The effect of grain boundary edges on grain growth and grain growth stagnation

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  • Materials Science and Engineering, A 108 (1989) 33-36 33

    The Effect of Grain Boundary Edges on Grain Growth and Grain Growth Stagnation

    T. O. SAETRE

    Norsk Hydro, Karmoy Fabrikker, 4265 Hdvik (Norway)

    N. RYUM

    Metallurgisk lnstitutt, NTH, 7034 Trondheim (Norway)

    O. HUNDERI

    lnstitutt Fysikk, NTH, 7034 Trondheim (Norway)

    (Received January 19, 1988; in revised form June 6, 1988)

    Abstract

    The effect of grain boundary edges on grain growth kinetics has been considered by an exten- sion of Beck's treatment of primary grain growth. It is found that the effect of edges is important only for very small grain sizes. The effect of edge parti- cles on grain growth stagnation is treated by a method analogous to Zener's treatment. It is shown that grain growth is completely suppressed at a critical volume fraction of particles. Finally, two-dimensional grain growth is considered and the results compared with recent results obtained by computer simulation.

    1. Introduction

    Grain growth has always been considered to be driven by the grain boundary energy. When the mean grain size/) increases, the volume-specific grain boundary area (grain boundary area per unit volume) decreases. Beck [1] assumed a linear relationship between the rate of growth of the mean grain size and the volume-specific grain boundary energy 7u//)

    dO) 7b (1) b=kb~

    where b is the specific grain boundary energy and k b a proportionality constant. This leads to a parabolic relationship between the mean grain size and time. This is also the main result of some of the more rigorous theories [2-6] but it is a result that is seldom, if ever, observed experi-

    mentally. The physical vagueness of Beck's model makes it very difficult to give a definite value to k b

    During grain growth the volume-specific grain edge length decreases. Since the edges have a certain specific energy y~, a reduction in the energy of the system, owing to a reduction in the volume-specific grain edge energy, takes place during grain growth and this energy must also contribute to the grain growth process. An evalu- ation of this effect, in a similar manner to Beck's treatment, can be made in a straightforward manner.

    When secondary particles are present in the material, grain growth stagnation is observed. The classical treatment of this important effect is due to Smith and Zener. They attributed the effect to the interaction between the particles and the grain boundaries. (For a recent review, see ref. 7.) Obviously there is a similar interaction between the particles and the grain edges and this interaction is likely to become important at large volume fractions of particles. An evaluation of this effect, by a method similar to that used by Zener and Smith, is also carried out.

    Experimental quantification of grain growth stagnation is rather difficult to achieve and the data show a great deal of scattering. In order to gain an insight into this process, two-dimensional grain growth stagnation has recently been simu- lated on a computer [8] and the results have also been extended to three dimensions [9]. These results are discussed in the following sections with special references to the results obtained here.

    0921-5093/89/$3.50 Elsevier Sequoia/Printed in The Netherlands

  • 34

    2. Grain growth We assume that the growth rate of the mean

    grain size owing to the reduction in grain edge length is proportional to the volume-specific grain edge energy to give

    where Ye is the specific grain edge energy. As with eqn. (1) we have difficulties in deter-

    mining the magnitude of the proportionality con- stant ke. Apart from a geometrical factor not far from unity, k e depends upon the mobility of the edges and thus on the atomic structure of the edges. The atoms in the edge region are likely to be somewhat more disordered than are the atoms in the grain boundary region, thus giving the edges a somewhat higher mobility than that of the boundaries. As a first-order approximation, we assume that the two growth rates given by eqns. (1) and (2)are additive. This gives

    ~-t = -~t b+ ~ e=kbD ke/~z (3a)

    the solution of which is

    1 {/)2 x(ZS-D0/

    +x 2 In ( D+~x/ / (3b) \/)o + x)]

    where x = k~ye/kb7 b and/)0 is the mean grain size at t= t0 .

    Equations (3a) and (3b) reduce to Beck's equa- tion (eqn. (1))when kb(Tb//))~> ke()'e//) 2) giving if) "~ t 1/2. On the other hand, if kb( Tb/ D ) ~. k~(Td/) 2) we obtain / )= t U3. The second term in the expression for the growth rate in eqn. (3a) thus causes the exponent to tend away from 1/2 and towards 1/3. The pertinent question now is how is important this second term. A rough estimate is obtained by modelling the edge structure as a tube of radius r 0. The edge energy is then equal to the surface energy of the tube. We take the spe- cific surface energy of the tube to be equal to the specific surface energy of a free surface, which is approximately twice the specific grain boundary energy. We thus obtain

    7~ ~ 4 zr o 7b

    The condition required for the edges to influence the growth rate to a marked degree is thus given by

    4ZroTb Yb kb ~- ke 1~ 2

    which gives

    ke ID-- 4nro kb (4)

    Taking r 0 = 2b, where b is the Burgers vector and ke/kb-~ 10 in eqn. (4), we obtain/3=/~min ~ 100 nrn.

    This analysis thus shows that the grain edge energy will influence the rate of grain growth in a marked way only in material with a very small grain size.

    3. Grain growth stagnation Such small grain sizes are not obtained except

    by means of pinning particles. Usually the pinning effect is expressed as a Zener restraining pressure Pz resulting from the interaction between the par- ticles and the grain boundary. The question to consider here is whether the interaction between the particles and the grain boundary edges can contribute to the pinning of the grain structure and to the stagnation of grain growth.

    A spherical particle of radius R situated on a grain boundary edge eliminates about (3/2)nR 2 of the grain boundary area. When the edge moves a distance of about R it breaks away from the par- ticle. As a first-order approximation for the pin- ning force K from one particle we obtain

    K = 23-nRyb ( 5 )

    If the density of particles is uniform through- out the material with nv particles per unit volume, the number of particles ns per unit length of the edge can be expressed as

    3f ns = ~R2nv - (6)

    4R

    where f is the volume fraction of particles. Com- bining eqns. (5) and (6) the total pinning force per unit edge length becomes

    9;rg Ktot= Kns=-~ - bf

  • The total pinning force on a grain of size/5 owing to particles at the edges now becomes

    KtotLt,,t = o~Ktot/5

    where Lto t (= (z/5) is the total edge length of a grain of size /5. We now distribute this force uniformly on the total surface area A to t of the mean grain to obtain the restraining pressure Pc due to the edge-particle interaction

    a/5 a/0 1 PC = Kto, = K,, , , - K,otS-~ (7 ) ,8/5 A,,, L)

    with Atot=f l / ) 2 and where a, fl and a/fl = S are stereologie constants that depend on the geo- metrical shape of the grain. For a tetrakaideca- hedron, which is the likely shape of the mean grain, S = 3.8. We can finally express P~ as

    9:r 3.8 Ybf (8)

    Stagnation of grain growth occurs when the sum of the restraining pressures (Pz + Pc) equals the driving pressure P such that

    P- t"z- P~=O (9)

    We have P=47b/ / ) and Pz=3fTb/2R. Putting these expressions and eqn. (8) into eqn. (9), and solving for/5, we find the limiting grain size to be

    8R D~im =~f (1 - 3.5f) =/5,Zm(1 -- 3.5f) (10)

    Here, we have assumed that the stagnated grain size /)lim ~ /~min, where /~min is derived earlier in this paper, and have thus omitted the driving pressure resulting from grain boundary edges.

    Equation (10) shows that the limiting grain size /)lim decreases more rapidly with the volume frac- tion f than does the Zener limiting grain size D~m. Moreover, at a critical volume fraction fc = 0.28, no grain growth is possible. This is physically more satisfying than is taking the asymptotic decrease of/)lim with increasing volume fraction, which is predicted by the standard Zener treat- ment.

    Grain growth stagnation is difficult to study experimentally because of several complicating factors: the particles are often inhomogeneously distributed; the particles coarsen at the annealing temperature and abnormal grain growth may develop. Recently, computer simulation has been

    35

    used in an attempt to gain insight into the ideal- ized grain growth stagnation process.

    Srolovitz et al. [8] have simulated particle- affected two-dimensional grain growth by means of the Monte Carlo technique on a computer. Their results indicate that

    R /5,ira = 3.5 f21/~ (11)

    holds between /51ira and f2 where f2 is the area fraction of particles. The absolute value of/)lim is also much smaller than is the value calculated by the Zener approach in two dimensions. In an accompanying paper, Doherty et al. [9] gave an analytic model for eqn. (11) and in addition, by analogous arguments, extended their calculations to three-dimensional grain growth stagnation to give

    4R Dlim fl/2 (12)

    where f is now the volume fraction of particles. The reason for the 1If 1/z (instead of l / f ) rela- tionship between the limiting grain size and the volume fraction is that most of the particles are situated on the grain boundaries. According to this research this leads to a breakdown of the Zener method of calculating the restraining pres- sure during grain growth. However, eqn. ( 11 ) can easily be derived by the Zener method.

    To do so, the number of particles fi needed to stop a grain of size/5 from shrinking is found by the pressure balance

    27 2n7

    /51i m Y'LDIi m

    which gives

    r i=n (13)

    Figure 1 shows that two particles cannot stop the shrinking, but three particles always can. Equa- tion (13) is thus a good approximation. The mean grain size will increase until it impinges on three particles. The stagnation thus occurs when the mean grain size is comparable with the inter- particle spacing, as was also pointed out by Doherty et al. [9]. This thus gives

    Jr n~ = 3

  • 36

    a b Fig. 1. The pinning of a shrinking grain by particles. Two particles cannot pin the grain boundary completely (a), but three particles can (b).

    where

    ns zt R 2 =f2

    and, on combination, this gives

    R / ) l im -~" 3.46 f2~/~ (14)

    which is close to the results of Srolovitz et al. [8] and to those derived in the analytic model by Doherty et aL [9]. However, this way of reasoning cannot be carried over to three dimensions, as was done by Doherty et al. [9]. For the three- dimensional case, the number of particles fi needed to halt grain shrinkage is given by

    4), ~Ry

    Olim 4 Yt'~)lim 2

    which gives

    16/)lim R

    In this case, r~ is no longer constant but depends on ~)lim and R. Also, Olim/R ~ 1 and thus the grain boundaries have to pass many particles to collect the required number ri of particles. We thus have to use the Zener method to calculate ri such that

    }~ = 4~r/)lim22Rnv

    where

    nv4grR 3 = f

    and the Zener expression

    8R /)lim -- (15)

    3f is again found for the stagnated grain size in three dimensions.

    4. Conclusions

    This investigation has demonstrated that the effect of grain edges on normal grain growth is only of importance for very small grain sizes (D = 100 nm). On the other hand, the edges contribute significantly to grain growth stagna- tion, particularly at large volume fractions owing to the interaction between edges and par- ticles. For small volume fractions, eqn. (15) is an acceptable approximation for the three- dimensional limiting grain size. For very small volume fractions, a better approximation can be obtained by using the expression for the Zener pressure derived by Nes et al. [7] (their eqn. (27)) to give/)lim ~" 3R/f0.98.

    When the volume fraction is large, the grain boundary edge effect becomes important and eqn. (10) is the best approximation. When two- dimensional grain growth stagnation is con- sidered (a rare if not completely unreafistic case) eqn. (14) is the best approximation.

    References

    1 P.A. Beck, Adv. Phys., 3 (1954) 245. 2 M. Hillert, Acta Metall., 13 (1965) 227. 3 G. Abbruzzese and K. Liicke, Acta Metall., 34 (1986) 905. 4 N. Louat, Philos. Mag. A, 47(1983) 903. 5 O. Hunderi, N. Ryum and H. Westengen, Acta Metall., 27

    (1979) 161. 6 M.P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni,

    Acta Metall., 32 (1984) 783. 7 E. Nes, N. Ryum and O. Hunderi, Acta Metall., 33

    (1985) 11. 8 D.J. Srolovitz, M. P. Anderson, G. S. Grest and P. S. Sahni,

    Acta Metall., 32 (1984) 1429. 9 R. D. Doherty, D. J. Srolovitz, R. D. Rollett and M. P.

    Anderson, Scr. Metall., 21 (1987) 675.

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