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Alternative length scales for polycrystalline materials—I. Microstructure evolution
Acta metall, mater. Vol. 39, No. 7, pp. 1657-1665, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
ALTERNATIVE LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I. MICROSTRUCTURE EVOLUTION C. S. NICHOLSI', R. F. COOK, D. R. CLARKE and D. A. S M I T H IBM Research Division, IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Received 7 September 1990) Abstract--It is a well-documented experimental observation that properties of grain boundaries depend on the atomic structure of the boundary. Yet constitutive relations for properties of polycrystalline materials containing a variety of grain boundaries currently do not take account of this boundary-toboundary variability. Instead, a single-length scale---the average grain diameter is utilized with the underlying assumption that all grain boundaries are the same. In this (I) and the following paper (II), we extend the accepted veiwpoint to encompass a binary classification of grain boundaries based on their misorientation angle. The resultant new length scale is that associated with clusters of grains linked by grain boundaries sharing misorientations in the same category. This first paper focuses on how a model polycrystal is generated, the energetics of the model, and its evolution under various external influences. Rrsumr---I1 est prouv6 exprrimentalement que les proprirtrs des joints de grains drpendent de la structure atomique du joint. Cependant les relations constitutives relatives aux proprirtrs des matrriaux polycristallins contenant un grand nombre de joints de grains ne tiennent grnrralement pas compte de la variabilit6 de joint a joint. A la place, on utilise une seule 6,chelle de longueur--le diamrtre moyen des grains--avec l'hypothrse sous-jacente que tousles joints de grains sont identiques. Dans les parties I e t II de cet article, nous 6tendons ce point de vue pour proposer une classification binaire des joints de grains basee sur leur angle de drsorientation. La nouvelle 6chelle de longueur rrsultante est associre aux amas de grains lirs par des joints partageant des drsorientations de mrme catrgorie. Le premier article prrsente un polycristal modele, les aspects 6nergrtiques du modrle et son 6volution sous diverses influences exterieures. Zusammenfassnng--Eine wohlbekannte experimentell Beobachtung ist, dab die Eigenschaften der Korngrenzen von der atomaren Struktur abh/ingen. Allerdings beriicksichtigen bisher die Grundbeziehungen zur Beschreibung der Eigenschaften polykristalliner Materialien, die eine Vielfalt von Korngrenzen enthalten, diese Zusammenh/inge nicht. Stattdessen wird einfacher L/ingenmaBstab, n/imlich der mittlere Korndurchmesser, benutzt; dahinter steht die Annahme, dab alle Komgrenzen gleich sind. In dieser (I) und der nachfolgenden (II) Arbeit erweitern wir den akzeptierten Standpunkt, um eine bin/ire Klassifikation der Korngrenzen auf der Basis ihrer Fehlorientierungswinkel einzuarbeiten. Der sich ergebende neue L/ingenmaBstab h/ingt zusammen mit Kornhaufen, die fiber Korngrenzen mit Fehlorientierungen derselben Kategorie zusammenh/ingen.
1. INTRODUCTION It is well established that the physical, mechanical and chemical properties of a polycrystalline material differ markedly from those of the same material in single-crystal form. Yet, historically, the relationship between the properties of a polycrystal and its consituent grains and grain boundaries has been somewhat obscure. There are two general divergent lines of research. One approach is to formulate scaling laws for how properties depend on a length scale linearly related to the average grain size and, often, assumed to be the average grain size, d, which implies that all grain boundaries are the same. Specific tPresent address: Cornell University, Department of Materials Science and Engineering, Ithaca, NY 14853, U.S,A.
examples of such relationships include the Hall-Petch relationship for plastic yield stress, a ocd-l/2; the Coble relationship for t h e strain rate in grainboundary-diffusion-controlled creep, ~ocd-3; the H e r r i n g - N a b a r r o relationship for the strain rate in lattice-diffusion-controlled creep, ~ ocd-2; and the Fuchs' expression for resistivity, p oc d - l . There exist, however, many failings of these expressions. In particular, there is ample experimental evidence that boundaries have distinctly different properties that depend on one or more interface degrees of freedom [1-14]. The second approach is to focus on "protot y p a r ' boundaries with the expectation that their properties are somehow generalizable to an arbitrary interface. The necessity remains, of course, to synthesize the information about individual boundaries and grains into a description of the polycrystal.
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NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS---I
In this and the following paper (II), our objective is to integrate these two lines of research by exploring the relevant length scale for polycrystalline materials when boundary-to-boundary variability in grain boundary property is explicitly considered. This work extends the earlier work reported by us [15]. The simplest extension of the assumption that all grain boundaries have the same properties is to divide them into two distinct categories. For example, Watanabe [14, 16-18] has pointed out from a survey of the expansive literature available, that low-angle and special grain boundaries generally have low energy, low mobility, low diffusivity, etc. whereas the remaining high-angle, random boundaries have the opposite properties. It is not entirely clear how to connect this binary classification with an experimentally accessible variable and ineluctably the demarcation between the two classes "low" and "high" is somewhat arbitrary. Here it is convenient to use the magnitude of the misorientation angle as the variable that distinguishes between the two categories of boundaries. In the present work, a two-dimensional polycrystal is modeled as an array of regular hexagonal grains consisting of randomly oriented material with a four-fold rotational axis perpendicular to the surface of the array. The relevant microstructural units are identified as clusters of grains that are linked by low-angle boundaries. These clusters are characterized by a mass and a linear dimension that are a function of the misorientation angle chosen to delineate between low- and high-angle boundaries. We discuss in this first paper both the generation of the model microstructure and its evolution as a whole under the influence of finite temperatures or externally applied fields coupled to the grain interiors. The following paper is concerned with how the measures of cluster dimension change with microstructure evolution. We also discuss at some length what the relevant cluster measure is for a given property and speculate on how the constitutive relations vary with cluster size. The remainder of the present paper is organized as follows. Section 2 is devoted to a discussion of tlae methodology of generation of the model microstructure, its energetics, and its temporal development. In Section 3 we consider the specific effects of temperature, applied strain, and applied field on microstructure evolution. We close in Section 4 with a summary and conclusions. 2. METHODOLOGY 2.1. Static model Euler's law [19] applied to two-dimensional microstructures requires that grains have, on average, six sides. In accordance with this, we idealize our polycrystalline system to consist of regular hexagons that tile the plane: .Each grain consists, in turn, of material with crystallographic four-fold rotational symmetry
in the plane. The microstructure is fiber-textured in that the grains share a common (001) axis of rotation which is perpendieu!ar te the plane. The in-plane axes of each grain are assigned a random orientation with respect to some fixed external coordinate system. In this work we have employed a random number generator which produces orientation angles with a uniform distribution. The misorientation angles 0 of adjoining grains are determined next. In doing so, the symmetry of the grains must be taken into account. In particular, because the grain boundaries in this model are [001] tilt boundaries in a four-fold symmetric system, only misorientation angles between 0 ° and 45 ° are unique. The misorientation angles generated form a continuous group and it can be shown from group theory [19] that in two dimensions the distribution function of the rotational angles is a constant. In order to minimize finite-size effects, periodic boundary conditions are applied in both directions. For the model microstructures used in the present work, we find a standard deviation of the misorientation angle distribution of <0.5% for samples consisting of 400 grains. The grain boundaries are next placed into two distinct classes. Most generally, grain boundaries are classified on the basis of geometrical criteria as low-angle/special or high-angle (random). As noted in the Introduction, numerous experiments demonstrate that low-angle and special boundaries have similar properties but are distinct from those of high-angle boundaries. We however make the assumption that there are only two distinct classes of boundary: those for which 0 is less than some chosen cut-off angle, 0c, and those for which 0¢ ~< 0 ~<45 °. This binary classification scheme can be generalized to any number of distinct categories and does not present a limitation of the methodology. Lastly, once this division of boundaries is made, clusters of grains that share like boundaries may be identified. A portion of a microstructure thus generated is shown in Fig. 1. The clusters of grains are identified in the figure by those grains whose centers are connected by heavy lines. 2.2. Energetics o f the model microstructure The total internal energy, Ug, of a single grain with Ng sides in a polycrystalline ensemble can be written as the sum of the grain boundary energy and the energy of interaction with an external field Ng
ug= ~ ,~,~,- Ja.~
(1)
i=l
where )-i is the length of the ith boundary, ~, is the interfacial energy per unit area of the same boundary, J is a coupling constant (or a modulus), a is one of the in-plane crystallographic axes, and £ is an external field. The summation in the first term runs over all the sides of the chosen grain, which is six in the
NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
.)
Fig. 1. A sample microstructure showing the hexagonal grains and the dusters of low-angle boundaries. Grains in the same cluster are connected by a heavy line through the center of the grain. The cut-off angle delineating low-angle and high-angle boundaries is 10°. present case. It is assumed that all grains have unit thickness so that the first term has the units of energy. The coupling of the field to the a-axis is arbitrary and our results would not qualitatively change if the field had been coupled to another crystallographic direction. The total internal energy of the polycrystal is given by the sum over all the individual grains, N, in equation (1)
U=
gffil
1N Ng ~'Jg--2gE E ~i~)i" ~1i=1
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system does not get trapped in local energy minima, the microstructure may evolve to a macroscopically inequivalent final state. If a strain is now applied to the polycrystal, it may deform in one of two ways. Either the lengths of the grain boundaries remain unchanged, with the consequence that the volume is not conserved and local triple junctions are not in equilibrium, or the boundaries do change length. In the latter case, the volume of the grains is conserved and local triple junctions are in equilibrium, but the total internal energy of the system may change because the total grain boundary length in the system changes. The polycrystal thus evolves so as to reduce the total interfacial grain boundary energy, consistent with changes in the grain boundary lengths. In addition to temperature and applied strain, an external field may couple to a grain's degrees of freedom to influence microstructural evolution. For example, an external electric field may couple to the polarization in a ferroelectric material and at temperatures close to the transition temperature, the orientation of the grain may be altered. Or, the field may be thought of as a gradient in some quantity, for example, temperature or defect concentration. Because this contribution to the total internal energy is negative (J >t 0), it competes with the interfacial grain boundary energy, which is always positive, in effecting microstructural evolution. 2.3. Temporal evolution o f the model
(2)
The second term appears so as to avoid doublecounting the grain boundary energy. We ignore the contribution to the internal energy of point defects within the individual grains. The as-generated microstructure has a spatially random distribution of low- and high-angle boundaries and clearly is not the lowest-energy state of the system. We allow the system to evolve, in a manner to be considered below, such that the total internal energy is minimized under the competing influences of the internal energy variables. If the coupling constant, J, is set to zero (or, equivalently, the external field is switched off) and all grain boundaries are forced to have the same length, 2 i = 2, then the only remaining external variable which may affect microstructure evolution is the temperature. For a polycrystal in contact with a thermal reservoir at a fixed temperature T~, the microstructure will evolve so as to minimize the total internal energy and will reach a minimum (which may only be a local minimum, however) given enough time. If we now raise the temperature of the thermal reservoir to T2, then one of two things will occur. In one instance, the microstructure will reach a macroscopically equivalent final state, although perhaps more rapidly and with greater fluctuations about this state. If, however, T2 is significantly higher than T~ such that the
The original experiments of Chaudhari and Matthews [21] and the later notable experiments of Gleiter and co-workers [22] on copper and silver single-crystal spheres sintered to single-crystal substrates demonstrated that the randomly oriented spheres rotated into specific low-energy orientations. A natural corollary of the unhindered rotation of the spheres is the existence of a torque on grains in a polycrystalline aggregate that provides the driving force for grains to adopt orientations with respect to their neighbors that minimizes the total internal energy. Experimental justification for the rotation of grains, or subgrains, has come from the electron microscopy work of Hu [23] and Li [24]. Specifically, Hu observed the gradual disappearance of subgrain boundaries and a loss of contrast between neighboring grains which indicated that these subgrains attained the same orientation. It was argued that subgrain rotation led to subgrain coalescence and was the precursor to recrystallization. In the present context, grain rotation is utilized as a means for lowering the total internal energy of the as-generated microstructure. Although we do not allow grain coalescence to occur, the distribution and nature of the clusters of grains that we use to characterize the polycrystal does change. This point is addressed more thoroughly in II. The microscopic mechanism of subgrain rotation is not understood and we consider it a peripheral issue beyond the scope
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of the present work. Li [24] presents some possible models for subgrain rotation. For our model polycrystals, grain rotation is facilitated by a Monte Carlo Metropolis algorithm [25]. A grain is selected at random and its total internal energy, Ug, is obtained. In the present work we use the Read-Shockley [26] form for the grain boundary energy per unit area 7 = ~oO(A - In 0)
(3)
where 70 is a prefactor that depends on material properties, 0 is the misorientation angle of adjacent grains, and A is a constant. We have chosen the value A = 0.231 which is the value originally used by Read and Shockley~ The energy vs misorientation angle for the range of angles considered here (0° ~< 0 ~<45 °) is plotted in Fig. 2. Although the Read-Shockley form is only strictly valid for misorientation angles below which dislocation cores do not overlap (0 < 15°), we have used it for larger values. We do this because it is generally recognized that grain boundary energy vs misorientation angle plots often show cusps corresponding to misorientations of particularly low energy. As is evident from Fig. 2. such a cusp exists for 0 in the vicinity of 45 °. It is thus possible to examine the evolution of the sytem as if it contained a special boundary at this angle. The selected grain is then allowed to rotate through a random angle [27] either clockwise or counterclockwise. The total internal energy of the rotated grain is calculated and compared to its total internal energy before rotation. If the energy is lower, the trial rotation is accepted. If the energy is higher, then a transition probability, W, is calculated W = exp( - AUg/k T)
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3. MICROSTRUCTURE EVOLUTION 3.1, Temperature effects
In this section, the boundary lengths are fixed, the coupling constant is set to zero and we explore the role played by temperature in microstructure evolution. In the Read-Shockley form of the grain boundary energy, equation (3), the prefactor is given explicitly by
(4)
where AUg = Ug(final)-Ug(initial), k is Boltzmann's constant, and T is the absolute temperature. If W is greater than some random number between zero and one, then the rotation is accepted, otherwise the move
0.6
is rejected. All energies are expressed in units of the Read-Shockley prefactor, 70, so that temperature is introduced through the ratio Vo/kT. Subsequently another grain is chosen at random and the process is repeated. The most ambiguous part of Monte Carlo simulations is the association of a time scale with each Monte Carlo step. We follow a typical procedure by defining one time step as the number of actual Monte Carlo steps divided by the number of grains in the system. Therefore, during one time step, each grain has the possibility of undergoing rotation. It should be further remarked that the time scale of a given Monte Carlo step is much larger than the time scale of local atomic motions, as also implied in the Potts model simulations of microstructure evolution [28]. All results reported in the present work are for samples of 20 grains on a side (400 grains total) and 125 time steps. Both of these variables have been extended to larger systems and longer times and we find no qualitatitive differences in the results. Furthermore, in all simulations we have chosen the cut-off angle, 0¢, delineating low-angle boundaries from high-angle boundaries, to be equal to 10°.
I
15 20 25 30 35 Misodentation angle (*)
[
40
45
Fig. 2. The Read-Shockley form of the grain boundary energy as a function of the misorientation angle across the boundary. The energy is in units of 70, the prefactor.
Y0=
#a(cos 0/2 + sin 0/2) 4~z(l - v)
(5)
where # is the rigidity modulus, v is the Poisson's ratio, a is the lattice constant, and 0 is the misorienration angle. In order to provide an appropriate temperature scale, a nominal value of 70 must he. found. Values for the unknown parameters are chosen such that they correspond roughly to values appropriate to ceramics. Thus, # = 100 GPa, v = 1/3, a---5 A, the sum of the trigonometric terms was chosen to be equal to be equal to one, and therefore, ~'0= 0.31 e V A -2. Defining the temperature scale as f l - 7 o / k T , room temperature then corresponds to fl ~ 12A -2. The units of A,-2 appear because the Read-Shockley grain boundary energy is an energy per unit area of boundary. Simulations were performed at three different temperatures, fl = 3, 12, and 4 0 A -~. Note that higher values of fl correspond to lower absolute temperatures. The microstructure of three samples allowed to evolve from the starting configuration shown in Fig. 1 at the above temperatures for 125 time steps are shown in Fig. 3(a), (b), and (c). Comparing the microstructures of Fig. 3(a)-(c) to that in Fig. 1, it can be seen that the fraction of
NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
(a)
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(b) ~-
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1661
here, the distribution o f grain orientations both before and after evolution has the characteristic that they are essentially flat, showing no preference for any particular orientation. There are striking and significant differences in the microstructures displayed in Fig. 3(a)-(c). For example, the density of low-angle boundaries is clearly much smaller for fl = 3 and fl = 40 than it is for fl = 12. This occurs because the fluctuations in grain orientation are larger for fl = 3 and tend to allow the dusters to form and break up over a shorter time scale than for lower temperatures. Figure 4(a) shows a smaller number of grains sharing low-angle misorientation relationships than the room temperature sample, Fig. 4(b). Figure 4(b) and, to a certain extent Fig. 4(c), is an inverted form of the Read-Shockley energy curve. In a study of grain boundary energies in halite, Moment and Gordon [29] made a similar observation for tricrystals annealed at 750°C. Their work is significant because it demonstrates that the grain boundary misorientation distributions evolve to reflect the grain boundary energy curve. Furthermore, Randle and Brown [30] observed in microtexture measurements on austenitic stainless steels that annealing at 0.72Tin increased the fraction of special (and low-energy) boundaries in the system. They also observed a clustering of special boundaries. In addition, other works have reported an increase in special boundaries on annealing in a
60
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(c) 40
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Fig. 3. Sample microstructures for systems allowed to evolve from the microstructure of Fig. 1 under the influence of finite temperatures. These examples are after 125 time steps. (a) f l = 3 , ~ b ) f l = 1 2 , ( c ) fl=40.
m°
0 60
(c)
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low-angle boundaries (0 ~< 0c) has significantly increased. This result is demonstrated quantitatively in Fig. 4(a)-(c) which are plots of the number of angles vs misorientation angle. In each plot, the dashed line is the distribution before evolution (corresponding to the microstructure in Fig. l) and is relatively flat, as discussed in Section 2.1 above. Although not shown
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5
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15 2 0
25
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35 4 o
45
Misorientation angle (*1 Fig. 4. Plots of the distribution of misorientation angle vs
misorientation angle for (a) fl = 3, (b)/~ = 12, and (c) fl = 40 for the corresponding microstructures in Fig. 3.
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variety of materials [31-32]. These experimental observations are corroborated by our present simulations. The low-energy cusp at 0 = 45 ° is reflected in Fig. 4(a), albeit only weakly. That Fig. 4(a) only weakly resembles the features of the Read-Shockley energy curve arises because of the relative magnitudes of the depth of the energy cusp and I//L The maxima in the interfacial energy curve may be thought of as energy barriers that the system must overcome in order to evolve. If the ratio of 1//~ to the depth of the energy cusp is greater than one, then the interfacial energy is sufficiently high so as not to allow accumulation in the low-energy cusp. Such a requirement is fulfilled for ]] = 3 for the cusp at 45 ° and to a lesser extent for the cusp at 0 °. F o r the/~ = 12 and/~ = 40 systems, this requirement is not met. The observations concerning the effect of temperature on microstructure evolution lead to the proposal of annealing criteria for polycrystalline systems. If the y(0) curve is known for a particular system, then a desired distribution of misorientation angles may be obtained by annealing at a temperature dictated by the depth of the cusps in the T(0) curve.
3.2. Applied strain effects In the present section we maintain the coupling constant at zero, but now allow the boundary lengths to change so as to simulate the application of an external uniaxial strain. In order to emphasize the effects of strain, we have used a relatively low temperature, fl = 40, in the present simulations. We make the further restriction that the polycrystal is plastically deformed, so that the area of the two-dimensional grains remains constant under either tension or compression. Using the Read-Shockley form for the grain boundary energy, strains of 5, 25 and 50% under tension and 25% under compression have been applied to the model polycrystai of Fig. 1. The resulting microstructures after 125 time steps are shown in Fig. 5(a)-(d). The direction of the applied strain is indicated on the figures. What is evident from the figures is that for higher applied tensile strains (>~25%), the distribution of low-angle boundaries, as shown by the heavy lines connecting the centers of neighboring grains, is predominantly parallel to the applied strain. For an applied compressive strain, the lowangle boundaries are predominantly in the directions not parallel to the applied strain. In either case, the driving force for development of a restricted fiber texture is that low-angle misorientation relationships appear along the boundaries of longest length since the interfacial energy is 7 times the boundary area. Plots of the number of angles with a given misorientation both before and after evolution are shown in Fig. 6(a)-(d) and correspond to the same strains as in Fig. 5(a)-(d). The striking feature of these plots is that they all display roughly the same distribution although the 5% strained example shows fewer low-
angle grain boundaries in the direction parallel to the applied strain. This last effect arises because the applied strain is not large enough to overcome temperature fluctuations. Indeed, Fig. 6(a)--(d) are very similar to Fig. 4(c), indicating that in these simulations strain plays a less significant role in microstructure evolution than does temperature.
3.3. Applied field effects Lastly, we consider microstructure evolution under the influence of an external field in the absence of applied strain. As was done for the case of applied strain, we adopt the position of minimizing the effects of temperature by setting /~ = 40 and vary the strength of coupling between the external field and one of the grain axes. The results reported here are for a field of magnitude [ E l = 1.0 oriented in the x-direction (see Fig. 1). Values of the coupling strength, J, were chosen so as to make E~,~ ,~ Ug ( J = 0 . 5 ) , E~cld"~ Ug (J = 1.5, 2.5), and E ~ > > Ug (J = 4.0) where E~c~ is the energy associated with the field ( = I - J a . E I). The microstructures generated after 125 time steps are displayed in Fig. 7(a)-(d). Figure 7(a), which was generated with J = 0.5, superficially resembles Fig. 3(c) so that the weakly coupled applied field has only a marginal effect on microstructure evolution. For the more strongly coupled systems, Fig. 7(b) (J --- 1.5) and Fig. 7(c) (J = 2.5), the microstructure appears to contain more low-angle boundaries than in Fig. 7(a). The very strongly coupled system, Fig. 7(d) (J = 4.0), shows definitive differences from the other systems, with a far greater fraction of low-angle boundaries. In the last system nearly one-third of the grains constitute a single cluster, suggesting that an external field may more readily nucleate clusters in this simulation than temperature alone. This is due in part to the fact that an external field can very effectively dampen the fluctuations associated with temperature. Plots of frequency of misorientation angle vs misorientation angle are shown in Fig. 8(a)-(d). The first three figures are all similar to one another and to Fig. 4(c), displaying the inverted shape of the energy vs misorientation angle curve. Figure 8(d) shows the effect of strong coupling through a greater fraction of low-angle boundaries as well as a flatter distribution for intermediate angles. Plots of the frequency of grain orientations vs orientation angle are not included here, but are all flat except for the very strong coupling case which shows a strong preference for orientations along the applied field direction. 4. SUMMARY AND CONCLUSIONS In summary, we have presented a model for a two-dimensional polycrystalline system and examined a possible mechanism for its evolution under the influence of finite temperatures, applied strains, and external fields. It was observed in the simulations that temperature had a strong effect on the resulting
NICHOLS et al.:
LENGTH SCALES F O R POLYCRYSTALLINE M A T E R I A L S - - I
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Fig. 5. Sample microstructures for systems allowed to evolve from the microstructure of Fig. 1 under the influence of various applied strains, The temperature was kept fixed at fl = 40 and these examples are after 125 time steps. The direction of the applied strain is indicated on each figure. (a) 5% tension, (b) 25% tension, (c) 50% tension, and (d) 25% compression.
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LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
microstructure, but that applied fields and, to a lesser extent, applied strain, tended to diminish the randomizing fluctuations associated with temperature. In all cases, lowering the total internal energy of the system resulted in an increase in the number of low-angle grain boundaries. There are a number of obvious generalizations of the present work. Firstly, all of these microstructure generation and evolution processes can be extended quite readily to three-dimensional systems. In this case, six-sided grains would be replaced by grains with an average of ~ 13 sides. Furthermore, the restriction of equiaxed grains may be dropped and more general grain morphologies adopted. It is also possible to examine systems with other fiber textures and with other than cubic crystallographic symmetry. Finally, for the model microstructure presented here, it is quite possible to extend the two-state grain boundary classification to a multi-state classification, differentiating coincidence site lattice boundaries, for example. Clusters might then be identified by lines of different colors connecting the centers of grains linked by like-property boundaries. Despite the number of generalizations possible, the model presented here exhibits a number of features observed experimentally and suggests others which may be of interest. The most notable observation is the increase in the number of low-angle
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(c)
Fig. 7. Sample microstructures for systems allowed to evolve from the microstructure of Fig. 1 under the influence of an external field with various coupling strengths, The temperature was kept fixed at ~ = 40 and these examples are after 125 time steps. (a) d = 0.5, (b) J = 1.5, (c) J = 2.5, and (d) J = 4.0.
NICHOLS et al.: 75
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LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I .
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aries, which has also been reported in various experimental systems [28-31]. While n o t conclusive, our simulations suggest t h a t grain r o t a t i o n is a possible m i c r o s t r u c t u r e evolution mechanism, for instance, w h e n grain b o u n d a r i e s are heavily pinned. F u r t h e r more, it was observed t h a t the m i s o r i e n t a t i o n angle distribution function exhibited a n inverted form of the characteristic features of the grain b o u n d a r y energy plot which m a y be used as a guide to annealing polycrystals. REFERENCES 1. V. Randle and B. Ralph, Rev. Phys. Appl. 23, 501 (1988). 2. J. D. Russell and A. T. Winter, Scripta metall. 19, 575 (1985). 3. J. D. Frize, C. Carry and A. Mocellin, in Advances in Ceramics, Vol. 10, Structure & Properties of MgO and Alz03 Ceramics (edited by W. D. Kingery), pp. 720. Am. Ceram. Sot., Cleveland, Ohio (1984). 4. L. A. Shvindlerman and B. B. Straumal, Acta metall. 33, 1735 (1985).
AM 39/7~S
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5. S. Lartigue and L. Priester, J. Am. Ceram. Soc. 71, 430 (1988). 6. I. Nakamichi, J. Sci. Hiroshima Univ., Ser. .4 54, 49 (1990). 7. A. P. Sutton and R. W. Balluffi, .4cta metall. 35, 2177 (1985). 8. J. D. Livingston, in Grain Boundaries in Engineering Materials (edited by J. L. Walter, J. H. Westbrook and D. A. Woodford), p. 165 Claitor's, Baton Rouge La. 9. G. A. Chadwick and D. A. Smith (editors), Grain Boundary Structure and Properties. Academic Press, New York (1976). 10. S. Hagege, G. Nouet, G. Sainfort and S. L. Sass (editors), J. Physique 43, C-6 (1982). 11. R. W. Ballutfi, Grain Boundary Structure and Kinetics, (Am. Soc. Metals, Metals Park, Ohio, 1980). 12. M. Rfihle, R. W. Balluffi, H. Fischmeister and S. L. Sass, (editors), J. Physique 46, C-4 (1985). 13. T. Endo, H. Fukutomi, M. Ohashi and T. Sakai, Japan Inst. Metals, Trans. Japan Inst. Metals 27, 571, (1986). 14. T. Watanabe, Mater. Forum 11, 284 (1988). 15. C. S. Nichols, D. R. Clarke and D. A. Smith, in Simulation and Theory of Evolving Microstructures and Textures (edited by M. P. Anderson and A. Rollett). T.M.S., Warrendale, Pa (1990). 16. T. Watanabe, Res. Mechanica 11, 47 (1984). 17. T. Watanabe, Trans. Japan Inst. Metals 27, suppl., 73 (1986). 18. T. Watanabe, Proc. Lake Placid Int. Conf. on Interface. Engng (1987). 19. C. S. Smith, in Metallnterfaces, p. 65. Am. Soc. Metals, Cleveland, Ohio (1952). 20. E. P. Wigner, Group Theory. Academic Press, New York (1959). 21. P. Chaudhari and J. W. Matthews, J. appl. Phys. 42, 3063 (1971). 22. W. Lojkowski, H. Gleiter and R. Maurer, Acta metall. 36, 69 (1988). 23. H. Hu, Trans. metall. Soc..4.I.M.E. 224, 75 (1962). 24. J. C. M. Li, J. appl. Phys. 33, 2958 (1962). 25. N. Metropolis et al., J. chem. Phys. 21, 1087 (1953). 26. W. Shockley and T. W. Read Jr, Phys. Rev. 78, 275 (1950). 27. A rotation through a maximum angle of 4 ° is allowed for each step. This maximum angle is chosen for both physical and computational reasons. It is unreasonable to allow a grain to rotate in one step through very large angles. Further, following Binder in "Introduction: Theory and "Technical" Aspects of Monte Carlo Simulations" [Monte Carlo Methods in Statistical Physics, p. 1. Springer, Berlin (1986)], we choose a small absolute maximum rotation angle so that the acceptance ratio of rotations is approximately 0.5. 28. M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, .4cta metall. 32, 783 (1984). 29. R. L. Moment and R. B. Gordon, J. Am. Ceram. Soc. 47, 570 (1964). 30. V. Randle and A. Brown, Phil. Mag. A 59, 1075 (1989). 31. V. Randle and B. Ralph, Proc. R. Soc..4 415, 239 (1988). 32. H. Grimmer, R. Bonnett, S. Lartigue and L. Priester, Phil. Mag. A 61, 493 (1990).
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
ALTERNATIVE LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I. MICROSTRUCTURE EVOLUTION C. S. NICHOLSI', R. F. COOK, D. R. CLARKE and D. A. S M I T H IBM Research Division, IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. (Received 7 September 1990) Abstract--It is a well-documented experimental observation that properties of grain boundaries depend on the atomic structure of the boundary. Yet constitutive relations for properties of polycrystalline materials containing a variety of grain boundaries currently do not take account of this boundary-toboundary variability. Instead, a single-length scale---the average grain diameter is utilized with the underlying assumption that all grain boundaries are the same. In this (I) and the following paper (II), we extend the accepted veiwpoint to encompass a binary classification of grain boundaries based on their misorientation angle. The resultant new length scale is that associated with clusters of grains linked by grain boundaries sharing misorientations in the same category. This first paper focuses on how a model polycrystal is generated, the energetics of the model, and its evolution under various external influences. Rrsumr---I1 est prouv6 exprrimentalement que les proprirtrs des joints de grains drpendent de la structure atomique du joint. Cependant les relations constitutives relatives aux proprirtrs des matrriaux polycristallins contenant un grand nombre de joints de grains ne tiennent grnrralement pas compte de la variabilit6 de joint a joint. A la place, on utilise une seule 6,chelle de longueur--le diamrtre moyen des grains--avec l'hypothrse sous-jacente que tousles joints de grains sont identiques. Dans les parties I e t II de cet article, nous 6tendons ce point de vue pour proposer une classification binaire des joints de grains basee sur leur angle de drsorientation. La nouvelle 6chelle de longueur rrsultante est associre aux amas de grains lirs par des joints partageant des drsorientations de mrme catrgorie. Le premier article prrsente un polycristal modele, les aspects 6nergrtiques du modrle et son 6volution sous diverses influences exterieures. Zusammenfassnng--Eine wohlbekannte experimentell Beobachtung ist, dab die Eigenschaften der Korngrenzen von der atomaren Struktur abh/ingen. Allerdings beriicksichtigen bisher die Grundbeziehungen zur Beschreibung der Eigenschaften polykristalliner Materialien, die eine Vielfalt von Korngrenzen enthalten, diese Zusammenh/inge nicht. Stattdessen wird einfacher L/ingenmaBstab, n/imlich der mittlere Korndurchmesser, benutzt; dahinter steht die Annahme, dab alle Komgrenzen gleich sind. In dieser (I) und der nachfolgenden (II) Arbeit erweitern wir den akzeptierten Standpunkt, um eine bin/ire Klassifikation der Korngrenzen auf der Basis ihrer Fehlorientierungswinkel einzuarbeiten. Der sich ergebende neue L/ingenmaBstab h/ingt zusammen mit Kornhaufen, die fiber Korngrenzen mit Fehlorientierungen derselben Kategorie zusammenh/ingen.
1. INTRODUCTION It is well established that the physical, mechanical and chemical properties of a polycrystalline material differ markedly from those of the same material in single-crystal form. Yet, historically, the relationship between the properties of a polycrystal and its consituent grains and grain boundaries has been somewhat obscure. There are two general divergent lines of research. One approach is to formulate scaling laws for how properties depend on a length scale linearly related to the average grain size and, often, assumed to be the average grain size, d, which implies that all grain boundaries are the same. Specific tPresent address: Cornell University, Department of Materials Science and Engineering, Ithaca, NY 14853, U.S,A.
examples of such relationships include the Hall-Petch relationship for plastic yield stress, a ocd-l/2; the Coble relationship for t h e strain rate in grainboundary-diffusion-controlled creep, ~ocd-3; the H e r r i n g - N a b a r r o relationship for the strain rate in lattice-diffusion-controlled creep, ~ ocd-2; and the Fuchs' expression for resistivity, p oc d - l . There exist, however, many failings of these expressions. In particular, there is ample experimental evidence that boundaries have distinctly different properties that depend on one or more interface degrees of freedom [1-14]. The second approach is to focus on "protot y p a r ' boundaries with the expectation that their properties are somehow generalizable to an arbitrary interface. The necessity remains, of course, to synthesize the information about individual boundaries and grains into a description of the polycrystal.
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NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS---I
In this and the following paper (II), our objective is to integrate these two lines of research by exploring the relevant length scale for polycrystalline materials when boundary-to-boundary variability in grain boundary property is explicitly considered. This work extends the earlier work reported by us [15]. The simplest extension of the assumption that all grain boundaries have the same properties is to divide them into two distinct categories. For example, Watanabe [14, 16-18] has pointed out from a survey of the expansive literature available, that low-angle and special grain boundaries generally have low energy, low mobility, low diffusivity, etc. whereas the remaining high-angle, random boundaries have the opposite properties. It is not entirely clear how to connect this binary classification with an experimentally accessible variable and ineluctably the demarcation between the two classes "low" and "high" is somewhat arbitrary. Here it is convenient to use the magnitude of the misorientation angle as the variable that distinguishes between the two categories of boundaries. In the present work, a two-dimensional polycrystal is modeled as an array of regular hexagonal grains consisting of randomly oriented material with a four-fold rotational axis perpendicular to the surface of the array. The relevant microstructural units are identified as clusters of grains that are linked by low-angle boundaries. These clusters are characterized by a mass and a linear dimension that are a function of the misorientation angle chosen to delineate between low- and high-angle boundaries. We discuss in this first paper both the generation of the model microstructure and its evolution as a whole under the influence of finite temperatures or externally applied fields coupled to the grain interiors. The following paper is concerned with how the measures of cluster dimension change with microstructure evolution. We also discuss at some length what the relevant cluster measure is for a given property and speculate on how the constitutive relations vary with cluster size. The remainder of the present paper is organized as follows. Section 2 is devoted to a discussion of tlae methodology of generation of the model microstructure, its energetics, and its temporal development. In Section 3 we consider the specific effects of temperature, applied strain, and applied field on microstructure evolution. We close in Section 4 with a summary and conclusions. 2. METHODOLOGY 2.1. Static model Euler's law [19] applied to two-dimensional microstructures requires that grains have, on average, six sides. In accordance with this, we idealize our polycrystalline system to consist of regular hexagons that tile the plane: .Each grain consists, in turn, of material with crystallographic four-fold rotational symmetry
in the plane. The microstructure is fiber-textured in that the grains share a common (001) axis of rotation which is perpendieu!ar te the plane. The in-plane axes of each grain are assigned a random orientation with respect to some fixed external coordinate system. In this work we have employed a random number generator which produces orientation angles with a uniform distribution. The misorientation angles 0 of adjoining grains are determined next. In doing so, the symmetry of the grains must be taken into account. In particular, because the grain boundaries in this model are [001] tilt boundaries in a four-fold symmetric system, only misorientation angles between 0 ° and 45 ° are unique. The misorientation angles generated form a continuous group and it can be shown from group theory [19] that in two dimensions the distribution function of the rotational angles is a constant. In order to minimize finite-size effects, periodic boundary conditions are applied in both directions. For the model microstructures used in the present work, we find a standard deviation of the misorientation angle distribution of <0.5% for samples consisting of 400 grains. The grain boundaries are next placed into two distinct classes. Most generally, grain boundaries are classified on the basis of geometrical criteria as low-angle/special or high-angle (random). As noted in the Introduction, numerous experiments demonstrate that low-angle and special boundaries have similar properties but are distinct from those of high-angle boundaries. We however make the assumption that there are only two distinct classes of boundary: those for which 0 is less than some chosen cut-off angle, 0c, and those for which 0¢ ~< 0 ~<45 °. This binary classification scheme can be generalized to any number of distinct categories and does not present a limitation of the methodology. Lastly, once this division of boundaries is made, clusters of grains that share like boundaries may be identified. A portion of a microstructure thus generated is shown in Fig. 1. The clusters of grains are identified in the figure by those grains whose centers are connected by heavy lines. 2.2. Energetics o f the model microstructure The total internal energy, Ug, of a single grain with Ng sides in a polycrystalline ensemble can be written as the sum of the grain boundary energy and the energy of interaction with an external field Ng
ug= ~ ,~,~,- Ja.~
(1)
i=l
where )-i is the length of the ith boundary, ~, is the interfacial energy per unit area of the same boundary, J is a coupling constant (or a modulus), a is one of the in-plane crystallographic axes, and £ is an external field. The summation in the first term runs over all the sides of the chosen grain, which is six in the
NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
.)
Fig. 1. A sample microstructure showing the hexagonal grains and the dusters of low-angle boundaries. Grains in the same cluster are connected by a heavy line through the center of the grain. The cut-off angle delineating low-angle and high-angle boundaries is 10°. present case. It is assumed that all grains have unit thickness so that the first term has the units of energy. The coupling of the field to the a-axis is arbitrary and our results would not qualitatively change if the field had been coupled to another crystallographic direction. The total internal energy of the polycrystal is given by the sum over all the individual grains, N, in equation (1)
U=
gffil
1N Ng ~'Jg--2gE E ~i~)i" ~1i=1
1659
system does not get trapped in local energy minima, the microstructure may evolve to a macroscopically inequivalent final state. If a strain is now applied to the polycrystal, it may deform in one of two ways. Either the lengths of the grain boundaries remain unchanged, with the consequence that the volume is not conserved and local triple junctions are not in equilibrium, or the boundaries do change length. In the latter case, the volume of the grains is conserved and local triple junctions are in equilibrium, but the total internal energy of the system may change because the total grain boundary length in the system changes. The polycrystal thus evolves so as to reduce the total interfacial grain boundary energy, consistent with changes in the grain boundary lengths. In addition to temperature and applied strain, an external field may couple to a grain's degrees of freedom to influence microstructural evolution. For example, an external electric field may couple to the polarization in a ferroelectric material and at temperatures close to the transition temperature, the orientation of the grain may be altered. Or, the field may be thought of as a gradient in some quantity, for example, temperature or defect concentration. Because this contribution to the total internal energy is negative (J >t 0), it competes with the interfacial grain boundary energy, which is always positive, in effecting microstructural evolution. 2.3. Temporal evolution o f the model
(2)
The second term appears so as to avoid doublecounting the grain boundary energy. We ignore the contribution to the internal energy of point defects within the individual grains. The as-generated microstructure has a spatially random distribution of low- and high-angle boundaries and clearly is not the lowest-energy state of the system. We allow the system to evolve, in a manner to be considered below, such that the total internal energy is minimized under the competing influences of the internal energy variables. If the coupling constant, J, is set to zero (or, equivalently, the external field is switched off) and all grain boundaries are forced to have the same length, 2 i = 2, then the only remaining external variable which may affect microstructure evolution is the temperature. For a polycrystal in contact with a thermal reservoir at a fixed temperature T~, the microstructure will evolve so as to minimize the total internal energy and will reach a minimum (which may only be a local minimum, however) given enough time. If we now raise the temperature of the thermal reservoir to T2, then one of two things will occur. In one instance, the microstructure will reach a macroscopically equivalent final state, although perhaps more rapidly and with greater fluctuations about this state. If, however, T2 is significantly higher than T~ such that the
The original experiments of Chaudhari and Matthews [21] and the later notable experiments of Gleiter and co-workers [22] on copper and silver single-crystal spheres sintered to single-crystal substrates demonstrated that the randomly oriented spheres rotated into specific low-energy orientations. A natural corollary of the unhindered rotation of the spheres is the existence of a torque on grains in a polycrystalline aggregate that provides the driving force for grains to adopt orientations with respect to their neighbors that minimizes the total internal energy. Experimental justification for the rotation of grains, or subgrains, has come from the electron microscopy work of Hu [23] and Li [24]. Specifically, Hu observed the gradual disappearance of subgrain boundaries and a loss of contrast between neighboring grains which indicated that these subgrains attained the same orientation. It was argued that subgrain rotation led to subgrain coalescence and was the precursor to recrystallization. In the present context, grain rotation is utilized as a means for lowering the total internal energy of the as-generated microstructure. Although we do not allow grain coalescence to occur, the distribution and nature of the clusters of grains that we use to characterize the polycrystal does change. This point is addressed more thoroughly in II. The microscopic mechanism of subgrain rotation is not understood and we consider it a peripheral issue beyond the scope
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NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
of the present work. Li [24] presents some possible models for subgrain rotation. For our model polycrystals, grain rotation is facilitated by a Monte Carlo Metropolis algorithm [25]. A grain is selected at random and its total internal energy, Ug, is obtained. In the present work we use the Read-Shockley [26] form for the grain boundary energy per unit area 7 = ~oO(A - In 0)
(3)
where 70 is a prefactor that depends on material properties, 0 is the misorientation angle of adjacent grains, and A is a constant. We have chosen the value A = 0.231 which is the value originally used by Read and Shockley~ The energy vs misorientation angle for the range of angles considered here (0° ~< 0 ~<45 °) is plotted in Fig. 2. Although the Read-Shockley form is only strictly valid for misorientation angles below which dislocation cores do not overlap (0 < 15°), we have used it for larger values. We do this because it is generally recognized that grain boundary energy vs misorientation angle plots often show cusps corresponding to misorientations of particularly low energy. As is evident from Fig. 2. such a cusp exists for 0 in the vicinity of 45 °. It is thus possible to examine the evolution of the sytem as if it contained a special boundary at this angle. The selected grain is then allowed to rotate through a random angle [27] either clockwise or counterclockwise. The total internal energy of the rotated grain is calculated and compared to its total internal energy before rotation. If the energy is lower, the trial rotation is accepted. If the energy is higher, then a transition probability, W, is calculated W = exp( - AUg/k T)
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3. MICROSTRUCTURE EVOLUTION 3.1, Temperature effects
In this section, the boundary lengths are fixed, the coupling constant is set to zero and we explore the role played by temperature in microstructure evolution. In the Read-Shockley form of the grain boundary energy, equation (3), the prefactor is given explicitly by
(4)
where AUg = Ug(final)-Ug(initial), k is Boltzmann's constant, and T is the absolute temperature. If W is greater than some random number between zero and one, then the rotation is accepted, otherwise the move
0.6
is rejected. All energies are expressed in units of the Read-Shockley prefactor, 70, so that temperature is introduced through the ratio Vo/kT. Subsequently another grain is chosen at random and the process is repeated. The most ambiguous part of Monte Carlo simulations is the association of a time scale with each Monte Carlo step. We follow a typical procedure by defining one time step as the number of actual Monte Carlo steps divided by the number of grains in the system. Therefore, during one time step, each grain has the possibility of undergoing rotation. It should be further remarked that the time scale of a given Monte Carlo step is much larger than the time scale of local atomic motions, as also implied in the Potts model simulations of microstructure evolution [28]. All results reported in the present work are for samples of 20 grains on a side (400 grains total) and 125 time steps. Both of these variables have been extended to larger systems and longer times and we find no qualitatitive differences in the results. Furthermore, in all simulations we have chosen the cut-off angle, 0¢, delineating low-angle boundaries from high-angle boundaries, to be equal to 10°.
I
15 20 25 30 35 Misodentation angle (*)
[
40
45
Fig. 2. The Read-Shockley form of the grain boundary energy as a function of the misorientation angle across the boundary. The energy is in units of 70, the prefactor.
Y0=
#a(cos 0/2 + sin 0/2) 4~z(l - v)
(5)
where # is the rigidity modulus, v is the Poisson's ratio, a is the lattice constant, and 0 is the misorienration angle. In order to provide an appropriate temperature scale, a nominal value of 70 must he. found. Values for the unknown parameters are chosen such that they correspond roughly to values appropriate to ceramics. Thus, # = 100 GPa, v = 1/3, a---5 A, the sum of the trigonometric terms was chosen to be equal to be equal to one, and therefore, ~'0= 0.31 e V A -2. Defining the temperature scale as f l - 7 o / k T , room temperature then corresponds to fl ~ 12A -2. The units of A,-2 appear because the Read-Shockley grain boundary energy is an energy per unit area of boundary. Simulations were performed at three different temperatures, fl = 3, 12, and 4 0 A -~. Note that higher values of fl correspond to lower absolute temperatures. The microstructure of three samples allowed to evolve from the starting configuration shown in Fig. 1 at the above temperatures for 125 time steps are shown in Fig. 3(a), (b), and (c). Comparing the microstructures of Fig. 3(a)-(c) to that in Fig. 1, it can be seen that the fraction of
NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
(a)
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here, the distribution o f grain orientations both before and after evolution has the characteristic that they are essentially flat, showing no preference for any particular orientation. There are striking and significant differences in the microstructures displayed in Fig. 3(a)-(c). For example, the density of low-angle boundaries is clearly much smaller for fl = 3 and fl = 40 than it is for fl = 12. This occurs because the fluctuations in grain orientation are larger for fl = 3 and tend to allow the dusters to form and break up over a shorter time scale than for lower temperatures. Figure 4(a) shows a smaller number of grains sharing low-angle misorientation relationships than the room temperature sample, Fig. 4(b). Figure 4(b) and, to a certain extent Fig. 4(c), is an inverted form of the Read-Shockley energy curve. In a study of grain boundary energies in halite, Moment and Gordon [29] made a similar observation for tricrystals annealed at 750°C. Their work is significant because it demonstrates that the grain boundary misorientation distributions evolve to reflect the grain boundary energy curve. Furthermore, Randle and Brown [30] observed in microtexture measurements on austenitic stainless steels that annealing at 0.72Tin increased the fraction of special (and low-energy) boundaries in the system. They also observed a clustering of special boundaries. In addition, other works have reported an increase in special boundaries on annealing in a
60
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Fig. 3. Sample microstructures for systems allowed to evolve from the microstructure of Fig. 1 under the influence of finite temperatures. These examples are after 125 time steps. (a) f l = 3 , ~ b ) f l = 1 2 , ( c ) fl=40.
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low-angle boundaries (0 ~< 0c) has significantly increased. This result is demonstrated quantitatively in Fig. 4(a)-(c) which are plots of the number of angles vs misorientation angle. In each plot, the dashed line is the distribution before evolution (corresponding to the microstructure in Fig. l) and is relatively flat, as discussed in Section 2.1 above. Although not shown
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Misorientation angle (*1 Fig. 4. Plots of the distribution of misorientation angle vs
misorientation angle for (a) fl = 3, (b)/~ = 12, and (c) fl = 40 for the corresponding microstructures in Fig. 3.
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NICHOLS et al.: LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
variety of materials [31-32]. These experimental observations are corroborated by our present simulations. The low-energy cusp at 0 = 45 ° is reflected in Fig. 4(a), albeit only weakly. That Fig. 4(a) only weakly resembles the features of the Read-Shockley energy curve arises because of the relative magnitudes of the depth of the energy cusp and I//L The maxima in the interfacial energy curve may be thought of as energy barriers that the system must overcome in order to evolve. If the ratio of 1//~ to the depth of the energy cusp is greater than one, then the interfacial energy is sufficiently high so as not to allow accumulation in the low-energy cusp. Such a requirement is fulfilled for ]] = 3 for the cusp at 45 ° and to a lesser extent for the cusp at 0 °. F o r the/~ = 12 and/~ = 40 systems, this requirement is not met. The observations concerning the effect of temperature on microstructure evolution lead to the proposal of annealing criteria for polycrystalline systems. If the y(0) curve is known for a particular system, then a desired distribution of misorientation angles may be obtained by annealing at a temperature dictated by the depth of the cusps in the T(0) curve.
3.2. Applied strain effects In the present section we maintain the coupling constant at zero, but now allow the boundary lengths to change so as to simulate the application of an external uniaxial strain. In order to emphasize the effects of strain, we have used a relatively low temperature, fl = 40, in the present simulations. We make the further restriction that the polycrystal is plastically deformed, so that the area of the two-dimensional grains remains constant under either tension or compression. Using the Read-Shockley form for the grain boundary energy, strains of 5, 25 and 50% under tension and 25% under compression have been applied to the model polycrystai of Fig. 1. The resulting microstructures after 125 time steps are shown in Fig. 5(a)-(d). The direction of the applied strain is indicated on the figures. What is evident from the figures is that for higher applied tensile strains (>~25%), the distribution of low-angle boundaries, as shown by the heavy lines connecting the centers of neighboring grains, is predominantly parallel to the applied strain. For an applied compressive strain, the lowangle boundaries are predominantly in the directions not parallel to the applied strain. In either case, the driving force for development of a restricted fiber texture is that low-angle misorientation relationships appear along the boundaries of longest length since the interfacial energy is 7 times the boundary area. Plots of the number of angles with a given misorientation both before and after evolution are shown in Fig. 6(a)-(d) and correspond to the same strains as in Fig. 5(a)-(d). The striking feature of these plots is that they all display roughly the same distribution although the 5% strained example shows fewer low-
angle grain boundaries in the direction parallel to the applied strain. This last effect arises because the applied strain is not large enough to overcome temperature fluctuations. Indeed, Fig. 6(a)--(d) are very similar to Fig. 4(c), indicating that in these simulations strain plays a less significant role in microstructure evolution than does temperature.
3.3. Applied field effects Lastly, we consider microstructure evolution under the influence of an external field in the absence of applied strain. As was done for the case of applied strain, we adopt the position of minimizing the effects of temperature by setting /~ = 40 and vary the strength of coupling between the external field and one of the grain axes. The results reported here are for a field of magnitude [ E l = 1.0 oriented in the x-direction (see Fig. 1). Values of the coupling strength, J, were chosen so as to make E~,~ ,~ Ug ( J = 0 . 5 ) , E~cld"~ Ug (J = 1.5, 2.5), and E ~ > > Ug (J = 4.0) where E~c~ is the energy associated with the field ( = I - J a . E I). The microstructures generated after 125 time steps are displayed in Fig. 7(a)-(d). Figure 7(a), which was generated with J = 0.5, superficially resembles Fig. 3(c) so that the weakly coupled applied field has only a marginal effect on microstructure evolution. For the more strongly coupled systems, Fig. 7(b) (J --- 1.5) and Fig. 7(c) (J = 2.5), the microstructure appears to contain more low-angle boundaries than in Fig. 7(a). The very strongly coupled system, Fig. 7(d) (J = 4.0), shows definitive differences from the other systems, with a far greater fraction of low-angle boundaries. In the last system nearly one-third of the grains constitute a single cluster, suggesting that an external field may more readily nucleate clusters in this simulation than temperature alone. This is due in part to the fact that an external field can very effectively dampen the fluctuations associated with temperature. Plots of frequency of misorientation angle vs misorientation angle are shown in Fig. 8(a)-(d). The first three figures are all similar to one another and to Fig. 4(c), displaying the inverted shape of the energy vs misorientation angle curve. Figure 8(d) shows the effect of strong coupling through a greater fraction of low-angle boundaries as well as a flatter distribution for intermediate angles. Plots of the frequency of grain orientations vs orientation angle are not included here, but are all flat except for the very strong coupling case which shows a strong preference for orientations along the applied field direction. 4. SUMMARY AND CONCLUSIONS In summary, we have presented a model for a two-dimensional polycrystalline system and examined a possible mechanism for its evolution under the influence of finite temperatures, applied strains, and external fields. It was observed in the simulations that temperature had a strong effect on the resulting
NICHOLS et al.:
LENGTH SCALES F O R POLYCRYSTALLINE M A T E R I A L S - - I
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Fig. 5. Sample microstructures for systems allowed to evolve from the microstructure of Fig. 1 under the influence of various applied strains, The temperature was kept fixed at fl = 40 and these examples are after 125 time steps. The direction of the applied strain is indicated on each figure. (a) 5% tension, (b) 25% tension, (c) 50% tension, and (d) 25% compression.
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NICHOLS et al.:
LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I
microstructure, but that applied fields and, to a lesser extent, applied strain, tended to diminish the randomizing fluctuations associated with temperature. In all cases, lowering the total internal energy of the system resulted in an increase in the number of low-angle grain boundaries. There are a number of obvious generalizations of the present work. Firstly, all of these microstructure generation and evolution processes can be extended quite readily to three-dimensional systems. In this case, six-sided grains would be replaced by grains with an average of ~ 13 sides. Furthermore, the restriction of equiaxed grains may be dropped and more general grain morphologies adopted. It is also possible to examine systems with other fiber textures and with other than cubic crystallographic symmetry. Finally, for the model microstructure presented here, it is quite possible to extend the two-state grain boundary classification to a multi-state classification, differentiating coincidence site lattice boundaries, for example. Clusters might then be identified by lines of different colors connecting the centers of grains linked by like-property boundaries. Despite the number of generalizations possible, the model presented here exhibits a number of features observed experimentally and suggests others which may be of interest. The most notable observation is the increase in the number of low-angle
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Z5
Misorientotion
3o 35
40
45
ongte (°)
Fig. 6. Plots of the distribution of mi~rientation angle vs misorientation angle for (a) 5% tension, (b) 25% tension, 511% tension, and (d) 25% compression for the corresponding microstructures of Fig. 5.
(c)
Fig. 7. Sample microstructures for systems allowed to evolve from the microstructure of Fig. 1 under the influence of an external field with various coupling strengths, The temperature was kept fixed at ~ = 40 and these examples are after 125 time steps. (a) d = 0.5, (b) J = 1.5, (c) J = 2.5, and (d) J = 4.0.
NICHOLS et al.: 75
.
.
.
.
.
LENGTH SCALES FOR POLYCRYSTALLINE MATERIALS--I .
5o~_._]
.
.
(a)
z5
o Il-----:--,---;--,-75
(
A
)'
5o z5 5
o 75
g 5o 25
O
0
5
10
15
20 25
50 55 40 45
M i s o r i e n t e t i o n ongle ( ° )
Fig. 8. Plots of the distribution of misorientation angle vs misorientation angle for (a) J = 0.5, (b) J = 1.5, (c) J = 2.5, and (d) J = 4.0.
aries, which has also been reported in various experimental systems [28-31]. While n o t conclusive, our simulations suggest t h a t grain r o t a t i o n is a possible m i c r o s t r u c t u r e evolution mechanism, for instance, w h e n grain b o u n d a r i e s are heavily pinned. F u r t h e r more, it was observed t h a t the m i s o r i e n t a t i o n angle distribution function exhibited a n inverted form of the characteristic features of the grain b o u n d a r y energy plot which m a y be used as a guide to annealing polycrystals. REFERENCES 1. V. Randle and B. Ralph, Rev. Phys. Appl. 23, 501 (1988). 2. J. D. Russell and A. T. Winter, Scripta metall. 19, 575 (1985). 3. J. D. Frize, C. Carry and A. Mocellin, in Advances in Ceramics, Vol. 10, Structure & Properties of MgO and Alz03 Ceramics (edited by W. D. Kingery), pp. 720. Am. Ceram. Sot., Cleveland, Ohio (1984). 4. L. A. Shvindlerman and B. B. Straumal, Acta metall. 33, 1735 (1985).
AM 39/7~S
1665
5. S. Lartigue and L. Priester, J. Am. Ceram. Soc. 71, 430 (1988). 6. I. Nakamichi, J. Sci. Hiroshima Univ., Ser. .4 54, 49 (1990). 7. A. P. Sutton and R. W. Balluffi, .4cta metall. 35, 2177 (1985). 8. J. D. Livingston, in Grain Boundaries in Engineering Materials (edited by J. L. Walter, J. H. Westbrook and D. A. Woodford), p. 165 Claitor's, Baton Rouge La. 9. G. A. Chadwick and D. A. Smith (editors), Grain Boundary Structure and Properties. Academic Press, New York (1976). 10. S. Hagege, G. Nouet, G. Sainfort and S. L. Sass (editors), J. Physique 43, C-6 (1982). 11. R. W. Ballutfi, Grain Boundary Structure and Kinetics, (Am. Soc. Metals, Metals Park, Ohio, 1980). 12. M. Rfihle, R. W. Balluffi, H. Fischmeister and S. L. Sass, (editors), J. Physique 46, C-4 (1985). 13. T. Endo, H. Fukutomi, M. Ohashi and T. Sakai, Japan Inst. Metals, Trans. Japan Inst. Metals 27, 571, (1986). 14. T. Watanabe, Mater. Forum 11, 284 (1988). 15. C. S. Nichols, D. R. Clarke and D. A. Smith, in Simulation and Theory of Evolving Microstructures and Textures (edited by M. P. Anderson and A. Rollett). T.M.S., Warrendale, Pa (1990). 16. T. Watanabe, Res. Mechanica 11, 47 (1984). 17. T. Watanabe, Trans. Japan Inst. Metals 27, suppl., 73 (1986). 18. T. Watanabe, Proc. Lake Placid Int. Conf. on Interface. Engng (1987). 19. C. S. Smith, in Metallnterfaces, p. 65. Am. Soc. Metals, Cleveland, Ohio (1952). 20. E. P. Wigner, Group Theory. Academic Press, New York (1959). 21. P. Chaudhari and J. W. Matthews, J. appl. Phys. 42, 3063 (1971). 22. W. Lojkowski, H. Gleiter and R. Maurer, Acta metall. 36, 69 (1988). 23. H. Hu, Trans. metall. Soc..4.I.M.E. 224, 75 (1962). 24. J. C. M. Li, J. appl. Phys. 33, 2958 (1962). 25. N. Metropolis et al., J. chem. Phys. 21, 1087 (1953). 26. W. Shockley and T. W. Read Jr, Phys. Rev. 78, 275 (1950). 27. A rotation through a maximum angle of 4 ° is allowed for each step. This maximum angle is chosen for both physical and computational reasons. It is unreasonable to allow a grain to rotate in one step through very large angles. Further, following Binder in "Introduction: Theory and "Technical" Aspects of Monte Carlo Simulations" [Monte Carlo Methods in Statistical Physics, p. 1. Springer, Berlin (1986)], we choose a small absolute maximum rotation angle so that the acceptance ratio of rotations is approximately 0.5. 28. M. P. Anderson, D. J. Srolovitz, G. S. Grest and P. S. Sahni, .4cta metall. 32, 783 (1984). 29. R. L. Moment and R. B. Gordon, J. Am. Ceram. Soc. 47, 570 (1964). 30. V. Randle and A. Brown, Phil. Mag. A 59, 1075 (1989). 31. V. Randle and B. Ralph, Proc. R. Soc..4 415, 239 (1988). 32. H. Grimmer, R. Bonnett, S. Lartigue and L. Priester, Phil. Mag. A 61, 493 (1990).