties of

shape of the deformed boundary by a numerical procedure, giving a graphical algorithm to obtain the Zener pressure. The approach of Rios32 is more drastic: he assumes that the preCipitates are com­pletely wetted by the grain boundary, so increasing the energy of the system-by creating new surfaces around each in­clusion-as the boundary sweeps through the material. From this energy increase he calculated the Zener pres­sure. In these two cases the estimates for the pressure do not differ significantly from Zener's original estimate.

In an attempt to take into account the whole distribution of sizes of the pre­cipitates, Haroun33 modified the treat­ment of the problem, showing good agreement with his experimental results in ceramics. In a different approach, Hazzledine et a1.16•23 calculated the Zener pressure using the analogy between this problem and the pinning of dislocations, as treated by Nabarr05 in the theory of solution hardening. Their findings give expressions of the same order of magni­tude as the previously cited results.

The effect of the shape of the pre­cipitates seems to be an important fac­tor. It is intuitive that if we have, say, a needle-shaped precipitate, the pinning action would be different if the bound­ary meets it frontally or laterally. Ryum et al.34 did careful calculations with an ellipSOidal precipitate. They point out that, although nonspherical precipitates can have a random distribution in the

tungslen·rhenium-thoria7' alioy; the effectof silican on precipitation in a Cu-Ni-Snalioy ,79,eD the reqystallization of AI-U,ll· and the slabilityotnanocrystalline ii(J-f'82all use the Zener drag phenomenon in their Interpretation .. Superplastic alloys also appear 10 this fjst.~r

The ceramic materials f~eratute also abounds with references to the Zener pinning effect. In fact, ceramic materials are usually made by sintering, so it is very common to have a pore distribution Inside the samples. Then, when a second phase is added, it is necessary 10 be careful in the analysis of the results, as it is nol easy to separate the effect of the pores Rom the effect of the' second-phase precipitates. The work of Olgaard and Evans· deserves special mention. They studied the pinning of boundaries in calCite by alumina particles and found an j-" ~ dependence lor the final grain size, This gave experimental support to the simulation resuHs discussed above.

Also In ceramics, theworkof Ikegarn~on MQOi:Ioped Alp$ and ZnO and Yoshizawa and Sakuma~ on zr02'

YP3 alloys must be noted. Langei1 mentioned theim· portance of Zener pinning in his review on powder prpeessing, Other materials appearing in the literature are Sb~03-doped Zno,at!) Ti02'~ zirconia·alumfna,9S zinc oxide-antimony oxide ,96 and Alp ,-Zr02 at high lemperatures.97

The Zener pinning phenomenon appears also In other materials, 8Pllrt from the traditional metals and ceramics. So, the geophysical characteriStics of marbles have been the subject of a paper by Holnessetal.,SIl and the behavior of Ice has been studied by Levi and Ceppi .9! To show the beautiful power of the Zener insight, the work of Hofstadter and Murray'OO on the Martian polar layered deposits must be m!)ntioned.

JOM • September 1992

matrix material, giving an average iso­tropic result, the effect is important when precipitates are arranged in a nonran­dom manner, as is the case in a rolled alloy. According to their calculations, the drag pressure in the transverse di­rection is two to four times greater than in the rolling direction.

Ringer et aJ.35 analyzed both theo­retically and experimentally the in­teraction of a growing boundary with a cubic-shaped precipitate, establishing that in this case, the enhancement of the drag is greater that in the ellipsoidal case, and even more pronounced if the precipitate is coherent with the matrix. In a recent paper, Li and EasterlingJ6 cal­culated the pinning action of an ellipsoid­al precipitate with any orientation and degree of coherence with the matrix.

As a last point in this account of the analytical theories of Zener pinning, it should be noted that Gore et al.37 and Worner and Olguin38 studied the effect of temperature on boundary drag. Al­though using different approaches, they stated that it is possible to obtain thermal detachment of the boundaries from the precipitates, if proper conditions of the involved parameters are achieved. This approach establishes that the depinning isa thermally activated process. This last insight may hint at the mechanism for starting the secondary recrystallization process (also called abnormal grain growth) usually associated with the pres­ence of second-phase particles inhibit­ing the growth of matrix grains. Tex­tured electrical steel is a typical example. As suggested recently,38.39 thermal de­pinning favors abnormal versus normal growth, and in this way it can start the secondary recrystallization process.

SIMULATION METHODS

Two simulation approaches are used in the literature on Zener pinning. The first one is based on the well-known analogy between the behavior of soap films and grain boundaries. Films mi­grate under the action of a pressure dif­ference. Their dynamiCS are ruled by a Laplace equation (for a review on soap films, see Isenberg'°), and they can in­teract with "precipitates" (more prop­erly, with a diversity of boundary condi­tions). Smith's original paperl used this analogy between grain boundaries and soap films in his study of microstructures, and Ashby et apl used this analogy to confirm their work on incoherent boundaries. The second simulation is modeling by computer the behavior of the material, usually using a Monte Carlo-type method (Le., simulating the real situation through an appropriate algorithm). Given the growing availabil­ity of computational capacity, its lower cost, and the increase in memory capac­ity, it is likely that more and more work will be done using this technique.

1992 September • JOM

Soap Film Simulation

Soap film experiments are ideally suited to the first of the three stages in the Zener calculation: measuring the force between a boundary and a single particle. The particle may have any shape and may be in any orientation relative to the boundary plane. Wold and Chal­mers41 were the first to measure the force between a boundary and a variety of disk- and needle-shaped particles by this technique. Ashby et aJ.31 used this anal­ogy to confirm their work on incoherent boundaries. More recently, Ringer et a1. 42 used the technique to study the force from ellipsoidal, cubic, or disk-shaped particles. The utility of this technique for tackling the problem of complex pre­cipitate shapes is evident. However, it should be noted that the soap film simu­lations consider a boundary that is pla­nar when not in contact with the particle; neither takes into account the macro­scopic curvature of a migrating bound­ary. No soap film simulation has been reported in which the second stage of the Zener calcula tion is tackled, although in principle it should be possible to mea­sure the force on a single boundary from a number of dispersed particles.

Computer Simulation Methods

In a series of papers on the computer modeling of grain growth, Srolovitz et al.43 analyzed the problem of particle­inhibited growth. They found that, in a two-dimensional system, the final stag­nated grain size depended on £-112. Haz­zledine and Oldershaw44 confirmed this result exactly. The result strongly differs from Zener's classical result (see Equa­tion 3), which predicts an £-1 dependence, a form not greatly modified by subse­quent theoretical treatments and unaf­fected by changes in dimensionality. Far from a theoretical curiosity, this fact implies that the pinning effect is much <;tronger (especially at low volume frac­tions) than the Zener model predicts. A possible explanation44 is that Friedel, not Zener, statistics apply to this problem, in which case an £-1/2 dependence of the grain size is expected in two dimen­sions. Subsequent computer calculations by Doherty et a1.,45.46 Anderson et a1.,47 and Hassold et aJ.48 confirmed the basic picture; by extending the simulation to three dimensions, they found an £-1/3 dependence of the stagnant grain size. Hazzledine and Oldershaw« found the same dependence at high volume frac­tions (f > 1 %), but at low volume frac­tions (f < 1 %) they considered the Zener or Friedel models, which agree in three dimensions and give an £-1 dependence, to be more reliable.

The situation may be summarized as follows. In two dimensions, both the computer simulations and the Friedel model give an £-1/2 dependence of the

stagnant grain size, whereas the Zener model gives an £-1 dependence. In three dimensions, both the Friedel and Zener models give an £-1 dependence, whereas the computer simulations give an £-1/3 dependence. The physical picture that comes out of the computer simulations is that grains mold themselves to the particles, becoming pinned by a critical number of particles. In this case, the stagnant grain size is proportional to £-1/n, where n is the dimenSionality of the system. This concept is quite different from Zener's concept, which is that grains are pinned by a constant number of par­ticles per unit length of grain boundary in two dimensions or per unit area of grain boundary in three dimensions. In both cases, the density of pinning points and the Zener pressure are proportional to f. When balanced against a driving force of approximately y/R, the result­ing stagnant grain size is proportional to £-1. A slight variation on this argument gives an £-112 dependence, as found ex­perimentally by Anand and Gurland49: the density of particles on a grain bound­ary is -3f/r2 and the total number of particles in contact with one grain is then proportional to f(R/r).2 If this total number is constant, then R is propor­tional to £-1/2.

The basic unanswered question in this controversy is whether the boundaries adopt a pre-established pattern follow­ing the anchoring points of the material, or whether they sweep the material "freely" until anchored by some pinning particles. As a matter of fact, these two views may be valid for different values of the volume fraction, as proposed by Hazzledine and Oldershaw.44 As for re­sults obtained by computer simulations, some of the conclusions can be doubtful, and it is necessary to have independent algorithms confirm the findings. The recent results on grain growth simulation published by Mulheran and Harding50

corroborate this apprehension. There are other simulation papers that

do not focus on the f-question. Abruz­zese51 used a mixture of deterministic and probabilistic methods to simulate stagnated grain growth with an empha­sis on predicting structures in silicon­iron alloys. Ceppi and Nasell052 simu­lated two-dimensional grain growth, including pinning, and obtained the ki­netics of the phenomenon.

CONCLUSION

Zener's original estimate of the stag­nant grain size in three dimensions re­mains valid, but the correction factor ~ in Equation 4 may vary by at least a factor of five in different circumstances. The volume fraction dependence of the stagnant grain size is a controversial question that requires more work before it is resolved. This is the crucial problem from an academic point of view. The

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scientific description of Zener pinning gives ample room for improvement, and it has direct applications to industrial practice in the processing of thin films, structural ceramics, and superplastic al­loys, both metallic and ceramic.

ACKNOWLEDGEMENTS C.H. W. acknowledges the partial finan­

cial support of the Programa de Tecnologia de Materiales (Organizacion de Estados Americanos) and the Fondo National para el Desarrollo de la Ciencia y la Tecnologia (Chile),grant92/0717.P.M.H.acknow/edges financial support from U.S. Air Force con­tract no. F 33615-39-C-5604.

References 1. C Zener as quoted by CS. Smith, "Grains, Phases and Interfaces: An Interpretation of Microstructure," Trans. AlME, 175 (1948), p. 15. 2. M. Hillerl, "Impact of Clarence Zener upon Metallurgy," J. Appl. Phys., 60 (1986), p. 1868. 3. F. Seitz, "On the Occasion of the SOth Birlhday Celebration for Clarence Zener," /. Appl. Phys., 60 (1986), p. 1865. 4. M. Hillerl, "Zener's Pinning Effect-Role of Dispersed Particles on Grain Size Control," Proc. THERMEC -88 (Tokyo: ISIl, 1988), p. 30. 5. F.R.N. Nabarro, "The Theory of Solution Hardening," Phil. Mag., 35 (1977), p. 613. 6. P. Gaunt, "Ferromagnetic Domain Wall Pinning by a Random Array oflnhomogeneities," Phil. Mag" 48 (983), p. 261. 7. CH. Womer and J.E. Valdes, "On the Pinning of Domain Walls in Low Magnetization Materials," J. Appl. Phys .• 63 (988), p. 4324. 8. P. Haasen,. "Mechanical, Magnetic and Superconductor Hardening by Precipitates," Mater. Sci. Eng., 9 (972), p. 191. 9. S. Jin, "Processing Techniques for Bulk High-1', Supercon­ductors," JOM, 43 (3) (J99]), p. 7. 10. R.E. Reed-Hill, Physical Metallurgy Principles, 2nd ed. (Monterey, CA: Brooks-Cole Engineering Division, 1973), p. 311. 11. P.G. Shewmon, Transformations in Metals (New York: McGraw-Hill, 1969), p. 122. 12. P. Haasen, Physical Metallurgy (New York: Van Nostrand, 1978), p. 362. 13. D.A. Porler and K.E. Easterling, Phase Transformations in Metals and Alloys (New York: Van Nostrand, 1983), p. 139. 14. A. Sarce, A. Cabo, and CH. Womer, "The Grain-Bound­ary Precipitate Interaction and Grain Growth Inhibition in Silicon-Iron Alloys" (Paper presented at the IX Simposio Latinoamericano de Fisica del Estado SOlido, 1985). 15. E. Nes, N. Ryum, and O. Hunderi, "On the Zener Drag," Acta Met., 33 (1985), p. 11. 16. 0. Hunderi, E. Nes, and N. Ryum, "On the Zener Drag­Addendum," Acta Met., 37 (989), p. 129. 17. P.M. Hazzledine, "Grain Boundary Pinning in Two­Phase Materials," Czech. J. Phys., B38 (1988), p. 431. 18. T. Gladman, "On the Theory of the Effect of Precipitate Particles on Grain Growth in Metals," Proc. Royal Soc" 294A (966), p. 298. 19. T. Gladman and F.B. Pickering, "Grain Coarsening of Austenite," f. lTOn Steel Inst., 205 (1967), p. 653. 20. CH. Worner and A. Cabo, "On the Shape of a Grain Boundary Pinned by a Spherical Particle," Scripta Met., 18 (1984), p. 865. 21. P. Hellman and M. Hillert, "On the Effect of Second­Phase Particles on Grain Growth, Scan. /. Metall" 4 (1975), p. 211. 22. M. Hillerl, "On the Theory of Normal and Abnormal Grain Growth," Acta Met., 13 (1965), p. 227. 23. P.M. Hazzledine, P.B. Hirsh, and N. Louat, "Migration of a Grain Boundary through a Dispersion of Particles," First Riso Int. Symp. (Roskilde, Denmark: Riso National Labora­tory, 19SO), p. 159. 24. M. Hiller!, "On the Estimation of the Zener Drag on Grain Boundaries," Scripta Met., 18 (1984), p. 1431. 25. CH. Womer, A. Cabo, and M. Hillerl, "On the Limit for Particle Attachment in Zener Drag," Scripta Met ., 20 (1986), p. 829. 26. CH. Womer and A. Cabo, "On the Grain Growth Inhibi­tion by Second Phase Particles," Acta Met., 35 (1987), p. 2S01. 27. N. Louat, "The Resistance to Normal Grain Growth from a Dispersion of Spherical Particles," Acta Met., 30 (1982), p. 1291. 28. N. Louat, 'The Inhibition of Grain-Boundary Motion by a Dispersion of Particles," Phil. Mag., 47 A (]983), p. 903. 29. M. Hillerl, "Inhibition of Grain Growth by Second-Phase Particles," Acta Met" 36 (1988), p. 3177. 30. CH. Worner, "Some Remarks on the Zener Drag," Scripta Met., 23 (1989), p. 1909. 31. F. Ashby, J. Harper, and J. Lewis, "Interaction of Crystal Boundaries with Second-Phase Particles," Trans. AIME, 245 (]969), p. 413. 32. P.R. Rios, "A Theory for Grain Boundary Pinning by Particles," Acta Met., 35 (]987), p. 2805. 33. N.A. Haroun, "Theory of Inclusion Controlled Grain Growth," f. Mater. Sci., 15 (19SO), p. 2816. 34. N. Ryum, O. Hunderi, and E. Nes, "On Grain Boundary

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Drag from Second Phase Particles," Scripta Met., 17 (1983), p. 1281. 35. S.P. Ringer, W.B. Li, and KE .Easterling. "On the Interac­tion and Pinning of Grain Boundaries by Cubic Shaped Precipitate Particles," Acla Met., 37 (1989), p. 831. 36. W-B. Li and KE. Easterling, 'The Influence of Particle Shape on Zener Drag," Acta Met., 38 (1990), p. 1045. 37. M.J. Gore et ai., "Thermally Activated Grain Boundary Unpinning," Acta Met., 37 (1989), p. 2849. 38. CH. Womer and A. Olguin, "On the Influence of the Temperature in Zener Pinning," Proc. Int. Con!. on Grain Growth in Polycrystalline Materiais (Zurich, Switzerland: Trans Tech,l99I). 39. CH. Womer, S. Romero, and P.M. Hazzledine, "Exten­sion of Gladman's Model for Abnormal Grain Growth," f. Mater. Res., 6 (1991), p. 1773. 40. C Isenberg, The Science of Soap Films and Soap Bubbles (Evon, U.K: Tieto, 1978). 41. K.G. Wold and F.M. Chalmers, "Grain-Size Control by Dispersed Phases," J. Aust. Inst. Metals, 13 (1968), p. 79. 42. S.P. Ringer, R.P. Kusiak, and KE. Easterling,"Liquid Film Simulation of Zener Grain Boundary Pinning by Second Phase Particles," Mater. Sci. Tech., 7 (199]), p. 193. 43. D.J. Srolovitz et ai., "Computer Simulations on Grain Growth: 1II1nfluence of a Particle Dispersion," Acta Met., 32 (1984), p. 783. 44. P.M. Hazzledine and R.D.J. Oldershaw, "Computer Simu­lation of Zener Pinning," Phil. Mag., A61 (1990), p. 579. 45. R.D. Dohertyetai., "On the Volume Fraction Dependenee of Particle Limited Grain Growth:' Scripta Met., 21 (1987), p. 675. 46. R.D. Doherty et al., "Computer Modelling of Particle Limited Grain Growth and its Experimental Verification," Proc. 10 Riso Int. Symp. (Roskilde, Denmark: Riso National Laboratory, 1989), p. 31. 47. M.P. Anderson et ai., "Inhibition of Grain Growth by Second Phase Particles: Three Dimensional Montecarlo Com­puter Simulations," Scripta Met., 23 (1989), p. 753. 48. GN. Hassold, E.A. Holm, and D.J. Srolovltz, "Effects of Particle Size on Inhibited Grain Growth," Scripta Met., 24 (1990), p. 101. 49. L. Anand and J. Gurland, "The Relationship between the Size of Cementite Particles and theSubgrainSizein Quenched and Tempered Steels," Met. Trans., 6A (1975), p. 928. 50. P. Mulheran and J.H. Harding, "A Simple Statistical Model for Grain Growth in Materials," Acta Met., 39 099]), p.2251. 51. G. Abruzzese, "Computer Simulated Grain GrowthStag­nation," Acla Mel .• 33 (1985), p. 1329. 52. E.A. Ceppi and O.B. Nasello, "Computer Simulation of Grain Growth," Alloy Theory and Phase Equilibria, ed. D. Farkas and F. Dyment (Materials Park, OH: ASM, 1986), p. 119. 53. P.M. Hazzledine, "Direct versus Indirect Dispersion Hardening," Scripta Met. Mater., 26 (1992), p. 57. 54. M.M. All Bepari, "Effects of Second-Phase Particles on Coarsening of Austenite in 0.15 Pct Carbon Steels," Met. Trans., 20A (989), p. 13. 55. D.J. L'Ecuyer and G. L'Esperanee, "Precipitation Interac­tions with Dynamic Recrystallization of an HSLA Steel," Acta Met" 37 (1989), p. 1023. 56. S.C Wang, 'The Effect of Titanium and Nitrogen Con­tents on the Austenite Grain Coarsening Temperature," ]. Mater. Sci., 24 (1989), p. 105. 57. R.H. Palma, V. Martinez, and J.J. Urcola, "Sintering Behavior of T42 Water Atomised High Speed Steel Powder under Vacuum and Industrial Almospheres with Free Car­bon Addition," Powder Mel., 32 (I989), p. 291. 58. P.P. Sinha et ai., "Grain Growth in 18Nl 1800 MPa Maraging Steel," J. Maler. Sci .• 26 (]99l), p. 4155. 59. O.M. Akselsen et ai., "AZ Grain Growth Mechanism in Welding of Low Carbon Microalloyed Steels," Acta Met., 34 (]986), p. 1807. 60. CV. Thompson, "Grain Growth in Thin Films," Ann. Rev. Mater. Sci., 20 (1990), p. 245. 61. H.P. Longworth and CV. Thompson, "Abnormal Grain Growth in Aluminum Alloy Thin Films," f. Appl. Phys., 69 (199]), p. 3929. 62. H. Yashiki and T. Kaneko, "Effects of Mn and 5 on the Grain Growth and Texture in Cold Rolled 0.5% Si Steel," lSI/ Int., 30 (1990), p. 325. 63. C-K. Hou, CT. Hou, and s. Lee, "Effect of Residual Aluminium on the Microstructure and Magnetic Properties of Low Carbon Electrical Steels," Mater. Sci. Eng" AI25 (1990), p. 241. 64. CK. Hou, CT. Hou, and S. Lee, "Effect of Aluminum on the Magnetic Properties of Lamination Steels," IEEE Trans. Mag., 27 (1991), p. 4305. 65. CJ. Tweed, N. Hansen, and B. Ralph, "Grain Growth in Samples of Aluminum Containing Alumina Particles," Met. Trans., 14A (1983), p. 2235. 66. CJ. Tweed, B. Ralph, and N. Hansen, "The Pinning by Particles of Low and High Angle Grain Boundaries During Grain Growth," Acta Met., 32 (1984), p. 1407. 67. H.M. Chan and F.J. Humphreys, 'The Recrystallization of Aluminium-Silicon Alloys Containing a Bimodal Particle Distribution," Acta Met .. 22 (1984), p. 235. 68. V. RandJe and B. Ralph, "Interactions of Grain Bound­aries with Coherent Precipitates During Grain Growth," Acta Met., 34 (1986), p. 891. 69. V. Raman, "An Estimate ofthe Driving Force for Bound­ary Migration," Scripta Met., 20 (1986), p. 217. 70. I. Beden, "Static and Dynamic Recrystallization of Cu­SiO, Alloys," Scripta Met., 20 (1986), p. 1. 71. R. Elst, J. van Humbeeck, and L. Delaey, "Evaluation of Grain Growth Criteria in Particle-Containing Materials," Acta Met., 36 (1988), p. 1723. 72. E. Batawi, M.A. Morris, and D.G. Morris, "Rapid Solidi­fication and Structural Stability of Copper-Boron Alloys,"

Acta Met" 36 (1988), p. 1755. 73. G.T. Grewal and S. Ankem, "Isothermal Particle Growth in Two-Phase Titanium Alloys," Met. Trans., 20A (1989), p. 39. 74. G.T. Grewal and S. Ankem, "Modeling Matrix Grain Growth in the Presence of Growing Second Phase Particles in Two Phase Alloys," Acta Met. Mater" 38 (1990), p. 1607. 75. B.R. Patterson, Y. Liu, and J.A. Griffin, "Degree of Pore­Grain Boundary Contact DuringSintering," Met. Trans., 21A (1990), p. 2137. 76. B.R. Patterson and Y. Liu, "Quantification of Grain Bound­ary-Pore Contact During Sintering," J. Am. Ceram. Soc., 73 (1990), p. 3703. 77. M.C Petri and A.A. Solomon, "Pore-Grain Boundary Interactions during Sintering of Pure Nickel under Variable Pressure," Scripta Met. Mater., 24 (1990), p. 45. 78. A. Luo, D.L. Jacobson, and KS. Shin, "Ultrahigh-Tem­perature Tensile Properties of a Powder Metallurgy Pro­ces~ed Tungsten-3.6 w / 0 Rhenium-1.0 w / 0 Thoria Alloy," SCTlpta Met. Mater., 25 (199]), p. 1811. 79. M. Miki and Y. Ogino, "Effect of Si Addition on the Cellular Precipitation in a Cu-lONi-8Sn Alloy," Mater. Trans. 11M, 31 (1990), p. 968. SO. M. Miki and Y. Ogino, "Influence of Solution-Treatment Conditions on the Cellular Precipitation in Si-Doped Cu­IDNi-8Sn Alloy," Maler. Trans. JIM, 32 (199]), p. 1135. 81. M. Gon~alves, "On the Grain Boundary Particle Pinning during Recrystallization of Al-Li Alloys," Scripta Met. Mater., 25 (199]), p. 835. 82. K. Boylan et ai., "An In-Situ TEM Study of the Thermal Stability of Nanocrystalline Ni-P," Scripta Met. Maler., 25 (199]), p. 2711. 83. R.J. Lindinger, R.C Gibson, and D.J. Brophy, "Quantita­tive Metallography of Microduplex Structures after Super­plastic Deformation," Trans. ASM, 62 (1969), p. 230. 84. B.M. Watts et aI., "SuperplasticityinAI-Cu-Zr Alloys Part II: Microstructural Study," Metal Sci. (1976), p. 198. 85. K Tsuzaki et al., "High-Strain Rate Superplasticity and Role of Dynamic Recrystallization in a Superplastic Duplex Stainless Steel," Mater. Trans. JIM, 31 (1990). p. 983. 86. A. Baldan, "Effects of Growth Rate on Carbides and Microporosity in DS200+Hf Superalloy," J. Mater. Sci., 26 (199]), p. 3879. 87. L.A. Xue and R. Raj, "Grain Growth in Superplastically Deformed Zinc Sulfide/Diamond Composites," J. Am. Ceram. Soc.,74 (1991), p. 1729. 88. D.L. OIgaard and B. Evans, "Effect of Second-Phase Particles on Grain Growth in Caldte," /. Am. Ceram. Soc., 69 (1986), p. C272. 89. T.lkegami, "Microstructural Development during Inter­mediate and Final Stage Sintering," Acta Met., 35 (1987), p. 667. 90. Y. Yoshizawa and T. Sakuma, "Evolution of Microstruc­ture and Grain Growth in zrO,-Y,o, Alloys," ISIJ Inl., 29 (1989), p. 746. 91. F.F. Lange, "Powder Processing Science and Technology for Increased Reliability," J. Am. Ceram. Soc., 72 (1989), p. 3. 92. J. Kim, T. Kimura, and T. Yamagushi, "Microstructure Development in Sb,o,-Doped ZnO," J. Mater. Sci., 24 (1989), p.2581. 93. J. Kim, T. Kimura, and T. Yamagushi, "Sintering of Sb,O,­Doped ZnO," J. Mater. Sci., 24 (1989), p. 213. 94. H.J. Hofler and R.S. Averback, "Grain Growth in Nano­crystalline TiO, and Its Relation to Vickers Hardness and Fracture Toughness," Scripta Met. Mater., 24 (1990), p. 2401. 95. R. Duclos and J. Crampon, "Grain Boundary-Inclusion Interactions in a Zirconia-Alumina Ceramic Composite," Scripta Met. Maler., 24 (1990), p. 1825. 96. T. Senda and R.C Bradt, "Grain Growth of Zinc Oxide during the Sintering of Zinc Oxide-Antimony Oxide Ceram­ics," J. Am. Ceram. Soc., 74 (199]), p. 1296. 97. K. Okada, Y. Yoshizawa, and T. Sakuma, "Grain-Size Distribution in AI,o,-ZrO, Generated by High-Temperature Annealing," J. Am. Ceram. Soc., 74 (1991), p. 2820. 98. M.B. Holness, M.J. Bickle, and B. Harte, "Textures of Fosferite-Calcite Marbles from the Bein and Dubhaich Aure­ole, Skye, and Implications for the Structure of Metamorphic Porosity," J. Ceol. Sci. (London), 146 (1989), p. 917. 99. L. Levi and E.A. Ceppi, "Grain Growth in lee," Nuuvo Cimento, 5C (1982), p. 445. 100. M.D. Hofstadter and B.C Murray, "Ice Sublimation and Rheology: Implications for the Martian Polar Layered De­posits," Icarus, 84 (1990), p. 352.

ABOUT THE AUTHORS ____ _

C.H. Worner earned his Dr. Fia. in physics at the Instituto Balseiro in Bariloche, Argentina. He is currently a professor at the Instituto de Fisica, Universidad Catolica de Valparaiso, Chile. Professor Worner is also a member of TMS.

P.M. Hazzledine earned his Ph.D_ in physics at Cambridge University, United Kingdom. He is currently with Universal Energy Systems in Dayton, Ohio. Dr. Hazzledine is also a mem­berof TMS.

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JOM • September 1992