“A method for grain size and grain size uniformity estimation - applications to polycrystalline materials”

Scripta METALLURGICA et MATERIALIA

"A METHOD

FOR

Vol. 27, pp. i17-120, 1992 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

GRAIN SIZE AND GRAIN SIZE UNIFORMITY ESTIMATION POLYCRYSTALLINE MATERIALS"

-

APPLICATIONS TO

K.J.Kurzydtowski and J.J.Bucki Institute of Materials Science and Engineering Warsaw University of Technology 02-524 Warszawa, Narbutta 85, POLAND (Received February ii, 1992) (Revised May 7, 1992) Introduction Grains in polycrystals differ in their size and shape. It has been suggested recently [1] that the degree of uniformity of size of grains might have an effect on plastic and more general on mechanical properties of polycrystalline materials. The sizes of grains, which are elements of 3-dimensional space, can be uniquely defined by their volume v. Grains in polycrystals form a population described by a volume distribution function f(v). In most material science applications the population is sufficiently well characterized by: a) the mean E(v) and b) the variance VAR(v) or standard deviation SD(v) of grain volume. The grain volume variance carries information on grain size uniformity. However, the degree of grain size uniformity can be better quantified by means of the grain volume coefficient of variation CV(v)" CV 2(v)- VAR(v) [E(v) ] 2

(I)

From an experimental point of view a distinction has to be made between t w o distribution functions: fN(V) and fv(V) - true volume and volume distribution functions, respectively, that can be used to describe a population of grains. The distinction between the "true volume" and the "volume", marked by the subscripts "N" and "V" is related to two different ways of sampling grains in the process of grain size measurements. The true volume distribution function fN(V) can be obtained if the grains are randomly sampled irrespective of their size. This condition however is rarely met in the case of metallographic observations which are characterized by a preference for large grains to be more frequently measured (more frequently cut by the section of observation) than small grains. Metallographic observations usually yield a volume weighted distribution function fv(V) which is briefly called volume distribution function. Experimental studies of true distribution function are much more difficult than studies of some weighted distributions. On the other hand, in most practical applications in materials science the volume distribution function is sufficiently well characterized if the weighted mean volume and the weighted volume variance are known. Certainly this is the case in the situation where the variance is used just to control the degree of grain size uniformity. Measurements of the grain volume impose technical difficulties. There is a little direct data concerning the distribution of grain volume available in the literature (see for example [2,3]). A number of stereological methods have been proposed that can be used to estimate E(v) and VAR(v) from 2-dimensional measurements on sections through polycrystals. These methods, however, often rely on certain assumptions regarding the shape of grains and/or the type of grain volume distribution function f(v). On the other hand, important progress in stereology has been made in recent years. New methods have been proposed that can be used to estimate the variance of particle volume [4-6]. These new results were obtained in the biological studies of cells. It is the aim of this paper to show how these new theoretical results can also be used in the field of materials science for the characterization of grain size in polycrystalline aggregates.

117 0956-716X/92 $5.00 + .00 Copyright (c) 1992 Pergamon Press

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Analysis The volume-weighted mean volume of grains Ev(v) can be relatively easily determined by the procedure proposed by Jensen and Gundersen [4]. This procedure requires random point sampling of the grains revealed on a cross-section and subsequent measurements of the length of I~intercepts drawn through the points P~ projected on the image of the grain structure. (The procedure is schematically shown in Fig.l). On the basis of such measurements, the mean volume Ev(v) can be estimated using the formula: Ev(v) =-~E(I 3) (2) The volume variance VARv(v) on the other hand can be calculated by means the equation:

VARv(v ) =Ev(V 2) _ (Ev(V) )2

(3)

Ev (v 2) =4xkE(a 3)

(4)

and the relationship derived in [6]: where: a k

is the area of the grains hit by the P~ points is a constant with a value in the range from 0.071 to 0.083 [6]. (The value of k depends on the shape of grains and is expected to be constant for a large group of polycrystalline materials. Its value can be determined either experimentally or by appropriate modelling).

From equations (2) to (4), one can obtain the following relationships: ~2 and:

VARy(v) =4~kE(a 3) --~- [E(I 3) ] 2

(s)

CV2v(V) - 36kE(a]) 1 ~ [E(13) ] 2

{6)

In practical implementation, the procedure based on the use of formulae (2)-(6) combines standard metallographic measurements such as the measurements of intercept length and the grain area. These measurements can be conducted with high precision and automatized by the use of automatic image analysis systems. In the case of geometrically isotropic grain structures, the measurements can be executed on one representative section of the polycrystal. Also, the constant k in formulae (4)-(6) varies in a narrow range of values and it does not need to be known for the purpose of simply grading the degree of grain size uniformity in similar grain structures. Application In order to examine the effectiveness of the procedure based on the above equations, measurements have been performed on Fe-u (BCC), Fe-F (austenitic stainless steel- FCC) and aluminium (FCC) polycrystals. All specimens represented "as annealed" material (after recrystallization). They were, however, heat treated at different temperatures and annealed for different times. Some of the selected materials have been investigated earlier in other studies [7-9]. The measurements have been performed on binary images of the grain boundary networks, examples of the microstructures selected for study are shown in Fig.2. A system for automatic image analysis has been employed for the measurements and a special program has been implemented which executes the following steps: 1 o sampling of the grains using a square point grid P~characterized by distance do 2 ° measurements of the grain area a i 3 ° generation of intercepts through the points P~ 4 ° measurements of the intercept length I~ 5 ° statistical analysis of the data - calculation of the grain volume variance. The results of the measurements are shown in Fig.3 in log-log coordinates in the form of a plot of the mean grain volume Ev(v) against the standard deviation SDv(v). Geometrically compatible polycrystals are characterized by a constancy of the coefficient of variation, i.e the constancy of the ratio of standard deviation SDv(v) to the mean volume Ev(v). In the

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coordinate system adopted in Fig.3, the points for a group of geometrically compatible polycrystals are expected to fall on a straight line inclined at an angle of 45 o to the axes and intersecting the SDv(v) axis at the point defining the value of Iog(CVv(v)). The plot in Fig.3 shows that, despite the differences in the heat treatment and microstructures of the studied materials, the values of CVv(v) generally fall into a relatively narrow range, from 0.5 to 2.0. On the other hand, this is not true for Fe-e polycrystals. For Fe-e, significant differences in the degree of grain size uniformity can be observed. In the range of small grain sizes, CVv(v) increases with increasing mean value of the grain size. Conclusions Recent progress in stereology makes it possible to estimate the grain volume weighted mean, and volume weighted variance by means of relatively simple 2-dimensional measurements of the intercept length and the grain area. The values of Ev(v) and VARv(v), or CVv(v) can be used to characterize the average grain size and its degree of uniformity. The degree of grain size uniformity can be described by the value of the variation coefficient CVv(v) or illustrated in a plot of the data in the way shown in Fig.3. The mean weighted volume of grains Ev(v) can be used as an alternative to other, more frequently employed, measures of size of grain such as, the mean intercept length and the mean grain area. The mean volume seems to be more convenient in a number of situations, where the volume of grains is more important than their linear dimensions (for example in an analysis of X-ray diffraction measurements). Measurements performed on the group of specimens representing structure of recrystallized metals have shown that although in general the value of CVv(v) varies in a narrow range, in certain cases significant differences are observed. These differences need to be accounted for in the analysis of the properties of polycrystalline materials. Finally, we suggest that the measurements of CVv(v) should be incorporated into standard practice in metallography and used for better characterization of grain size in polycrystals. Acknowledgments This work had been supported by grant from Polish Committee for Scientific Research. References 1. K.J.Kurzydtowski: Scripta Met., v. 24, 879, (1990) 2. F.N.Rhines, B.R.Patterson and R.A.Ellis: Metalurgia i Odlewnictwo, 5, 281, (1979 3. E.E.Underwood: "Quantitative Stereology", Adison Wesley Publishing Company, Reading, p.109 4. E.B.Jensen and H.J.G.Gunderson, J.AppI.Prob., 22, 82, (1985) 5. F.B.Sorensen, J.Microscopy, 162, 202, (1991) 6. E.B.Jensen and F.B.Sorensen, J.Microscopy, 164, 21, (1991) 7. K.J.Kurzydtowski, K.J.McTaggart and K.Tangri, Phil.Magazine A, 81, 61 (1990) 8. K.J.Kurzyd{owski, J.J.Bucki and J.W.Wyrzykowski, Archiwum Nauki o Materialach, 12, 3 (1991) 9. S.Sangal, K.J.Kurzydtowski and K.Tangri, Acta Met., 39, 1281, (1991)

FIG. 1. Schematic illustration of the employed method for volume mean and variance measurement.

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FIG.2. Examples of the studied mictostructures (aluminium recrystallized at 300 and 350°C).

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FIG.3. Results of the measurements for Fe-o (@), Fe-y (--) and aluminium (u) polycrystalso