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Dislocation patterning—a statistical mechanics perspective
Scripta Materialia 52 (2005) 1005–1010 www.actamat-journals.com
Dislocation patterning—a statistical mechanics perspective A.H.W. Ngan
*
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China Received 11 December 2004; received in revised form 11 January 2005; accepted 19 January 2005
Abstract Dislocation patterning is investigated by balancing energy and entropy through a noise factor. The following sequence is predicted for low-density situations: homogeneous distribution at low noise levels, regular cell formation at intermediate noise levels, and irregular cell formation at high noise levels. In high-density situations, homogeneous distribution does not occur. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dislocations; Dislocation theory; Deformation structure; Statistical mechanics; Self-organization and patterning
1. Introduction It is well known that as materials deform, dislocations will seldom distribute uniformly but will instead form patterns. By considering local balance between dislocation generation, annihilation and growth, Ha¨hner and co-workers proposed a theory in which dislocation patterning is driven by noise [1,2]. Their predicted distribution function for the dislocation density q is P ðqÞ qð14=r
2 Þ=2
pffiffiffi exp 4 q=r2 ;
ð1Þ
where r2 represents the noise. At small values of r2 such as 0.1, P(q) exhibits a single sharp peak at about the mean density value, corresponding to the situation where the dislocation distribution is highly homogeneous, as shown schematically in Fig. 1(a). This peak becomes less sharp as the noise increases, and at r2 > 4, P(q) becomes monotonically decreasing with a singularity at zero density. This form of P(q) is interpreted by
*
Tel.: +852 2859 7900; fax: +852 2858 5415. E-mail address: [email protected]
Ha¨hner et al. to correspond to a situation of irregular or fractal cell formation [2,3], illustrated schematically in Fig. 1(c). There are, however, situations where cell formation tends to be regular; an example is dislocation patterning in fatigued samples. A pattern of regular cells should have a twin-peaked dislocation density distribution, as illustrated schematically in Fig. 1(b), but, as noted by Ha¨hner et al. [2], this cannot be explained by Eq. (1). In fact, from a configurational point of view, a state in which regular dislocation cells form may be regarded as more random than the homogeneous state in which dislocations are equidistantly distributed, but is less random than a state of irregular or even fractal cell formation. Hence, on increasing noise, one would expect the transformation sequence: homogeneous distribution ! regular cell formation ! irregular cell formation. The distribution in Eq. (1), however, transits directly from the homogeneous state at small noise to irregular cell formation at large noise, and the regime of regular cell formation is not accounted for. This paper aims at proposing an alternative derivation of the P(q) function based on consideration of both energetics and noise, where regular as well as irregular cell formation are both allowed.
1359-6462/$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.01.027
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A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
of considering both energetics and noise (entropy) in the prediction of dislocation patterning has been stipulated recently by Thomson [4]. Another way to interpret Eq. (2) is to say that when dislocations settle to an equilibrium configuration, energy U has to be minimized but because of the existence of noise in the form of, for example, spatially fluctuating lattice friction or formation of locks by interaction with other lattice defects, U cannot attain the fully minimized value but instead, a residual degree of randomness, represented by S, has to be preserved. In other words, the search for the minimum in U has got to take place along a constant-S surface, and so the parameter h in Eq. (2) has the meaning of a Lagrange multiplier for the constraint S when U is minimized. In this sense, U is a potential energy whose value depends on the positions of the dislocations, or the function P(q), but the existence of random lattice friction, or other locking mechanisms, prevents the dislocations from settling into their fully relaxed positions. Finally, we note in passing that the principle of minimizing the generalized free energy as in Eq. (2) was found to be applicable in describing the static internal force distributions in macroscopic random structures such as granular packings [5] and open-cell foams [6].
P(ρ)
ρ
0
(a)
λ
t
P( ρ)
cell wall cell interior
0
ρ
0
ρ
(b)
P( ρ)
cell wall with varying density
(c) Fig. 1. Schematic illustrating different forms of dislocation patterning: (a) homogeneous distribution; (b) regular cell formation; (c) irregular cell formation.
2. Generalized free energy The key assumption in the present treatment is that when a dislocation distribution settles to an equilibrium state, it is describable by the minimization of a functional F analogous to the thermodynamic free energy: F ¼ U hS:
ð2Þ
Here, U is the average energy of a dislocation, including the self energy and the share of the interaction energy between the dislocation energy concerned and all other dislocations situated in the neighborhood, S is an entropy functional representing the randomness of the dislocation arrangement, and h is a parameter analogous to the thermal temperature which also represents the noise level. The main idea is to represent U and S as functionals of the dislocation density distribution P(q); minimization of F in Eq. (2) then leads to the establishment of the equilibrium form of P(q). A few words are required before we proceed to derive the functionals U[P(q)] and S[P(q)]. One way to interpret Eq. (2) is to say that a given equilibrium state P(q) is represented by the balance between energy minimization and entropy maximization, and the relative importance between energy and entropy is controlled by the magnitude of the parameter h. The importance
3. Low dislocation density systems To derive the functional U[P(q)] in situations with low dislocation densities, we note that the interaction energy between a pair of parallel dislocations is ½alð~ b1 ~ nÞð~ b2 ~ nÞ=2p lnðRo =‘Þ, where l is the shear mod~ ulus, bi the Burgers vectors, ~ n the dislocation direction, Ro the outer cut-off radius, ‘ the dislocation spacing, and a is a configurational factor to account for the dislocation character. Thus, in a 2-D array, the share of the interaction energy of a dislocation of unit length situated at a site with local dislocation density q is approximately given by: 1 nalð~ b1 ~ nÞð~ b2 ~ nÞ Ro EðqÞ ln ; ð3Þ 2 2p ‘ pffiffiffi where ‘ 1= q is the average dislocation spacing, and n the number of nearest dislocations interacting with the dislocation concerned, and a now also takes into account the mixity of the signs of the interacting dislocations. In Eq. (3), the factor 1/2 is needed to ascribe half of the interaction energy between a pair of parallel dislocations to one of the dislocations, to avoid double counting when the interaction energies are summed up to get the average value later on. Eq. (3) is a crude approximation because (i) a 2-D array of parallel dislocations with identical Burgers vector is assumed, and (ii) it neglects stress screening effects amongst the dislocations, which, nevertheless, is not important in a low-
A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
density environment. In particular, Eq. (3) cannot be used to describe high-density dislocation arrangements such as low-angle walls, for which the energy scales with the local density as E ln(‘/b) ln(1/q) [7]. In such cases, the energy per dislocation length is actually lowered by clustering, and this will be considered in the next section. In Eq. (3), by assuming that the interacting neighbors are equidistantly situated in a circular shell around the central dislocation, n 2p‘/‘ = 2p. By defining a minimum density qmin ¼ 1=R2o ! 0, Eq. (3) can be rewritten as alb2 q EðqÞ ln : ð3aÞ qmin 4 Suppose we divide up the region in consideration into a large number (L) of pixels within each the dislocation density can be measured to form the distribution P(q). The interaction energy per dislocation length inside these pixels {E1, E2, . . . , EL} form a distribution Pe(E), which corresponds to P(q) by Pe(E)dE = P(q)dq. From Eq. (3a), P ðqÞ ¼
alb2 P e ðEÞ: 4q
ð4Þ
We assume that the energy space {E1, E2, . . . , EL} is ergodic, so that each pixel has equal probability to adopt any value of energy. This assumption is analogous to that used in the statistical mechanics description of an ideal gas, in which the gas molecules are assumed to have equal chance to take up any energy, as a result of the frequent collisions amongst them. In the present case of a collection of dislocations, we can justify the assumption of ergodicity in the energy space by the frequent interactions between dislocations in a deforming solid, so that at different sampling instances along a continuously deformation process, each pixel in the volume has equal chance of taking up any energy value. The logarithm of the number of indistinguishable ways to partition the energy amongst the pixels is then given by the entropy functional Z 1 S ¼ k P e ðEÞ ln½P e ðEÞ dE; ð5Þ
tion function for the normalized dislocation density hqi ¼ q= q, satisfying P(hqi)dhqi = P(q)dq, as A hqi exp ðc lnhqiÞ : ð7Þ P ðhqiÞ ¼ hqi h Here, A and c are normalizing constants corresponding to the constraints Z 1 Z 1 P ðhqiÞ dhqi ¼ 1; and hqiP ðhqiÞ dhqi ¼ 1: 0
in which Eo represents the self energy which is indepen is the average dent of the local dislocation density, and q dislocation density. Substituting Eq. (4) into Eqs. (5) and (6), and then into Eq. (2), F can be minimized using standard variational calculus means to give the distribu-
0
ð8Þ The second constraint above is to ensure that the minimization of F is done at constant quantity of dislocations. The P(hqi) is expressed into the dimensionless form in Eq. (7) by choosing the normalizing constant k in the entropy definition in Eq. (5) to be k = alb2/4, the pre-logarithmic factor in Eq. (3a). This is admissible since the choice of k affects only the scale of h in Eq. (2), which can be arbitrary. Fig. 2 shows the log–log plot of P(hqi) at different h. It can be seen that at h < 392, a singularity occurs at hqi = 0 and a broad peak occurs at a larger value of hqi. Such a form of P(hqi) corresponds to the case of regular cell formation in Fig. 1(b), since the singularity at zero density represents the cell interior where almost no dislocations can be found, and the broad peak at a finite hqi represents the cell walls where high densities of dislocations reside. This twin-peak form of P(hqi) in fact prevails for h < 392, although at very low values of h, say below 0.1, the zero peak is so narrow that it seems to have disappeared. For h > 392, the peak at high hqi vanishes, and only the zero-density singularity remains. This corresponds to irregular cell formation in Fig. 1(c). A simple measure of the extent of cell formation can be devised by considering the idealized cell structure shown in Fig. 1(b), in which k and t are, respectively,
P (<ρ>)
0
where k is a normalizing parameter analogous to the Boltzmann factor in thermodynamics. The average energy of a dislocation of unit length in the entire deformation volume is Z 1 1 U¼ EqP e ðEÞ dE þ Eo ð6Þ 0 q
1007
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
0.01
Note: At θ = 392, high-density peak just disappears.
θ = 0.01 θ = 0.1 θ= 1
θ = 10 θ = 100 θ = 1000
0.1
1
10
100
1000
10000
<ρ> Fig. 2. Log–log plot of dislocation density distribution functions at different noise levels (h) from Eq. (7) for low dislocation density situations. Normalized dislocation density hqi ¼ q= q.
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A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
alb2 qmax ln ; EðqÞ 8p q
the mean cell size and cell wall thickness. In this case, the volume ratio of the cell interiors, vi, and that of the cell walls, vb, are approximately v i N k2 ;
and;
Nt2 ; vb 2N kt þ 4
ð12Þ
where a is a geometrical factor, and qmax = 1/b2 [7]. A similar minimization procedure as above yields the following form of the dislocation density distribution: A0 hqi 0 ðlnhqi c Þ ; exp ð13Þ P ðhqiÞ ¼ h hqi
ð9Þ
where N is the number of cells per unit cross-sectional area. Eliminating N between the two equations in (9) yields s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k vi vi vi ¼ þ : ð10Þ þ t vb vb 4vb
where the normalization factor in the entropy definition is now set to k = alb2/8p. A 0 and c 0 are normalization constants corresponding to the constraints in Eq. (8), but because the integrals in (8) now diverge, the upper limits are replaced by a finite value of hqmax i ¼ qmax = q. At small h values, this form of P(hqi) has a singularity at q = 0, and then diverges at high densities, as shown by the curves for h = 0.1 and 1 in Fig. 3, for the case of hqmaxi = 10. This corresponds to regular cell formation. At large h values, only the zero-density singularity prevails, as shown by the curve for h = 10 in Fig. 3, and this corresponds to irregular cell formation. The regime of homogeneous distribution is predicted not to happen in the dense situation, because the energy function in Eq. (12) favors clustering.
The volume fractions vi and vb can be evaluated as Z hqc i P ðhqiÞ dhqi; and vi Z0 1 vb P ðhqiÞ dhqi 1 vi ; ð11Þ hqc i
where the integration limit hqci is the local minimum of the P(hqi) plot. The values of hqci, vi, vb and k/t numerically calculated from Eqs. (10) and (11) at different h, are listed in Table 1. It can be seen that for h smaller than about 0.1, k/t is less than 1, and so cell formation cannot be regarded as unambiguous. In this case, the P(hqi) function is dominated by a sharp peak at hqi 1, as Fig. 2 shows. This corresponds to homogeneous dislocation distribution illustrated in Fig. 1(a). For h between 1 and 329, the k/t ratio increases with h, which means increasing noise level promotes the formation of larger regular cells. At h > 329, no solution for hqci exists since P(hqi) becomes monotonically decreasing as discussed above, and so cell formation becomes irregular, as illustrated in Fig. 1(c). In other words, at increasing noise, the following sequence is established for low-density situations: homogeneous distribution ! regular cell formation ! irregular cell formation.
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
<ρmax> = 10
P (<ρ>)
θ = 10 θ=1
θ = 0.1
0
1
2
3
4
5 6 <ρ>
7
8
9
Fig. 3. Semi-log plot of dislocation density distribution functions at different noise levels (h) from Eq. (13) for high dislocation density situations. Normalized dislocation density hqi ¼ q= q. Normalization by the integrations in Eq. (8) is only carried out up to hqmaxi = 10, since the integrals diverge as q ! 1.
4. High dislocation density systems In a dense dislocation array, the energy per unit length of dislocation can be expressed as the form
Table 1 Cell size to wall thickness ratios calculated from Eq. (10) h 0.01 0.1
hqci 0.00154351 0.259907
10
vi
vb
0 0.1405
1 0.8595
k/t 0 0.423
No cell formation
1 10 100
0.302412 3.53168 47.5338
0.8697 0.9814 0.9970
0.1303 0.0186 0.0030
45.38 105.7 664.8
Regular cell formation
>329
No solution
–
–
–
Irregular cell formation
A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
5. Discussions The present treatment is based on a number of assumptions which need to be discussed further. First, dislocation density is regarded as a state variable and there is no provision to distinguish between mobile and immobile dislocations. This is not an unreasonable assumption, because interaction energy between dislocations in general does not depend on dislocation mobility but only on character and instantaneous positions of dislocations, and hence on local density. The presence of immobile dislocations will prevent the system from collapsing into a global minimum energy state, and this is one reason why entropy needs to be incorporated into the minimization functional in Eq. (2), as discussed before. However, because of the reference to the dislocation-density domain, the present treatment gives no information on other aspects of patterning, such as the scaling behaviour of velocities [8], mean waiting time or jump distances [9–11] of dislocations. Such scaling behaviours are found to exhibit close-to-criticality features in materials with high dislocation mobility [8]. Secondly, it is important to realise that the present treatment is a quasi-equilibrium theory. The second constraint in Eq. (8) also requires the search for quasiequilibrium to be performed at a constant dislocation quantity. This constraint, however, does not preclude the dislocation density from increasing during an actual straining process, so long as the process is approximately quasi-static. In the initial stage of deformation, the mean dislocation density and noise are both small, and so the dislocation distribution should be homogeneous according to the solution for the low-density scenario discussed above. As strain develops, dislocation density increases and so does noise, since interactions amongst dislocations are more frequent, and one expects regular or irregular cell formation depending on the noise level, as allowed by the solution for either the low or high density scenario. In fatigue samples, it is likely that the repetitious stress conditions help to regulate the growth of noise, and hence the distribution can be maintained at a regular cellular pattern. The quasi-static assumption here is likely to hold for small strain rates under which relaxation kinetics are much faster than multiplication kinetics. Another important question to ask is whether the perturbations, both temporal and spatial, during the average waiting time before massive dislocation generation can be well presented by the entropy constraint in the present approach. One key assumption here is that the various dynamics effects in terms of Peierls barriers in materials with thermally activated dislocation motion or viscous drag in solute-rich systems can be well presented, on average, by the entropy constraint in Eq. (2). This remains an assumption to be further investigated, with a view to reconcile with available experimental data on the relationship between dislocation pattern
1009
formation and quantities such as mobile dislocation densities, mean jump distances and waiting times, strain rates, level of impurities, and so forth [9–11]. Thirdly, the energy expressions in Eqs. (3a) and (12) are oversimplifications but are nevertheless approximations required to obtain analytical solutions. Long-range interactions between dislocations are ignored, but such interactions are heavily shielded by near neighbours and so this assumption is fair. On a local scale, the dislocation positions are assumed to be uncorrelated, i.e. dislocations are assumed to distribute like a uniform ‘‘gas’’ on a local scale, but on a larger scale, the ‘‘gas’’ density is allowed to fluctuate to form patterns. The true picture is, of course, much more complicated with the possibility of forming 3-D networks and walls, but these cannot be modelled analytically.
6. Conclusions Dislocation patterning is interpreted as a result of the balance between energy and entropy. The equilibrium dislocation density is obtained by minimizing a generalized free energy which links energy and entropy through a noise factor, which is analogous to temperature in a thermal system. The energy function, however, is different between the low-density and high-density situations. In the low-density case, energy drives the dislocation arrangement towards a homogeneously distributed state, while entropy drives it towards a random state. In this situation, the following sequence is predicted at increasing noise levels: homogeneous distribution ! regular cell formation ! irregular cell formation. In the high-density situation, energy drives towards regular cell formation rather than a homogeneous distribution. The sequence at increasing noise becomes: regular cell formation ! irregular cell formation, and homogeneous distribution does not exist at all.
Acknowledgments This work was motivated by two excellent presentations by Peter Ha¨hner and Robb Thomson at the Dislocations 2004 Conference, for which the author was fortunate to be in the audience. The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, PR China (Project no. HKU7201/03E).
References [1] Ha¨hner P. Acta Mater 1996;44:2345. [2] Zaiser M, Bay K, Ha¨hner P. Acta Mater 1999;47:2463. [3] Ha¨hner P, Bay K, Zaiser M. Phys Rev Lett 1998;81:2470.
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[4] Thomson RM, Oral Presentation in Dislocations 2004 Conference, La Colle-sur-Loup, France, September 13–17, 2004. [5] Ngan AHW. Phys Rev E 2003;68:011301. [6] Ngan AHW. Proc Roy Soc Lond Ser A 2005;461:433. [7] Hansen H, Kuhlmann-Wilsdorf D. Mater Sci Eng 1986;81:141. [8] Miguel MC, Vespignani A, Zapperi S, Weiss J, Grasso J-R. Nature 2001;410:667.
[9] De Hosson JTM, Kanert O, Sleeswyk AW. In: Nabarro FRN, editor. Dislocations in solids, vol. 6. Amsterdam: North-Holland; 1983. p. 441–534. [10] De Hosson JTM, Kanert O, Boom GJ. Mater Res 1988;3: 645. [11] De Hosson JTM, Boom G, Schlagowski U, Kanert O. Acta Metall 1986;34:1571.
Dislocation patterning—a statistical mechanics perspective A.H.W. Ngan
*
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China Received 11 December 2004; received in revised form 11 January 2005; accepted 19 January 2005
Abstract Dislocation patterning is investigated by balancing energy and entropy through a noise factor. The following sequence is predicted for low-density situations: homogeneous distribution at low noise levels, regular cell formation at intermediate noise levels, and irregular cell formation at high noise levels. In high-density situations, homogeneous distribution does not occur. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dislocations; Dislocation theory; Deformation structure; Statistical mechanics; Self-organization and patterning
1. Introduction It is well known that as materials deform, dislocations will seldom distribute uniformly but will instead form patterns. By considering local balance between dislocation generation, annihilation and growth, Ha¨hner and co-workers proposed a theory in which dislocation patterning is driven by noise [1,2]. Their predicted distribution function for the dislocation density q is P ðqÞ qð14=r
2 Þ=2
pffiffiffi exp 4 q=r2 ;
ð1Þ
where r2 represents the noise. At small values of r2 such as 0.1, P(q) exhibits a single sharp peak at about the mean density value, corresponding to the situation where the dislocation distribution is highly homogeneous, as shown schematically in Fig. 1(a). This peak becomes less sharp as the noise increases, and at r2 > 4, P(q) becomes monotonically decreasing with a singularity at zero density. This form of P(q) is interpreted by
*
Tel.: +852 2859 7900; fax: +852 2858 5415. E-mail address: [email protected]
Ha¨hner et al. to correspond to a situation of irregular or fractal cell formation [2,3], illustrated schematically in Fig. 1(c). There are, however, situations where cell formation tends to be regular; an example is dislocation patterning in fatigued samples. A pattern of regular cells should have a twin-peaked dislocation density distribution, as illustrated schematically in Fig. 1(b), but, as noted by Ha¨hner et al. [2], this cannot be explained by Eq. (1). In fact, from a configurational point of view, a state in which regular dislocation cells form may be regarded as more random than the homogeneous state in which dislocations are equidistantly distributed, but is less random than a state of irregular or even fractal cell formation. Hence, on increasing noise, one would expect the transformation sequence: homogeneous distribution ! regular cell formation ! irregular cell formation. The distribution in Eq. (1), however, transits directly from the homogeneous state at small noise to irregular cell formation at large noise, and the regime of regular cell formation is not accounted for. This paper aims at proposing an alternative derivation of the P(q) function based on consideration of both energetics and noise, where regular as well as irregular cell formation are both allowed.
1359-6462/$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.01.027
1006
A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
of considering both energetics and noise (entropy) in the prediction of dislocation patterning has been stipulated recently by Thomson [4]. Another way to interpret Eq. (2) is to say that when dislocations settle to an equilibrium configuration, energy U has to be minimized but because of the existence of noise in the form of, for example, spatially fluctuating lattice friction or formation of locks by interaction with other lattice defects, U cannot attain the fully minimized value but instead, a residual degree of randomness, represented by S, has to be preserved. In other words, the search for the minimum in U has got to take place along a constant-S surface, and so the parameter h in Eq. (2) has the meaning of a Lagrange multiplier for the constraint S when U is minimized. In this sense, U is a potential energy whose value depends on the positions of the dislocations, or the function P(q), but the existence of random lattice friction, or other locking mechanisms, prevents the dislocations from settling into their fully relaxed positions. Finally, we note in passing that the principle of minimizing the generalized free energy as in Eq. (2) was found to be applicable in describing the static internal force distributions in macroscopic random structures such as granular packings [5] and open-cell foams [6].
P(ρ)
ρ
0
(a)
λ
t
P( ρ)
cell wall cell interior
0
ρ
0
ρ
(b)
P( ρ)
cell wall with varying density
(c) Fig. 1. Schematic illustrating different forms of dislocation patterning: (a) homogeneous distribution; (b) regular cell formation; (c) irregular cell formation.
2. Generalized free energy The key assumption in the present treatment is that when a dislocation distribution settles to an equilibrium state, it is describable by the minimization of a functional F analogous to the thermodynamic free energy: F ¼ U hS:
ð2Þ
Here, U is the average energy of a dislocation, including the self energy and the share of the interaction energy between the dislocation energy concerned and all other dislocations situated in the neighborhood, S is an entropy functional representing the randomness of the dislocation arrangement, and h is a parameter analogous to the thermal temperature which also represents the noise level. The main idea is to represent U and S as functionals of the dislocation density distribution P(q); minimization of F in Eq. (2) then leads to the establishment of the equilibrium form of P(q). A few words are required before we proceed to derive the functionals U[P(q)] and S[P(q)]. One way to interpret Eq. (2) is to say that a given equilibrium state P(q) is represented by the balance between energy minimization and entropy maximization, and the relative importance between energy and entropy is controlled by the magnitude of the parameter h. The importance
3. Low dislocation density systems To derive the functional U[P(q)] in situations with low dislocation densities, we note that the interaction energy between a pair of parallel dislocations is ½alð~ b1 ~ nÞð~ b2 ~ nÞ=2p lnðRo =‘Þ, where l is the shear mod~ ulus, bi the Burgers vectors, ~ n the dislocation direction, Ro the outer cut-off radius, ‘ the dislocation spacing, and a is a configurational factor to account for the dislocation character. Thus, in a 2-D array, the share of the interaction energy of a dislocation of unit length situated at a site with local dislocation density q is approximately given by: 1 nalð~ b1 ~ nÞð~ b2 ~ nÞ Ro EðqÞ ln ; ð3Þ 2 2p ‘ pffiffiffi where ‘ 1= q is the average dislocation spacing, and n the number of nearest dislocations interacting with the dislocation concerned, and a now also takes into account the mixity of the signs of the interacting dislocations. In Eq. (3), the factor 1/2 is needed to ascribe half of the interaction energy between a pair of parallel dislocations to one of the dislocations, to avoid double counting when the interaction energies are summed up to get the average value later on. Eq. (3) is a crude approximation because (i) a 2-D array of parallel dislocations with identical Burgers vector is assumed, and (ii) it neglects stress screening effects amongst the dislocations, which, nevertheless, is not important in a low-
A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
density environment. In particular, Eq. (3) cannot be used to describe high-density dislocation arrangements such as low-angle walls, for which the energy scales with the local density as E ln(‘/b) ln(1/q) [7]. In such cases, the energy per dislocation length is actually lowered by clustering, and this will be considered in the next section. In Eq. (3), by assuming that the interacting neighbors are equidistantly situated in a circular shell around the central dislocation, n 2p‘/‘ = 2p. By defining a minimum density qmin ¼ 1=R2o ! 0, Eq. (3) can be rewritten as alb2 q EðqÞ ln : ð3aÞ qmin 4 Suppose we divide up the region in consideration into a large number (L) of pixels within each the dislocation density can be measured to form the distribution P(q). The interaction energy per dislocation length inside these pixels {E1, E2, . . . , EL} form a distribution Pe(E), which corresponds to P(q) by Pe(E)dE = P(q)dq. From Eq. (3a), P ðqÞ ¼
alb2 P e ðEÞ: 4q
ð4Þ
We assume that the energy space {E1, E2, . . . , EL} is ergodic, so that each pixel has equal probability to adopt any value of energy. This assumption is analogous to that used in the statistical mechanics description of an ideal gas, in which the gas molecules are assumed to have equal chance to take up any energy, as a result of the frequent collisions amongst them. In the present case of a collection of dislocations, we can justify the assumption of ergodicity in the energy space by the frequent interactions between dislocations in a deforming solid, so that at different sampling instances along a continuously deformation process, each pixel in the volume has equal chance of taking up any energy value. The logarithm of the number of indistinguishable ways to partition the energy amongst the pixels is then given by the entropy functional Z 1 S ¼ k P e ðEÞ ln½P e ðEÞ dE; ð5Þ
tion function for the normalized dislocation density hqi ¼ q= q, satisfying P(hqi)dhqi = P(q)dq, as A hqi exp ðc lnhqiÞ : ð7Þ P ðhqiÞ ¼ hqi h Here, A and c are normalizing constants corresponding to the constraints Z 1 Z 1 P ðhqiÞ dhqi ¼ 1; and hqiP ðhqiÞ dhqi ¼ 1: 0
in which Eo represents the self energy which is indepen is the average dent of the local dislocation density, and q dislocation density. Substituting Eq. (4) into Eqs. (5) and (6), and then into Eq. (2), F can be minimized using standard variational calculus means to give the distribu-
0
ð8Þ The second constraint above is to ensure that the minimization of F is done at constant quantity of dislocations. The P(hqi) is expressed into the dimensionless form in Eq. (7) by choosing the normalizing constant k in the entropy definition in Eq. (5) to be k = alb2/4, the pre-logarithmic factor in Eq. (3a). This is admissible since the choice of k affects only the scale of h in Eq. (2), which can be arbitrary. Fig. 2 shows the log–log plot of P(hqi) at different h. It can be seen that at h < 392, a singularity occurs at hqi = 0 and a broad peak occurs at a larger value of hqi. Such a form of P(hqi) corresponds to the case of regular cell formation in Fig. 1(b), since the singularity at zero density represents the cell interior where almost no dislocations can be found, and the broad peak at a finite hqi represents the cell walls where high densities of dislocations reside. This twin-peak form of P(hqi) in fact prevails for h < 392, although at very low values of h, say below 0.1, the zero peak is so narrow that it seems to have disappeared. For h > 392, the peak at high hqi vanishes, and only the zero-density singularity remains. This corresponds to irregular cell formation in Fig. 1(c). A simple measure of the extent of cell formation can be devised by considering the idealized cell structure shown in Fig. 1(b), in which k and t are, respectively,
P (<ρ>)
0
where k is a normalizing parameter analogous to the Boltzmann factor in thermodynamics. The average energy of a dislocation of unit length in the entire deformation volume is Z 1 1 U¼ EqP e ðEÞ dE þ Eo ð6Þ 0 q
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10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
0.01
Note: At θ = 392, high-density peak just disappears.
θ = 0.01 θ = 0.1 θ= 1
θ = 10 θ = 100 θ = 1000
0.1
1
10
100
1000
10000
<ρ> Fig. 2. Log–log plot of dislocation density distribution functions at different noise levels (h) from Eq. (7) for low dislocation density situations. Normalized dislocation density hqi ¼ q= q.
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alb2 qmax ln ; EðqÞ 8p q
the mean cell size and cell wall thickness. In this case, the volume ratio of the cell interiors, vi, and that of the cell walls, vb, are approximately v i N k2 ;
and;
Nt2 ; vb 2N kt þ 4
ð12Þ
where a is a geometrical factor, and qmax = 1/b2 [7]. A similar minimization procedure as above yields the following form of the dislocation density distribution: A0 hqi 0 ðlnhqi c Þ ; exp ð13Þ P ðhqiÞ ¼ h hqi
ð9Þ
where N is the number of cells per unit cross-sectional area. Eliminating N between the two equations in (9) yields s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k vi vi vi ¼ þ : ð10Þ þ t vb vb 4vb
where the normalization factor in the entropy definition is now set to k = alb2/8p. A 0 and c 0 are normalization constants corresponding to the constraints in Eq. (8), but because the integrals in (8) now diverge, the upper limits are replaced by a finite value of hqmax i ¼ qmax = q. At small h values, this form of P(hqi) has a singularity at q = 0, and then diverges at high densities, as shown by the curves for h = 0.1 and 1 in Fig. 3, for the case of hqmaxi = 10. This corresponds to regular cell formation. At large h values, only the zero-density singularity prevails, as shown by the curve for h = 10 in Fig. 3, and this corresponds to irregular cell formation. The regime of homogeneous distribution is predicted not to happen in the dense situation, because the energy function in Eq. (12) favors clustering.
The volume fractions vi and vb can be evaluated as Z hqc i P ðhqiÞ dhqi; and vi Z0 1 vb P ðhqiÞ dhqi 1 vi ; ð11Þ hqc i
where the integration limit hqci is the local minimum of the P(hqi) plot. The values of hqci, vi, vb and k/t numerically calculated from Eqs. (10) and (11) at different h, are listed in Table 1. It can be seen that for h smaller than about 0.1, k/t is less than 1, and so cell formation cannot be regarded as unambiguous. In this case, the P(hqi) function is dominated by a sharp peak at hqi 1, as Fig. 2 shows. This corresponds to homogeneous dislocation distribution illustrated in Fig. 1(a). For h between 1 and 329, the k/t ratio increases with h, which means increasing noise level promotes the formation of larger regular cells. At h > 329, no solution for hqci exists since P(hqi) becomes monotonically decreasing as discussed above, and so cell formation becomes irregular, as illustrated in Fig. 1(c). In other words, at increasing noise, the following sequence is established for low-density situations: homogeneous distribution ! regular cell formation ! irregular cell formation.
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
<ρmax> = 10
P (<ρ>)
θ = 10 θ=1
θ = 0.1
0
1
2
3
4
5 6 <ρ>
7
8
9
Fig. 3. Semi-log plot of dislocation density distribution functions at different noise levels (h) from Eq. (13) for high dislocation density situations. Normalized dislocation density hqi ¼ q= q. Normalization by the integrations in Eq. (8) is only carried out up to hqmaxi = 10, since the integrals diverge as q ! 1.
4. High dislocation density systems In a dense dislocation array, the energy per unit length of dislocation can be expressed as the form
Table 1 Cell size to wall thickness ratios calculated from Eq. (10) h 0.01 0.1
hqci 0.00154351 0.259907
10
vi
vb
0 0.1405
1 0.8595
k/t 0 0.423
No cell formation
1 10 100
0.302412 3.53168 47.5338
0.8697 0.9814 0.9970
0.1303 0.0186 0.0030
45.38 105.7 664.8
Regular cell formation
>329
No solution
–
–
–
Irregular cell formation
A.H.W. Ngan / Scripta Materialia 52 (2005) 1005–1010
5. Discussions The present treatment is based on a number of assumptions which need to be discussed further. First, dislocation density is regarded as a state variable and there is no provision to distinguish between mobile and immobile dislocations. This is not an unreasonable assumption, because interaction energy between dislocations in general does not depend on dislocation mobility but only on character and instantaneous positions of dislocations, and hence on local density. The presence of immobile dislocations will prevent the system from collapsing into a global minimum energy state, and this is one reason why entropy needs to be incorporated into the minimization functional in Eq. (2), as discussed before. However, because of the reference to the dislocation-density domain, the present treatment gives no information on other aspects of patterning, such as the scaling behaviour of velocities [8], mean waiting time or jump distances [9–11] of dislocations. Such scaling behaviours are found to exhibit close-to-criticality features in materials with high dislocation mobility [8]. Secondly, it is important to realise that the present treatment is a quasi-equilibrium theory. The second constraint in Eq. (8) also requires the search for quasiequilibrium to be performed at a constant dislocation quantity. This constraint, however, does not preclude the dislocation density from increasing during an actual straining process, so long as the process is approximately quasi-static. In the initial stage of deformation, the mean dislocation density and noise are both small, and so the dislocation distribution should be homogeneous according to the solution for the low-density scenario discussed above. As strain develops, dislocation density increases and so does noise, since interactions amongst dislocations are more frequent, and one expects regular or irregular cell formation depending on the noise level, as allowed by the solution for either the low or high density scenario. In fatigue samples, it is likely that the repetitious stress conditions help to regulate the growth of noise, and hence the distribution can be maintained at a regular cellular pattern. The quasi-static assumption here is likely to hold for small strain rates under which relaxation kinetics are much faster than multiplication kinetics. Another important question to ask is whether the perturbations, both temporal and spatial, during the average waiting time before massive dislocation generation can be well presented by the entropy constraint in the present approach. One key assumption here is that the various dynamics effects in terms of Peierls barriers in materials with thermally activated dislocation motion or viscous drag in solute-rich systems can be well presented, on average, by the entropy constraint in Eq. (2). This remains an assumption to be further investigated, with a view to reconcile with available experimental data on the relationship between dislocation pattern
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formation and quantities such as mobile dislocation densities, mean jump distances and waiting times, strain rates, level of impurities, and so forth [9–11]. Thirdly, the energy expressions in Eqs. (3a) and (12) are oversimplifications but are nevertheless approximations required to obtain analytical solutions. Long-range interactions between dislocations are ignored, but such interactions are heavily shielded by near neighbours and so this assumption is fair. On a local scale, the dislocation positions are assumed to be uncorrelated, i.e. dislocations are assumed to distribute like a uniform ‘‘gas’’ on a local scale, but on a larger scale, the ‘‘gas’’ density is allowed to fluctuate to form patterns. The true picture is, of course, much more complicated with the possibility of forming 3-D networks and walls, but these cannot be modelled analytically.
6. Conclusions Dislocation patterning is interpreted as a result of the balance between energy and entropy. The equilibrium dislocation density is obtained by minimizing a generalized free energy which links energy and entropy through a noise factor, which is analogous to temperature in a thermal system. The energy function, however, is different between the low-density and high-density situations. In the low-density case, energy drives the dislocation arrangement towards a homogeneously distributed state, while entropy drives it towards a random state. In this situation, the following sequence is predicted at increasing noise levels: homogeneous distribution ! regular cell formation ! irregular cell formation. In the high-density situation, energy drives towards regular cell formation rather than a homogeneous distribution. The sequence at increasing noise becomes: regular cell formation ! irregular cell formation, and homogeneous distribution does not exist at all.
Acknowledgments This work was motivated by two excellent presentations by Peter Ha¨hner and Robb Thomson at the Dislocations 2004 Conference, for which the author was fortunate to be in the audience. The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region, PR China (Project no. HKU7201/03E).
References [1] Ha¨hner P. Acta Mater 1996;44:2345. [2] Zaiser M, Bay K, Ha¨hner P. Acta Mater 1999;47:2463. [3] Ha¨hner P, Bay K, Zaiser M. Phys Rev Lett 1998;81:2470.
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[4] Thomson RM, Oral Presentation in Dislocations 2004 Conference, La Colle-sur-Loup, France, September 13–17, 2004. [5] Ngan AHW. Phys Rev E 2003;68:011301. [6] Ngan AHW. Proc Roy Soc Lond Ser A 2005;461:433. [7] Hansen H, Kuhlmann-Wilsdorf D. Mater Sci Eng 1986;81:141. [8] Miguel MC, Vespignani A, Zapperi S, Weiss J, Grasso J-R. Nature 2001;410:667.
[9] De Hosson JTM, Kanert O, Sleeswyk AW. In: Nabarro FRN, editor. Dislocations in solids, vol. 6. Amsterdam: North-Holland; 1983. p. 441–534. [10] De Hosson JTM, Kanert O, Boom GJ. Mater Res 1988;3: 645. [11] De Hosson JTM, Boom G, Schlagowski U, Kanert O. Acta Metall 1986;34:1571.