- Email: [email protected]
Microstructural path concept applied to normal grain growth
Acta metall, mater. Vol. 40, No. 6, pp. 1159--1166,1992
0956-7151/92$5.00+ 0.00 Pergamon Press Ltd
Printed in Great Britain
MICROSTRUCTURAL PATH CONCEPT APPLIED TO NORMAL GRAIN GROWTH R. A. VANDERMEER Naval Research Laboratory, Materials Science and Technology Division, Washington, DC 20375-5000, U.S.A. (Received 15 March 1991; in revised form II November 1991)
Abstract--Normal grain growth in solids is considered in the light of a microstructural path concept. The topological model first presented by Rhines and Craig is corrected and reformulated so as to include the Doherty sweep constant. The reformulated model predicts the experimentally observed linear time dependence for volumetric grain growth only when the average curvature per grain remains constant during growth. A geometrical path function for normal grain growth relating average microstructural properties on a per grain basis is proposed and compared with experimental observations on aluminum and titanium. By combining the geometrical path function with the topological model, further comprehension of normal grain growth kinetics results, specifically the role of the average grain shape. R6sttm6---On consid6re la croissance normale du grain dans les solides fi la lumi6re du concept de trajet microstructural. Le mod61e topologique pr6sent6 pour la premi6re fois par Rhines et Craig est corrig6 et reformul6 de fagon fi inclure la constante de balayage de Doherty. Le mod61e reformul6 ne pr6voit la d6pendance lin6aire observ6e en fonction du temps de la croissance en volume du grain que lorsque la courbure moyenne par grain reste constante durant la croissance. Une fonction de trajet g6om6trique pour la croissanee normale des grains, liant les propri6t6s microstructurales moyennes par grain, est propos6e et compar6e fi des observations exp6rimentales sur l'aluminium et It titane. En comparant la fonction de trajet g6om6trique avec le mod61e topologique, on comprend mieux la cin6tique de croissance normale du grain, et en particulier le r61e de la forme moyenne du grain. Zusmnmenfassnng--Das normale Kornwachstum in Festk6rpem wird im Bilde tines mikrostrukturellen Pfades betrachtet. Das zuerst von Rhines und Craig vorgelegte topologische Modell wird berichtigt und neu formuliert, um die Bewegungskonstante yon Doherty einzubauen. Das neu formulierte Modell sagt die experimentell beobachtete lineare Zeitabh~ingigkeit des Volumwachstums der K6rner nur dann voraus, wenn die mittlere Kriimmung pro Korn w/ihrend des Wachstums konstant bleibt. Eine geometrische Pfadfunktion des normalen Kornwachstums, die die mittleren Mikrostruktureigenschaften pro Korn beschreibt, wird vorgeschlagen und mit experimentellen Beobachtungen an Aluminium und Titan verglichen. Aus der Kombination dieser geometrischen Pfadfunktion mit dem topologischen Modell folgt ein erweitertes Verst/indnis der Kinetik des normalen Kornwachstums, insbesondere der Rolle der mittleren Kornform.
1. INTRODUCTION In the solid state, the microstructure of metallic materials is normally observed to be polycrystalline in nature. This means that the mass is composed of an aggregate of contiguous crystals called grains, each of a different orientation from its neighbors. Physically, each grain is separated from its neighbours by a thin grain boundary layer or interface region having a thickness of atomic dimensions. On a microstructural scale, this interface forms an interconnected array that may be regarded as a three dimensional topological network, i.e. it consists of nodes (grain corners), branches (grain edges) and faces (grain boundaries). In a thermodynamic sense this interface array imparts an excess energy to the material which would not be present were the mass a single crystal instead. The material perceives the excess energy through the AMM 40/6---D
topological grain boundary network as an interfacial tension in the sense discussed by Gibbs [1]. If all the grain boundaries in the system have the same energy, i.e. interfacial energy does not vary with the misorientation between grains (an assumption used throughout the paper), then as pointed out by Smith [2], the surface tension has the impetus of causing the grain boundaries coming together to form triple junctions (grain edges) to tend to meet in dihedral angles of 120 ° . Likewise, it is the tendency for the four triple junctions meeting to form the quadruple points (grain corners) to approach mutually included angles of 109 ° 28' and 16". Given sufficient thermal activation and atomic mobility, the a m o u n t of grain boundary area in the polycrystalline material can be reduced, thus decreasing the excess energy of the system. This reduction is accomplished by the systematic elimination of some
1159
1160
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH
of the grains from the material and the concomitant enlargement of the remaining grains, coordinated with such grain size and shape adjustments as are needed to maintain perfect shape filling at all times. This evolutionary process is known as grain growth. The mechanism for grain growth is generally thought to involve the migration of the grain boundary segments normal to themselves by atomic jumps across the interface as discussed by Turnbull [3]. When the grain growth process is characterized microstructurally by a progressive increase in the average size of the grains without seeming to change their geometric form, i.e. shape, number of sides per grain, size distribution etc., the process is referred to as normal or steady state grain growth. By contrast, the growth process is designated abnormal or discontinuous grain growth when normal grain growth stagnates for one reason or another and a select few grains, by virtue of having highly mobile interfaces, emerge from and grow into the matrix of smaller stagnant grains until they are consumed. By analogy with recrystallization, abnormal grain growth may be treated phenomenologically as a nucleation and growth process. In this paper microstructural path concepts are introduced and applied to normal grain growth. A model of grain growth based on the average microstructural properties of the material is presented. Abnormal grain growth will be discussed in a subsequent paper. 2. MICROSTRUCTURAL PROPERTIES
The grain size of a material is usually characterized in terms of the mean intercept-free grain length, 2, which is measured by counting, on a two dimensional plane of polish, the number of grain boundary intersections with a unit length of test line, NI. The 2 is equal to 1/N1. In principle, however, what is determined by measuring N~ is the important stereological property, Sv, the grain boundary area per unit volume because by definition [4], Sv = 2. Nl. As stressed by Rhines [5], Sv and hence 2 are very sensitive to the shape of the grains in the microstructure. Rhines [5] also pointed out that there are several other measurable, global microstructural properties which are critical to a quantitative description of the microstructure of a single phase, polycrystalline material and the evolution of normal grain growth. They are, M~, the total grain boundary curvature per unit volume and Nv, the number of grains per unit volume. All of the microstructural properties mentioned can be measured using stereological procedures [4]. While SI and Mv can be measured with relative ease on a random, two dimensional plane of polish, the measurement of Nv can only be accomplished using laborious, serial sectioning techniques, and unfortunately it's measurement is rarely carried out. Assuming that the three global properties discussed above are sufficient to characterize the microstructure,
the normal grain growth process can be embodied in the function
F{Sv(t), My(t), Nv(/)} = 0
(1)
where t is the annealing time. Because of the definition noted above, 2 may be substituted for Sv in equation (1). This function may be recognized as a microstructural path function because it defines the instantaneous state of the microstructure and how the state microstructural variables change as grain growth takes place with time. The way the path function is deduced for normal grain growth is the subject of this paper. Partial microstructural path functions can also be formulated. An example is the kinetic function 2 =f(t). This is the function usually sought experimentally and is the one for which a number of theories of normal grain growth have been developed. Also in this paper another partial function, Sv = f(1/N~), which is a geometrical path function, will be proposed. 3. THEORIES OF NORMAL GRAIN GROWTH
The status of the theories of normal grain growth in pure single phase material systems was reviewed very recently by Atkinson [6]. The earliest theory of grain growth was first annunciated clearly by Burke and Turnbull [7]. Based on the idea that grain boundaries tend to migrate towards their centres of curvature and that the radius of curvature is proportional to the mean grain radius, they assumed that the grain boundary migration rate was proportional to the pressure difference across the boundary. From what may be considered to be essentially simple dimensional arguments, Burke and Turnbull deduced parabolic grain growth kinetics of the form 2 2 - 2 2 = x" t
(2)
where 20 is the mean intercept-free grain length at t = 0, 2 and 2o are assumed to be proportional to the appropriate mean grain radii and r is a rate constant. Beginning with Feltham [8], a number of more sophisticated theories were developed over the years. Most of these theories take a mean field approach and view the process of grain growth in terms of the change in the grain size distribution as the number of grains in the system decreases and the mean grain size increases. The growth or shrinkage of individual grains is considered amid an environment characterized by an average over the entire ensemble of grains. The flux of grains through grain size space is either curvature-driven by a drift term as in Hillert's theory [9] or treated statistically as a diffusion-like process without the need for curvature-driven drift in the case of Louat's theory [10]. Both approaches predict parabolic kinetics like equation (2) but for different reasons, Hillert's because the curvature is taken to be inversely proportional to grain size, and Louat's because of the diffusional aspect of the formulation.
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH An interesting blending of these two ideas was presented recently by Pande [11] who assumes the flux through grain size space to be a super position of a statistical, diffusion-like noise term upon the curvature-driven drift term. Both the Pande theory and the Louat theory yield more realistic predictions for the grain size distributions observed experimentally than does the Hillert theory. One criticism of all these theories is the lack of concern regarding topological constraints and their role in grain growth. To address that issue, the work of Rhines and Craig [12], hereafter referred to as RC, and the followup considerations of it by Doherty [13, 14], stress the topological aspects of grain growth. The RC approach is predicted on microstructural and topological property measurements made during steady state grain growth in aluminium in which it was found by a serial sectioning technique that the number of corners, edges and faces (boundaries) all on a per grain basis approached and then maintained, on average, the numbers 6:7:12 during the growth process. Topologically these numbers describe a tetrakaidecahedron (sometimes referred to as a truncated octahedron). RC view the toplogical path in grain growth as a degrading of complex grains (more than four faces) to simple grains (four faces or fewer) by the loss, one at a time, of triangular faces. To maintain the 6:7:12 ratio during grain growth, the elimination of grains must be such as to occur by the removal or corners, faces and edges in exactly this ratio. This could be achieved, for example, by the collapse of one tetrahedral grain and three grain faces in the material. All such processes are equivalent to removing a tetrakaidecahedral grain without changing the average topology of its neighbors. These considerations led RC to postulate that steady-state grain growth can be described in terms of the elimination, one at a time, of grains of average volume each having the average topological characteristics of a tetrakaidecahedron. Strictly speaking the RC approach is more a model of grain growth than a theory because, unlike the theories of Hillbert, Louat and Pande, the RC formulation cannot predict a grain size distribution. RC assume that the mechanism by which grains in a material shrink and vanish under the driving force of surface tension, thereby transferring their volume to the grains remaining in the system, is by the sweeping of grain boundary. This sweeping action occurs on a global scale and over an extended time period. Anyone observing grain growth taking place in the photoemission electron microscope is struck by the activeness and extent of these boundary sweeps. Using several simple, albeit somewhat unrealistic models as analogs, RC deduced that, irrespective of the size of the average grain, the total volume swept by the grain boundary is always the same for the disappearance of one average grain including the concomitant adjustment of the remaining grains to being equiaxed. Thus, RC define a sweep parameter O,
116l
which they assumed to be a constant, such that O is the number of grains lost from the system when grain boundary sweeps out a volume of material equal to a unit volume. In this definition, the units of O are dimensionless. RC express grain growth kinetics in terms of the topological parameter Nv, the number of grains per unit volume of material. The inverse of Nv is the average volume per grain, Vg, which is a hybrid topological-geometrical property of the microstructure. The kinetic relationship derived by RC is given by 1
1
N--~= ~ ( 1 + O "#Rc'7 '¢y • t)
(3)
where N o is the number of grains per unit volume initially, O is the sweep parameter, #RC is the mobility (velocity per unit force), y is the grain boundary energy (surface tension), a is the product of the average curvature per grain, Mv/Nv, and the grain boundary area per unit volume, Sv, and t is again the annealing time. The parameter a is regarded as the dimensionless structural gradient of the material. Experimentally, RC found that o was a constant for grain growth in aluminum. As Fig. 1 indicates, equation (3) describes the experimental results of RC and Rhines and Patterson [15] for aluminum very well. For reasons discussed later in the paper, 1/Nv can be deduced to be proportional to 2 3, so that equation (3) does not predict parabolic grain growth kinetics as do other theories and models. Why this is so has never been fully understood. In a discussion to the RC paper [12], Doherty [13, 14], hereafter called Dh, took issue with the definition of the sweep constant and pointed out that O would not be a true constant in grain growth. Dh proposed another sweep parameter, O*, defined slightly differently, as the number of grains lost when grain boundaries sweep through a volume equal to the volume of the average grain, Vg. More recently Hunderi [16] showed how to calculate the sweep parameter within the framework of the Hillert grain growth theory. He also concluded on the basis of computer simulations of grain growth in two dimensions that for the sweep parameter to be truly a ,
5 10 .4
3 10 .4
-
2 10 .4
_ -
,
i
I 10 .4
. . . . . .
,
Grain Growth in Aluminum 635 °C
~r~-G-r
R
h
•
.
.
i
~
0 0, 20
40
Time
60
80
100
(min)
Fig. 1. Average grain volume as a function of time for grain growth in aluminum.
1162
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH 0.25
0.2 ........................................................ i............ ' " - / " ~ 0.15
.
.
.
.
.
.
.
/--
o1
..................... :.......................... -
.
.
.
"
:7
.....................
/ . . . . . . . . . . . . . . . .
0 0
~l,~,r 10
. . . . . . . . . . . . . . . . . . . . . . . . . .
i .... 20
I .... 30
. . . . . . . . . . . . . . . . . .
i
i ........ 40 50
i .... 60
70
Time (rain) Fig. 2. Average curvature per grain plotted vs time for steady-state grain growth in aluminum. Data of Rhines and Patterson.
constant, it must be defined in terms of the mean grain volume as proposed by Dh. Furthermore, Dh noted a mathematical error in the derivation of equation (3) which cast doubt on its validity. Dh recast the RC model in terms of a different mobility, one based on the velocity per unit pressure. In so doing, he found that in differential form, dI
Nv
= O*
'~D'r
. M~ dt
Nv
(4)
where #D is the new mobility. For an integration of equation (4) to yield the same linear dependence on time as equation (3), the average curvature per grain, M,/N~, must be a constant. Dh stated that such may likely be the case. However, the experiments of RC on aluminum, the only available data with which to test the idea, do not substantiate this view as Fig. 2 demonstrates. If these results are correct, then under the assumptions of the model, the integrated form of equation (4) will not lead to a linear dependence of volumetric grain growth on time. As noted above, the microstructural modeling approaches of RC and Dh lack an explicit consideration of the grain size distribution and how it changes during grain growth. On this issue, the analysis of Kurz and Carpay [17] seems to be relevant. These workers assumed that the grain size distribution and the topological feature distribution are log-normal in nature in both two dimensions and three dimensions. They were able, by expanding on the work of Feltham and combining it with the approach of RC, to show that the RC structure gradient, tr, is indeed a constant and, furthermore, that tr is related to the standard deviation of the grain face distribution. In contrast to the 23 vs t kinetic relationship derived from the RC theory, Kurz and Carpay deduced the customary parabolic grain growth kinetics, i.e. 22 vs t. 4. A REFORMULATION OF THE RHINES AND CRAIG MODEL This reformulation of steady-state grain growth kinetics seeks to understand the origin of the 23 vs t
behavior. It retains the basic postulates of RC except when they are in error. Consider a system of unit volume at time, t(s). Let #v be the mobility associated with the moving grain boundary, i.e. velocity per unit driving force acting at a point (cm/Ns) and ,l the surface tension (N. cm/cm2). This definition of the mobility retains the logic of RC; it differs from the mobility, #o, proposed by Dh, which is based on a force per unit area (pressure) definition. It can be shown from reaction rate theory that the two approaches should be essentially equivalent and that #v differs from /~D by only a constant. What the constant is will be shown later. RC and Dh take the average pressure, P, acting on a grain boundary segment to be given by the product of the surface tension and the curvature such that p = 7" My (N/cm2).
(5)
s~
At this point in the development, Dh writes the average grain boundary velocity, ~, in terms of the driving force, P, making use of the proper assumption that velocity is the product of mobility and driving force. On the other hand, RC consider the driving force as P ' S v which unitwise is a force per unit volume (N/cm3). The implication of the RC approach is that it essentially spreads the driving force through the entire unit of volume. Physically this does not seem correct. It seems to be more realistic to suppose that the driving force is sensed only by those atoms at the grain boundaries, the ones participating in the migration process. The driving force per grain boundary atom is given approximately by P . fl2/3, where fl is the atomic volume. With this correction to the RC model, the average grain boundary velocity can be written as f = / ~ . p . f~2/3= f~2/3. #v" Y " My
(cm/s).
where the average is taken over all the grain boundary area in the system. In the interval of time between t and t + d t , the total volume swept out b y all the grain boundary segments in a unit volume of material is given by dV swept-- S v " 6" dt = f~2/3. #v '7 " My" dt
(cm3/cm3).
A number of grains are lost in this time increment. This number, dnv, on a per unit volume basis can be obtained by introducing the Doherty sweep constant, O*, defined earlier. Thus dnv
O* 7s dVv
"It~'r'N,'Mv'dt.
According to the postulate of the RC model that, on average, each of the grains lost is of average volume,
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH the total volume lost in the increment, dt, per unit volume of material can be written as d V l v°st =
~g"
10-I
• ' '"'"!
......
"1
.......
1
......
10. . . . . . . . . . . . . . . .
dnv
1
......
........
1163 "1
.......
1
.......
1
.....
....................
Aluminum
dnv
Nv
8"
D2/3.#~ y My dt
10.3
(cm3/cm3). Z
But volume must be conserved so that the volume lost by the disappearing grains must be gained by the grains remaining in the system, i.e. dV l°st= dV gained. The volume gained is averaged over all the grains left in the system giving, on a per grain basis, the equation 1
d V~ ~i"~
Nv
d ~ = dfrs=
= O * ' ~'~2/3 . # v ' Y " ~ "
dt
(cm3).
(6)
which except for the mobility and the atomic volume, both of which are constants, is identical in form to the Dh equation, equation (4). It also indicates that ~/D= f~2/3 . /'/v" For equation (6) to be integrated to the form of the RC equation, equation (3), the average curvature per grain must remain constant during grain growth. 5. A GEOMETRICAL PATH FUNCTION F O R STEADY STATE GRAIN GROWTH RC observed that during normal grain growth, the average grain possessed the topological parameters roughly equivalent to that of a tetrakaidecahedron. Assuming that the global microstructural properties of the grains in a material can be described equivalently in terms of the properties of the average grain, a relationhip between Sv and Nv can be expressed following Kaiser [18], (see also Ref. [4]) as (7)
S v = F . Nlv/3
where F is a constant that depends on the specific geoemetric shape. Table 1 lists the values of F calculated by Kaiser for four different space-filling polyhedral shapes. If equation (7) is divided by N~, then the following formula is obtained Sv = F "
NZ
= F " (~g)2/3
(8)
which is essentially a function describing how the average interfacial area per grain varies as a function of its volume. In a geometrical sense this equation can be regarded as a grain shape or geometrical path function for steady-state grain growth. Its similarity
T a b l e 1. S h a p e factors for spacefilling p o l y h e d r a Polyhedron Cube Hexagonal prism Rhombic dodecahedron Tetrakaidecahedron
F 3.000 2.866 2.673 2.659
10-4
lO.S
. . . . . . . . . .
1 O" 6
.......
~
10"to
Tetrakaid~abedra Model
....
.......
J
.......
J
.......
10"*
I
......
.t
......
10.6
a
.......
a
10 .4
.....
10 .2
1 / N v (cm 3)
Fig. 3. Comparison of the geometrical path model for steady state grain growth in aluminum with experiment. • Rhines and Craig; C) Rhines and Patterson; • Okazaki and Conrad. to a path function proposed for recrystallization by Vandermeer, Masumura and Rath [19], namely, Sv = f ( X v ) , where Xv is the volume fraction recrystallized, should be noted. Regrettably most experimental studies of grain growth only measure Sv. But, the grain growth investigations on aluminum at 635°C by RC and Rhines and Patterson [15] and the studies by Okazaki and Conrad [20] on titanium measured both Sv and Nv. These studies thus afford the possibility to test whether equation (8) may be applicable. In Fig. 3 the average interfacial area per gain is plotted vs the average volume per grain on log-log scales. As Fig. 3 depicts, describing steady-state grain growth by means of a tetrakaidecahedron model in accord with the topological observations, i.e. F = 2.659, is very reasonable, especially for aluminum. Note that the experimental data for aluminum extend over an average grain volume range of almost two and a half orders of magnitude. In most studies of steady-state grain growth, the microstructural parameter, 2, is measured and taken to be 'an average grain size. Under the assumptions that the grain shape remains constant and the grain structure is equiaxed, the geometrical path function given by equation (8) can be combined with reformulated RC theory [equation (6)] to find a relationship between ). and time as follows: Inserting equation (8) into equation (6), substituting for Sv by 2, differentiating and rearranging, the equation can be written as d22 =--16- O* • f~2/3. Pv" )' My ,dt 3 F3 2 • Nv where F is considered to be constant. Replacing the 2 on the right hand side of this equation with Sv, yields the very interesting result d 2 2 = 8 0 * ' ~ ' ~ 2 / 3 " ] / v ' y S~ " M v d t 3 F3 Nv 8
~ * " ~"~2/3 ' ]./v ' ~ " O"
3
F3
dt
1164
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH
where the right hand side now contains the RC structural gradient constant, a which experimentally was found to be constant. Upon integration, this equation becomes 8
22 -- 22 = 3
~ *
" ~'~2/3
" /d,v ' '~
F3
" O"
t
(9)
which is equivalent to equation (2) in giving parabolic kinetics. If 20 in equation (9) is small and negligible, 2 is proportional to t ~/2.
4.0
10 "a
....
TT,.?, ............
i ....
v ....
i ....
i ....
II
l
I
/
'
........
.....
_2Oo
.
.
.
. . . . . . . .
i 10 °
.... 0
The form of the steady-state grain growth kinetics function [equation (9)] derived from the microstructural path approach agrees with most theories [6-11]. Very recent work by Mullins and Vinals [21] and Mullins [22] have demonstrated that the two conditions necessary for the parabolic growth law to be obeyed are (1) a statistical self-similarity holds, i.e. consecutive configurations developed by the structure as growth proceeds are geometrically similar in a statistical sense, and (2) the grain boundary velocity is proportional to the local mean curvature. In this vein, the Monte Carlo simulations of three dimensional grain growth by Anderson et al. [23, 24] are also of interest. These investigators found that the grain growth exponent, i.e. the power of t in a bulk grain growth experiment, determined from the grain size growth kinetics was close to 1/2. The physics of the Monte Carlo simulation is based on lattice site switching such that when the energy of the system is lowered by the switch, the switch gives rise to a local movement of the boundary by one lattice spacing. The simulation, therefore, seems to localize the driving force at the grain boundary lattice sites as the reformulated microstructural path model does. From the analysis carried out in this work, it is apparent that the 23 dependence on time implicit in the RC model [equation (3)] arises from an improper consideration of the driving force for grain growth, specifically how that force is distributed among the atoms in the material. Indeed, it can be readily shown that by following the RC logic and distributing the driving force over every atom in the volume rather than over just the grain boundary atoms and carrying through the analysis presented in Section 4 above, a 23 rather than a 22-dependence on time results. As alluded to earlier, the stereological measurement yielding 2 is more a measure of the amount of grain boundary area per unit volume than it is grain size. RC argue that I/Nv is the only real measure of grain size that is not shape dependent. They suggest that 2 should be very sensitive to grain shape and equation (9) reveals this to be true. This point will now be addressed. Using the experimental Sv data of RC, a plot of 22 vs time is shown in Fig. 4 for aluminum. Also illustrated in Fig. 4 is a curve calculated from the tetrakaidecahedral model with the constants estimated from the 1 / N v vs time data of Fig. 1. The
i ....
~.o ,o. . . . . .
0.0
6. DISCUSSION
r ....
[ Volumetric
i .... 10
, .... 20
~ . . . . . . . . . 30
40
i . . . . . . . 50
60
, 70
Time (min)
Fig. 4. Plot of 2 2 vS time for steady state grain growth in aluminum, Data adopted from Rhines and Craig where 2 = 2/Sv.
poor fit of the data in Fig. 4 to a straight line as predicted by equation (9) or to a line determined from the volumetric model of RC is readily apparent. The letters beside each data point refer to the topological properties found experimentally by RC (extracted from Fig. 6 of their paper). Only at the longest annealing times (data points labeled T) did the topological relationship approach that descriptive of a tetrakaidecahedron. At shorter annealing times (data points labelled D), the topological parameters were close to ones describing a decahedron. (The ratio of corners to faces to edges on a per grain basis are 4: 5: 8 for a decahedron while for a tetrakaidecahedron they are 6:7:12, respectively.) Thus, it would appear that there was grain shape changes taking place during the grain growth in aluminum at 635°C investigated by RC. Upon reviewing the RC grain growth results, Atkinson [6] alluded to this possibility also. Further demonstration of the role of grain shape change on grain growth kinetics will now be presented. From Table 1 it is apparent that F is a decreasing function of the number of faces comprising the polyhedron. Thus, if during grain growth the topological characteristics of the average grain changes from say a lower number of faces to a higher number, as is experimentally observed, then the value of F will decrease, introducing deviations in the linear 22 vs t plot. However when the average grain contains 12 or greater number of faces the value of F is fairly insensitive to the polyhedon's shape. Thus, the later stages of normal grain growth should approach nearly linear 22 vs t behavior but with a 20 that is only an apparent one. The data in Fig. 4 seem to conform to this tendency. A similar argument can be made if on average, grains deviate from being equiaxed. Consider an hexagonal prism shape. The value of F for this shape depends on the c/a ratio. Thus for that ratio being 1/2 (pancake tendency), 1 (equiaxed) or 2 (needle tendency), the Fvalue is calculated to be 2.962, 2.866 or 3.065, respectively [4]. Again assuming that the initial grain structure deviates from being equiaxed but approaches such during grain growth, the value
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH of F will decrease during growth again causing a deviation in the 22 vs t plot. Clearly grain shape changes can influence the time dependence of 22 significantly, leading to a non-linear time dependence. It is concluded that 22 vs time which contains F must be very sensitive to shape changes, while 1/Nv vs time which does not depend on F is relatively insensitive to shape change, thus confirming the validity of RC's concern noted earlier. There is one issue that remains unresolved regarding the RC experiments on aluminum. That pertains to the inconsistency of the experimental data shown in Figs 1 and 2 when examined in light of the reformulated model. If 1/Nv is linear with time as appears to be the case (Fig. 1), then according to the model [equation (6)], Mv/Nv should be a constant. This was not observed for aluminum (Fig. 2). Alternatively, if M~/Nv is not a constant, then the model predicts that the 1/Nv should not be linear with time. Two possibilities can be put forward to account for this discrepancy. Doherty [13] suggested that if the velocity~lriving force relationship is modified by a velocity independent drag term, then under certain conditions, presumeably impurity related, an approximately linear 1/N v vs time relationship could result. Another possibility is that the velocity may not be a linear function of driving force due to impurity drag effects as envisioned by Cahn [25] or in a recent new theory by by Louat and Imam [26]. Further studies are necessary to decide this issue. A few comments concerning the role of a on grain growth kinetics seem appropriate. The grain growth studies of RC and Rhines and Patterson [15] were both carried out on aluminum of the same purity at 635°C. Yet as Fig. 1 illustrates, the rates of volumetric grain growth are radically different. On the other hand, the geometrical path function for the two studies is essentially the same (Fig. 3). Since in both studies the volumetric growth kinetics are observed to be linear with time, the difference in rates must be associated with the structural gradient parameter, a. This should be apparent from consideration of equation (9). Unfortunately Rhines and Patterson did not report a value for a for the grain growth they observed. They did show convincingly, however, that, other factors being equal, grain growth kinetics are strongly influenced by the width of the grain volume (face) distribution which is largely determined by the degree of cold work during the deformation process prior to recrystallization and grain growth. Wider distributions were demonstrated to promote faster grain growth. Kurz and Carpay [17] have shown a connection between a and the character of the grain volume (face) distribution. Thus, it is concluded that RC used deformation procedures that were significantly different from the Rhines and Patterson procedures (for the data presented here the aluminum was compressed 80%). Unfortunately RC did not report the complete details of their deformation procedure.
1165
Finally, an implicit message emanating from the topological approach is that a thorough characterization of the microstructure and the microstructural path is essential for an improved understanding of normal grain growth. In other words, it is not enough to measure only the property 2 (or Sv). Additional microstructural property measurements are demanded if a more complete comprehension of the process is desired. 7. CONCLUSIONS 1. The concept of microstructural path can be developed for steady-state grain growth and is shown to be applicable to grain growth in aluminum for which a geometrical path function is also deduced. 2. The topological grain growth model of Rhines and Craig can be corrected and reformulated using the Doherty sweep constant O* to yield a linear dependence of volumetric growth on time, but only when the average curvature per grain remains constant during growth. 3. Introducing the geometrical path function into the modified model has allowed clarification of the kinetics of grain growth as follows: (a) A linear ;~2 dependence on time is predicted provided the average grain shape does not change during growth. (b) If the average grain shape changes, deviations from the predicted parabolic time dependence of 2 can be understood, at least qualitatively, in terms of the role of the grain shape factor, F. (c) The 23 vs time behavior implicit in the earlier RC model appears to result from an improper distribution of the driving force for grain growth among all the atoms in the system rather than those at the grain boundaries where the migration process takes place. Acknowledgements--The author would like to thank Drs C. S. Pande, B. B. Rath, J. C. Ayers, R. A. Masumura and Professor R. Doherty for helpful discussions and for reading the manuscript. This work was performed at the Naval Research Laboratory under the sponsorship of the Ot~ce of Naval Research of the U.S. Department of Navy whose support is gratefully acknowledged.
REFERENCES
1. J. W. Gibbs, On the Equilibrium of Heterogeneous Substances, The Scientific Papers of J. Willard Gibbs, p. 1. Dover, New York (1961). 2. C. S. Smith, Metal Interfaces, p. 65. Am. Soc. Metals, Metals Park, Ohio (1952). 3. D. Turnbull, Trans. Am. Inst. Metall. Engrs 191, 661 (1951). 4. Quantitative Microscopy (edited by R. T. DeHoff and F. N. R_hines), p. 77. McGraw-Hill, New York (1968). 5. F. N. R_hines, Metall. Trans. 8A, 127 (1977). 6. H. V. Atkinson, Acta metall. 36, 469 (1988). 7. J. E. Burke and D. Turnbull, Prog. Metal Phys. 3, 220 (1952).
1166 8. 9. I0. 11. 12. 13. 14. 15. 16. 17. 18.
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH
P. Feltham, Aeta metall. 5, 97 (1957). M. Hillert, Aeta metall. 13, 227 (1965). N. P. Louat, Acta metall. 22, 721 (1974). C. S. Pande, Acta metall. 35, 2671 (1987). F. N. Rhines and K. R. Craig, Metall. Trans. 5, 413 (1974). R. D. Doherty, Metall. Trans. 6A, 588 (1975). R. D. Doherty, J. Mater. Educ. 6, 841 (1984). F. N. Rhines and B. R. Patterson, Metall. Trans. 13A, 985 (1982). O. Hunderi, Acta metall. 27, 167 (1979). S. K. Kurtz and F. M. A. Carpay, J. appl. Phys. 51, 5725 (1980): ibid. 51, 5745 (1980). H. F. Kaiser, Metals Alloy 9, 23 (1938).
19. R. A. Vandermeer, R. A. Masumura and B. B. Rath, Acta metall. 39, 383 (1991). 20. K. Okazaki and H. Conrad, Trans. Japan Inst. Metals 13, 198 (1972). 21. W. W. MuUins and J. Vinals, Acta metall. 37, 991 (1989). 22. W. W. Mullins, Acta metall. 37, 2979 (1989). 23. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Scripta metall. 19, 225 (1985). 24. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Phil. Mag. B 59, 293 (1989). 25. J. W. Cahn, Acta metalL 10, 789 (1962). 26. N. P. Louat and M. A. Imam, Recrystallization '90 (edited by T. Chandra) p. 387. TMS, Warrendale, Pa (1990).
0956-7151/92$5.00+ 0.00 Pergamon Press Ltd
Printed in Great Britain
MICROSTRUCTURAL PATH CONCEPT APPLIED TO NORMAL GRAIN GROWTH R. A. VANDERMEER Naval Research Laboratory, Materials Science and Technology Division, Washington, DC 20375-5000, U.S.A. (Received 15 March 1991; in revised form II November 1991)
Abstract--Normal grain growth in solids is considered in the light of a microstructural path concept. The topological model first presented by Rhines and Craig is corrected and reformulated so as to include the Doherty sweep constant. The reformulated model predicts the experimentally observed linear time dependence for volumetric grain growth only when the average curvature per grain remains constant during growth. A geometrical path function for normal grain growth relating average microstructural properties on a per grain basis is proposed and compared with experimental observations on aluminum and titanium. By combining the geometrical path function with the topological model, further comprehension of normal grain growth kinetics results, specifically the role of the average grain shape. R6sttm6---On consid6re la croissance normale du grain dans les solides fi la lumi6re du concept de trajet microstructural. Le mod61e topologique pr6sent6 pour la premi6re fois par Rhines et Craig est corrig6 et reformul6 de fagon fi inclure la constante de balayage de Doherty. Le mod61e reformul6 ne pr6voit la d6pendance lin6aire observ6e en fonction du temps de la croissance en volume du grain que lorsque la courbure moyenne par grain reste constante durant la croissance. Une fonction de trajet g6om6trique pour la croissanee normale des grains, liant les propri6t6s microstructurales moyennes par grain, est propos6e et compar6e fi des observations exp6rimentales sur l'aluminium et It titane. En comparant la fonction de trajet g6om6trique avec le mod61e topologique, on comprend mieux la cin6tique de croissance normale du grain, et en particulier le r61e de la forme moyenne du grain. Zusmnmenfassnng--Das normale Kornwachstum in Festk6rpem wird im Bilde tines mikrostrukturellen Pfades betrachtet. Das zuerst von Rhines und Craig vorgelegte topologische Modell wird berichtigt und neu formuliert, um die Bewegungskonstante yon Doherty einzubauen. Das neu formulierte Modell sagt die experimentell beobachtete lineare Zeitabh~ingigkeit des Volumwachstums der K6rner nur dann voraus, wenn die mittlere Kriimmung pro Korn w/ihrend des Wachstums konstant bleibt. Eine geometrische Pfadfunktion des normalen Kornwachstums, die die mittleren Mikrostruktureigenschaften pro Korn beschreibt, wird vorgeschlagen und mit experimentellen Beobachtungen an Aluminium und Titan verglichen. Aus der Kombination dieser geometrischen Pfadfunktion mit dem topologischen Modell folgt ein erweitertes Verst/indnis der Kinetik des normalen Kornwachstums, insbesondere der Rolle der mittleren Kornform.
1. INTRODUCTION In the solid state, the microstructure of metallic materials is normally observed to be polycrystalline in nature. This means that the mass is composed of an aggregate of contiguous crystals called grains, each of a different orientation from its neighbors. Physically, each grain is separated from its neighbours by a thin grain boundary layer or interface region having a thickness of atomic dimensions. On a microstructural scale, this interface forms an interconnected array that may be regarded as a three dimensional topological network, i.e. it consists of nodes (grain corners), branches (grain edges) and faces (grain boundaries). In a thermodynamic sense this interface array imparts an excess energy to the material which would not be present were the mass a single crystal instead. The material perceives the excess energy through the AMM 40/6---D
topological grain boundary network as an interfacial tension in the sense discussed by Gibbs [1]. If all the grain boundaries in the system have the same energy, i.e. interfacial energy does not vary with the misorientation between grains (an assumption used throughout the paper), then as pointed out by Smith [2], the surface tension has the impetus of causing the grain boundaries coming together to form triple junctions (grain edges) to tend to meet in dihedral angles of 120 ° . Likewise, it is the tendency for the four triple junctions meeting to form the quadruple points (grain corners) to approach mutually included angles of 109 ° 28' and 16". Given sufficient thermal activation and atomic mobility, the a m o u n t of grain boundary area in the polycrystalline material can be reduced, thus decreasing the excess energy of the system. This reduction is accomplished by the systematic elimination of some
1159
1160
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH
of the grains from the material and the concomitant enlargement of the remaining grains, coordinated with such grain size and shape adjustments as are needed to maintain perfect shape filling at all times. This evolutionary process is known as grain growth. The mechanism for grain growth is generally thought to involve the migration of the grain boundary segments normal to themselves by atomic jumps across the interface as discussed by Turnbull [3]. When the grain growth process is characterized microstructurally by a progressive increase in the average size of the grains without seeming to change their geometric form, i.e. shape, number of sides per grain, size distribution etc., the process is referred to as normal or steady state grain growth. By contrast, the growth process is designated abnormal or discontinuous grain growth when normal grain growth stagnates for one reason or another and a select few grains, by virtue of having highly mobile interfaces, emerge from and grow into the matrix of smaller stagnant grains until they are consumed. By analogy with recrystallization, abnormal grain growth may be treated phenomenologically as a nucleation and growth process. In this paper microstructural path concepts are introduced and applied to normal grain growth. A model of grain growth based on the average microstructural properties of the material is presented. Abnormal grain growth will be discussed in a subsequent paper. 2. MICROSTRUCTURAL PROPERTIES
The grain size of a material is usually characterized in terms of the mean intercept-free grain length, 2, which is measured by counting, on a two dimensional plane of polish, the number of grain boundary intersections with a unit length of test line, NI. The 2 is equal to 1/N1. In principle, however, what is determined by measuring N~ is the important stereological property, Sv, the grain boundary area per unit volume because by definition [4], Sv = 2. Nl. As stressed by Rhines [5], Sv and hence 2 are very sensitive to the shape of the grains in the microstructure. Rhines [5] also pointed out that there are several other measurable, global microstructural properties which are critical to a quantitative description of the microstructure of a single phase, polycrystalline material and the evolution of normal grain growth. They are, M~, the total grain boundary curvature per unit volume and Nv, the number of grains per unit volume. All of the microstructural properties mentioned can be measured using stereological procedures [4]. While SI and Mv can be measured with relative ease on a random, two dimensional plane of polish, the measurement of Nv can only be accomplished using laborious, serial sectioning techniques, and unfortunately it's measurement is rarely carried out. Assuming that the three global properties discussed above are sufficient to characterize the microstructure,
the normal grain growth process can be embodied in the function
F{Sv(t), My(t), Nv(/)} = 0
(1)
where t is the annealing time. Because of the definition noted above, 2 may be substituted for Sv in equation (1). This function may be recognized as a microstructural path function because it defines the instantaneous state of the microstructure and how the state microstructural variables change as grain growth takes place with time. The way the path function is deduced for normal grain growth is the subject of this paper. Partial microstructural path functions can also be formulated. An example is the kinetic function 2 =f(t). This is the function usually sought experimentally and is the one for which a number of theories of normal grain growth have been developed. Also in this paper another partial function, Sv = f(1/N~), which is a geometrical path function, will be proposed. 3. THEORIES OF NORMAL GRAIN GROWTH
The status of the theories of normal grain growth in pure single phase material systems was reviewed very recently by Atkinson [6]. The earliest theory of grain growth was first annunciated clearly by Burke and Turnbull [7]. Based on the idea that grain boundaries tend to migrate towards their centres of curvature and that the radius of curvature is proportional to the mean grain radius, they assumed that the grain boundary migration rate was proportional to the pressure difference across the boundary. From what may be considered to be essentially simple dimensional arguments, Burke and Turnbull deduced parabolic grain growth kinetics of the form 2 2 - 2 2 = x" t
(2)
where 20 is the mean intercept-free grain length at t = 0, 2 and 2o are assumed to be proportional to the appropriate mean grain radii and r is a rate constant. Beginning with Feltham [8], a number of more sophisticated theories were developed over the years. Most of these theories take a mean field approach and view the process of grain growth in terms of the change in the grain size distribution as the number of grains in the system decreases and the mean grain size increases. The growth or shrinkage of individual grains is considered amid an environment characterized by an average over the entire ensemble of grains. The flux of grains through grain size space is either curvature-driven by a drift term as in Hillert's theory [9] or treated statistically as a diffusion-like process without the need for curvature-driven drift in the case of Louat's theory [10]. Both approaches predict parabolic kinetics like equation (2) but for different reasons, Hillert's because the curvature is taken to be inversely proportional to grain size, and Louat's because of the diffusional aspect of the formulation.
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH An interesting blending of these two ideas was presented recently by Pande [11] who assumes the flux through grain size space to be a super position of a statistical, diffusion-like noise term upon the curvature-driven drift term. Both the Pande theory and the Louat theory yield more realistic predictions for the grain size distributions observed experimentally than does the Hillert theory. One criticism of all these theories is the lack of concern regarding topological constraints and their role in grain growth. To address that issue, the work of Rhines and Craig [12], hereafter referred to as RC, and the followup considerations of it by Doherty [13, 14], stress the topological aspects of grain growth. The RC approach is predicted on microstructural and topological property measurements made during steady state grain growth in aluminium in which it was found by a serial sectioning technique that the number of corners, edges and faces (boundaries) all on a per grain basis approached and then maintained, on average, the numbers 6:7:12 during the growth process. Topologically these numbers describe a tetrakaidecahedron (sometimes referred to as a truncated octahedron). RC view the toplogical path in grain growth as a degrading of complex grains (more than four faces) to simple grains (four faces or fewer) by the loss, one at a time, of triangular faces. To maintain the 6:7:12 ratio during grain growth, the elimination of grains must be such as to occur by the removal or corners, faces and edges in exactly this ratio. This could be achieved, for example, by the collapse of one tetrahedral grain and three grain faces in the material. All such processes are equivalent to removing a tetrakaidecahedral grain without changing the average topology of its neighbors. These considerations led RC to postulate that steady-state grain growth can be described in terms of the elimination, one at a time, of grains of average volume each having the average topological characteristics of a tetrakaidecahedron. Strictly speaking the RC approach is more a model of grain growth than a theory because, unlike the theories of Hillbert, Louat and Pande, the RC formulation cannot predict a grain size distribution. RC assume that the mechanism by which grains in a material shrink and vanish under the driving force of surface tension, thereby transferring their volume to the grains remaining in the system, is by the sweeping of grain boundary. This sweeping action occurs on a global scale and over an extended time period. Anyone observing grain growth taking place in the photoemission electron microscope is struck by the activeness and extent of these boundary sweeps. Using several simple, albeit somewhat unrealistic models as analogs, RC deduced that, irrespective of the size of the average grain, the total volume swept by the grain boundary is always the same for the disappearance of one average grain including the concomitant adjustment of the remaining grains to being equiaxed. Thus, RC define a sweep parameter O,
116l
which they assumed to be a constant, such that O is the number of grains lost from the system when grain boundary sweeps out a volume of material equal to a unit volume. In this definition, the units of O are dimensionless. RC express grain growth kinetics in terms of the topological parameter Nv, the number of grains per unit volume of material. The inverse of Nv is the average volume per grain, Vg, which is a hybrid topological-geometrical property of the microstructure. The kinetic relationship derived by RC is given by 1
1
N--~= ~ ( 1 + O "#Rc'7 '¢y • t)
(3)
where N o is the number of grains per unit volume initially, O is the sweep parameter, #RC is the mobility (velocity per unit force), y is the grain boundary energy (surface tension), a is the product of the average curvature per grain, Mv/Nv, and the grain boundary area per unit volume, Sv, and t is again the annealing time. The parameter a is regarded as the dimensionless structural gradient of the material. Experimentally, RC found that o was a constant for grain growth in aluminum. As Fig. 1 indicates, equation (3) describes the experimental results of RC and Rhines and Patterson [15] for aluminum very well. For reasons discussed later in the paper, 1/Nv can be deduced to be proportional to 2 3, so that equation (3) does not predict parabolic grain growth kinetics as do other theories and models. Why this is so has never been fully understood. In a discussion to the RC paper [12], Doherty [13, 14], hereafter called Dh, took issue with the definition of the sweep constant and pointed out that O would not be a true constant in grain growth. Dh proposed another sweep parameter, O*, defined slightly differently, as the number of grains lost when grain boundaries sweep through a volume equal to the volume of the average grain, Vg. More recently Hunderi [16] showed how to calculate the sweep parameter within the framework of the Hillert grain growth theory. He also concluded on the basis of computer simulations of grain growth in two dimensions that for the sweep parameter to be truly a ,
5 10 .4
3 10 .4
-
2 10 .4
_ -
,
i
I 10 .4
. . . . . .
,
Grain Growth in Aluminum 635 °C
~r~-G-r
R
h
•
.
.
i
~
0 0, 20
40
Time
60
80
100
(min)
Fig. 1. Average grain volume as a function of time for grain growth in aluminum.
1162
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH 0.25
0.2 ........................................................ i............ ' " - / " ~ 0.15
.
.
.
.
.
.
.
/--
o1
..................... :.......................... -
.
.
.
"
:7
.....................
/ . . . . . . . . . . . . . . . .
0 0
~l,~,r 10
. . . . . . . . . . . . . . . . . . . . . . . . . .
i .... 20
I .... 30
. . . . . . . . . . . . . . . . . .
i
i ........ 40 50
i .... 60
70
Time (rain) Fig. 2. Average curvature per grain plotted vs time for steady-state grain growth in aluminum. Data of Rhines and Patterson.
constant, it must be defined in terms of the mean grain volume as proposed by Dh. Furthermore, Dh noted a mathematical error in the derivation of equation (3) which cast doubt on its validity. Dh recast the RC model in terms of a different mobility, one based on the velocity per unit pressure. In so doing, he found that in differential form, dI
Nv
= O*
'~D'r
. M~ dt
Nv
(4)
where #D is the new mobility. For an integration of equation (4) to yield the same linear dependence on time as equation (3), the average curvature per grain, M,/N~, must be a constant. Dh stated that such may likely be the case. However, the experiments of RC on aluminum, the only available data with which to test the idea, do not substantiate this view as Fig. 2 demonstrates. If these results are correct, then under the assumptions of the model, the integrated form of equation (4) will not lead to a linear dependence of volumetric grain growth on time. As noted above, the microstructural modeling approaches of RC and Dh lack an explicit consideration of the grain size distribution and how it changes during grain growth. On this issue, the analysis of Kurz and Carpay [17] seems to be relevant. These workers assumed that the grain size distribution and the topological feature distribution are log-normal in nature in both two dimensions and three dimensions. They were able, by expanding on the work of Feltham and combining it with the approach of RC, to show that the RC structure gradient, tr, is indeed a constant and, furthermore, that tr is related to the standard deviation of the grain face distribution. In contrast to the 23 vs t kinetic relationship derived from the RC theory, Kurz and Carpay deduced the customary parabolic grain growth kinetics, i.e. 22 vs t. 4. A REFORMULATION OF THE RHINES AND CRAIG MODEL This reformulation of steady-state grain growth kinetics seeks to understand the origin of the 23 vs t
behavior. It retains the basic postulates of RC except when they are in error. Consider a system of unit volume at time, t(s). Let #v be the mobility associated with the moving grain boundary, i.e. velocity per unit driving force acting at a point (cm/Ns) and ,l the surface tension (N. cm/cm2). This definition of the mobility retains the logic of RC; it differs from the mobility, #o, proposed by Dh, which is based on a force per unit area (pressure) definition. It can be shown from reaction rate theory that the two approaches should be essentially equivalent and that #v differs from /~D by only a constant. What the constant is will be shown later. RC and Dh take the average pressure, P, acting on a grain boundary segment to be given by the product of the surface tension and the curvature such that p = 7" My (N/cm2).
(5)
s~
At this point in the development, Dh writes the average grain boundary velocity, ~, in terms of the driving force, P, making use of the proper assumption that velocity is the product of mobility and driving force. On the other hand, RC consider the driving force as P ' S v which unitwise is a force per unit volume (N/cm3). The implication of the RC approach is that it essentially spreads the driving force through the entire unit of volume. Physically this does not seem correct. It seems to be more realistic to suppose that the driving force is sensed only by those atoms at the grain boundaries, the ones participating in the migration process. The driving force per grain boundary atom is given approximately by P . fl2/3, where fl is the atomic volume. With this correction to the RC model, the average grain boundary velocity can be written as f = / ~ . p . f~2/3= f~2/3. #v" Y " My
(cm/s).
where the average is taken over all the grain boundary area in the system. In the interval of time between t and t + d t , the total volume swept out b y all the grain boundary segments in a unit volume of material is given by dV swept-- S v " 6" dt = f~2/3. #v '7 " My" dt
(cm3/cm3).
A number of grains are lost in this time increment. This number, dnv, on a per unit volume basis can be obtained by introducing the Doherty sweep constant, O*, defined earlier. Thus dnv
O* 7s dVv
"It~'r'N,'Mv'dt.
According to the postulate of the RC model that, on average, each of the grains lost is of average volume,
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH the total volume lost in the increment, dt, per unit volume of material can be written as d V l v°st =
~g"
10-I
• ' '"'"!
......
"1
.......
1
......
10. . . . . . . . . . . . . . . .
dnv
1
......
........
1163 "1
.......
1
.......
1
.....
....................
Aluminum
dnv
Nv
8"
D2/3.#~ y My dt
10.3
(cm3/cm3). Z
But volume must be conserved so that the volume lost by the disappearing grains must be gained by the grains remaining in the system, i.e. dV l°st= dV gained. The volume gained is averaged over all the grains left in the system giving, on a per grain basis, the equation 1
d V~ ~i"~
Nv
d ~ = dfrs=
= O * ' ~'~2/3 . # v ' Y " ~ "
dt
(cm3).
(6)
which except for the mobility and the atomic volume, both of which are constants, is identical in form to the Dh equation, equation (4). It also indicates that ~/D= f~2/3 . /'/v" For equation (6) to be integrated to the form of the RC equation, equation (3), the average curvature per grain must remain constant during grain growth. 5. A GEOMETRICAL PATH FUNCTION F O R STEADY STATE GRAIN GROWTH RC observed that during normal grain growth, the average grain possessed the topological parameters roughly equivalent to that of a tetrakaidecahedron. Assuming that the global microstructural properties of the grains in a material can be described equivalently in terms of the properties of the average grain, a relationhip between Sv and Nv can be expressed following Kaiser [18], (see also Ref. [4]) as (7)
S v = F . Nlv/3
where F is a constant that depends on the specific geoemetric shape. Table 1 lists the values of F calculated by Kaiser for four different space-filling polyhedral shapes. If equation (7) is divided by N~, then the following formula is obtained Sv = F "
NZ
= F " (~g)2/3
(8)
which is essentially a function describing how the average interfacial area per grain varies as a function of its volume. In a geometrical sense this equation can be regarded as a grain shape or geometrical path function for steady-state grain growth. Its similarity
T a b l e 1. S h a p e factors for spacefilling p o l y h e d r a Polyhedron Cube Hexagonal prism Rhombic dodecahedron Tetrakaidecahedron
F 3.000 2.866 2.673 2.659
10-4
lO.S
. . . . . . . . . .
1 O" 6
.......
~
10"to
Tetrakaid~abedra Model
....
.......
J
.......
J
.......
10"*
I
......
.t
......
10.6
a
.......
a
10 .4
.....
10 .2
1 / N v (cm 3)
Fig. 3. Comparison of the geometrical path model for steady state grain growth in aluminum with experiment. • Rhines and Craig; C) Rhines and Patterson; • Okazaki and Conrad. to a path function proposed for recrystallization by Vandermeer, Masumura and Rath [19], namely, Sv = f ( X v ) , where Xv is the volume fraction recrystallized, should be noted. Regrettably most experimental studies of grain growth only measure Sv. But, the grain growth investigations on aluminum at 635°C by RC and Rhines and Patterson [15] and the studies by Okazaki and Conrad [20] on titanium measured both Sv and Nv. These studies thus afford the possibility to test whether equation (8) may be applicable. In Fig. 3 the average interfacial area per gain is plotted vs the average volume per grain on log-log scales. As Fig. 3 depicts, describing steady-state grain growth by means of a tetrakaidecahedron model in accord with the topological observations, i.e. F = 2.659, is very reasonable, especially for aluminum. Note that the experimental data for aluminum extend over an average grain volume range of almost two and a half orders of magnitude. In most studies of steady-state grain growth, the microstructural parameter, 2, is measured and taken to be 'an average grain size. Under the assumptions that the grain shape remains constant and the grain structure is equiaxed, the geometrical path function given by equation (8) can be combined with reformulated RC theory [equation (6)] to find a relationship between ). and time as follows: Inserting equation (8) into equation (6), substituting for Sv by 2, differentiating and rearranging, the equation can be written as d22 =--16- O* • f~2/3. Pv" )' My ,dt 3 F3 2 • Nv where F is considered to be constant. Replacing the 2 on the right hand side of this equation with Sv, yields the very interesting result d 2 2 = 8 0 * ' ~ ' ~ 2 / 3 " ] / v ' y S~ " M v d t 3 F3 Nv 8
~ * " ~"~2/3 ' ]./v ' ~ " O"
3
F3
dt
1164
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH
where the right hand side now contains the RC structural gradient constant, a which experimentally was found to be constant. Upon integration, this equation becomes 8
22 -- 22 = 3
~ *
" ~'~2/3
" /d,v ' '~
F3
" O"
t
(9)
which is equivalent to equation (2) in giving parabolic kinetics. If 20 in equation (9) is small and negligible, 2 is proportional to t ~/2.
4.0
10 "a
....
TT,.?, ............
i ....
v ....
i ....
i ....
II
l
I
/
'
........
.....
_2Oo
.
.
.
. . . . . . . .
i 10 °
.... 0
The form of the steady-state grain growth kinetics function [equation (9)] derived from the microstructural path approach agrees with most theories [6-11]. Very recent work by Mullins and Vinals [21] and Mullins [22] have demonstrated that the two conditions necessary for the parabolic growth law to be obeyed are (1) a statistical self-similarity holds, i.e. consecutive configurations developed by the structure as growth proceeds are geometrically similar in a statistical sense, and (2) the grain boundary velocity is proportional to the local mean curvature. In this vein, the Monte Carlo simulations of three dimensional grain growth by Anderson et al. [23, 24] are also of interest. These investigators found that the grain growth exponent, i.e. the power of t in a bulk grain growth experiment, determined from the grain size growth kinetics was close to 1/2. The physics of the Monte Carlo simulation is based on lattice site switching such that when the energy of the system is lowered by the switch, the switch gives rise to a local movement of the boundary by one lattice spacing. The simulation, therefore, seems to localize the driving force at the grain boundary lattice sites as the reformulated microstructural path model does. From the analysis carried out in this work, it is apparent that the 23 dependence on time implicit in the RC model [equation (3)] arises from an improper consideration of the driving force for grain growth, specifically how that force is distributed among the atoms in the material. Indeed, it can be readily shown that by following the RC logic and distributing the driving force over every atom in the volume rather than over just the grain boundary atoms and carrying through the analysis presented in Section 4 above, a 23 rather than a 22-dependence on time results. As alluded to earlier, the stereological measurement yielding 2 is more a measure of the amount of grain boundary area per unit volume than it is grain size. RC argue that I/Nv is the only real measure of grain size that is not shape dependent. They suggest that 2 should be very sensitive to grain shape and equation (9) reveals this to be true. This point will now be addressed. Using the experimental Sv data of RC, a plot of 22 vs time is shown in Fig. 4 for aluminum. Also illustrated in Fig. 4 is a curve calculated from the tetrakaidecahedral model with the constants estimated from the 1 / N v vs time data of Fig. 1. The
i ....
~.o ,o. . . . . .
0.0
6. DISCUSSION
r ....
[ Volumetric
i .... 10
, .... 20
~ . . . . . . . . . 30
40
i . . . . . . . 50
60
, 70
Time (min)
Fig. 4. Plot of 2 2 vS time for steady state grain growth in aluminum, Data adopted from Rhines and Craig where 2 = 2/Sv.
poor fit of the data in Fig. 4 to a straight line as predicted by equation (9) or to a line determined from the volumetric model of RC is readily apparent. The letters beside each data point refer to the topological properties found experimentally by RC (extracted from Fig. 6 of their paper). Only at the longest annealing times (data points labeled T) did the topological relationship approach that descriptive of a tetrakaidecahedron. At shorter annealing times (data points labelled D), the topological parameters were close to ones describing a decahedron. (The ratio of corners to faces to edges on a per grain basis are 4: 5: 8 for a decahedron while for a tetrakaidecahedron they are 6:7:12, respectively.) Thus, it would appear that there was grain shape changes taking place during the grain growth in aluminum at 635°C investigated by RC. Upon reviewing the RC grain growth results, Atkinson [6] alluded to this possibility also. Further demonstration of the role of grain shape change on grain growth kinetics will now be presented. From Table 1 it is apparent that F is a decreasing function of the number of faces comprising the polyhedron. Thus, if during grain growth the topological characteristics of the average grain changes from say a lower number of faces to a higher number, as is experimentally observed, then the value of F will decrease, introducing deviations in the linear 22 vs t plot. However when the average grain contains 12 or greater number of faces the value of F is fairly insensitive to the polyhedon's shape. Thus, the later stages of normal grain growth should approach nearly linear 22 vs t behavior but with a 20 that is only an apparent one. The data in Fig. 4 seem to conform to this tendency. A similar argument can be made if on average, grains deviate from being equiaxed. Consider an hexagonal prism shape. The value of F for this shape depends on the c/a ratio. Thus for that ratio being 1/2 (pancake tendency), 1 (equiaxed) or 2 (needle tendency), the Fvalue is calculated to be 2.962, 2.866 or 3.065, respectively [4]. Again assuming that the initial grain structure deviates from being equiaxed but approaches such during grain growth, the value
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH of F will decrease during growth again causing a deviation in the 22 vs t plot. Clearly grain shape changes can influence the time dependence of 22 significantly, leading to a non-linear time dependence. It is concluded that 22 vs time which contains F must be very sensitive to shape changes, while 1/Nv vs time which does not depend on F is relatively insensitive to shape change, thus confirming the validity of RC's concern noted earlier. There is one issue that remains unresolved regarding the RC experiments on aluminum. That pertains to the inconsistency of the experimental data shown in Figs 1 and 2 when examined in light of the reformulated model. If 1/Nv is linear with time as appears to be the case (Fig. 1), then according to the model [equation (6)], Mv/Nv should be a constant. This was not observed for aluminum (Fig. 2). Alternatively, if M~/Nv is not a constant, then the model predicts that the 1/Nv should not be linear with time. Two possibilities can be put forward to account for this discrepancy. Doherty [13] suggested that if the velocity~lriving force relationship is modified by a velocity independent drag term, then under certain conditions, presumeably impurity related, an approximately linear 1/N v vs time relationship could result. Another possibility is that the velocity may not be a linear function of driving force due to impurity drag effects as envisioned by Cahn [25] or in a recent new theory by by Louat and Imam [26]. Further studies are necessary to decide this issue. A few comments concerning the role of a on grain growth kinetics seem appropriate. The grain growth studies of RC and Rhines and Patterson [15] were both carried out on aluminum of the same purity at 635°C. Yet as Fig. 1 illustrates, the rates of volumetric grain growth are radically different. On the other hand, the geometrical path function for the two studies is essentially the same (Fig. 3). Since in both studies the volumetric growth kinetics are observed to be linear with time, the difference in rates must be associated with the structural gradient parameter, a. This should be apparent from consideration of equation (9). Unfortunately Rhines and Patterson did not report a value for a for the grain growth they observed. They did show convincingly, however, that, other factors being equal, grain growth kinetics are strongly influenced by the width of the grain volume (face) distribution which is largely determined by the degree of cold work during the deformation process prior to recrystallization and grain growth. Wider distributions were demonstrated to promote faster grain growth. Kurz and Carpay [17] have shown a connection between a and the character of the grain volume (face) distribution. Thus, it is concluded that RC used deformation procedures that were significantly different from the Rhines and Patterson procedures (for the data presented here the aluminum was compressed 80%). Unfortunately RC did not report the complete details of their deformation procedure.
1165
Finally, an implicit message emanating from the topological approach is that a thorough characterization of the microstructure and the microstructural path is essential for an improved understanding of normal grain growth. In other words, it is not enough to measure only the property 2 (or Sv). Additional microstructural property measurements are demanded if a more complete comprehension of the process is desired. 7. CONCLUSIONS 1. The concept of microstructural path can be developed for steady-state grain growth and is shown to be applicable to grain growth in aluminum for which a geometrical path function is also deduced. 2. The topological grain growth model of Rhines and Craig can be corrected and reformulated using the Doherty sweep constant O* to yield a linear dependence of volumetric growth on time, but only when the average curvature per grain remains constant during growth. 3. Introducing the geometrical path function into the modified model has allowed clarification of the kinetics of grain growth as follows: (a) A linear ;~2 dependence on time is predicted provided the average grain shape does not change during growth. (b) If the average grain shape changes, deviations from the predicted parabolic time dependence of 2 can be understood, at least qualitatively, in terms of the role of the grain shape factor, F. (c) The 23 vs time behavior implicit in the earlier RC model appears to result from an improper distribution of the driving force for grain growth among all the atoms in the system rather than those at the grain boundaries where the migration process takes place. Acknowledgements--The author would like to thank Drs C. S. Pande, B. B. Rath, J. C. Ayers, R. A. Masumura and Professor R. Doherty for helpful discussions and for reading the manuscript. This work was performed at the Naval Research Laboratory under the sponsorship of the Ot~ce of Naval Research of the U.S. Department of Navy whose support is gratefully acknowledged.
REFERENCES
1. J. W. Gibbs, On the Equilibrium of Heterogeneous Substances, The Scientific Papers of J. Willard Gibbs, p. 1. Dover, New York (1961). 2. C. S. Smith, Metal Interfaces, p. 65. Am. Soc. Metals, Metals Park, Ohio (1952). 3. D. Turnbull, Trans. Am. Inst. Metall. Engrs 191, 661 (1951). 4. Quantitative Microscopy (edited by R. T. DeHoff and F. N. R_hines), p. 77. McGraw-Hill, New York (1968). 5. F. N. R_hines, Metall. Trans. 8A, 127 (1977). 6. H. V. Atkinson, Acta metall. 36, 469 (1988). 7. J. E. Burke and D. Turnbull, Prog. Metal Phys. 3, 220 (1952).
1166 8. 9. I0. 11. 12. 13. 14. 15. 16. 17. 18.
VANDERMEER: MICROSTRUCTURAL PATH OF GRAIN GROWTH
P. Feltham, Aeta metall. 5, 97 (1957). M. Hillert, Aeta metall. 13, 227 (1965). N. P. Louat, Acta metall. 22, 721 (1974). C. S. Pande, Acta metall. 35, 2671 (1987). F. N. Rhines and K. R. Craig, Metall. Trans. 5, 413 (1974). R. D. Doherty, Metall. Trans. 6A, 588 (1975). R. D. Doherty, J. Mater. Educ. 6, 841 (1984). F. N. Rhines and B. R. Patterson, Metall. Trans. 13A, 985 (1982). O. Hunderi, Acta metall. 27, 167 (1979). S. K. Kurtz and F. M. A. Carpay, J. appl. Phys. 51, 5725 (1980): ibid. 51, 5745 (1980). H. F. Kaiser, Metals Alloy 9, 23 (1938).
19. R. A. Vandermeer, R. A. Masumura and B. B. Rath, Acta metall. 39, 383 (1991). 20. K. Okazaki and H. Conrad, Trans. Japan Inst. Metals 13, 198 (1972). 21. W. W. MuUins and J. Vinals, Acta metall. 37, 991 (1989). 22. W. W. Mullins, Acta metall. 37, 2979 (1989). 23. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Scripta metall. 19, 225 (1985). 24. M. P. Anderson, G. S. Grest and D. J. Srolovitz, Phil. Mag. B 59, 293 (1989). 25. J. W. Cahn, Acta metalL 10, 789 (1962). 26. N. P. Louat and M. A. Imam, Recrystallization '90 (edited by T. Chandra) p. 387. TMS, Warrendale, Pa (1990).