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Correlations between disorientations in neighbouring dislocation boundaries
Acta mater. 48 (2000) 3005±3014 www.elsevier.com/locate/actamat
CORRELATIONS BETWEEN DISORIENTATIONS IN NEIGHBOURING DISLOCATION BOUNDARIES W. PANTLEON 1{ and D. STOYAN 2 1
Materials Research Department, Risù National Laboratory, P.O. Box 49, 4000 Roskilde, Denmark and 2Institute of Stochastics, Freiberg University of Mining and Technology, Bernhard-von-CottaStraûe 2, 09596 Freiberg, Germany (Received 8 December 1999; accepted 23 February 2000)
AbstractÐDuring plastic deformation, dislocation structures develop with regions of dierent orientations separated by dislocation boundaries. Experimental investigations by means of electron back-scattering diffraction or rocking curves from X-ray diraction give some evidence for the existence of correlations between the disorientations in neighbouring boundaries. Based on a straightforward statistical model an occurrence of anti-correlations between disorientation angles is predicted which can be traced to the limited free path of mobile dislocations carrying the plastic deformation. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. ZusammenfassungÐBei plastischer Verformung bilden sich Versetzungsstrukturen aus, die durch VersetzungswaÈnde voneinander getrennte Gebiete unterschiedlicher Orientierung aufweisen. Aus experimentellen Untersuchungen mit Hilfe der Elektronen-RuÈckstreu-Diraktometrie oder Rocking-Kurven der RoÈntgendiraktometrie lassen sich Hinweise auf Korrelationen zwischen den Desorientierungen in benachbarten Grenzen gewinnen. Auf Grundlage eines einfachen statistischen Modells wird das Auftreten von Anti-Korrelationen zwischen den Fehlorientierungswinkeln vorhergesagt und auf die endliche freie WeglaÈnge der mobilen Versetzungen, den TraÈgern der plastischen Verformung, zuruÈckgefuÈhrt. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Dislocations; Microstructure; Disorientations; Mechanical properties (plastic); Theory & modelling
1. INTRODUCTION
During plastic deformation, dislocation structures develop with dislocation boundaries of high dislocation density and nearly dislocation-free regions (see, e.g. Refs [1, 2]). Generally, two dierent dislocation boundaries can be identi®ed [3]: ordinary dislocation walls arising from statistical mutual trapping of dislocations (incidental dislocation boundaries, IDBs) and straight and parallel dislocation boundaries caused by a dierence in the activation of slip systems on both sides (geometrically necessary boundaries, GNBs). Orientation dierences between the regions separated by a dislocation boundary arise in connection with both types of boundaries [4, 5]. For incidental
{ To whom all correspondence should be addressed. { The alternating disorientations observed in adjacent GNBs (dense dislocation boundaries) are attributed to a dierent activation of slip systems on both sides of GNBs (e.g. Refs [10, 11]) and are not considered here.
dislocation boundaries disorientations are caused by statistical ¯uctuations in the dislocation ¯uxes passing through the cell walls [6]. Geometrically necessary boundaries show a stronger increase of the disorientation angles related to the dierent activity of slip systems on either side of the boundary [7]. Certain experimental results on deformed single crystals obtained by electron back-scattering diraction [8] and X-ray diraction [9] cannot be explained assuming that disorientations of neighbouring boundaries are independent of each other. Both techniques show evidence for the existence of correlations between disorientation angles in adjacent boundaries. The occurrence of correlations is investigated here in case of a dislocation cell structure with only one kind of boundary, IDBs{, based on a simple statistical model for a dislocation cell structure with parallel dislocation walls. In contrast to the previous model [6] for the statistical formation of disorientations in IDBs, the consideration of a chain of parallel dislocation boundaries allows the revelation of spatial correlations.
1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 0 8 3 - 5
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PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
Fig. 1. Mean disorientation angle determined with EBSD line scans (along 350 mm in dierent directions with a step size Dx 1 mm as a function of the eective step size m Dx 0 =Dx for aluminium single crystals of three dierent orientations after plane strain compression to E 1:5 (courtesy A. Godfrey) [8]. 2. EXPERIMENTAL EVIDENCE FOR CORRELATIONS
2.1. Electron back-scattering diraction With a scanning electron microscope local orientations can be determined at certain points on the surface of a (e.g. deformed) material with electron back-scattering diraction (EBSD, cf. Ref. [12]). Usually a constant step size Dx is used for the distances between the measuring points in line (or mesh) scans and the disorientation angles are calculated between adjacent points. From the same set of orientations the disorientation angle can also be evaluated between next-nearest or even more distant neighbours and the mean disorientation angle depends on the corresponding eective step size Dx 0 mDx [8]. In Fig. 1 the mean disorientation angle of aluminium single crystals deformed in plane strain compression at room temperature increases with the step size at small distances between the measuring points. But for large step sizes Dx' the average disorientation angle will level-o and show a saturation value. From the mean chord length l (the mean distance
{ The angular brackets h.i denote averages over all boundaries or spatial averages, respectively. The notation a(n ) with the index in parentheses refers to an angle across n boundaries (the distant disorientation angle), whereas ai is the angle of a speci®c boundary i.
of boundaries along a straight testing line) of the underlying dislocation structure, the number of boundaries n
Dx 0 l
1
between two measuring points in a distance Dx' can be estimated. Consequently, the observed levelling o of the mean disorientation angle for large step sizes Dx' corresponds to a saturation of the mean disorientation angle across a large number of boundaries n as well. As will be shown in the following Section 2.2, such a saturation behaviour is in con¯ict with an assumption of independent disorientation angles and correlations must occur. 2.2. Independent disorientations For uncovering possible correlations between the disorientations in neighbouring boundaries a chain of parallel boundaries with independent disorientations is taken as a point of reference. If the disorientation angles ai of all dislocation boundaries i correspond to the same rotation axis, the disorientation angle{ across n neighbouring boundaries (``distant disorientation angle'') a n
n X ai
2
i1
is the sum of the individual disorientation angles ai
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
with mean value ha n i
n X hai i nhai:
3
i1
For completely uncorrelated disorientation angles hai aj i 0 for all i6j the second-order moment{ n
D E X
2 ai2 n a 2 a n
4
i1
increases linearly with the number n of the boundaries. For a statistical accumulation of excess dislocations (of one sign of the Burgers vector) from ¯uctuations in the dislocation ¯uxes [6], the disorientation angle of each individual boundary i shows a Gaussian distribution with vanishing mean value hai i 0 for all i ) and a standard deviation ha 2 i sa2 : As for all Gaussian distributions with vanishing mean value the standard deviation determines the mean modulus r 2 2 ha i: 5 hjaji p In the same manner, the distant disorientation has a Gaussian distribution with vanishing mean value ha n i 0
6
and the mean modulus of the disorientation angle across n boundaries r r p 2D 2 E 2 p a a n nha 2 i nhjaji 7 hj n ji p p
3007
crystal spectrometer the sample is rocked around an axis perpendicular to the diraction plane and the diracted intensity from a monochromatic primary X-ray beam is detected as a function of the rocking angle o. In investigations on copper single crystals after hot compression along [001] at about 6008C only a slight dependence (cf. Fig. 2) of the angular width Do of rocking curves on the size T of the X-ray beam in the diraction plane (width of the beam at the position of the sample perpendicular to the rocking axis) was found [9]. For a theoretical modelling of the angular width of rocking curves, a set of parallel dislocation tilt boundaries in a mutual distance d with disorientations around the same rotation axis is considered [13, 14] similar to that discussed in Section 2.2. For disorientation angles of either a or ÿa with the same probability and no short-range correlation between the disorientation angles in adjacent boundaries, the (angular) width of rocking curves r p T DoAa a n 8 d is determined by the disorientation angle a and the number n of dislocation boundaries under investigation [13, 14]. The square root behaviour is a direct consequence of the random distribution of the sign of the disorientation angle. According to equation (8) the angular width Do should depend on the beam size T determining the number of dislocation cells or boundaries, respectively, n T=d in the beam. A similar result can be obtained if the disorientations are distributed according to a nor-
increases with the square root of the number of dislocation boundaries. Contrary to the prediction of equation (7) for independent disorientation angles no unlimited increase with the number of boundaries or the step size, respectively, is seen in the experimental data of Fig. 1. The saturation clearly reveals the existence of correlations between disorientation angles in neighbouring boundaries. 2.3. Broadening of X-ray rocking curves In a more integrative manner information about disorientations in crystals can be gained from rocking curves (azimuthal intensity distributions) in Xray diraction (e.g. Ref. [13])Ðin an X-ray double
{ The result given by equation (4) will not change if the disorientation angles do not have the same rotation axis, but the rotation axes are distributed randomly. The square P 2 of the distant disorientation angles a n ni1 ai2 will still be determined by the sum of the squares of the individual disorientation angles.
Fig. 2. Dependence of the angular width Do (peak broadening) of rocking curves on the beam size T in the diraction plane for copper single crystals after hot compression 6008C, E_ 10 ÿ2 =s, rocking axis parallel to the compression axis) [9]. (The mean chord length l 1 mm of the higher deformed single crystal leads, even for the smallest beam size T 1 mm, to a large number of dislocation boundaries n 1000 in the beam in the diraction plane.)
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PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
mal distribution; only the proportionality factor will change [9, 14]. If all dislocation boundaries have disorientations of the same sign corresponding to a strong (positive) correlation instead of a random distribution of the signs, a long-range curvature evolves and the width DoAna becomes proportional to the number of boundaries [15]. The experimentally observed slight dependence in Fig. 2 is in contrast to the predicted square root behaviour of equation (8) for uncorrelated disorientations (or even a linear increase of DoAna for positively correlated disorientations of the same sign). Owing to the weak dependence of Do on T or n, respectively, the disorientation angles in neighbouring boundaries cannot be independent and must be anti-correlated signi®cantly, i.e. disorientations in neighbouring boundaries are more likely of opposite than of the same sign. 3. MODEL
3.1. Outline An elementary model (illustrated in Fig. 3) is considered for the investigation of possible correlations between disorientations: a strip of material with height H deforming in simple shear. The deformation is carried by mobile edge dislocations of a single slip system which are emitted pairwise from dislocation sources. The pairs of edge dislocations of opposite sign of the Burgers vector (_ and ~) and in®nite length represent dislocation loops in this essentially two-dimensional model. The dislo-
cation sources are assumed to be randomly and uniformly distributed (equal probability everywhere). A constant (linear) density of sources nsc along the length L of the strip leads to a total number Nsc nsc L of dislocation sources. A continuous distribution of the dislocation sources (and dislocations) along the strip is considered only for modelling purposes. For a more realistic description of a cell structure (containing cell walls of width w with a high dislocation density and nearly dislocation-free cell interiors) all dislocations in the interval from x to x d can be assumed to gather into a dislocation boundary located somewhere in the bin, e.g. at the centre. In this manner, a chain of dislocation cells with equidistant boundaries and a cell size d is obtained. Alternatively, one may interpret the continuous arrangement of Fig. 3 as a chain of dislocation cells of size d where the interiors have been cut out and a chain of directly adjoining walls has been obtained. The passage of the mobile dislocations through cell interiors is immaterial for the model and only the walls of width w must be taken into account. In this interpretation the dislocation sources are quite reasonably restricted to the dislocation walls. A similar model was treated numerically [16] for dierent deformation conditions, but without considering statistical ¯uctuations in the spatial distribution of the dislocation sources. In the model only tilt boundaries arise, because the Burgers vectors are always perpendicular to the boundary plane. The parallel line vector of all dislocations determines the rotation axis common for all boundaries. 3.2. Mean free path Each dislocation source emits only a single pair of mobile dislocations with opposite sign. Both dislocations move in opposite directions owing to the applied stress. Each mobile dislocation travels along a path l before it is stopped. The individual paths of the edge dislocations are assumed to be distributed according to an exponential distribution f l
Fig. 3. Strip of material deformed in simple shear by a single slip system: only seven bins numbered from j ÿ 3 to j 3 are shown. Pairs of edge dislocations of opposite sign (_ or ~) are emitted from dislocation sources (1±4). The straight dislocation lines are perpendicular to the drawing plane. From the dislocation loops labelled 5±8 only one dislocation is immobilized within the ®gure; the dislocations of opposite sign are deposited somewhere outside. At the borders of each bin, Z dislocation loops contribute to the plastic strain. The disorientation connected with the dislocations in each bin (or with the boundary located in the bin, respectively) is determined by the dierences DZ. As is obvious from bin j 1 an annihilation of the two encircled dislocations of opposite sign will not change the disorientation connected with this bin.
1 l exp ÿ l l
9
with a mean free path l. The exponential distribution re¯ects an immobilization of mobile dislocations along their travel distance proportional to their density drm rm ÿ : dx l
10
A continuous depletion of the mobile dislocation can be best understood from the interpretation of the cell structure ignoring the cell interiors and assuming the cell walls are adjoined [16]. Mobile
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
dislocations passing through a cell wall are trapped with a constant immobilization probability P in a wall limiting the free path l of the mobile dislocations. From slip line measurements [17] on copper single crystals after tensile deformation along [100] direction, for instance, it can be concluded [18] that for this particular orientation the slip line length is about six times as large as the cell size. Thus, on average, a mobile dislocation is stopped in the third dislocation wall and the mean free path l d=P of the mobile dislocations will be l 3d corresponding to an immobilization probability P 1=3: The mean free path l of the mobile dislocations determines also the average plastic strain: gpl 2bl
Nsc nsc 2bl : LH H
11
3.3. Disorientations Re¯ecting a cell structure the dislocations distributed along the x coordinate are grouped into bins of size d and assumed to gather in a tilt boundary in (the centre of) the bin. From the number of dis> locations N? i and Ni of each sign in each bin i, the > excess dislocations DNi N? i ÿ Ni and the disorientation angle bÿ ? ai N ÿ N> i H i
12
associated with the boundary located in bin i are determined. In the following the height H of the strip will be chosen equal to the dislocation cell size d assuming an equiaxed cell structure. For the evolution of the disorientations it is sucient to consider storage of dislocations only. Possible annihilation events between dislocations of opposite sign will not aect the disorientations as long as the mean free path remains unaltered, because in each annihilation event two dislocations of opposite sign cancel each other and no change in the number of excess dislocations occurs (compare bin j 1 in Fig. 3). On the other hand, the total number of dislocations is strongly aected by annihilation. 4. ANALYTICAL RESULTS
4.1. Basic considerations At any point x along the strip the number of dislocation loops Z(x ) which contributes to the local
{ Resulting in the average plastic strain gpl given in equation (11). { Again, the parentheses of the index in a(n ) indicate the disorientation angle across n boundaries, not the disorientation of the wall n.
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plastic deformation can be determined by an analogy with a problem in applied probability (as outlined in the Appendix). As a result, at any point the number Z is Poisson distributed with a mean value
sc Z x hZ i 2ln
13
and the covariance function between the number of dislocation loops at two points is
jx ÿ y j Z xZ y ÿ hZ i 2 hZ iexp ÿ : 14 2l
4.2. Disorientation angles The disorientation angle between two points x and y can be obtained directly from the dierence between the number of dislocation loops at these points. As is obvious from Fig. 3, the dierence DZ Z x ÿ Z y speci®es directly the number of excess dislocations (or geometrically necessary dislocations) between both points. The mean distance h H= Z x ÿ Z y of geometrically necessary dislocations along the height H determines the disorientation angle a x,y
b bÿ Z x ÿ Z y h H
15
between the two points. The same result can be deduced alternatively from the local plastic strain{ gpl x b
Z x : H
16
Gradients in the local plastic strain are related to a geometrically necessary dislocation density [19] rgn x ÿ
1 @ 1 @ g x ÿ Z x b @ x pl H @x
17
which has to be integrated over the distance between x and y to get the (linear) density (1/h ) of geometrically necessary dislocations leading to the disorientation angle of equation (15). Following the interpretation of the strip as a chain of equiaxed cell walls with cell size d the disorientation angles are determined for a single wall a
bÿ Z x ÿ Z x d d
18
or across n adjacent walls{ a n
bÿ Z x ÿ Z x nd d
19
For large hZ xiw1 the Poisson distributions of the numbers of dislocation loops Z(x ) can be approximated by Gaussian distributions with the same mean value and the same standard deviation (central limit theorem). Consequently, all pairwise
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PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
dierences DZ Z x ÿ Z x Dx (and all disorientation angles) are described reasonably well by Gaussian distributions with the moments
Z x ÿ Z x Dx 0 20 Dÿ
Z x ÿ Z x Dx
2E
jDx j 2hZ i 1 ÿ exp ÿ : 2l
4.4. Disorientation across several boundaries 21
Owing to the vanishing mean value of the Gaussian distributions, relations of the type given by equation (5) are valid for DZ, a, and a(n ). 4.3. Disorientation angle of a single boundary For a single boundary a mean disorientation angle hai ha 1 i 0
22
and a standard deviation D
2E b2 ÿ a 2 2 Z x ÿ Z x d d b2 d 2 2hZ i 1 ÿ exp ÿ d 2l
assumption of totally anti-correlated ¯uctuations between the ¯uxes in both directions in order to get an exactly constant strain rate everywhere. If instead of a constant strain rate, ¯uctuations in the strain rate are also allowed as in the model proposed here, the factor of 2 vanishes.
The disorientation angle between distant cells (across n boundaries) will show a Gaussian distribution with ha n i 0 and a standard deviation D E b2 nd 2 a n 2 2hZ i 1 ÿ exp ÿ d 2l
1 ÿ exp ÿnP=2 a2 27 1 ÿ exp ÿP=2 which directly relates to the average modulus of the disorientation angle r 2D 2 E a 28 hja n ji p n
23
follows. With mean free path l d=P and plastic strain gpl of equation (11) (with H d for equiaxed cells) the following results:
b P a 2 2 gpl 1 ÿ exp ÿ 24 d 2 The average modulus of the disorientation angle rs 4bgpl P 25 1 ÿ exp ÿ hjaji 2 pd increases with the square root of the plastic strain. Such a square root dependence was proposed to arise from any random ¯uctuation in the dislocation ¯uxes [20] and the proportionality coecient was derived for the ®rst time in Ref. [21] for impenetrable boundaries (cf. the discussion in Ref. [6]). Taking into account the immobilization probability P a slightly dierent coecient was obtained based on a rigorous treatment of the distribution of disorientation angles [6]. Comparing the approximation
bP a 2 1 gpl d
26
of equation (24) for small immobilization probabil ities PW1 with the previous result ha 2 i 2bPgpl d of Ref. [6] for a constant cell size, a factor of 2 is missing. In Ref. [6] this factor was due to an
Fig. 4. Analytical results: (a) disorientation angle across n boundaries as predicted from the statistical model for P 1=3 [equation (29)] with a saturation value .p hja 1 ji hjaji 1 ÿ exp ÿP=2 and, alternatively, from equation (7) for independent disorientation angles; (b) correlation coecient r(n ) for the disorientation angles in neighbouring boundaries according to the statistical model for P 1=3 [equation (34)].
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
s 1 ÿ exp ÿnP=2 hja n ji hjaji 1 ÿ exp ÿP=2
29
shown in Fig. 4(a). For a large number n of boundaries a saturation value r 4bgpl hjaji 30 hja 1 ji p pd 1 ÿ exp ÿP=2 for the distant disorientation angle is expected, in contrast to the case of uncorrelated disorientations where the average modulus of the angle is proportional to the square root of n and therefore increases unlimited [cf. Fig. 4(a)].
Table 1. Data for the numerical simulation of Fig. 5 Nsc Nbin Nbin
ÿ b 2 Dÿ Z x ÿ Z x d Z y d2 E ÿ Z y d
hai ain i
hai ain i 2
31
b2 d jx ÿ y j exp ÿ 1 ÿ cosh 2hl i 2hl i d2 32
hai ain i 2
b2 nP P 1 ÿ cosh 33 exp ÿ 2 2 2 d
can be characterized by the correlation coecient{ hai ain i r n q
2 2 ai ain nP 1 ÿ cosh P=2 exp ÿ : 2 1 ÿ cosh ÿ P=2
34
As is obvious from Fig. 4(b) the correlation coecient becomes negative r n < 0 for all n. The disorientation angles in neighbouring boundaries are always anti-correlated and it is more likely that disorientations in neighbouring boundaries are of opposite rather than of the same sign.
ai
The shear deformation of a strip of material with a single active slip system is simulated numerically:
x y
requires
13 1 10ÿ6 m 3 10ÿ10 m
bÿ ? N ÿ N> i : d i
35
Table 2. Parameters obtained by curve ®tting from the numerical simulations of Fig. 5 and analytical values
(a)
(b) (c)
{ The auto-correlation coecient separate evaluation: r 0 1:
P d b
is calculated. Only the inner Nbin bins were considered for the evaluation of the disorientation angles or the correlation coecient in order to avoid (boundary) eects from both rims of the strip. The data used in the simulation given in Table 1 result in a total plastic strain gpl 13:62: The results of the simulations are summarized in Fig. 5. For comparison with the results of Section 4, the analytical formulae were ®tted to the simulation results. The ®tting parameters obtained are displayed in Table 2 together with the analytical predictions. The accumulated frequencies P of the disorientation angles ai of the individual bins (i.e. the probability P of ®nding a disorientation angle less than a in an individual boundary) in Fig. 5(a) show that the disorientation angles can be described quite well by a Gaussian distribution as expected from Section 4.2. The slight deviation of the mean value hai from 08 is most likely due to the limited statistics. On the other hand, the standard deviation sa is close to the predicted value. In Fig. 5(b) the square root dependence given by equation (25) of the mean modulus of the disorientation angle hjaji in the boundaries on the plastic strain gpl is reproduced clearly with the proper proportionality coecient. The simulated distant disorientations hja n ji across several boundaries [Fig. 5(c)] are in obvious
Fig. 5
5. SIMULATIONS
106 497 415
First, Nsc sources are distributed randomly and uniformly along the entire length L of the strip. Then each source emits a dislocation pair where both dislocations travel in opposite directions along the same free path l distributed according to equation (9). The whole strip is subdivided in Nbin bins of cell size d L=Nbin : The numbers of positive and negative dislocations deposited are counted separately for each bin and the disorientation angle
4.5. Correlation between disorientation angles The correlation between the disorientation angles of a boundary (between x and x d and a distant boundary (between y and y d with y x nd, where n 1 marks the nearest neighbour, n 2 the next nearest, etc.)
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Parameter hai q sa 2bgpl 1 ÿ exp ÿP=2 d q 4b 1 ÿ exp ÿP=2 pd q hja 1 ji 4bgpl pd P/2
Analytical value Simulation 08
ÿ0.028
1.058
1.128
0.448
0.468
2.138
1.998
0.167
0.197
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PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
con¯ict with a square root law given by equation (7) proposed for independent disorientations; instead they show a behaviour according to equation (29). The saturation [in agreement with the expected value hja 1 ji indicates the existence of correlations between the disorientation in neighbouring boundaries. This is further con®rmed by the correlation coef®cient: Fig. 5(d) gives some evidence for a negative correlation of the disorientation angles between close neighbours comparable to the theoretical predictions. The rather strong noise does not allow a ®tting; instead the expression (34) is shown again in Fig. 5(d) for P 1=3 for comparison purposes.
Obviously, the simulations are able to re¯ect the analytical results reasonably well and the parameters obtained by ®tting in Table 2 are in overall agreement with the analytical predictions. But still there are some dierences owing to limitations of the numerical treatment arising from the used random numbers, the discretization into bins, and the limited numbers of bins (boundaries) and dislocations in a bin, respectively. These problems cause the noise in the correlation coecient and aect, in particular, the disorientations across several boundaries. This may explain the large deviation in the value obtained for P/2 (see Table 2) and the arti®cial oscillation for n > 20 in Fig. 5(c).
Fig. 5. Results of a numerical simulation of the activation of Nsc 106 dislocation sources in a strip containing Nbin 497 cells compared with analytical formulae: (a) accumulated frequency of the disorientation angles in individual boundaries and ®t of a Gaussian distribution; (b) evolution of average modulus of the disorientation angle hjaji with plastic strain gpl and square root ®t given by equation (25); (c) mean disorientation angle across n boundaries and ®t according to equation (29) [alternatively, square root ®t given by equation (7)]; (d) simulated correlation coecient r(n ) and analytical expression given by equation (34).
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS 6. DISCUSSION
The evaluation of the correlation coecient given by equation (34) shows the occurrence of a negative correlation r n < 0 between neighbouring boundaries which is con®rmed also by the simulation for the ®rst neighbours. This anti-correlation causes a saturation behaviour of the mean modulus of the disorientation angle across several boundaries, in contrast to the unlimited increase in the case of independent disorientations. As a second consequence of the anti-correlation the width of X-ray broadening curves does not increase with the beam size as predicted from uncorrelated disorientations [equation (8)]. The saturation of hja n ji is clearly visible in the EBSD scans of Fig. 1 where for large step sizes the average disorientation angle is levelling o. The overall dependence of the average disorientation angle on the step size seems to resemble the theoretical ®nding of equation (29) in Fig. 4(a) rather well, and one might be tempted to obtain the saturation value and the slope for small step sizes from the experimental data. For an interpretation of the saturation value in terms of the model presented here, one has to take into account that the dislocation structure in the deformed aluminium single crystals after plane strain compression does not consist of a simple dislocation cell structure with ordinary dislocation walls (IDBs) only, but that a hierarchical structure exists with other types of boundaries (GNBs) present also [22, 23]. An evaluation of the immobilization probability from the slope of the curve in Fig. 1 at small step sizes is limited by the problem of determining the disorientation angles for step sizes smaller than the mean chord length l of the dislocation structureÐ only the average number of boundaries between the measuring points can be obtained from the chord length. For small eective step sizes Dx 0 l11 some point pairs will not include any boundary at all and an apparent disorientation angle is only due to uncertainties in the orientation determination. The measured mean disorientation angle is aected by irrelevant data where no boundary is present between the measuring points. The irrelevant contribution has to be eliminated ®rst [24], before the initial slope can be used for an estimation of the trapping probability or the mean free path. 6.1. A dierent type of correlation The anti-correlation between the disorientations of adjacent boundaries discussed here diers from correlations reported from investigations of the microstructure after creep in LiF [25, 26]. In Ref. [26] the dislocation sources are distributed quite inhomogeneously and slip zones exist: dislocations emitted by sources in the centre deform the entire
3013
slip zone and the dislocations of opposite sign of the Burgers vector are on the opposite sides of the central area containing the sources. Consequently, adjacent boundaries on the same side show the same sign of the disorientation angle [25, 26] and a positive correlation between neighbouring boundaries can be observed. These dislocation structures are connected with curvatures of the slip planes and the mean disorientation angle should increase strongly with the distance as long as the distance does not exceed the correlation length. 7. CONCLUSIONS
For deformation-induced dislocation structures the occurrence of correlations between disorientations in neighbouring boundaries is established. . A comparison of experimental investigations by means of electron back-scattering diraction and X-ray diraction on single crystals with model predictions for independent disorientations revealed that disorientations in neighbouring boundaries cannot be independent and correlations must exist. . The proposed model of dislocation pairs emitted by a spatially random distribution of dislocation sources and travelling along a free path in opposite direction covers certain features of a dislocation (boundary) structure, if all dislocations in a region corresponding to the size of a cell are gathered into a dislocation boundary. . A preferred occurrence of disorientations of opposite sign in neighbouring boundaries is predicted by the model. This negative correlation leads to a levelling o of the disorientation angle across several boundaries which is supported by experimental evidence. . The anti-correlation between the disorientation angles of neighbouring boundaries is caused by the limited free path of the dislocations. The pairs of dislocations of opposite sign of the Burgers vectors are separated only by a distance 2l determined by the mean free path l of the individual dislocations. If both dislocations should become totally independent, any correlation would vanish.
AcknowledgementsÐHelpful discussions with Dr N. Hansen and Professor P. Klimanek are gratefully acknowledged. The work was partially performed within the Engineering Science Center for Structural Characterization and Modelling of Materials at Risù National Laboratory. REFERENCES 1. Hansen, N. and Kuhlmann-Wilsdorf, D., Mater. Sci. Engng, 1986, 81, 141. 2. Bay, B., Hansen, N., Hughes, D. A. and KuhlmannWilsdorf, D., Acta metall. mater., 1992, 40, 205.
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PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
3. Kuhlmann-Wilsdorf, D. and Hansen, N., Scripta metall. mater., 1991, 25, 1557. 4. Liu, Q. and Hansen, N., Scripta metall. mater., 1995, 32, 1289. 5. Hughes, D. A., Liu, Q., Chrzan, D. C. and Hansen, N., Acta mater., 1997, 45, 105. 6. Pantleon, W., Acta mater., 1998, 46, 451. 7. Pantleon, W., in Proc. 20th Risù Symposium on Materials Science: Deformation-Induced Microstructures: Analysis and Relation to Properties, ed. J. B. Bilde-Sorensen. Risù National Laboratory, Roskilde, Denmark, 1999, pp. 123±124. 8. Juul Jensen, D., Ultramicroscopy, 1997, 67, 25. 9. Breuer, D., Klimanek, P. and Pantleon, W., J. appl. Cryst., submitted. 10. Bay, B., Hansen, N. and Kuhlmann-Wilsdorf, D., Mater. Sci. Engng, 1989, A113, 385. 11. Driver, J. H., Juul Jensen, D. and Hansen, N., Acta metall. mater., 1994, 42, 3105. 12. Randle, V., Microtexture Determination and its Applications. The Institute of Materials, London, 1992. 13. Krivoglaz, M. A., Ryaboshapka, K. P. and Barabash, R. I., Fizika Metall., 1970, 30, 1134. 14. Barabash, R. I. and Klimanek, P., J. appl. Cryst., 1999, 32, 1050. 15. Barabash, R. I., Krivoglaz, M. A. and Ryaboshapka, K. P., Fizika Metall., 1976, 41, 33. 16. Francke, D., Pantleon, W. and Klimanek, P., Comput. Mater. Sci., 1996, 5, 111. 17. Ambrosi, P. and Schwink, C., Scripta metall., 1978, 12, 303. 18. Kawasaki, Y. and Takeuchi, T., Scripta metall., 1980, 14, 183. 19. Ashby, M. F., Phil. Mag., 1970, 21, 399. 20. Argon, A. S. and Haasen, P., Acta metall. mater., 1993, 41, 3289. 21. Nabarro, F. R. N., Scripta metall. mater., 1994, 30, 1085. 22. Godfrey, A., Juul Jensen, D. and Hansen, N., Acta mater., 1998, 46, 823. 23. Godfrey, A., Juul Jensen, D. and Hansen, N., Acta mater., 1998, 46, 835. 24. TroÈtzschel, J., Pantleon, W., Haberjahn, M. and Klimanek, P., Mater. Sci. Engng, 1997, A234±236, 842. 25. Smirnov, B. I., ShpeiÆzman, V. V., Ivanov, S. A., Mal'chuzenko, K. V. and Chudnova, R. S., Soviet Phys. Solid State, 1978, 20, 2158. 26. Biberger, M. and Blum, W., Phil. Mag. A, 1992, 65, 757. 27. BenesÏ , V. E., Bell System Technol. J., 1957, 36, 965.
28. Reynolds, J. F., Adv. appl. Prob., 1975, 7, 383.
APPENDIX A
A.1. Relationship to queueing theory Consider a stationary Poisson process on a line (along the x-axis) of an intensity n (equal to nsc). In each point of the process a random interval begins, which has a length following an exponential distriÿ1 bution with parameter m [equal to (2l) ]. Let Z(x ) be the random number of intervals covering the point x of the line. Then Z(x ) has a Poisson distribution with parameter hZ xi n=m: The covariance function is given by: ÿ n ÿ cov Z x, Z x X exp ÿ mjX j : m
A1
A generalization is also possible: if the random intervals follow the general length distribution function B(x ) with a density b x dB x=dx and a ÿ1 mean m , then Z(x ) follows a Poisson distribution with parameter n/m as above. The covariance function then takes the general form ÿ cov Z x, Z x X ! jX j ÿ n exp 1 ÿ m 1 ÿ B t dt : m 0
A2
These formulae seem to be found ®rst in the context of queueing theory [27, 28] in the study of the model M/M/1 (exponential service times with parameter m ) and M/G/1 (general service times). The interpretation in queueing theory is as follows: the points of the Poisson process represent the arrival instants of customers and the random interval lengths correspond to the service time duration leading to a number of customers Z(x ) in the system at a time t x: The system has in®nitely many servers; for each customer, service begins immediately after arrival without waiting time.
CORRELATIONS BETWEEN DISORIENTATIONS IN NEIGHBOURING DISLOCATION BOUNDARIES W. PANTLEON 1{ and D. STOYAN 2 1
Materials Research Department, Risù National Laboratory, P.O. Box 49, 4000 Roskilde, Denmark and 2Institute of Stochastics, Freiberg University of Mining and Technology, Bernhard-von-CottaStraûe 2, 09596 Freiberg, Germany (Received 8 December 1999; accepted 23 February 2000)
AbstractÐDuring plastic deformation, dislocation structures develop with regions of dierent orientations separated by dislocation boundaries. Experimental investigations by means of electron back-scattering diffraction or rocking curves from X-ray diraction give some evidence for the existence of correlations between the disorientations in neighbouring boundaries. Based on a straightforward statistical model an occurrence of anti-correlations between disorientation angles is predicted which can be traced to the limited free path of mobile dislocations carrying the plastic deformation. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. ZusammenfassungÐBei plastischer Verformung bilden sich Versetzungsstrukturen aus, die durch VersetzungswaÈnde voneinander getrennte Gebiete unterschiedlicher Orientierung aufweisen. Aus experimentellen Untersuchungen mit Hilfe der Elektronen-RuÈckstreu-Diraktometrie oder Rocking-Kurven der RoÈntgendiraktometrie lassen sich Hinweise auf Korrelationen zwischen den Desorientierungen in benachbarten Grenzen gewinnen. Auf Grundlage eines einfachen statistischen Modells wird das Auftreten von Anti-Korrelationen zwischen den Fehlorientierungswinkeln vorhergesagt und auf die endliche freie WeglaÈnge der mobilen Versetzungen, den TraÈgern der plastischen Verformung, zuruÈckgefuÈhrt. 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Dislocations; Microstructure; Disorientations; Mechanical properties (plastic); Theory & modelling
1. INTRODUCTION
During plastic deformation, dislocation structures develop with dislocation boundaries of high dislocation density and nearly dislocation-free regions (see, e.g. Refs [1, 2]). Generally, two dierent dislocation boundaries can be identi®ed [3]: ordinary dislocation walls arising from statistical mutual trapping of dislocations (incidental dislocation boundaries, IDBs) and straight and parallel dislocation boundaries caused by a dierence in the activation of slip systems on both sides (geometrically necessary boundaries, GNBs). Orientation dierences between the regions separated by a dislocation boundary arise in connection with both types of boundaries [4, 5]. For incidental
{ To whom all correspondence should be addressed. { The alternating disorientations observed in adjacent GNBs (dense dislocation boundaries) are attributed to a dierent activation of slip systems on both sides of GNBs (e.g. Refs [10, 11]) and are not considered here.
dislocation boundaries disorientations are caused by statistical ¯uctuations in the dislocation ¯uxes passing through the cell walls [6]. Geometrically necessary boundaries show a stronger increase of the disorientation angles related to the dierent activity of slip systems on either side of the boundary [7]. Certain experimental results on deformed single crystals obtained by electron back-scattering diraction [8] and X-ray diraction [9] cannot be explained assuming that disorientations of neighbouring boundaries are independent of each other. Both techniques show evidence for the existence of correlations between disorientation angles in adjacent boundaries. The occurrence of correlations is investigated here in case of a dislocation cell structure with only one kind of boundary, IDBs{, based on a simple statistical model for a dislocation cell structure with parallel dislocation walls. In contrast to the previous model [6] for the statistical formation of disorientations in IDBs, the consideration of a chain of parallel dislocation boundaries allows the revelation of spatial correlations.
1359-6454/00/$20.00 7 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 0 8 3 - 5
3006
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
Fig. 1. Mean disorientation angle determined with EBSD line scans (along 350 mm in dierent directions with a step size Dx 1 mm as a function of the eective step size m Dx 0 =Dx for aluminium single crystals of three dierent orientations after plane strain compression to E 1:5 (courtesy A. Godfrey) [8]. 2. EXPERIMENTAL EVIDENCE FOR CORRELATIONS
2.1. Electron back-scattering diraction With a scanning electron microscope local orientations can be determined at certain points on the surface of a (e.g. deformed) material with electron back-scattering diraction (EBSD, cf. Ref. [12]). Usually a constant step size Dx is used for the distances between the measuring points in line (or mesh) scans and the disorientation angles are calculated between adjacent points. From the same set of orientations the disorientation angle can also be evaluated between next-nearest or even more distant neighbours and the mean disorientation angle depends on the corresponding eective step size Dx 0 mDx [8]. In Fig. 1 the mean disorientation angle of aluminium single crystals deformed in plane strain compression at room temperature increases with the step size at small distances between the measuring points. But for large step sizes Dx' the average disorientation angle will level-o and show a saturation value. From the mean chord length l (the mean distance
{ The angular brackets h.i denote averages over all boundaries or spatial averages, respectively. The notation a(n ) with the index in parentheses refers to an angle across n boundaries (the distant disorientation angle), whereas ai is the angle of a speci®c boundary i.
of boundaries along a straight testing line) of the underlying dislocation structure, the number of boundaries n
Dx 0 l
1
between two measuring points in a distance Dx' can be estimated. Consequently, the observed levelling o of the mean disorientation angle for large step sizes Dx' corresponds to a saturation of the mean disorientation angle across a large number of boundaries n as well. As will be shown in the following Section 2.2, such a saturation behaviour is in con¯ict with an assumption of independent disorientation angles and correlations must occur. 2.2. Independent disorientations For uncovering possible correlations between the disorientations in neighbouring boundaries a chain of parallel boundaries with independent disorientations is taken as a point of reference. If the disorientation angles ai of all dislocation boundaries i correspond to the same rotation axis, the disorientation angle{ across n neighbouring boundaries (``distant disorientation angle'') a n
n X ai
2
i1
is the sum of the individual disorientation angles ai
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
with mean value ha n i
n X hai i nhai:
3
i1
For completely uncorrelated disorientation angles hai aj i 0 for all i6j the second-order moment{ n
D E X
2 ai2 n a 2 a n
4
i1
increases linearly with the number n of the boundaries. For a statistical accumulation of excess dislocations (of one sign of the Burgers vector) from ¯uctuations in the dislocation ¯uxes [6], the disorientation angle of each individual boundary i shows a Gaussian distribution with vanishing mean value hai i 0 for all i ) and a standard deviation ha 2 i sa2 : As for all Gaussian distributions with vanishing mean value the standard deviation determines the mean modulus r 2 2 ha i: 5 hjaji p In the same manner, the distant disorientation has a Gaussian distribution with vanishing mean value ha n i 0
6
and the mean modulus of the disorientation angle across n boundaries r r p 2D 2 E 2 p a a n nha 2 i nhjaji 7 hj n ji p p
3007
crystal spectrometer the sample is rocked around an axis perpendicular to the diraction plane and the diracted intensity from a monochromatic primary X-ray beam is detected as a function of the rocking angle o. In investigations on copper single crystals after hot compression along [001] at about 6008C only a slight dependence (cf. Fig. 2) of the angular width Do of rocking curves on the size T of the X-ray beam in the diraction plane (width of the beam at the position of the sample perpendicular to the rocking axis) was found [9]. For a theoretical modelling of the angular width of rocking curves, a set of parallel dislocation tilt boundaries in a mutual distance d with disorientations around the same rotation axis is considered [13, 14] similar to that discussed in Section 2.2. For disorientation angles of either a or ÿa with the same probability and no short-range correlation between the disorientation angles in adjacent boundaries, the (angular) width of rocking curves r p T DoAa a n 8 d is determined by the disorientation angle a and the number n of dislocation boundaries under investigation [13, 14]. The square root behaviour is a direct consequence of the random distribution of the sign of the disorientation angle. According to equation (8) the angular width Do should depend on the beam size T determining the number of dislocation cells or boundaries, respectively, n T=d in the beam. A similar result can be obtained if the disorientations are distributed according to a nor-
increases with the square root of the number of dislocation boundaries. Contrary to the prediction of equation (7) for independent disorientation angles no unlimited increase with the number of boundaries or the step size, respectively, is seen in the experimental data of Fig. 1. The saturation clearly reveals the existence of correlations between disorientation angles in neighbouring boundaries. 2.3. Broadening of X-ray rocking curves In a more integrative manner information about disorientations in crystals can be gained from rocking curves (azimuthal intensity distributions) in Xray diraction (e.g. Ref. [13])Ðin an X-ray double
{ The result given by equation (4) will not change if the disorientation angles do not have the same rotation axis, but the rotation axes are distributed randomly. The square P 2 of the distant disorientation angles a n ni1 ai2 will still be determined by the sum of the squares of the individual disorientation angles.
Fig. 2. Dependence of the angular width Do (peak broadening) of rocking curves on the beam size T in the diraction plane for copper single crystals after hot compression 6008C, E_ 10 ÿ2 =s, rocking axis parallel to the compression axis) [9]. (The mean chord length l 1 mm of the higher deformed single crystal leads, even for the smallest beam size T 1 mm, to a large number of dislocation boundaries n 1000 in the beam in the diraction plane.)
3008
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
mal distribution; only the proportionality factor will change [9, 14]. If all dislocation boundaries have disorientations of the same sign corresponding to a strong (positive) correlation instead of a random distribution of the signs, a long-range curvature evolves and the width DoAna becomes proportional to the number of boundaries [15]. The experimentally observed slight dependence in Fig. 2 is in contrast to the predicted square root behaviour of equation (8) for uncorrelated disorientations (or even a linear increase of DoAna for positively correlated disorientations of the same sign). Owing to the weak dependence of Do on T or n, respectively, the disorientation angles in neighbouring boundaries cannot be independent and must be anti-correlated signi®cantly, i.e. disorientations in neighbouring boundaries are more likely of opposite than of the same sign. 3. MODEL
3.1. Outline An elementary model (illustrated in Fig. 3) is considered for the investigation of possible correlations between disorientations: a strip of material with height H deforming in simple shear. The deformation is carried by mobile edge dislocations of a single slip system which are emitted pairwise from dislocation sources. The pairs of edge dislocations of opposite sign of the Burgers vector (_ and ~) and in®nite length represent dislocation loops in this essentially two-dimensional model. The dislo-
cation sources are assumed to be randomly and uniformly distributed (equal probability everywhere). A constant (linear) density of sources nsc along the length L of the strip leads to a total number Nsc nsc L of dislocation sources. A continuous distribution of the dislocation sources (and dislocations) along the strip is considered only for modelling purposes. For a more realistic description of a cell structure (containing cell walls of width w with a high dislocation density and nearly dislocation-free cell interiors) all dislocations in the interval from x to x d can be assumed to gather into a dislocation boundary located somewhere in the bin, e.g. at the centre. In this manner, a chain of dislocation cells with equidistant boundaries and a cell size d is obtained. Alternatively, one may interpret the continuous arrangement of Fig. 3 as a chain of dislocation cells of size d where the interiors have been cut out and a chain of directly adjoining walls has been obtained. The passage of the mobile dislocations through cell interiors is immaterial for the model and only the walls of width w must be taken into account. In this interpretation the dislocation sources are quite reasonably restricted to the dislocation walls. A similar model was treated numerically [16] for dierent deformation conditions, but without considering statistical ¯uctuations in the spatial distribution of the dislocation sources. In the model only tilt boundaries arise, because the Burgers vectors are always perpendicular to the boundary plane. The parallel line vector of all dislocations determines the rotation axis common for all boundaries. 3.2. Mean free path Each dislocation source emits only a single pair of mobile dislocations with opposite sign. Both dislocations move in opposite directions owing to the applied stress. Each mobile dislocation travels along a path l before it is stopped. The individual paths of the edge dislocations are assumed to be distributed according to an exponential distribution f l
Fig. 3. Strip of material deformed in simple shear by a single slip system: only seven bins numbered from j ÿ 3 to j 3 are shown. Pairs of edge dislocations of opposite sign (_ or ~) are emitted from dislocation sources (1±4). The straight dislocation lines are perpendicular to the drawing plane. From the dislocation loops labelled 5±8 only one dislocation is immobilized within the ®gure; the dislocations of opposite sign are deposited somewhere outside. At the borders of each bin, Z dislocation loops contribute to the plastic strain. The disorientation connected with the dislocations in each bin (or with the boundary located in the bin, respectively) is determined by the dierences DZ. As is obvious from bin j 1 an annihilation of the two encircled dislocations of opposite sign will not change the disorientation connected with this bin.
1 l exp ÿ l l
9
with a mean free path l. The exponential distribution re¯ects an immobilization of mobile dislocations along their travel distance proportional to their density drm rm ÿ : dx l
10
A continuous depletion of the mobile dislocation can be best understood from the interpretation of the cell structure ignoring the cell interiors and assuming the cell walls are adjoined [16]. Mobile
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
dislocations passing through a cell wall are trapped with a constant immobilization probability P in a wall limiting the free path l of the mobile dislocations. From slip line measurements [17] on copper single crystals after tensile deformation along [100] direction, for instance, it can be concluded [18] that for this particular orientation the slip line length is about six times as large as the cell size. Thus, on average, a mobile dislocation is stopped in the third dislocation wall and the mean free path l d=P of the mobile dislocations will be l 3d corresponding to an immobilization probability P 1=3: The mean free path l of the mobile dislocations determines also the average plastic strain: gpl 2bl
Nsc nsc 2bl : LH H
11
3.3. Disorientations Re¯ecting a cell structure the dislocations distributed along the x coordinate are grouped into bins of size d and assumed to gather in a tilt boundary in (the centre of) the bin. From the number of dis> locations N? i and Ni of each sign in each bin i, the > excess dislocations DNi N? i ÿ Ni and the disorientation angle bÿ ? ai N ÿ N> i H i
12
associated with the boundary located in bin i are determined. In the following the height H of the strip will be chosen equal to the dislocation cell size d assuming an equiaxed cell structure. For the evolution of the disorientations it is sucient to consider storage of dislocations only. Possible annihilation events between dislocations of opposite sign will not aect the disorientations as long as the mean free path remains unaltered, because in each annihilation event two dislocations of opposite sign cancel each other and no change in the number of excess dislocations occurs (compare bin j 1 in Fig. 3). On the other hand, the total number of dislocations is strongly aected by annihilation. 4. ANALYTICAL RESULTS
4.1. Basic considerations At any point x along the strip the number of dislocation loops Z(x ) which contributes to the local
{ Resulting in the average plastic strain gpl given in equation (11). { Again, the parentheses of the index in a(n ) indicate the disorientation angle across n boundaries, not the disorientation of the wall n.
3009
plastic deformation can be determined by an analogy with a problem in applied probability (as outlined in the Appendix). As a result, at any point the number Z is Poisson distributed with a mean value
sc Z x hZ i 2ln
13
and the covariance function between the number of dislocation loops at two points is
jx ÿ y j Z xZ y ÿ hZ i 2 hZ iexp ÿ : 14 2l
4.2. Disorientation angles The disorientation angle between two points x and y can be obtained directly from the dierence between the number of dislocation loops at these points. As is obvious from Fig. 3, the dierence DZ Z x ÿ Z y speci®es directly the number of excess dislocations (or geometrically necessary dislocations) between both points. The mean distance h H= Z x ÿ Z y of geometrically necessary dislocations along the height H determines the disorientation angle a x,y
b bÿ Z x ÿ Z y h H
15
between the two points. The same result can be deduced alternatively from the local plastic strain{ gpl x b
Z x : H
16
Gradients in the local plastic strain are related to a geometrically necessary dislocation density [19] rgn x ÿ
1 @ 1 @ g x ÿ Z x b @ x pl H @x
17
which has to be integrated over the distance between x and y to get the (linear) density (1/h ) of geometrically necessary dislocations leading to the disorientation angle of equation (15). Following the interpretation of the strip as a chain of equiaxed cell walls with cell size d the disorientation angles are determined for a single wall a
bÿ Z x ÿ Z x d d
18
or across n adjacent walls{ a n
bÿ Z x ÿ Z x nd d
19
For large hZ xiw1 the Poisson distributions of the numbers of dislocation loops Z(x ) can be approximated by Gaussian distributions with the same mean value and the same standard deviation (central limit theorem). Consequently, all pairwise
3010
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
dierences DZ Z x ÿ Z x Dx (and all disorientation angles) are described reasonably well by Gaussian distributions with the moments
Z x ÿ Z x Dx 0 20 Dÿ
Z x ÿ Z x Dx
2E
jDx j 2hZ i 1 ÿ exp ÿ : 2l
4.4. Disorientation across several boundaries 21
Owing to the vanishing mean value of the Gaussian distributions, relations of the type given by equation (5) are valid for DZ, a, and a(n ). 4.3. Disorientation angle of a single boundary For a single boundary a mean disorientation angle hai ha 1 i 0
22
and a standard deviation D
2E b2 ÿ a 2 2 Z x ÿ Z x d d b2 d 2 2hZ i 1 ÿ exp ÿ d 2l
assumption of totally anti-correlated ¯uctuations between the ¯uxes in both directions in order to get an exactly constant strain rate everywhere. If instead of a constant strain rate, ¯uctuations in the strain rate are also allowed as in the model proposed here, the factor of 2 vanishes.
The disorientation angle between distant cells (across n boundaries) will show a Gaussian distribution with ha n i 0 and a standard deviation D E b2 nd 2 a n 2 2hZ i 1 ÿ exp ÿ d 2l
1 ÿ exp ÿnP=2 a2 27 1 ÿ exp ÿP=2 which directly relates to the average modulus of the disorientation angle r 2D 2 E a 28 hja n ji p n
23
follows. With mean free path l d=P and plastic strain gpl of equation (11) (with H d for equiaxed cells) the following results:
b P a 2 2 gpl 1 ÿ exp ÿ 24 d 2 The average modulus of the disorientation angle rs 4bgpl P 25 1 ÿ exp ÿ hjaji 2 pd increases with the square root of the plastic strain. Such a square root dependence was proposed to arise from any random ¯uctuation in the dislocation ¯uxes [20] and the proportionality coecient was derived for the ®rst time in Ref. [21] for impenetrable boundaries (cf. the discussion in Ref. [6]). Taking into account the immobilization probability P a slightly dierent coecient was obtained based on a rigorous treatment of the distribution of disorientation angles [6]. Comparing the approximation
bP a 2 1 gpl d
26
of equation (24) for small immobilization probabil ities PW1 with the previous result ha 2 i 2bPgpl d of Ref. [6] for a constant cell size, a factor of 2 is missing. In Ref. [6] this factor was due to an
Fig. 4. Analytical results: (a) disorientation angle across n boundaries as predicted from the statistical model for P 1=3 [equation (29)] with a saturation value .p hja 1 ji hjaji 1 ÿ exp ÿP=2 and, alternatively, from equation (7) for independent disorientation angles; (b) correlation coecient r(n ) for the disorientation angles in neighbouring boundaries according to the statistical model for P 1=3 [equation (34)].
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
s 1 ÿ exp ÿnP=2 hja n ji hjaji 1 ÿ exp ÿP=2
29
shown in Fig. 4(a). For a large number n of boundaries a saturation value r 4bgpl hjaji 30 hja 1 ji p pd 1 ÿ exp ÿP=2 for the distant disorientation angle is expected, in contrast to the case of uncorrelated disorientations where the average modulus of the angle is proportional to the square root of n and therefore increases unlimited [cf. Fig. 4(a)].
Table 1. Data for the numerical simulation of Fig. 5 Nsc Nbin Nbin
ÿ b 2 Dÿ Z x ÿ Z x d Z y d2 E ÿ Z y d
hai ain i
hai ain i 2
31
b2 d jx ÿ y j exp ÿ 1 ÿ cosh 2hl i 2hl i d2 32
hai ain i 2
b2 nP P 1 ÿ cosh 33 exp ÿ 2 2 2 d
can be characterized by the correlation coecient{ hai ain i r n q
2 2 ai ain nP 1 ÿ cosh P=2 exp ÿ : 2 1 ÿ cosh ÿ P=2
34
As is obvious from Fig. 4(b) the correlation coecient becomes negative r n < 0 for all n. The disorientation angles in neighbouring boundaries are always anti-correlated and it is more likely that disorientations in neighbouring boundaries are of opposite rather than of the same sign.
ai
The shear deformation of a strip of material with a single active slip system is simulated numerically:
x y
requires
13 1 10ÿ6 m 3 10ÿ10 m
bÿ ? N ÿ N> i : d i
35
Table 2. Parameters obtained by curve ®tting from the numerical simulations of Fig. 5 and analytical values
(a)
(b) (c)
{ The auto-correlation coecient separate evaluation: r 0 1:
P d b
is calculated. Only the inner Nbin bins were considered for the evaluation of the disorientation angles or the correlation coecient in order to avoid (boundary) eects from both rims of the strip. The data used in the simulation given in Table 1 result in a total plastic strain gpl 13:62: The results of the simulations are summarized in Fig. 5. For comparison with the results of Section 4, the analytical formulae were ®tted to the simulation results. The ®tting parameters obtained are displayed in Table 2 together with the analytical predictions. The accumulated frequencies P of the disorientation angles ai of the individual bins (i.e. the probability P of ®nding a disorientation angle less than a in an individual boundary) in Fig. 5(a) show that the disorientation angles can be described quite well by a Gaussian distribution as expected from Section 4.2. The slight deviation of the mean value hai from 08 is most likely due to the limited statistics. On the other hand, the standard deviation sa is close to the predicted value. In Fig. 5(b) the square root dependence given by equation (25) of the mean modulus of the disorientation angle hjaji in the boundaries on the plastic strain gpl is reproduced clearly with the proper proportionality coecient. The simulated distant disorientations hja n ji across several boundaries [Fig. 5(c)] are in obvious
Fig. 5
5. SIMULATIONS
106 497 415
First, Nsc sources are distributed randomly and uniformly along the entire length L of the strip. Then each source emits a dislocation pair where both dislocations travel in opposite directions along the same free path l distributed according to equation (9). The whole strip is subdivided in Nbin bins of cell size d L=Nbin : The numbers of positive and negative dislocations deposited are counted separately for each bin and the disorientation angle
4.5. Correlation between disorientation angles The correlation between the disorientation angles of a boundary (between x and x d and a distant boundary (between y and y d with y x nd, where n 1 marks the nearest neighbour, n 2 the next nearest, etc.)
3011
Parameter hai q sa 2bgpl 1 ÿ exp ÿP=2 d q 4b 1 ÿ exp ÿP=2 pd q hja 1 ji 4bgpl pd P/2
Analytical value Simulation 08
ÿ0.028
1.058
1.128
0.448
0.468
2.138
1.998
0.167
0.197
3012
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
con¯ict with a square root law given by equation (7) proposed for independent disorientations; instead they show a behaviour according to equation (29). The saturation [in agreement with the expected value hja 1 ji indicates the existence of correlations between the disorientation in neighbouring boundaries. This is further con®rmed by the correlation coef®cient: Fig. 5(d) gives some evidence for a negative correlation of the disorientation angles between close neighbours comparable to the theoretical predictions. The rather strong noise does not allow a ®tting; instead the expression (34) is shown again in Fig. 5(d) for P 1=3 for comparison purposes.
Obviously, the simulations are able to re¯ect the analytical results reasonably well and the parameters obtained by ®tting in Table 2 are in overall agreement with the analytical predictions. But still there are some dierences owing to limitations of the numerical treatment arising from the used random numbers, the discretization into bins, and the limited numbers of bins (boundaries) and dislocations in a bin, respectively. These problems cause the noise in the correlation coecient and aect, in particular, the disorientations across several boundaries. This may explain the large deviation in the value obtained for P/2 (see Table 2) and the arti®cial oscillation for n > 20 in Fig. 5(c).
Fig. 5. Results of a numerical simulation of the activation of Nsc 106 dislocation sources in a strip containing Nbin 497 cells compared with analytical formulae: (a) accumulated frequency of the disorientation angles in individual boundaries and ®t of a Gaussian distribution; (b) evolution of average modulus of the disorientation angle hjaji with plastic strain gpl and square root ®t given by equation (25); (c) mean disorientation angle across n boundaries and ®t according to equation (29) [alternatively, square root ®t given by equation (7)]; (d) simulated correlation coecient r(n ) and analytical expression given by equation (34).
PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS 6. DISCUSSION
The evaluation of the correlation coecient given by equation (34) shows the occurrence of a negative correlation r n < 0 between neighbouring boundaries which is con®rmed also by the simulation for the ®rst neighbours. This anti-correlation causes a saturation behaviour of the mean modulus of the disorientation angle across several boundaries, in contrast to the unlimited increase in the case of independent disorientations. As a second consequence of the anti-correlation the width of X-ray broadening curves does not increase with the beam size as predicted from uncorrelated disorientations [equation (8)]. The saturation of hja n ji is clearly visible in the EBSD scans of Fig. 1 where for large step sizes the average disorientation angle is levelling o. The overall dependence of the average disorientation angle on the step size seems to resemble the theoretical ®nding of equation (29) in Fig. 4(a) rather well, and one might be tempted to obtain the saturation value and the slope for small step sizes from the experimental data. For an interpretation of the saturation value in terms of the model presented here, one has to take into account that the dislocation structure in the deformed aluminium single crystals after plane strain compression does not consist of a simple dislocation cell structure with ordinary dislocation walls (IDBs) only, but that a hierarchical structure exists with other types of boundaries (GNBs) present also [22, 23]. An evaluation of the immobilization probability from the slope of the curve in Fig. 1 at small step sizes is limited by the problem of determining the disorientation angles for step sizes smaller than the mean chord length l of the dislocation structureÐ only the average number of boundaries between the measuring points can be obtained from the chord length. For small eective step sizes Dx 0 l11 some point pairs will not include any boundary at all and an apparent disorientation angle is only due to uncertainties in the orientation determination. The measured mean disorientation angle is aected by irrelevant data where no boundary is present between the measuring points. The irrelevant contribution has to be eliminated ®rst [24], before the initial slope can be used for an estimation of the trapping probability or the mean free path. 6.1. A dierent type of correlation The anti-correlation between the disorientations of adjacent boundaries discussed here diers from correlations reported from investigations of the microstructure after creep in LiF [25, 26]. In Ref. [26] the dislocation sources are distributed quite inhomogeneously and slip zones exist: dislocations emitted by sources in the centre deform the entire
3013
slip zone and the dislocations of opposite sign of the Burgers vector are on the opposite sides of the central area containing the sources. Consequently, adjacent boundaries on the same side show the same sign of the disorientation angle [25, 26] and a positive correlation between neighbouring boundaries can be observed. These dislocation structures are connected with curvatures of the slip planes and the mean disorientation angle should increase strongly with the distance as long as the distance does not exceed the correlation length. 7. CONCLUSIONS
For deformation-induced dislocation structures the occurrence of correlations between disorientations in neighbouring boundaries is established. . A comparison of experimental investigations by means of electron back-scattering diraction and X-ray diraction on single crystals with model predictions for independent disorientations revealed that disorientations in neighbouring boundaries cannot be independent and correlations must exist. . The proposed model of dislocation pairs emitted by a spatially random distribution of dislocation sources and travelling along a free path in opposite direction covers certain features of a dislocation (boundary) structure, if all dislocations in a region corresponding to the size of a cell are gathered into a dislocation boundary. . A preferred occurrence of disorientations of opposite sign in neighbouring boundaries is predicted by the model. This negative correlation leads to a levelling o of the disorientation angle across several boundaries which is supported by experimental evidence. . The anti-correlation between the disorientation angles of neighbouring boundaries is caused by the limited free path of the dislocations. The pairs of dislocations of opposite sign of the Burgers vectors are separated only by a distance 2l determined by the mean free path l of the individual dislocations. If both dislocations should become totally independent, any correlation would vanish.
AcknowledgementsÐHelpful discussions with Dr N. Hansen and Professor P. Klimanek are gratefully acknowledged. The work was partially performed within the Engineering Science Center for Structural Characterization and Modelling of Materials at Risù National Laboratory. REFERENCES 1. Hansen, N. and Kuhlmann-Wilsdorf, D., Mater. Sci. Engng, 1986, 81, 141. 2. Bay, B., Hansen, N., Hughes, D. A. and KuhlmannWilsdorf, D., Acta metall. mater., 1992, 40, 205.
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PANTLEON and STOYAN: CORRELATIONS BETWEEN DISORIENTATIONS
3. Kuhlmann-Wilsdorf, D. and Hansen, N., Scripta metall. mater., 1991, 25, 1557. 4. Liu, Q. and Hansen, N., Scripta metall. mater., 1995, 32, 1289. 5. Hughes, D. A., Liu, Q., Chrzan, D. C. and Hansen, N., Acta mater., 1997, 45, 105. 6. Pantleon, W., Acta mater., 1998, 46, 451. 7. Pantleon, W., in Proc. 20th Risù Symposium on Materials Science: Deformation-Induced Microstructures: Analysis and Relation to Properties, ed. J. B. Bilde-Sorensen. Risù National Laboratory, Roskilde, Denmark, 1999, pp. 123±124. 8. Juul Jensen, D., Ultramicroscopy, 1997, 67, 25. 9. Breuer, D., Klimanek, P. and Pantleon, W., J. appl. Cryst., submitted. 10. Bay, B., Hansen, N. and Kuhlmann-Wilsdorf, D., Mater. Sci. Engng, 1989, A113, 385. 11. Driver, J. H., Juul Jensen, D. and Hansen, N., Acta metall. mater., 1994, 42, 3105. 12. Randle, V., Microtexture Determination and its Applications. The Institute of Materials, London, 1992. 13. Krivoglaz, M. A., Ryaboshapka, K. P. and Barabash, R. I., Fizika Metall., 1970, 30, 1134. 14. Barabash, R. I. and Klimanek, P., J. appl. Cryst., 1999, 32, 1050. 15. Barabash, R. I., Krivoglaz, M. A. and Ryaboshapka, K. P., Fizika Metall., 1976, 41, 33. 16. Francke, D., Pantleon, W. and Klimanek, P., Comput. Mater. Sci., 1996, 5, 111. 17. Ambrosi, P. and Schwink, C., Scripta metall., 1978, 12, 303. 18. Kawasaki, Y. and Takeuchi, T., Scripta metall., 1980, 14, 183. 19. Ashby, M. F., Phil. Mag., 1970, 21, 399. 20. Argon, A. S. and Haasen, P., Acta metall. mater., 1993, 41, 3289. 21. Nabarro, F. R. N., Scripta metall. mater., 1994, 30, 1085. 22. Godfrey, A., Juul Jensen, D. and Hansen, N., Acta mater., 1998, 46, 823. 23. Godfrey, A., Juul Jensen, D. and Hansen, N., Acta mater., 1998, 46, 835. 24. TroÈtzschel, J., Pantleon, W., Haberjahn, M. and Klimanek, P., Mater. Sci. Engng, 1997, A234±236, 842. 25. Smirnov, B. I., ShpeiÆzman, V. V., Ivanov, S. A., Mal'chuzenko, K. V. and Chudnova, R. S., Soviet Phys. Solid State, 1978, 20, 2158. 26. Biberger, M. and Blum, W., Phil. Mag. A, 1992, 65, 757. 27. BenesÏ , V. E., Bell System Technol. J., 1957, 36, 965.
28. Reynolds, J. F., Adv. appl. Prob., 1975, 7, 383.
APPENDIX A
A.1. Relationship to queueing theory Consider a stationary Poisson process on a line (along the x-axis) of an intensity n (equal to nsc). In each point of the process a random interval begins, which has a length following an exponential distriÿ1 bution with parameter m [equal to (2l) ]. Let Z(x ) be the random number of intervals covering the point x of the line. Then Z(x ) has a Poisson distribution with parameter hZ xi n=m: The covariance function is given by: ÿ n ÿ cov Z x, Z x X exp ÿ mjX j : m
A1
A generalization is also possible: if the random intervals follow the general length distribution function B(x ) with a density b x dB x=dx and a ÿ1 mean m , then Z(x ) follows a Poisson distribution with parameter n/m as above. The covariance function then takes the general form ÿ cov Z x, Z x X ! jX j ÿ n exp 1 ÿ m 1 ÿ B t dt : m 0
A2
These formulae seem to be found ®rst in the context of queueing theory [27, 28] in the study of the model M/M/1 (exponential service times with parameter m ) and M/G/1 (general service times). The interpretation in queueing theory is as follows: the points of the Poisson process represent the arrival instants of customers and the random interval lengths correspond to the service time duration leading to a number of customers Z(x ) in the system at a time t x: The system has in®nitely many servers; for each customer, service begins immediately after arrival without waiting time.