Effect of Recrystallized Grain Size on Grain GrowthPaul A. Beck Citation: Journal of Applied Physics 19, 507 (1948); doi: 10.1063/1.1698164 View online: http://dx.doi.org/10.1063/1.1698164 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/19/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of size distribution and grain growth on the formation of molecules in star forming regions AIP Conf. Proc. 1543, 195 (2013); 10.1063/1.4812614 Effect of grain size on cluster size J. Appl. Phys. 97, 10E312 (2005); 10.1063/1.1852331 The grain growth blocking effect of polycrystalline silicon film by thin native silicon oxide barrierduring the excimer laser recrystallization Appl. Phys. Lett. 75, 460 (1999); 10.1063/1.124400 Grain growth and grain size distributions in thin germanium films J. Appl. Phys. 62, 2492 (1987); 10.1063/1.339460 Reducing the grain size of polycrystalline lead films by the use of barriers to grain growth J. Appl. Phys. 54, 652 (1983); 10.1063/1.332071

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Letters to the Editor

Convenient Methods for Obtaining djn Values from X-Ray Diffraction Patterns

C. H. "-fEWART ANlJ K S. LUTTON

'Thf Proett'( and Gamble Company. h1oyy<iult', Ohio Jannary 8, 1948

FOR deriving din values from x-ray powder patterns multiple graphs or elahorate tables or scales must

generally be prepared unless the Bragg calculation is to be carried through in detail for each spacing, Relations accurate to O,OlkX(1kX = 1.00202A)1 have been found which permit rapid slide rule calculations of d/n values for both (i) flat films, in conjunction with a single graph for d/n values below 3,OkX and (ii) cylindrical films,

(i) Flat ,lilms.-The well-known Bragg formula may be expressed

din = Aj2sin8. (1)

This relation can be approximated by the following simple equation

din = [l()),/(RD,)]+ A(RD,), (2 )

since 2 sin8= lan20 for small angles, and tan20= (RD 5)/1O, (RD,) heing the diffraction ring diameter in em for sample:to-film distance of S em. A is a small constant calculated empirically, If Cu Ka-radiation is used and )..= l.S39kX,

d!n = [lS,39/(RD.1)]+0,OS5(RD 5V (3)

The most convenient form for use is

d/n= [15.39/(RD5)]+0,10X [(RD 5)/2] +0,01 X [(RD 5)/2]. (4)

The first term is obtained by a single setting of an ordinary slide rule; the last two terms are readily added mentally' with a minimum of practice. Relations (3) and (4) are accurate to O,OlkX down to 3.0kX. In many cases, notably with long-chain compounds, there is little need for lower values. \Vhen they are required, a single graph, as in Fig, 1, will cover the range from 3.0 to L~SkX.

" " ... '" c .~

3,0

~ 2.0

Q

1.5

~ FOR FLAT rlLIl

FUll> Dillacca - 5 Om.

* from ::1.0 10 L3li tX

A.1.539 lX

10 15 20 ROt} or Rlng Diameter (em)

FIG. 1.

VOLUME 19, MAY, 1948

25

(ii) Cyl-indrical films.-For cylindrical films with fixed camera radius = 7.16 cm, the basic equation is

din = ),,/[2 sin(4r)], (5)

where r is the diffraction ring radius measured in em, and 4r is the value of the angle /J measured in degrees.

Here an approximate relationship can be put in the form

).. 7.16}. d/n=2X(4r/57.30)+Br=~r -+Br, (6)

since, for small angles, 2 sinO=20 and 0 in radian~=4r/ 57.30; B is a small constant evaluated empirically. For CuK a-radiation this becomes

d/n = (1 1.02/r)+0.01Or} (7 )

Here, too, a single slide rule setting is sufficient; a simple mental addition follows. The equation is accurate to 0.006kX for d/n values from 0.9 to 22kX(r=0,50 em), Normally the upper limitation is without practical im­portance since cylindrical cameras are usually built so as to exclude from the film any lines corresponding to spacings greater than 22kX.

For radiation other than CuK a, Eqs. (4) and (7) should be used and the results multiplied hy Aj1.S39 so that Eq. (4) for fiat film becomes

d/n = [(15.39/ RD 5)+ (RD./2)X (0.1O+0.01)J X (}./1.539), (8)

Error :::::0.010 X ()"/1.539) down to 3,OkXX (}./1.539),

and Eq. (7), for cylindrical film with 7.16-cm camera radius, becomes

dln= [(11.02/r)+0,010r JX ()"/1.539), (9) Error :::::0.006 X ()"/1.539).

1 E. A, Wood. Phys. Rev. 72,436 (1947). Z As has been pointed out to the authors by Dr. A. S. Richardson of

this laboratory. the values of the small constants in relations (3) and (7) may be determined mathematically. To accomplish this. 2 sin8 and tan20 are expressed in terms of 8 by means of infinite series. After the necessary operations. substitutions and elimination of smaller terms, equations almost identical with (3) and (7) appear.

Effect of Recrystallized Grain Size on Grain Growth

PAUL A. BECK

Department of Metallurgy. University of Notre Dame Noire Dame, Indiana •

March 12. 1948

I N the discussion of a previous paperl it was briefly mentioned and in a recent note2 it was stated in more

detail that, according to results obtained with high purity aluminum, the instantaneous rate of grain growth at a given temperature depends only on the grain size D at the instant considered. The instantaneous rate of grain growth was found to be independent of the particular time-tem­perature ~chedule used previously to produce that grain size.

It would be of considerable fundamental interest to as­certain under what conditions or within what limits this principle retains it~ validity. An interesting question in

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this connection is whether or not isothermal grain growth in specimens of the same material, starting from different recrystallized grain sizes, Dr, has the same instantaneous rate dD/dt at corresponding grain-size levels. The assump­tion that it has may be readily expressed in a form suitable for experimental verification, as shown in the present note.

It has been shown!,3 that isothermal grain growth in a pure metal and in certain pure solid solutions follows the equation

(1 )

Here tg is that portion of the total annealing time t which is available for grain growth, i.e., the total annealing time less the time R necessary for complete recrystallization: tg'=t-R: The parameters nand K are functions of the temperature. When recrystallization is just completed 10 = 0, and from Eq. (1)

(2 )

It is possible to eliminate A from Eqs. (1) and (2). Bya slight rearrangement one then obtains

D!/n-Dr!/n= C·t. where C=K!/n.

(3 )

Here nand C are parameters which depend on the tem­perature. Equation (3) is, of course, equivalent to Eqs. (1) and (2), which have already been published.! However, its presentation is warranted by the convenience it offers in certain applications.

Equation (2) leads to the interpretation of parameter A as the period of imaginary grain growth which would in­crease the grain size from zero to the recrystallized grain size Dr in accordance with the extrapolation of the for­mula describing grain growth subsequent to recrystalliza­tion, Eq. (1). These conditions are illustrated in Fig. 1. Here point 1 represents recrystallization just completed, and point 2 represents a condition obtained by recrystalliza­tion followed by grain growth along the solid curve 1-2 during the period t.. This curve, a parabola of the nth order, corresponds to Eq. (1). According to Eq. (2) the value of A is determined so as to bring the apex of the dotted extension of the parabola exactly to the common origin of the time and the grain-size scales. The sum t.+A, appearing in Eq. (1), is the total grain-growth time which would produce a final grain size D from an imaginary starting grain size of zero, according to the simple power law, Eq. (1). Actually, the dotted portion of the parabola

D

a

508

/' , .' I

" I ,. I I I

.. I I

II Grain-Growth Time

FIG. 1.

(0-1) is not operative. Instead of grain growth, the process resulting in grain size Dr is recrystallization, a process in­volving nucleation and growth, during the period R, of virtually strain-free grains in a strained matrix.

Now, if the recrystallization process is so conducted as to lead to a different grain size Dr', the grain sizes obtained by grain growth starting with this Dr' cannot be predicted without some additional relationship. In the following the assumption is made that here too, as in the case previously investigated' where different starting grain sizes were pro­duced by grain growth at different temperatures, the in­stantaneous rate of grain growth dD Idt at a certain tem­perature depends only on the instantaneous grain size D (see the first paragraph). With this assumption the D vs. t curve will be as shown in Fig. 2. The tangent of curve b at 3 (grain size D/) will be the same as that of curve a at 2 (the same grain size). It is easy to see that curve b can be derived from curve a by a parallel displacement along the time axis. In other words, grain growth starting from a certain recrystallized grain size will, under such conditions, follow the same equation as grain growth starting from any other recrystallized grain size, provided that the parameter A is properly determined from Eq. (2) for each starting grain size. This implies that in Eq. (1) the param­eters K and n are independent of the starting grain size. The same is true for parameter C in Eq. (3); the conveni­ence in using this equation is now apparent. If D!/~-Dr!/n is plotted against ta, the same straight line describes grain growth regardless of the starting grain size. In a case like that of 70-30 brass,where n is independent of the tempera­ture, the same ordinate scale can be used throughout, and the lines representing grain growth at various tem­peratures will be straight lines of different slope, intersect­ing each other at the origin.

In the log D us. log t plot which was used in previous publications to represent grain-growth data for high purity aluminum, for an AI+2 percent Mg alloy and for 70-30 brass, the consequences of the above assumption can be expressed as shown in Fig. 3. For a certain grain size Dr the time for complete recrystallization R may be the same as parameter A calculated from Eq. (2). For that par­ticular starting grain size Eq. (1) is simplified to

D=K·tn, (4)

where t=tD+R=I.+A, and the logarithmic grain-growth line is a straight line (a in Fig. 3). If Dr' is such that A > R a curve is obtained (b in Fig. 3), which starts higher than

Annealing Time

FIG. 2.

JOURNAL OF APPLIED PHYSICS

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~ "iii );i OIl

6 .~ Ci D: .~ D,.

\5 ;; It -<~

Annealing Time (log scale)

FIG. 3.

the straight line, but approaches it asymptotically. If for a certain D/'A <R, the curve approaches the straight line from below (line C, Fig. 3). Examples of all three types of logarithmic grain-growth lines have actually been recorded in previous work,'·3 This may now be interpreted to indi­cate that the time for complete recrystallization R, leading to a recrystallized grain size of Dr, may be either larger or smaller than or equal to the time A during which a grain size Dr would develop from zero size by means of the hy­pothetical "extrapolated grain growth."

For 70-30 brass of a grain size of 0.028 mm, reduced 33 percent by rolling, the log D vs. log t plot 'of the grain sizes produced by annealing at 450 0 very closely approximates a straight line all the way to the recrystallization point. In this case, therefore, A is approximately equal to R, i.e., the hypothetical grain-growth mechanism would lead to the same grain size in the same period of annealing as recrystallization does. \Vork is at present under way in this laboratory to study grain growth in brass after re­crystallization to various grain sites. The results of these experiments, plotted in either one of the two ways dis­cussed, should tell whether or not the above assumption is correct.

1 Paul A. Beck, J. C. Kremer. L. J. Derner. and M. L. Holzworth, "Grain growth in high purity aluminum and in an aluminum-mag­nesium alloy," AIME Tech. Pub. No. 2280 Metals Tech. (September 1947).

2 Paul A. Beck, M. L. Holzworth, and Haun H u, "Instantaneous rates of grain growth," Phys. Rev. 73, 526 (1948).

3 Paul A. Beck, John Towers, Jr., and William D. Manly, "Grain growth in 70-30 brass," AIME Tech. Pub. No. 2326, Metals Tech. (February 1948).

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VOLUME 19, MAY, 1948

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