On the statistical origin of disorientations in dislocation structures

PII:

Acta mater. Vol. 46, No. 2, pp. 451±456, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00286-3

ON THE STATISTICAL ORIGIN OF DISORIENTATIONS IN DISLOCATION STRUCTURES W. PANTLEON{ Institute of Physical Metallurgy, Freiberg University of Mining and Technology, Gustav-Zeuner-Straûe 5, D-09596 Freiberg, Germany (Received 31 January 1997; accepted 29 July 1997) AbstractÐThe formation of disorientations in dislocation structures during plastic deformation is considered as a random accumulation process of excess dislocations in the dislocation boundaries. For a single slip system the assumption of a Gaussian white noise for the bias of dislocation ¯uxes leads to a Gaussian distribution for the disorientation angles with a vanishing mean value. The corresponding standard deviation is determined by the dependence of the cell size on plastic strain, if the number of mobile dislocations of one sign of the Burgers vector at the same time present in a dislocation cell can be described by a Poisson distribution. For a constant cell size the square root dependence of the mean disorientation angle on plastic strainÐas previously proposed by Argon and Haasen and derived by NabarroÐis con®rmed simultaneously showing the intrinsic restriction of this dependence to a constant cell size. The necessary modi®cations for symmetrical double slip as well as for a non-constant cell size are presented. # 1998 Acta Metallurgica Inc. AbstractÐDie Bildung von Fehlorientierungen in Versetzungsstrukturen waÈhrend plastischer Verformung wird auf Grundlage des stochastischen Akkumulationsprozesses von UÈberschuûversetzungen eines Vorzeichens des Burgers-Vektors in einer Versetzungswand behandelt. Im Falle eines einzigen Gleitsystems fuÈhrt die Annahme eines (Gaussschen) weiûen Rauschens des UÈberschuûstromes mobiler Versetzungen auf eine Normal-Verteilung der auftretenden Desorientierungswinkel mit verschwindendem Mittelwert. Die dazugehoÈrige Standardabweichung ist durch die AbhaÈngigkeit des Zelldurchmessers von der plastischen Dehnung gegeben, sofern die Anzahl mobiler Versetzungen eines Vorzeichens, die sich gleichzeitig in einer Versetzungszelle be®nden, einer Poisson-Verteilung genuÈgt. FuÈr eine konstante ZellgroÈûe wird die bereits fruÈher von Argon und Haasen vorgeschlagene und von Nabarro abgeleitete Quadratwurzel-AbhaÈngigkeit des mittleren Desorientierungswinkels von der plastischen Dehnung erhalten und dabei zugleich die intrinsische BeschraÈnkung dieser Beziehung auf konstante ZellgroÈûe aufgezeigt. Die notwendigen Modi®kationen fuÈr symmetrische Doppel-Gleitung und auch fuÈr nicht-konstante Zelldurchmesser werden vorgestellt.

1. INTRODUCTION

During plastic deformation in dislocation structures disorientations are developed between adjacent cells with a monotonically increasing mean disorientation angle. From experimental data it is concluded in Ref. [1, Section 3.2.1.2] that the mean disorientation carried by a single cell wall or a subgrain boundary depends almost linearily on strain, but recent studies [2, 3] show, that for incidental dislocation boundaries (geometrically necessary boundaries will not be considered here) a square root dependence of the mean disorientation angle on plastic strain is also in agreement with experimental data on cold-rolled aluminium [4, 5]. Argon and Haasen [6] argued that a square root law for the dependence of the mean disorientation angle p yˆB g …1† {Present address: Risù National Laboratory, Materials Research Department, P.O. Box 49, DK-4000 Roskilde, Denmark. 451

on plastic strain will arise due to an accumulation of random errors caused by random ¯uctuations in the ¯uxes of mobile dislocations passing a cell wall and coming from the cells on either side. A ®rst theoretical derivation of the proportionality coecient B was given by Nabarro [7] taking into account that in each cell of a cell structure the expansion of Z dislocation loops with Burgers vector b to the cell size d will lead to a plastic strain g = Zb/d. If each of these loops will lead to the storage of one dislocation in the cell walls, each of the walls will consist of Z dislocations. From a statistical excess of DZ = ZZ dislocations of a preferred sign in each wall Nabarro expected a disorientation r p b b p g: …2† yˆ Z ˆ d d Beside this more suggestive approach, at present no rigorous treatment of the statistical origin of the disorientations in dislocations structures is available. Therefore, the aim of the paper is to solve the problem of accumulation of disorientations on the

452

PANTELEON: ORIGIN OF DISORIENTATIONS

Fig. 1. Fluxes of mobile edge dislocations of a single slip system traversing a chain of cell walls.

basis of elementary statistics and to ®gure out the consequences for the distribution function of the disorientation angles. 2. MODELLING

2.1. Accumulation of dislocations and disorientations Mobile dislocations carrying the plastic deformation are partially trapped in dislocation walls during their passage through the walls as shown schematically for a single slip system with perpendicular cell walls in Fig. 1. The immobilization probability P for mobile dislocations is given by the fact that the slip line lengths L on surfaces are larger than the cell diameter d; e.g. in copper single crystals deformed along the [001]-direction each mobile dislocation will pass three cell walls before their immobilization leading to an immobilization probability P = 2d/L = 1/3 [8, 9]. For a single slip system the evolution of the dislocation density{ of one sign (_) of the Burgers vector b in a cell wall dr? P ! ˆ j dt d

…3†

m of mobile dislocations is given by the ¯ux j 4=vr_ m r_ moving with the velocity v to the right. A combination with the analogous expression for the dislocations of opposite sign of the Burgers vector (~) leads to the evolution of the total dislocation density

dr P ! P ˆ … j ‡ j3 † ˆ j dt d d

…4†

da ˆ bPDj dt

…6†

results obviously from (5) for a constant cell size d, but will remain valid for varying cell sizes also, if again the assumption of a slow variation is taken into account as for the derivation of the equation (3) governing the dislocation accumulation. Consequently, the disorientation angle of a speci®c wall Zt a…t† ˆ bPDj…t 0 †dt 0 …7† 0

can be found by integration of the bias of dislocation ¯uxes Dj (t) = j 4ÿj3 in this wall. 2.2. Fluctuating dislocation ¯uxes

and the excess dislocation density dDr P ! P ˆ … j ÿ j 3 † ˆ Dj; dt d d

m m (r_ +r~ ) is related to the strain rate gÇ =bvrm=bj by the Orowan equation. It should be mentioned, that in derivation of (3) resp. (4, 5) it is tacitly assumed, that the cell diameter d evolves slowly in comparison with the transition of a mobile dislocation through the dislocation cell (cf. [6]). If a certain amount of dislocations may annihilate each other, this will a€ect only the total dislocation density r = r_+r~ (4), whereas the evolution of excess dislocations Dr = r_ÿr~ (5) remains unchanged, because each annihilation event involves two dislocations of opposite sign of the Burgers vector [10, 11]. From the mean distance 1/dDr of excess dislocations in a single cell wall or a subgrain boundary the disorientation angle a = bdDr can be obtained from the Read±Shockley-relation [12]. The corresponding evolution equation of the disorientation angle

…5†

where the total dislocation ¯ux j = j 4ÿj3 =v {Despite the concentration of the stored dislocations in the cell walls all dislocation densities are related to the whole cell size here.

In deterministic systems under homogeneous deformation conditions there will be no reason for such a bias of the dislocation ¯uxes Dj and no disorientation should appear. However, from statistical ¯uctuations such a bias will arise incidently in some of the walls, but it will disappear immediately again. If a random noise is assumed for the ¯uctuating dislocation ¯uxes, the ensemble averages

PANTELEON: ORIGIN OF DISORIENTATIONS

(denoted by h.i) of an ensemble of equivalent walls can be determined. For instance, the mean value hDj…t†i ˆ 0

…8†

will disappear and, if the correlation time t of the ¯uctuations will be shorter than the typical time scale for the accumulation of dislocations or disorientations, the (second order) correlation function hDj…t†Dj…t 0 †i ˆ …Dj0 †2 td…t ÿ t 0 †

…9†

is approximately given by the instantaneous ¯uctuation amplitude Dj0 and their correlation time t. Then the mean disorientation angle Z t  Zt hai ˆ bPDj…t 0 †dt 0 ˆ bPhDj…t 0 †idt 0 ˆ 0 …10† 0

0

vanishes as expected, whereas the moment of the second order Z t Z t  bPDj…t 0 †bPDj…t}†dt 0 dt } ha2 i ˆ 0

ˆ

Z tZ 0

0

ˆ…bP†2

453

equal to the square root of relevant numbers the amplitude of the bias of dislocation ¯uxes is given by r r v v Z 2v ? > ˆ Dj0 ˆ 2 …DZ ‡ DZ † ˆ 2 2 j : …15† d d 2 d2 Using the life time t = l/v of the mobile dislocations as the (maximal possible) correlation time of the dislocation movement and the mean free path l = L/2 = d/P the integration kernel of (11) 2v d 2 2 g
…Dj0 †2 t ˆ 2 j …16† ˆ jˆ d Pv Pd Pd b is completely determined. Therefore, the evaluation of the standard deviation Zt Zg 2bP
0 2bP 0 2 …17† gdt ˆ dg sa ˆ d d 0 0 will be strongly a€ected by the time (or strain) dependence of the cell size.

0

t

…bP†2 hDj…t 0 †Dj…t }†idt 0 dt }

Z

t 0

def

…Dj0 †2 tdt 0 ˆ s2a

remains ®nite andÐdue to the vanishing mean valueÐwill be directly connected with the standard deviation sa of the distribution function for the disoriention angles. Further investigation of sa requires the speci®cation of the functional dependence on time of the integration kernel (Dj0)2 t ®rst, which can be obtained from the statistics of mobile dislocations. 2.3. Considerations on the number of mobile dislocations In equiaxed cells with a diameter d the number of mobile dislocations simultaneously present in a single cell will be j Zˆd r ˆd v 2 m

2

…12†

resp. the average number of mobile dislocations with one sign of b hZ? i ˆ hZ > i ˆ

Z 1 2 m ˆ d r : 2 2

3. RESULTS

…11†

…13†

Assuming a density of mobile dislocations of m m rm = r_ + r~ 11013 mÿ2 and a typical cell diameter of d = 2 mm this number can be estimated to be Z 1 40 resp. hZ_i = hZ~i = 20. These (integer) numbers are a result of rare and independent emission processes at dislocation scources and should be Poisson distributed. With the standard deviation r p Z ˆ DZ> DZ > ˆ hZ ? i ˆ …14† 2

3.1. Distribution function Following the arguments above, a random Gaussian white noise [13, 14] can be expected for the ¯uctuating bias of the ¯uxes. This assumption determines not only the ®rst two but all higher order correlation functions [13, 14] hDj…t1 †Dj…t2 † . . . Dj…t2nÿ1 †i ˆ0 hDj…t1 †Dj…t2 † . . . Dj…t2n †i ˆ‰…Dj0 †2 tŠn

X

d…t1 ÿ t2 †

. . . d…t2nÿ1 ÿ t2n †

…18†

where the sum extends over all relevant permutations of the ti. Consequently, all higher order moments of the disorientation angle ha2nÿ1 iˆ 0

…19† Z

ha2n iˆ …bP†2n

t

…Dj…t 0 ††2 tdt 0

n

0

…2n†! …2n†! 2n ˆ s 2n n! 2n n! a

…20†

can be evaluated. With the help of the characteristic function (cf. [13, 14]) def f~a …u† ˆ 1‡

1 X …iu†m mˆ1

m!

ˆ1‡

ham i ˆ 1 ‡

nˆ1

1 X …ÿu2 s2 †n a

nˆ1

1 X …iu†2n …2n†!

2n n!

…2n†! 2n n!

s2n a

 2 2 u sa ˆ exp ÿ 2

…21†

the distribution function can be obtained immediately by Fourier back transformation. The resulting distribution function for the disorientation angle a of cell walls or subgrain boundaries caused by a single slip system

454

PANTELEON: ORIGIN OF DISORIENTATIONS

f …a† ˆ

 2 2 u sa du exp…ÿiua† exp ÿ 2 0   1 a2 ˆ p exp ÿ 2 2sa 2psa

1 2p

Z

1

…22†

is a normal distribution with a mean value of 08 and a standard deviation sa. Remarkably, for this normal distribution the mean value of the modulus of the disorientation angle r r 2 2 2 hjaji ˆ …23† ha i ˆ sa p p is directly connected with the standard deviation sa. This rigorous treatment justi®es earlier results [15, 16] where a normal distribution of the disorientation angle in cell walls resulting from a single set of dislocations is found from computer simulations of a linear chain of cell walls. 3.2. Constant cell size 3.2.1. Single slip system. In the case of a constant cell size and a single slip system the standard deviation of the normal distribution (17) can be easily found by integration leading to 2bP s2a ˆ ha2 i ˆ g …24† d and a square root dependence of the mean disorientation angle r 2 2bPp p …25† hjaji ˆ g ˆ B g p d on plastic strain. The resulting coecient r bP B ˆ 2 pd

…26†

di€ers from the expression B = Zb/d of Nabarro just by a factor of 2ZP/p. This slight di€erence is due to the fact that here the Poisson statistics is applied to the numbers of dislocations of each sign of Burgers vector and not to the total amount of dislocations and that the mean value is determined taking into account the complete distribution function. Furthermore, the considered probability factor P re¯ects a partial penetration of mobile dislocations through the walls. 3.2.2. Double slip. Experimentally determined distribution functions for the disorientation angles (e.g. [4, 5, 15]) show a signi®cant lack of small disorientation angles and can not be described satisfactorily by the normal distribution with vanishing mean value derived above. According to Pantleon [17] this can be attributed to the fact that in each cell wall not only a single set of dislocations is responsible for the total disorientation angle y, but two dislocation sets will contribute in an equal manner. These two dislocation sets, which may arise, e.g. from symmetrical double slip, are

assumed to be located in the same boundary and inclined towards one another by an inclination angle b. Following the considerations above, both sets should have normal distributed disorientation angles with a vanishing mean value and the same standard deviation sa. Then, the distribution function ! ! dP y y2 y2 cos b fb …y† ˆ ˆ 2 exp ÿ I0 dy sa sin b 2s2a sin2 b 2s2a sin2 b …27† for the positive de®nite disorientation angle y can be derived analytically [17] using the Bessel function of the ®rst kind I0. For example, for two perpendicular sets, resp. an inclination angle b = 908, the disorientation angles are distributed according to a Rayleigh distribution   y y2 …28† f90 …y† ˆ 2 exp ÿ 2 2sa sa with a mean value hyi ˆ

r p sa : 2

…29†

Finally, in connection with (17) the mean disorientation angle r pbPp hyi ˆ g …30† d will depend on the square root of the strain g in each of the two equivalent slip systems.

3.3. Non-constant cell size As usually observed the cell size d changes during deformation (at least at the beginning) and this alteration has to be taken into account. For symmetrical multiple slip conditions where the macroscopic plastic strain rate Çe=mngÇ is related to the strain rates gÇ of each of the n equivalent slip systems with the same Schmid-factor m this leads to a second order moment Z 2bP e de s2a ˆ ha2 i ˆ : …31† mn 0 d…e† In order to specify the variation of the cell diameter d(e) more precisely the well known scaling law [18±20] of the cell diameter d with the dislocation density or the stress ~ K Kmb d ˆ p ˆ r Ds

…32†

is considered here in the more appropriate second form where Ds = s ÿ s0 denotes the contribution of the dislocations to the ¯ow stress s (s0 summarizes all contributions not related to dislocations [20]).

PANTELEON: ORIGIN OF DISORIENTATIONS

Consequently, the second order moment (17) Z 2P 1 t
ˆ 2P Dwpl …33† Dsedt s2a ˆ ha2 i ˆ ~ ~ mn Km Kmmn 0 |‚‚‚‚‚{z‚‚‚‚‚} ˆDwpl

depends directly on the contribution of the plastic work related with the ¯ow resistance of the dislocations Ze Ze Dwpl ˆ Ds…e 0 †de 0 ˆ ÿs0 e ‡ s…e 0 †de 0 : …34† 0

0

Under the assumption of two (n = 2) independent slip systems inclined by an angle b = 908 contributing in an equal manner to the total disorientation the mean value of the disorientation angle  rs Ze pPb de …35† hyi ˆ 2m 0 d…e† s pP p …36† ˆ Dwpl ~ 2Kmm can be determined using again the Rayleigh distribution (28).

3.4. Comparison with experimental data The proposed mechanism of random accumulation processes should be especially capable to explain the formation of disorientations of dislocation cell walls or other dislocation boundaries where no genuine geometric reason is accountable for the appearing disorientations. Here, experimental results of Liu and Hansen [5, 21] on cold-rolled polycrystalline aluminium are utilized for a comparison with the theoretical predictions. By transmission electron microscopy they have investigated the mutual spacings d of incidental dislocation boundaries as well as the disorientation angles hyi related to them. From the data on the boundary spacings (cf. [21]) a dependence of the cell diameter d = 0.9 mm*e0.42 on the (van Mises) strain e can be found. Using this pure empirical relation for the evaluation of equation (35) together with a Burgers vector of b = 2.86*10ÿ10 m, an assumed Schmid-factor of m 11/2 and an immobilization probability P = 1/3, a mean disorientation angle of hyi 10.98*e0.72 should be expected. If, alternatively, the relation hyi=BÄen with a ®xed n = 0.72 is ®tted to the experimentally determined mean disorientations angles for the incidental dislocation boundaries [5], a proportionality factor BÄ=2.08 can be obtained, which is of the same order of magnitude as the predicted value 0.98. The agreement within a factor of 2 seems rather close, remembering the crude assumptions made on the number and the Schmid-factor of the slip systems and the immobilization probability and can be

455

improved signi®cantly by more properly chosen values. But it should be mentioned, that the ®tting of the disorientation angles with a constant exponent n is, of course, intended for comparison purposes only and other approximations on the experimental data as, e.g. hyi11.88 e0.5 are also possible (cf. [2, 3]).

4. CONCLUSION

The formation of disorientations during plastic deformation is treated on the basis of ¯uctations in the number of mobile dislocations. The assumption of a single slip system and a Gaussian white noise for the ¯uctuating bias of dislocation ¯uxes leads to a normal distribution for the disorientation angles with a vanishing mean disorientation. The dependence of the average modulus of the disorientation angle on the plastic deformation is calculated for several di€erent cases: for a constant cell size of the dislocation structure the dependence of the disorientation angle on the square root of plastic strain as proposed by [6, 7] is retained with an exact determination of the proportionality coecient. On the other hand, if a dependence of the cell size on the ¯ow stress is taken into account, the mean disorientation will depend on the square root of the plastic work done against the ¯ow stress contribution of the dislocations. Finally, an example for the correspondence of the developed theoretical ideas with experimental data is presented.

AcknowledgementsÐThis work was partially performed during a visit at the Materials Research Department of the Risù National Laboratory. The author gratefully acknowledges N. Hansen for stimulating discussions and the ®nancial support from the Risù. REFERENCES 1. Gil Sevillano, J., van Houtte, P. and Aernoudt, E., Progr. Mater. Sci., 1982, 25, 69. 2. Hughes, D. A., in Proc. 16th Risù Intern. Symp. Mater. Sci., Microstructural and Crystallographic Aspects of Recrystallisation, ed. N. Hansen et al.. Risù National Laboratory, Roskilde, Denmark, 1995, p. 63. 3. Hughes, D. A., Liu, Q., Chrzan, D. C. and Hansen, N., Acta Mater., 1997, 45, 105. 4. Liu, Q. and Hansen, N., Phys. Stat. Sol., 1995, 149, 187. 5. Liu, Q. and Hansen, N., Scripta metall. mater., 1995, 32, 1289. 6. Argon, A. S. and Haasen, P., Acta metall. mater., 1993, 41, 3289. 7. Nabarro, F. R. N., Scripta metall. mater., 1994, 30, 1085. 8. Ambrosi, P. and Schwink, C., Scripta metall., 1978, 12, 303. 9. Kawasaki, Y. and Takeuchi, T., Scripta metall., 1980, 14, 183. 10. Francke, D., Pantleon, W. and Klimanek, P., Comput. Mater. Sci., 1996, 5, 111.

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PANTELEON: ORIGIN OF DISORIENTATIONS

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