On the possible correlation between grain size distribution and distribution of CSL boundaries in polycrystals

Vol. 43, No. 4, pp. 1541-1547, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0956-7151/95 $9.50 + 0.00

Acta metall, mater.

~

Pergamon

0956-7151(94)00371kX

ON THE POSSIBLE CORRELATION BETWEEN GRAIN SIZE DISTRIBUTION A N D DISTRIBUTION OF CSL BOUNDARIES IN POLYCRYSTALS A. GARBACZ1, B. RALPH2 and K. J. KURZYDLOWSKI3t qnstitute of Technology and Organization of Building Production, Warsaw University of Technology, A1. Ludowej 16, 00-637 Warsaw, Poland, 2Department of Materials Technology, Brunel University of West London, Uxbridge, Middlesex UB8 3PH, U.K. and 3Faculty of Materials Science and Engineering, Warsaw University of Technology, Narbutta 85, 02-524 Warsaw, Poland (Received 24 February 1994; in revised forrn 20 June 1994)

Abstract--This paper addresses the possible correlation between grain size distribution and the geometrical properties of grain boundaries. A series of functions are used to describe the grain size and the shape distribution, and considerations given to the population of grain edges and corners which result. Of particular interest are the dihedral angles at grain edges and how these are expected to vary with a change in the grain size distribution. In turn this is then correlated with fraction of special and general (random) grain boundaries. The concepts introduced here are then tested by looking at some microstructural and textural data from a grain growth experiment on high purity aluminum. In parallel with this a model of the distribution function of the grain boundary misorientation parameter is employed. The outcome of this series of investigations is the ability shown to establish a correlation between parameters describing the distribution of size and shape of the grains, and other parameters describing the fraction of grain edges which join boundaries having random and special character.

1.

INTRODUCTION

Grain boundaries in polycrystals are characterized by differences in their microstructure and as a result in their properties. This phenomenon is termed grain boundary character distribution [1, 2]. The differences in the structure of individual grain boundaries are usually described quantitatively in terms of the distribution of the misorientation parameters [3-5]. The distribution function of their misorientation parameters has attracted a great deal of interest since it has been postulated that this parameter controls the energy, diffusivity and other physical properties of grain boundaries [1-5]. However, it should be appreciated that the grain boundary plane may also have an influence, as discussed later. Grain boundaries, and as a result the grains, differ also in their geometrical properties which are described in terms of the distribution functions of the size of grains. Both physical and geometrical properties of grain boundaries have an influence on properties of polycrystals. In the past these two types of grain boundary properties have been studied independently one of the other. On the other hand there are arguments, for instance referring to the energy of the system, which suggest that the geometry of grains is related to the physical properties of the grain boundaries and vice versa [6]. Such a correlation would have important implications to tTo whom all correspondence should be addressed.

the modelling of the properties of polycrystalline materials. The existing models for the properties of polycrystalline materials concentrate attention either on the physical or geometrical properties of the grain boundaries. The concept of "grain boundary engineering" and "grain size effect" seem to be developed with a neglect one of the other. However, if the geometrical properties (size of grains) are correlated with their microstructure, and in turn other properties, the conclusion drawn with regard to the role of one or the other factor are likely to be incomplete or unsubstantiated. It should be noted also that geometrical properties in general are much easier to measure and quantify than the physical properties of grain boundaries. This means that in some conditions the measurements of geometrical properties can help in the assessment of the physical properties of the grain boundaries. The aim of the present paper is to describe the results of a study of a possible correlation between the geometry of grains and distribution of grain boundary microstructure. In this study the geometry of grains has been described in terms of the grain size and shape distribution functions and the microstructure of the grain boundaries using a Coincidence Site Lattice (CSL) formalism. A model has been used which relates the texture of the polycrystal to the distribution of grain boundary misorientation and the distribution function of the characteristics of the grain boundary common edges. These distributions

1541

1542

GARBACZ et al.: CSL BOUNDARIES IN POLYCRYSTALS

are used to explain observed variations in the degree of grain size uniformity and shape of grains. The computations have been carried out for the data obtained on a series of aluminum specimens. 2. DISTRIBUTION FUNCTION OF GRAIN BOUNDARY MICROSTRUCTURE

Individual grain boundaries are typically identified by a set of parameters that include: misorientation angle, O, misorientation axis, !, and the orientation of the grain boundary plane normal, n. Among these parameters, the misorientation angle and axis are most frequently related to the properties of the grain boundaries. In principle these parameters, and the others listed, can be considered to be variables having a continuous character. However, in some applications it is more convenient to analyze the set of grain boundary characteristic parameters, for example possible misorientation angles, as consisting of classes (misorientation intervals) each represented by a single value. Such a situation is exemplified by the procedure based on the CSL model [7]. In this case the structure of grain boundaries is characterized by the E parameter which is computed for each misorientation angle-axis pair: (0, I). Many experimental data confirm that grain boundaries with ~ ~<29 have special properties [8, 9] and therefore the grain character distribution is frequently understood as a Coincidence Grain Boundary Distribution. Special properties of grain boundaries are retained despite small deviations from the exact misorientation characteristic of a grain boundary with a given value of E. The maximum value of this deviation, A0m~, or in other words the size of the 0 intervals represented by given Y~values, is commonly calculated from the Brandon criterion [10]: A0max= 15°X -½

2.1. Individual grain boundaries and systems of grain boundaries Grain boundaries in polycrystals form a system of surfaces connected along threefold edges, TEs, and four-fold points, FPs [Fig. l(a)]. These three-fold edges and four-fold points play an important role in processes taking place at grain boundaries, especially in micro- and nano-grain sized polycrystals. On a cross-section, TEs are revealed as so-called triple points, TPs [Fig. l(b)]. Grain corners, FPs, which would only very rarely be seen and are not identified on a section from a polycrystal, may be observed rarely on projected images obtained in the TEM. A system of grain boundaries can be described by a function defining the properties (distribution of properties) of triple edges. In a simplified approach,

o)

Aa

L,

groinsecllon edge

(1)

In the extreme case the population of grain boundaries is divided into 2 broad categories: 1. random boundaries (R) which can be assigned ~z values larger than 29; 2. special boundaries (S) which can be assigned values of Y~< 29. It should be appreciated that establishing a cut off at E29 with maximum deviation, A0, given by Brandon criterion ([10] and equation (1)) is somewhat arbitrary. Other means of dividing a population of grain boundaries into 2 sets, special and random, are certainly possible. The random grain boundaries defined in this way have been found to have generally higher energy and lower sensitivity to the orientation of their grain boundary planes. Special grain boundaries have lower energy, which is usually strongly dependent on the orientation of the boundary plane.

c)

CV(A)=O

d)

CV(A)~.O

Fig. 1. Definition of the threefold edges, fourfold points, triple points, and grain section edges (a) and schematic, 2-dimensional arguments (b) for the effect of the dihedral angles on the degree of the grain size uniformity. A grid of hexagons of a constant size is characterized by CV(A ) = 0, CV(n) = 0 and all angles equal to 120° (c); combination of regular polygons with 8 and 4 edges results in increased values of CV(n) and CV(A )---in this case the dihedral angles differ from 120° (d).

GARBACZ et al.: CSL BOUNDARIES IN POLYCRYSTALS based on the definition of random and special grain boundaries, four specific types of TEs were distinguished [11]: (a) R R R or random-random-random, ( q = 0 ) ; (b) RRS or random-random-special, (q = 1); (c) RSS or random-special-special, (q = 2); (d) SSS or special-special-special, (q =3); where q = 0 , 1, 2 or 3 is the number of special grain boundaries at a given TE. 3. CHARACTERIZATION OF GRAIN SIZE

Grain size is known to be an important microstructural parameter which can have a strong influence, direct and indirect, on the physical properties of materials. This influence is often reflected in the formation of the Hall-Petch relationship [12] and its many modifications [e.g. 13]. Indirectly, these models lead to the development of the procedures for the characterization of the geometry of grains in terms of their size and shape [14-17]. Grains in polycrystals can be described in terms of their distribution of grain volume, f ( V ) , which describes the size of individual grains. This distribution of grain volume can be characterized by parameters such as the mean value, E (V), standard deviation, SD(V), and coefficient of variation, CV(V) = S D ( V ) / E ( V ) . The process of grain growth results in a systematic increase in the mean volume of grains: d

dSE(v)> 0.

(21

An increase in the mean size of grains can be accomplished through evolutions of the initial distribution function of grain volume, f ( V , t =0). A special theoretical meaning is given to so-called normal growth which preserves the shape of the distribution function. This implies the constancy of the normalized volume distribution function: Normal:f

,t

= f ( V , t = 0)

(3)

and the constancy of the coefficient of variation CV(V). Grain growth in experimental situations frequently departs from the condition given by equation (3). The two-dimensional space E ( V ) - C V ( V ) can be used to discriminate the grain growth character [18]. If CV(V) is constant with time the process is termed "normal". If CV(V) increases with time the process can be called "abnormal" and if CV(V) decreases with time--"supernormal". Due to the technical difficulties involved in grain volume measurements, the grain geometry is usually described by other parameters such as the grain area, A, or an equivalent grain diameter, deq (calculated as the diameter of the circle that has the same area as a given grain section). The grain growth path can be plotted in the space E ( A )-CV(A) o r E(deq)-C(deq ) [18]. Stereological rules link the grain area (diameter)

1543

distribution function with the volume distribution. These relationships depend on the resolution limit characteristic of the observation technique employed [19]. However, under some additional assumptions, a monotonic dependence of CV(V) on CV(A ) can be expected. The size of grains is uniquely described by their volume. However, the same volume can be ascribed to grains of a different shape. Hence the geometry of grains can be described more precisely by parameters which also define their shape. The following geometrical parameters and shape factors have been suggested for a description of the geometry of grains [20]: (a) maximum chord, dm and the shape factor, = dm/deq; (b) grain perimeter, p, and the shape factor,/3 = p / ~ • deq; where d~q refers to the equivalent circle diameter. The latter of these shape factors, /~, is a sensitive measure of the grain boundary undulations. The shape factor ct depends on the level of grain equixiality. It is equal to 1 for circles and increases its value with increasing elongation of the grains (strictly in 2-dimensions, grain sections). For equiax grain structures the mean value of ~ is in the range 1.25-1.30 [20]. 4. THE PHYSICAL AND GEOMETRICAL PROPERTIES OF GRAIN BOUNDARIES

Under quasi-equilibrium conditions, for example in annealed polycrystals, the geometry of grains can be expected to be related to their physical properties. Simple considerations show that in a polycrystal with all grain boundaries with the same properties, grains assume regular geometry which can be approximated by a set of equal-volume tetrakaidekahedra [15]. The system of such grains is distinguished by the condition C V ( V ) = 0. In this case the grain boundaries are joined along the common edges at an angle close to 120 ° . The angles between the grain boundaries, termed frequently dihedral angles, in near equilibrium conditions tend to depend on the differences in the energy of the grain boundaries [20]. The larger are the differences in this energy the larger is the departure of the angles from 120 ° . It can be further shown, on purely geometrical grounds, that a departure from 120 ° in equiaxed structures implies a larger variation in the size of grains [compare Fig. l(d) with l(c)]. For polycrystals characterized by a variation in the energy of the grain boundaries it should be expected that under near equilibrium conditions grains will differ in their size; the effect reflected by the larger values of CV(V). This can be rationalized by the following fact that the energy of the grain boundaries is related to the misorientation parameters. Therefore, a relationship can be expected to exist between the distribution function of the misorientation parameter and the value of CV(V), which measures the degree of uniformity of the size of grains. It is

1544

GARBACZ et al.: CSL BOUNDARIES IN POLYCRYSTALS

suggested here that the linking element between the distribution of the misorientation parameter and the distribution of the size of grains is the distribution function of the character of the grain boundary common edges. This idea can be schematically described in the following way: Texture ---~f(O, 1) ---*f(E) ---~f(TE) D-angles ~ CV(V)

(4)

where D-angles stands for dihedral angles. In the present study the elements at the ends of the schematic, the texture and CV(V), have been established experimentally. At the present stage measurements of the distribution of dihedral angles have not been made. However, ignoring torque terms the character of the grain boundary edges, f(TE) is related to distribution of dihedral angles. The internal elements in this scheme have been computed using a model of a polycrystal described in [21].

5. MATERIAL AND MODEL The relation between the texture and CV(V) has been studied for high purity aluminum (99.99%). Polycrystalline specimens of this material were extruded and then recrystaltized, for 1 hour at various temperatures. The microstructures obtained differed in grain size distributions [22], annealing textures and the distribution of the diffusivity properties of their grain boundaries [23]. A detailed description of the material and the distributions of grain boundary diffusivity for each type of sample are given elsewhere [23].

The texture of the material has been analyzed earlier [24]. The texture analysis for each type of sample can be approximated by a normal distribution--N(0, a) where a = 5° stands for the standard deviation and determines the sharpness of texture in terms of the deviation from an exact (hkl) direction (see Table 1). The method developed by Garbacz and Grabski [21] was used to model the distribution function of grain boundary misorientation parameters and values. In the model it was assumed that the orientation of a given grain was not correlated with its position within a polycrystalline aggregate. No clustering of texture components was allowed. This means that the texture was taken to be homogeneous and deviations from an ideal direction (hkl) were randomly ascribed to particular grains. The modelled distributions of grain orientation were used previously by Garbacz and Wyrzykowski [25] to simulate distributions of grain boundary misorientations and distributions of CSL boundaries. The main conclusions of the Garbacz and Wyrzykowski paper were that there exists a relationship between grain boundary diffusivity and grain boundary coincidence distribution functions. This indirectly confirms the adequateness of the model used to compute the distribution function of the grain boundary misorientation parameter. In the present study, the approach proposed by Garbacz and Grabski [21] has been extended to obtain the characteristics of the triple edges, TEs, of the grain boundary system for each texture. For each type of texture, associated with a given annealing temperature, three independent simulations of TEs were performed for a model of the polycrystal

Table 1. Parameters of grain size distribution, coincidence grain boundary and three-fold edges distribution for the tested AI polyerystal Annealing temperature (K)

533

553

573

593

623

648

673

Texture Component %

80(100) 20(111)

70(100) 30(111)

55(100) 45(111)

45(100) 55(111)

15(100) 65(111) 20(733)

60(111) 40(733)

No data

Sv/~m - 1 E (d)~)#m CV(A ) CV(n) CV(dcq) E(ct)

--------

--------

0.077 28.0 1.43 0.464 0.67 1.44 4.32

0.038 62.7 0.98 0.343 0.52 1.26 3.66

0.032 75.2 1.05 0.361 0.55 1.24 3.63

0.0247 102.6 / 0.89 .J 0.328 0.49 "] 1.26 / 3.57 ]

--1 =J

Ref. 24

Grain size distribution

E(fl)

0.053 42.7 1.18 0.405 0.59 1.37 3.96

19 26 27

Coincidence grain boundary distribution

0132 0.,32

0.105 0124

0102 0.119

00 7 011,

0.080 0110

0.229

0.221

0.201

0.190

0.513 0.382 0.095 0.011 0.487

0.536 0.367 0.087 0.010 0.464

LAB + CGB

0.301

0.264

RRR RRS RSS SSS RRS + RSS + SSS

0.362 0.416 0.176 0.046 0.638

0.403 0.416 0.152 0.029 0.597

25

Distribution of three-fold edges 0.463 0.408 0.114 0.015 0.537

0.478 0.401 0.111 0.013 0.522

--7 ~J

This work

LAB stands for Low Angle grain Boundaries; CGB stands for Coincidence Grain Boundaries; LAB + CGB stands for Low Angle grain Boundaries plus Coincidence Grain Boundaries.

GARBACZ et al.: CSL BOUNDARIES IN POLYCRYSTALS

1545

0.48 0.44

> o

0.40-

0.36-

0.32 280 300 320 340 360 380 400 420 Ta [°C]

Fig. 4. The dependence of the coefficient of variation of the number of grain boundary edges, CV(n) on the temperature of annealing. Fig. 2. Schematic explanation of the parameters used for The other parameters, as explained in Fig. 2, have description of the geometry of individual grains: A is an area of a grain section, deqis the diameter of circle of area A, dmaX been measured for individual grains. Histograms of is the length of the maximum length chord, p is the grain grain section area, .4, for the specimens recrystallized section perimeter and n is the number of sides. at various temperatures are shown in Fig. 3. The mean values of A, ~ and fl, E(A ), E(ct), E(fl), are consisting of 10 layers of 20 × 20 grains which given in Table 1. It should be noted that the mean permitted examination of a total of 24,000 TEs. Each value of n, E (n), for the grain boundary network CSL boundary with coincidence, E, ranging from 1 to revealed on sections of polycrystals equals 6, disre29 was counted into an "S" category. The common garding the details of the grain geometry. Table 1 also edges in the system of grain boundaries have been lists values of the coefficients of variation, CV(A ), examined in terms of the number of special/random CV(deq) and CV(n). The dependence of CV(n) on the grain boundaries which they connect. annealing temperature is shown in Fig. 4. The results of these measurements indicate that the 6. RESULTS AND DISCUSSION polycrystals annealed at various temperatures differ The geometry of the grains in the series of alumi- in size. This is reflected by a systematic increase in the num polycrystals have been determined with an equivalent grain area, E (deq), and in the grain boundautomatic image analyzer [26, 27]. The grain geo- ary area per unit volume, Sv. The systematic increase metry was described, as a function of the annealing in the mean size of grains is accompanied by changes in the normalized grain size distribution function, temperature of the polycrystal, using the following f ( V / E ( V ) ) . This is reflected by alterations in the parameters: (1) Sv--grain boundary area in unit histograms of grain section area, A, and in changes volume; (2) .4 and deq; (3) shape factors cc and fl; (4) in the coefficients of variation CV(A ) and CV(d~q) number of grain section edges, n. The values of S,r have been estimated from the mean which are monotonic functions of CV(V). Values of these two coefficients decrease with increasing grain intercept length, E(/), using the well-known stereosize. This means that the series studied achieves a logical relationship in the following form: higher degree of grain size uniformity with increasing 2 S, (5) grain size. E(/) The results of the measurements of grain geometry also show that the increase in the size of grains is 0.6

ml 533 K I

0.5

.

.

.

.

I-'553 K I

0,4

573 K I

g

o.4

~

0.3

0.3

" 3K

0.2

N 0.2

~

0.1-

o.1

o

1

2

3

4

5

6

A/E(A)

Fig. 3. Histograms of grain section area, A, for specimens annealed in the temperatures in the range 533~48 K. AM 43/4--P

.

Mi

0.5

o.o

.

O---

RRR

RRS RSS TYPE OF TEs

SSS

Fig. 5. Histograms of the TE characteristics as a function of the annealing temperature.

1546

GARBACZ et al.: 120

O O o

CSL BOUNDARIES IN POLYCRYSTALS 1.6-

a)

CV(de q ) ~

80

1.2

T-

x > U)

g "o

E

40

0.8

E(A)

g

CV(A)*~""~..~

.

W 0.4 4.0

0.8

E(~)

C)

3.5 (n

Cn rr

3.0

2O

2.5

,< u. ILl n < "r (n

0.7

2.0

TE

(q=1,2,3)

d)

~-- 0.6 "6 ~

E(~)

0.5 0.4

U.

1.5

0.3 1.0

I

0.2

0,5 (/) W

0.4"

!

e) 0.3

m

¢, z

0.2

g < If.

0.1

;ao

"o e60 s4o ANNEALING TEMPERATURE, K

8ao

Fig. 6. Temperature of annealing effect on the parameters describing the geometry of grain and the characteristics of the threefold edges: (a) alterations in the mean size of grains; (b) alterations in the coefficients of variation of the grain area and equivalent diameter, CV(A ) and CV(d~); (c) alterations in shape factors; (d) alterations in the fractions of TEs; (e) variation of the fraction of CSL boundaries. associated with an equilibration of their shape. There is a clear tendency for the mean values of E (~) and E (fl) to reduce as the mean grain size, E (d~q), rises. The grains become less elongated, less undulated and more uniform in terms of the number of edges. It may be noted that extensive changes in the shape of grains and the degree of grain size uniformity take place as a function the annealing temperature in the range 533-625 K. These changes are accompanied by a systematic decrease in the sharpness of the material texture. Higher temperatures of annealing seem to lead to some steady state in the development of the texture, shape of grains and the degree of the grain size uniformity.

The results obtained from modelling the TE distribution functions for different annealing textures/ temperatures are presented in Fig. 5. This figure presents histograms of TEs of a specified number of random/special boundaries. It may be noted that there is a clear dependence of the TE character distribution function on the texture of the polycrystals. Generally, the fraction of TEs without special grain boundaries, (q = 0), increases with increasing mean grain size while the fractions of all other TEs, (q = 1, 2, 3), decrease. The results of these computations show, that within the framework of the model adopted, the observed evolution in the texture of the material is

GARBACZ et al.: CSL BOUNDARIES IN POLYCRYSTALS accompanied by systematic changes in the distribution function of the TE character--see Fig. 6. It may be noted that the evolution in the texture of the material causes a systematic increase in the number of TEs joining 3 random boundaries TE(q = 0 ) (RRR). The computed variation in the relative numbers of different types of TE agrees well with the observed variation in the character of the grain size distribution function. The increasing fraction of RRR common edges is accompanied by an increasing degree of grain size uniformity; as reflected by decreasing values of the coefficient of variation of grain volume. The decreasing fraction of the common edges with at least one special grain boundary correlates also with variations in the shape of grains. There is a clear tendency for the grains to assume a more equiax shape with a decreasing factor of (SSS) + (SSR) + (SRR) common edges.

7. CONCLUDING REMARKS

In this work an assumption has been made that the pattern of changes in the geometrical features of the grain boundaries can be explained in terms of the changes in their physical properties which, in turn, are functionally related to the polycrystalline texture. In particular, it has been shown that there are logical arguments leading one to expect that the distribution function of grain boundary misorientations and the characteristics of the common edges of the grain boundaries have an influence on the degree of the grain size uniformity. The suggested relationship between the geometry and physical properties of grain boundaries has been studied using a series of aluminum specimens. The specimens were recrystallized at various temperatures and differed in their texture and in the geometry of their grain boundaries. The measured textures of the specimens were used to compute the distribution functions of the grain boundary misorientation and the characteristics of the common edges, TEs. These two modelled distributions and, in particular, the distribution functions of TEs, have been found to be in agreement with the observed evolution in the distribution function of the grain size. The degree of grain size uniformity has been found to correlate with the fraction of common edges joining three grain boundaries with a random misorientation. There are two interesting implications of the results of the present study. The first is related to the widely-accepted contention that the geometrical and physical properties of grain boundaries control the properties of polycrystals. The fact that these classes of properties, physical and geometrical, are related implies that they should not be studied separately (with the exception of situations where the properties of one of the classes vary to a very small degree).

1547

The second implication is related to the process of the evolution of the microstructure in polycrystals. The approach presented in the present paper underlines the need for an integral analysis of the changes taking place in the system of grain boundaries. Grain boundaries are seen to change their geometry as well as their physical characteristics. As a result, studies of the driving forces controlling microstructural evolution should consider the variations in the physical properties of grain boundaries as well as grain geometry. Acknowledgement--This work has been supported by a

grant from Polish Committee for Scientific Research. REFERENCES

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