- Email: [email protected]
Dislocation Patterns in Crystalline Solids—Phenomenology and Modelling
Crystal Growth - from Fundamentals to Technology G. Muller, J.-J. Metois and P. Rudolph (Editors) © 2004 Elsevier B.V. All rights reserved.
215
Dislocation Patterns in Crystalline Solids - Phenomenology and Modelling Michael Zaiser^ *The University of Edinburgh, Center for Materials Science and Engineering, The King's Buildings, Sanderson Building, Edinburgh EH93JL, United Kingdom The paper gives an overview of dislocation patterning phenomena in metals and semiconductors and their modelling using dislocation dynamics approaches. After a brief introduction of some fundamental concepts, experimental observations are reviewed and some fundamental scahng relations are discussed which characterize dislocation patterns under widely varying conditions. The emergence of dislocation patterns is related to the non-equilibrium dynamics of the system of interacting dislocation lines. Modelling approaches include discrete dislocation dynamics simulations as well as models based on ideas from non-equilibrium thermodynamics and statistical mechanics.
1. I N T R O D U C T I O N Plastic deformation of crystalline solids is intimately related to the motion and proliferation of linear lattice defects, i.e. dislocations. Preservation of the lattice structure during plastic deformation implies that plastic displacements can occur only by shifting adjacent lattice planes against each other by a lattice vector ('slip vector') b. If the slipped area is bounded, the boundary line corresponds to a lattice dislocation and the slip vector is the Burgers vector of this dislocation. For energetic reasons, in a given lattice structure slip occurs preferentially on the closest-packed lattice planes and the slip vector is usually a vector connecting nearest neighbors (i.e. b = \b\ is the interatomic spacing). A set of parallel lattice planes on which slip occurs ('slip planes') is characterized by their normal vector n; a slip plane normal plus slip vector define a slip system. In the following we will formally distinguish different slip systems by an index f3; each value of P corresponds to a particular pair (6, n). As crystalline solids deform plastically, the expansion of slipped areas leads to an increase in the dislocation line length within the crystal ('dislocation multiplication'). This proliferation of dislocations goes along with their spatial re-distribution and, more often than not, the formation of heterogeneous dislocation patterns (Figure 1). The characteristics of such patterns depend on material properties and deformation conditions. Two extreme cases are illustrated in Figure 1 (a) and (b). Although in both cases the material (monocrystalline Cu), stress state (uniaxial tension/compression) and crystal orientation
216
M. Zaiser
Figure 1. Dislocation patterns formed in different materials and under different deformation conditions, (a) FVactal dislocation cell structure formed in a Cu single crystal deformed in tension along the [100] axis; deformation was carried out at room temperature to a stress (resolved shear stress in each of the 8 active slip systems) of 75.6 MPs; transmission electron micrograph (TEM micrograph) after Mughrabi et. al. [1]. (b) TEM micrograph of the 'labyrinth' structure formed in a [100] oriented Cu single crystal after cyclic deformation (tension/compresssion) at a strain amplitude 7 = 1.8 x 10~^ to a saturation stress amplitude of 40 MPa, after Gong et. al. [2]; (c) Dislocation cell structure in a GaAs single crystals; the pattern has formed after plastic relaxation of thermal stresses during crystal growth; dislocation positions are mapped by etch pits, courtesy of P. Rudolph, (d) Dislocation cell structure in an Al polycrystal after rolling, TEM micrograph after Liu et. al. [3].
Dislocation patterns in crystalline solids - phenomenology and modelling
111
(stress axis along the [100] lattice direction) are the same, two different deformation modes (unidirectional versus cyclic straining) yield completely different morphologies of the dislocation pattern: while cyclic deformation leads to highly regular 'labyrinth' patterns of intersecting dislocation-dense walls with a well defined spacing, unidirectional deformation yields a very irregular pattern which is characterized by fractal scale-invariance rather than any characteristic length scale [4]. Such scale-invariance is, however, an exception rather than the rule - the majority of dislocation patterns in unidirectionally deformed crystals pertains to the cellular type illustrated in Figure 1 (c) and (d); these patterns, while irregular, exhibit a well-defined characteristic length scale (mean cell size A).
2. D I S L O C A T I O N D Y N A M I C S : F U N D A M E N T A L S 2.1. Forces and interactions in dislocation systems Dislocations move under the action of stresses. Specifically, the so-called Peach-Koehler force acting on a dislocation segment of unit length, Burgers vector h and tangent vector t is given by /PK - {ha) X t
(1)
where cr is the stress tensor at the location of the dislocation segment. From a dynamical point of view it is important to distinguish glide motion of a dislocation line within its slip plane (which corresponds to the plane spanned by h and t^ with normal vector n) from out-of-plane 'climb' motions. The latter cannot take place without inserting or removing material, and therefore dislocation climb is possible only at temperatures where longrange self-diffusion may occur. The glide component of the Peach-Koehler force can be written as /G = Tfe6(nxi)
.
(2)
Tg^J = sari is called the resolved shear stress in the slip system (6, n); s = h/h is the unit sUp vector. The key problem of dislocation dynamics is to evaluate the stresses at the positions of all dislocation line segments. The local stress at a given point in the crystal lattice is a sum of externally applied and internal stresses. As 'external' we classify any stresses acting from outside on the dislocation system - due to tractions applied to the surface of the dislocated body, temperature gradients giving rise to thermal stresses, etc. Such stresses act as external driving forces which may induce dislocation motion and plastic flow (see below). Dislocation interactions, on the other hand, are embodied within the internal stress field created by the dislocation system itself. Dislocations are sources of internal stresses. The stress field of a straight dislocation with Burgers vector h and tangent vector t is given by
'SA^ = T "i^n^-^f^ ,
(3)
where r is a vector of modulus |f| = r in the plane normal to t, and G is the shear modulus of the material. The dislocation stress field decays radially like b/r and may be strongly
218
M. Zaiser
anisotropic; its dependence on the azimuthal angle 0 is governed by the angle 0 between the Burgers vector h and line direction t^ which defines the type of the dislocation. For instance, dislocations with 6 parallel to Tare referred to a screw dislocations, dislocations with 6 perpendicular to Tedge dislocations, and dislocations with 9 = ±7r/3 or ±27r/3 are called 60° dislocations. Explicit expressions for the stress fields of edge, screw and general ('mixed') dislocations in isotropic media can be found in standard works (e.g. [5]). Here we only note that the stress field of a straight dislocation is antisymmetric in both b and For a quasi-two-dimensional system of straight parallel dislocations with common line direction ^, the internal stress at the point rin the plane perpendicular to the dislocation lines is simply obtained by summation over all dislocations, J
For a three-dimensional system of curved dislocation lines, the situation is slightly more complicated. A dislocation segment of tangent vector T, Burgers vector 6, and unit length situated at the origin creates at the point f = (r, 0, ^ ) a stress
.(M>-) = ^^M|LM) .
(5)
Explicit expressions for the stress fields of short straight segments of general orientation can be found in [6]. The internal stress field at r is obtained by summation over all segments, ^'^'(r) = E
[ or(6,,r(.),f - f(s))ds
.
(6)
Here, r{s) is the parametrization of a three-dimensionally curved dislocation Ci with Burgers vector fei, and t{s) is the corresponding bundle of tangent vectors. Again, the summation runs over all dislocation lines ^. 2.2. Dislocation motion and plastic flow Dislocations move under the action of Peach-Koehler forces. Moving dislocations are subject to electron and phonon drag leading to a friction force per unit length / F = X^ which is proportional to the dislocation velocity. Since the effective mass of a dislocation is small, dislocation glide often takes place in an over-damped manner and can therefore be described by 'Aristotelian' dynamics, v = h{r^lx
•
(7)
This description ceases to be correct when dislocation motion is controlled by thermal activation processes. Thermal activation of dislocations may be important for the overcoming of localized obstacles such as solute atoms, radiation- or deformation-induced ^For topological reasons, dislocations cannot terminate inside the crystal and the Burgers vector is constant along the dislocation line. Any dislocation must either form a closed loop or connect two triple nodes where J^j ^i = 0. In a finite crystal, one or both of these nodes may be replaced by a point at the surface, and in Etn infinite crystal by the point at infinity.
Dislocation patterns in crystalline solids - phenomenology and modelling
219
point defects and their agglomerates, or small precipitates. Thermal activation is also crucial for the motion of dislocations in crystals with high Peierls barriers: If a dislocation moves along its slip plane, its core configuration and hence its energy changes in a lattice-periodic manner. In certain materials the resulting energy barriers to dislocation motion ('Peierls barriers') are significant and at sufficiently low temperatures, dislocation motion is governed by their overcoming via a thermally activated double-kink mechanism. In cases where dislocation motion is controlled by thermal activation, the velocity (defined on a scale which is large in comparison with the spacing of the relevant obstacles) is given by an Arrhenius-type expression. H{T)
1/6 exp
(8)
kT
where z/ is a characteristic frequency of the order of magnitude of the Debye frequency. The stress dependence of the activation energy H{T) reflects the way how the energy profile of the relevant obstacles is tilted by the resolved shear stress acting on the dislocation. Whatever the stress dependence of its velocity, a gUding dislocation (dislocation segment) of tangent vector t moves in the direction n x f of the glide component of the Peach-Koehler force. For dynamic purposes it is often convenient to distinguish dislocations according to their direction of motion into different 'populations'. For a 2D dislocation system of straight parallel dislocations, t can take only two values which define two possible 'signs' s := sign(fe.f) of a dislocation. For each slip system f3 and sign s we may formally introduce a discrete dislocation density pj^^ by
p^D'{^ = j:^ssMAr~r^)
,
(9)
J
where the sum runs over all dislocations. The local shear strain rates on the different slip systems are related to the dislocation densities and dislocation velocities by Orowan's relation,
7^(0 = EV''(0K^xt>(r) .
(10)
s
Depending on the level of description (discrete vs. continuum), p^'^{r) in this equation is understood as the discrete density given by Eq. (9), or an appropriate coarse-grained version thereof. Analogous expressions can be formulated in 3D. The plastic strain rate tensor is obtained from the shear strain rates on the different slip systems via i{r)^J^M^^^(r)
(11)
where the projection tensors M^ = [s^ ^ ^^Jsym are the symmetrized tensor products of the respective unit slip vectors and slip plane normals. 2.3. Scaling relations for dislocation patterns Irrespective of their morphology, dislocation patterns in plastically deformed crystals obey some fairly universal scahng relations. We discuss these scaHng relations first for
220
M. Zaiser
the simplified case of a system of straight parallel dislocations which can be envisaged as point particles in a plane normal to their line direction. We assume that temperatures are sufficiently low such that dislocations move only by gUde, and that Peierls barriers are low such that glide may take place at arbitrarily small stresses. Under these circumstances a necessary and sufficient condition for a system of N dislocations to be in static equilibrium at an applied stress (Text is that the resolved shear stresses acting on all dislocations must be zero, i.e. ni = 0 ,
hi
iG\l...N]
.
(12)
Now assume that we increase the applied stress by a factor r/ and simultaneously decrease all dislocation spacings by the same factor. Prom Eqs. (3) and (12) it follows immediately that the new dislocation arrangement is also in static equilibrium. This simple scaling invariance also holds in the limit AT ~> cx) in an infinite crystal. In this case the re-scaling by a factor r; simply increases the dislocation density (number of dislocations per unit area) by a factor of r/^. Prom this scaling property the following conclusions derive: • If a given dislocation arrangement of average density p is stable under a certain applied stress (Text', then there exists a one-parameter manifold of dislocation arrangements of density p' with the same morphology that are stable under stresses (p'/p)VVe«. • If a dislocation arrangement exhibits inhomogeneous patterns with a characteristic length A (for examples see (b)-(d) in Pigure 1) and is stable under the applied stress o-ext, then morphologically equivalent arrangements with characteristic length A' are stable under stresses (X/X')(Tcxt• Assume that the applied stress is slowly increased until sustained dislocation motion occurs on a slip system (6, n). The corresponding critical resolved shear stress Tg^ is also called the flow stress of this slip system, and the critical resolved shear stress of the slip system(s) where long-range dislocation motion first occurs is simply called the flow stress r^ — min[Tg^]. Prom scaling it follows that, for dislocation patterns of similar morphology but different total density, flow stresses are proportional to the square root of dislocation density. This is often expressed in terms of Taylor's relation r' = aGby/p
,
(13)
where G is the shear modulus of the material, and the constant a depends weakly on the morphology of the dislocation arrangement (typically 0.2 < a < 0.4). In general, because of dislocation multiplication (cf. below) plastic deformation goes along with an increase in dislocation density, and this in turn implies an increasing flow stress (work hardening).
Dislocation patterns in crystalline solids - phenomenology and modelling
iu|
r-, 10'
^
I I I miif
111
I I I iini|
10'i ••*-.»
2, io°i •
Fe Cu Al
•
• o
g 10"
10"'
E 10'
)30
GaAs
iiiiiiH
^ 10"
^. oc
I iiiinn
I I I mm
I I limn
i
iiiiiilj
10° 10' 10' 10' Mean cell size Jl [^im] "I i i i i i i |
o
I I iiiiii|
I I limn
GaAs
•
Fe
•
CuMn
I I iiini|
10' I I iiiiij
,90)
,9'
cd
^10° c
g
1 ^°' i 10-^10"
I inn
III!
I I I linn
I I I nun
10^ 10^ 10' Mean cell size A [\xm]
10°
' ' ''""I
10^
Figure 2. Scaling of dislocation cell patterns in metals and semiconductors. Top figure: cell size vs. flow stress (resolved shear stress in the active slip system(s)). Bottom figure: cell size vs. mean dislocation spacing. Data for pure metals after Holt [7], data for CuMn after Neuhaus and Schwink [8], data for GaAs after Rudolph et. al. [9].
The above arguments can be generalized to three-dimensional systems of curved dislocation lines. For a 3D dislocation arrangement in static equilibrium, the internal stress field (6) must balance the externally applied stress everywhere along the dislocation lines. Again, by increasing the applied stress by a factor r/, and shrinking all distances including the length ds of a line element (cf, Eq. (6)) by the same factor, we obtain another static dislocation arrangement. This procedure increases the dislocation density (now defined as line length per unit volume) by a factor of 77^, and we recover the same scaling relations as above. The experimental data in Figure 2 illustrate the relations between dislocation densities, charsu^teristic lengths of dislocation cell patterns, and the flow stresses at which these patterns have formed. It is seen that the general scaling relations r^ a y/p oc 1/A are
222
M. Zaiser
fulfilled over a wide range of materials and deformation conditions, with proportionality factors that are remarkably insensitive to all peculiarities of the deformation process. Often, in the course of deformation the flow stress and dislocation density increase and the characteristic length of the dislocation pattern decreases in accordance with these scaling relations, while the morphology of the dislocation arrangement remains more or less unchanged. Such self-similar behavior of the dislocation pattern during hardening has been termed the 'law of similitude' [10]. At this point we have to address an apparent paradox: The scaling arguments we have formulated pertain to static dislocation arrangements while the formation of heterogeneous dislocation patterns necessarily involves the motion of dislocations. By looking at Eqs. (7) or (8) we see, however, that the dynamic behavior of dislocations is not invariant under the re-scaling procedure discussed above. For situations where the motion of dislocations is governed by drag forces or, more generally, the dislocations are highly mobile (e.g. at high temperatures), this paradox is readily resolved. In this case, numerical estimates show that at each given moment plastic deformation is carried by only a few dislocations (dislocation segments). These 'active' segments move rapidly in regions where the dislocation-induced internal stresses assist the externally applied stress. At the same time, the overwhelming majority of the dislocations is practically at rest. It has been estimated that, in a typical fee crystal deforming at a characteristic experimental strain rate of 10"^ s~\ only a fraction of 10~^ of all dislocations at each given moment appreciably contribute to the deformation [11,12]. Under these circumstances, scaling relations for static dislocation arrangements retain their validity even during plastic flow. The situation is more complicated in crystals where the motion of dislocations is limited by localized barriers that are not dislocation-related, such as Peierls barriers or localized obstacles. At low temperatures the overcoming of such obstacles may require substantial stresses (cf. Eq. (7); for appreciable dislocation motion to occur on typical experimental timescales, stresses must be high enough to reduce the activation enthalpy H{T) to about 25/cT). This may lead to a temperature-dependent flow stress contribution that is independent on dislocation density, and in such situations the scaling relationships of the 'law of similitude' do not apply.
3. D I S C R E T E DISLOCATION D Y N A M I C S ( D D D ) SIMULATIONS The most straightforward way of dealing with the dynamics of a many-dislocation system is to directly solve the equations of motion of the dislocation lines, keeping track of all the dislocation interactions. We first briefly discuss fully three-dimensional dislocation dynamics simulations which attempt to give a true representation of the evolution of the system of interacting dislocation lines, and then the less realistic but computationally much less intensive alternative of studying idealized quasi-two-dimensional systems consisting of parallel dislocations. 3.1. D D D simulation of 3-diinensional dislocation systems Several different schemes have been proposed for simulating the 3D dynamics of a system of interacting dislocations. These schemes mainly differ in the way how the dislocation
Dislocation patterns in crystalline solids - phenomenology and modelling
223
Figure 3. Dislocation pattern observed in a three-dimensional DDD simulation (simulation of an fee crystal deformed along a [100] axis to a total strain of 0.003; the image shows the dislocations contained within a 'TEM foil' of 5 //m thickness and [HI] orientation, after Madec et. al. [16].
lines are represented. Representations in terms of parametric splines have been adapted, using the spline parameters as generalized line coordinates [13]. Other simulations represent a dislocation line by a sequence of straight segments of variable orientation [14], using the nodes between segments as line coordinates. In the following we discuss in more detail the approach by Kubin, Devincre and co-workers [6,15-17] who use a representation of dislocation lines in terms of sequences of linear segments for which only a limited number of discrete orientations are allowed. Elastic interactions between dislocation segments are mediated by the corresponding segment stress fields which according to Eq. (5) decay in space like 1/r^. In this sense the computational problem is very similar to the molecular dynamics simulation of a system of particles with long-range (e.g. gravitational or electrostatic) interactions^. Once the stress acting on a dislocation segment is known, the Peach-Koehler force and the segment velocity can be calculated. In moving the segments, however, it is important to maintain connectivity of the dislocation line which for curved dislocations requires continuous adjustment of the segment length and/or the insertion and removal of segments. Additional problems arise if two dislocation segments get very close. Two dislocation segments of the same slip system but opposite sign may annihilate spontaneously if they ^Since piecewise linear representations of the dislocation lines give rise to artificial stress singularities at the 'corners', it may be necessary to apply corrections in order to correctly evaluate the interaction of adjacent segments.
224
M. Zaiser
approach each other below a critical distance (typically a few interatomic spacings)^. Another process may take place if two dislocation lines gliding on different slip planes intersect. If the Burgers vectors 61 and 62 and line configuration of the dislocations are such that the two lines attract each other, then upon contact they will form a new 'junction' segment with Burgers vector hi +62, thereby reducing the total elastic energy and dislocation line length. As a consequence, the intersection takes an H- rather than an X-shape and two triple nodes emerge"^. The junction segment initially runs along the direction of intersection of the slip planes of the two dislocations. If only discrete segment orientations are allowed, it is important that these include the orientation of potential junction segments. For instance, in fee crystals where the Burgers vectors are directed along [110] lattice directions and the slip planes are of [111] type, junction segments can have screw or 60° orientations and these orientations should be included in the discretization [17]. There are two reasons why junction formation is of crucial importance for the dynamical behavior of a dislocation system. On the one hand, junctions act as dislocation obsta^^les. A dislocation moving on its slip plane will continuously form junctions with 'forest' dislocations threading through this plane, and dislocation motion can only proceed if the stress acting on the dislocation is high enough to destroy these junctions. In metals deforming in multiple slip, this mechanism yields a major contribution to the flow stress. On the other hand, dislocation segments pinned at two anchoring points may bulge out and nucleate dislocation loops that expand during deformation. Expansion of dislocation loops increases the total dislocation line length and is thereby responsible for the increasing dislocation density in a deforming crystal. Through this mechanism junction formation (or, more general, the 'knitting' of a 3D dislocation network) is also crucial for dislocation multiplication and work hardening. Three-dimensional dislocation dynamics simulations correctly represent the processes of dislocation multiplication and junction formation. They account without additional assumptions for the multitude of conceivable bowing and intersection processes which may occur in the dislocation network. However, the neccessity to continuously re-evaluate the stresses acting on all dislocation segments tends to make such simulations computationally extremely intensive. Because of their high computational cost, present-day simulations of dislocation dynamics in bulk metals are restricted to comparatively small system sizes (typically about (10//m)^, though periodic boundary conditions may be imposed to mimic a bulk crystal), small strains (typically far less than 1% total deformation), and low dislocation densities (pL^ < 5 x 10^ where L is the characteristic linear dimension of the simulated volume). Under these circumstances, only the very first stages of dislocation cell patterning can be studied (Figure 3). Since dislocation densities are low, the cell size of the incipient dislocation cell patterns is of the same order as the size of the simulation cell (e.g.. Figure 3 shows a section of the simulation cell plus its first periodic images), and therefore not much information about the spatial morphology of the emergent patterns can be obtained.
^In case of two edge segments, annihilation may leave an agglomerate of point defects in the crystal. ^Only triple nodes are stable in a dislocation system; intersections which do not lead to junction formation imply that the two dislocations repel each other and, hence, no quadruple nodes are formed.
Dislocation patterns in crystalline solids - phenomenology and modelling
225
0.75 ixm
Figure 4. Dislocation pattern observed in a two-dimensional DDD simulation of deformation in symmetrical double slip; shear strain in each of the two sHp systems 7 = 0.073, resolved shear stress r = 2.6x 10~^G, total dislocation density 2 x lO^^m"^; after Benzerga et. al. [18].
3.2. D D D simulation of 2-dimensional dislocation systems Because of the tremendous reduction in degrees of freedom, two-dimensional dislocation systems can be simulated to much larger strains and dislocation densities. Since the basic scaling relations for dislocation patterns are similar in two and three dimensions, one may ask whether such simulations are a way of modelling the later stages of dislocation pattern formation and to the evolution of these patterns during sustained deformation. A 2D dislocation system is essentially a system of interacting point particles moving in a plane. For 2D dislocation dynamics to be representative of the behavior of a real, three-dimensional crystal, two conceptual problems must be addressed: (i) In a system of straight dislocations there is no mechanism for increasing the dislocation density (dislocation multiplication) which in 3D occurs by bulging of segments in the dislocation network and expansion of the ensuing dislocation loops. Since straight dislocations can neither bulge nor expand, one has instead to devise some phenomenological rule for throwing new dislocations into the system, (ii) Since dislocations cannot intersect in 2D, the formation/destruction of junctions and the corresponding work hardening have to be accounted for in a phenomenological manner. The problem is complicated by the fact that dislocation loop generation and destruction of junctions in a 3D dislocation system may be 'many-body' processes which in general simultaneously involve several dislocation lines. The approach adopted in the most recent 2D simulations is to model junction formation as a dislocation pair reaction: as soon as two dislocations of different slip systems approach beyond a certain collision radius, they form a sessile junction. This junction can be broken if a critical stress is exceeded which is related to the dislocation configuration in its surrounding. Source formation and operation are modelled in a similar spirit. This procedure makes it necessary to isolate and parameterize a limited set of dislocation mechanisms. It introduces into the models a
226
M. Zaiser
number of phenomenological parameters which have to be determined either by matching to 3D simulations or to experimental data [18]. The advantage of 2D simulations lies in the possibility to achieve strains and dislocation densities sufficiently high such that at least the early stages of dislocation pattern formation and evolution can be observed (Figure 4) and statistically significant information on the morphology of the dislocation patterns can be deduced. However, even if the problems relating to mapping 3D dislocation processes onto a 2D geometry can be solved, the approach is restricted to certain glide geometries since it is difficult to represent simultaneous dislocation glide on more than two intersecting slip planes in a 2D model.
4. C O N T I N U U M D I S L O C A T I O N D Y N A M I C S A P P R O A C H E S In view of the computational complexity of discrete dislocation dynamics simulations, repeated attempts have been made to arrive at some kind of coarse-grained description of dislocation dynamics. There are two fundamental problems: a) how to define coarsegrained dislocation densities in a manner which retains the relevant information about the kinetic properties of the discrete dislocation lines, and b) how to obtain the corresponding equations of motion. Two main approaches may be followed - an 'energetic' approach which starts out from the elastic energy associated with the dislocation system, and a 'dynamic' approach which determines equations of motion for dislocation densities in a phenomenological manner. 4.1. Linear irreversible thermodynamics and energy minimization Dislocations move under the action of forces, which implies that they move 'downhill' in an energy functional. Any static dislocation arrangement necessarily corresponds to a local minimum of the elastic energy^. It is therefore tempting to address dislocation patterning in terms of energy minimization arguments. An early attempt in this direction was made by Holt [7] who studied the stability of a spatially homogeneous dislocation system with conserved total dislocation density. In the following we present a slightly generalized version of this model which works analogous to the Cahn-Hilliard theory of spinodal decomposition [19]. The elastic energy of the internal stress field is given by E,^ = / ( T ( r ) C - V ( r ) d V
(14)
where C~^ is the tensor of elastic compliances and V is the crystal volume. In the following we consider a system of straight parallel dislocations. If we denote by
(15)
where the integration is carried over the plane normal to the dislocation. The dislocation stress field is given by Eq. (3), and accordingly the dislocation self-energy diverges with ^Note that the elastic energies associated with the dislocation system are usually so large that entropy effects can safely be neglected.
Dislocation patterns in crystalline solids - phenomenology and modelling
227
increasing crystal radius R like E^^^ = K^Gb\n{R/b) where K^ is a constant of the order of unity which depends on the type of the dislocation. The energy divergence which occurs according to Eqs. (3,15) as one approaches the dislocation core may be truncated at radius b since, on scales comparable to the atomic spacing, local linear elasticity ceases to be valid and Eq. (3) no longer applies. Far away from the dislocation, on the other hand, linear elasticity is perfectly valid and, hence, the logarithmic divergence of the self-energy is limited only by the finite crystal size. The interaction energy of two dislocations of slip systems f3 and /?' and signs s and s' at the respective positions r and r*' is given by E^nt""\r-
r') = ss' j a^{f- f")C-^(T^\r'
- r")d?r"
.
(16)
This interaction energy can be written as E^""^ = ss'K^^'{(j))Gb\n{R/\r - r'\) where K^^ {(j)) is a non-dimensional function that depends on the relative orientation of the two dislocations in the plane. Note that the dislocation interaction is strongly anisotropic; its angular average vanishes, J K^^\(f)d(f) — 0. Now the energy of a discrete dislocation system characterized by the discrete densities PD{^ as defined in Eq. (9) is readily written as
E = / ^-(r)^i,dV + E ^ - 7 / P'ii'i^Pn'inE^^'''(^-
^"'OdVdV .
(17)
To arrive at a continuum description we average over an ensemble of statistically equivalent dislocation systems. This leads to continuous densities p^'^{r) — (PD (r))- Averaging the product of discrete densities in the interaction term of Eq. (17) leads to pair densities p f | ' ' " " V , f ' ) - / ' ^ ( r ) / ' ' " ' ( r ' ) [ l +
(18)
jp[r')EUr-r'W
Here the average self- and interaction energies of a dislocation axe given by E^M = T.f"EL I3,s
EUr^= E ss'f-f-'E^,-'{r)d^^'-'(r^ /3l3',ss'
,
(19)
228
M. Zaiser
where /^'"^ = p^'^/p- In a random dislocation arrangement without correlations, the average interaction energies vanish and the energy density (18) diverges logarithmically. The presence of dislocation pair correlations re-normalizes this divergence [12]. To derive the temporal evolution of the dislocation pattern from the elastic energy functional we make the following assumptions: (i) only the total dislocation density changes whereas the 'composition' of the dislocation arrangement as expressed by the density fractions /^'* remains invariant, (ii) the total number of dislocations is preserved, i.e., the time evolution of p is given by the continuity equation dtp{r^ = ~VJ{r)
(20)
(iii) the dislocation flux J is proportional to the gradient of the average dislocation energy, J = —XP^E(f}/x where x is an effective friction coefficient. The problem is very similar to the linear irreversible thermodynamics treatment of a system with conserved order parameter as developed, for instance, in the Cahn-Hilliard theory of spinodal decomposition. Now we consider a homogeneous dislocation arrangement of total density po and study the time evolution of small space-dependent density fluctuations Sp{r). Retaining in the equation of evolution only terms of linear order in 6p^ we obtain Xdt6p(r) - poA J6p{r')EUr-
r')dh'
« poAp^'^<^/^(r) + VD^^^V6p{f)]
(21)
where
D^'^ ^ J EUr)d'r
, Z>(^) = | ^ 5 ^ , ( r l d V .
(22)
In the last step we have performed a Taylor expansion of p(f') around f^ — f, using the fact that the dislocation pair correlation functions and therefore also the average interaction energy £'mt(^— ^ 0 are short-ranged functions, cf. Eq. (19). Re-writing the evolution equation for Sp in Fourier space we get xdtSp(k) = /OoP[-/)(°^ + kD^^^k]6p{k)
(23)
Since dislocations arrange such as to reduce their total energy, the coefficients D^^^ as well as the components of the matrix D^^^ are negative. This implies a long-wavelength instability similar to the instability encountered in spinodal decomposition. The amplification of fluctuations is in general anisotropic; the direction(s) of maximum amplification corresponds to the Eigenvector(s) of JD^^^ with the largest Eigenvalue D^j^. The corresponding wavelength of dislocation density fluctuations is given by A = TT Note that, since the range of the dislocation pair correlations is proportional to the dislocation spacing, Z)^°^ oc l/po and D^^^ a l/(po)^- Hence the wavelength of maximum instability which characterizes the emergent patterns is proportional to (but in general much larger than) the dislocation spacing, in agreement with the principle of similitude. This 'energetic' approach to dislocation patterning has certain attrax^tive features, as the treatment of dislocation patterning within the general framework of linear irreversible thermodynamics leads to a simple continuum theory which correctly accounts for the
Dislocation patterns in crystalline solids - phenomenology and modelling
229
observed scaling relation A oc p~^/^, while the possibility of describing the entire dislocation population in terms of a single partial differential equation significantly reduces the computational complexity. Furthermore, the approach offers a simple explanation for dislocation patterning: the elastic energy of a dislocation is reduced if the dislocation density increases, and therefore the dislocation system segregates into regions of high and low dislocation density. On the other hand, the approach rises several problems: • The assumption that the dislocation flux follows the gradient of the elastic energy may not be warranted. The motion of dislocations is subject to dynamic constraints, e.g., at low temperatures their glide is confined to the sUp planes. Even at higher temperatures, where out-of-plane motions are possible, the effective mobilities for in-plane and out-of-plane motions are rather different.
• As dislocations of different slip systems may move in different slip planes, their fluxes have different directions.
• The approach cannot account for dislocation motions driven by an external stress. Under an externally applied stress, positive and negative dislocations may move in opposite directions, and the evolution of the dislocation arrangement cannot be described in terms of a single dislocation flux even if only dislocations of a single slip system are present. • During plastic deformation the dislocation density continually increases, and hence the assumption that the dislocation density behaves like a conserved order parameter is not justified. • It has also been criticized that the predicted proportionality of the pattern wavelength and the dislocation spacing is actually a consequence of the assumed scaling properties of the pair correlation functions. However, a recent study has to a certain extent justified the energy-based approach. By statistically averaging the equations of motion of a discrete dislocation system, Zaiser and co-workers arrived at evolution equations which under certain restrictive assumptions (dislocations of a single slip system, no external stress) are similar to Eq. (22). The dynamic constraints to dislocation motion (glide motion can occur on a single slip plane only) introduce additional anisotropics; for the case of a system of straight parallel edge dislocations it has been found that the instabiUty leads to the formation of periodic walls perpendicular to the dislocation glide direction [20]. Such walls are indeed observed in cycfically deformed metals (Figure 1 (b)). In prolonged cyclic deformation the dislocation system reaches a steady state where the dislocation density remains practically constant, and one may conjecture that periodic reversal of the external stress reduces the influence of directed dislocation fluxes on the dislocation patterning. In unidirectional deformation, on the other hand, evolution of the dislocation microstructure is characterized by directed dislocation fluxes and increasing dislocation densities, and it is difficult to see how these features can be accommodated within the 'energetic' approach.
230
M. Zaiser
4.2. S y n e r g e t i c m o d e l s A plastically deforming crystal is a driven system which by the continuous influx and dissipation of mechanical work is kept in an out-of-equilibrium state. This observation has prompted attempts to model dislocation patterning as a 'synergetic' pattern formation phenomenon, in analogy with spatial patterning phenomena observed in many driven far-from-equilibrium systems, e.g., the Belushov-Zhabotinskii reaction in chemical kinetics or the Taylor instability in hydrodynamics (for overviews of synergetic patterning, see [21,22]). In the theoretical description of these phenomena, spatial patterns emerge from symmetry-breaking instabilities in systems of nonlinear reaction-diffusiontransport equations. It has been proposed to adopt this framework to dislocation systems by distinguishing dislocation populations pk of different specification and writing down reaction-transport equations of the type [23-25] dtPk^VJk
= fk[[pi]) .
(24)
Here, the non-linear reaction terms fk model local interactions between dislocation populations and the flux terms describe long-range dislocation transport. Several phenomenological models of this type have been proposed in the literature, assuming different transport mechanisms to be operative. For instance, Kratochvil considered the flux of edge dislocation dipoles induced by their 'sweeping' by moving screw dislocations [26]. Walgraef and Aifantis [24] consider single-slip cyclic deformation and describe the net transport arising from the forward-backward motion of dislocations during a stress cycle by diffusion-like terms of the form Jk = \^Dpk. In its simplest one-dimensional form, the Walgraef Aifantis model reads ^
^
= ^ - ^ ^ + /(Pi,pJ
,
^ A 0 + p(Pi)-/(A,Pm) .
(25)
Here, p ^ and pi are the densities of mobile and immobile (dipole) dislocations, respectively, each population consisting of dislocations of both signs. The non-linear reaction term is assumed as f{puPm) = Cipi — C2PmP\ where Ci and C^ were introduced as phenomenological coefficients. The Walgraef-Aifantis model allows to obtain periodic solutions corresponding to the ladder-like dislocation arrangement in persistent slip bands (PSB, see Figure 5, right). An extension of the model to more than one dimension has been also used to investigate the competition between matrix and PSB patterns as observed experimentally (Figure 5, left) [24]. In either case, the model leads to estimates of the pattern wavelength in terms of the diffusionlike coefficients D^^ and Di entering the reaction-diffusion scheme (25). A general problem with the phenomenological synergetic approach is that, in the absence of a well-defined procedure for coarse-graining dislocation dynamics, the definition of the dislocation populations in Eq. (24), the determination of the functional form of the reaction terms, and the mathematical specification of the dislocation fluxes are all left to educated guess. With some mathematical training it is possible to devise nonUnear equations which produce a particular type of spatial patterns but, since the results are in a sense pre-determined by the phenomenological 'input', it is arguable whether this really
Dislocation patterns in crystalline solids - phenomenology and modelling
M
PSB M PSB
cT"*
25-r
1
1
1
1
231
1
1
j
Til
H
\ X)
1 ^H 0
54
CO
b 0 r 0
u
1 2
1 4
u
1 6
1 8
uu 1 10
1 12
\
1 1
Distance [M.m]
Figure 5. Left: dislocation structure in single-slip cyclic deformation; the ladder-like structure corresponds to a persistent slip band (PSB); the surrounding patchy structure is referred to as 'matrix' (M). Right: One-dimensional dislocation density pattern in a persistent slip band obtained from the Walgraef-Aifantis model; Courtesy of E.G. Aifantis.
solves the problem of dislocation patterning. Nevertheless the synergetic approax^h has been important from a conceptual point of view: If the dislocation system is envisaged as a driven out-of-equilibrium system, one cannot simply take for granted the appUcabiUty of energy minimization concepts or of linear irreversible thermodynamics, and the driven dynamics of the dislocation ensemble is put into the focus of interest.
5. STOCHASTIC APPROACHES 5.1. Discrete stochastic dislocation dynamics The discrete stochastic dislocation d5niamics approach proposed by Groma and coworkers [27,28] considers the stress-driven motion of discrete dislocations but uses a statistical description of the force created by the internal stress field. As discussed in detail in [12], in many situations the internal stress field associated with a dislocation arrangement can be split into a long-wavelength part ('mean-field stress') which varies in spax^e on the scale A of dislocation density variations and can be expressed as a functional of the dislocation densities [12,27], plus a short-wavelength part which fluctuates on the scale p""^/^ of the average local dislocation spacing. The probability distribution of this fluctuating part of the internal stress has been evaluated in [27], and the short-range nature of its spatial correlations has been demonstrated in [12]. The basic idea of a discrete stochastic dislocation dynamics simulation is to calculate explicitly only the mean-field stresses, while replacing the short-wavelength flucutations of the internal stress field by a statistically equivalent stochastic process. The meanfield stresses are functional of the dislocation densities which change slowly in space and
232
M. Zaiser
time. Since these stresses need not be updated in every time step and, moreover, can be evaluated from the dislocation densities by convolution using Fast Fourier TVansformation (FFT), a very considerable reduction in the computational cost of the simulation can in principle be achieved.
Figure 6. Simulated dislocation pattern using a stochastic dislocation dynamics code; left: dislocation density pattern, right: greyscale pattern of the excess Burgers vector density; after Groma [28]
The crucial problem with this method is to correctly represent the statistical properties of the fluctuating stresses acting on the dislocations. Groma et. al. use a strong approximation by representing the temporal statistics of the internal stresses at the dislocation positions in terms of the statistics of the internal stress fluctuations at a random point in space. This is feasible if (a) only a small fraction of the dislocations is mobile at each given moment and (b) these mobile dislocations move as individuals, each of them individually scanning the internal stress 'landscape' created by the majority of the dislocation arrangement which remains at rest. While assertion (a) is presumably valid, it is rather difficult to assess the validity of assumption (b). In fact, analytical arguments [11,12,29] as well as simulations [30,31] and experiment [30,32] indicate that dislocations move in a strongly correlated, avalanche-like manner where dynamic interactions among the moving dislocations may be of crucial importance. An adequate theoretical understanding of these collective effects and an appropriate mathematical description of the corresponding
Dislocation patterns in crystalline solids - phenomenology and modelling
233
fluctuation phenomena are still missing. In spite of these problems, simulations of deformation-induced dislocation patterns using the discrete stochastic dislocation dynamics approach yield interesting results. An example is given in Figure 6 which shows the dislocation pattern forming in a 2D dislocation system deforming in symmetrical double slip. The hierarchical nature of the dislocation pattern with long straight walls carrying substantial excess Burgers vector (and, hence, substantial lattice rotations) compares well with certain experimentally observed cellular microstructures (see e.g. Figure 1 (d)). 5.2. Continuum stochastic dislocation dynamics The continuum stochastic dislocation dynamics method formulated by Hahner [29] and elaborated by Hahner and Zaiser [4,11,12,33] approaches the problem of dislocation patterning on a much more phenomenological level. This approach starts out from phenomenological differential equations which describe dislocation density evolution and work hardening on a macroscopic scale. The complex dynamics of plastic flow on mesoscopic scales is then taken into account by considering the shear strain rates on the different slip systems [i.e. the dislocation fluxes, cf. Eq. (10)] as fluctuating functions of space and time. Since dislocation accumulation and dislocation reactions are driven by the dislocation fluxes, this impUes that dislocation density evolution is described in terms of a set of stochastic differential equations. For illustration, we discuss the simplest model of this type [34]. We assume deformation to occur in symmetrical double slip. Dislocations are stored in the crystal after travelling a mean ghde path L = B/y/p that is proportional to, but much larger than, the average dislocation spacing. The increase of the total dislocation density p is then given by
dtP^fvrt = §Vpm + s^]
(26)
where in the second step we have split the (local) strain rate in either of the sUp systems into the mean strain rate and a spatio-temporally fluctuating contribution. Fluctuations in the local strain rates are modelled by assuming that straining proceeds in the form of a shot noise where the strain rate is composed of discrete random 'events', J(r,t) = 'ZjS{t-U)g(r-ri)
(27)
i
The times ti and locations rj of the different 'slip events' are assumed as statistically independent random variables. The shape function g{f) and event amplitude 7 are chosen to reflect general scaling relations and phenomenological observations of collective dislocation motion: a) In agreement with the scaling relations underlying the 'law of similitude' we assume that the range of spatial correlations in plastic flow decreases in inverse proportion with the flow stress. This implies that 5' is a function of f/r^. b) The event amplitude 7 increases in proportion with the flow stress. This reflects the scaling relation 7 = pbL [cf. Orowan's relation, Eq. (10)] where L can be understood as the mean path travelled by a dislocation during a collective slip event. Since p oc (r^)^ and L oc l/r^ it follows that 7 oc r ^ c) Correlations in plastic flow are strongly anisotropic, their range in the direction of dislocation motion significantly exceeds their range normal to that direction.
234
M. Zaiser
Relevant information on dislocation patterning can be directly obtained by studying the stochastic differential equation (26). As discussed in detail in [34], the flow stress of an inhomogeneous dislocation arrangement can be written as r^ = aGb{y/p) where (...) denotes an average over the inhomogeneous microstructure. It then follows from Eq. (26) that the flow stress increases linearly with average strain, r^ = {aGB/2){y). Equation (26) is conveniently re-written by introducing the transformation of variables
I
1
2
yo_t>_oo-yo-g.^..^-^ 3
scaled dislocation density p l
215
Dislocation Patterns in Crystalline Solids - Phenomenology and Modelling Michael Zaiser^ *The University of Edinburgh, Center for Materials Science and Engineering, The King's Buildings, Sanderson Building, Edinburgh EH93JL, United Kingdom The paper gives an overview of dislocation patterning phenomena in metals and semiconductors and their modelling using dislocation dynamics approaches. After a brief introduction of some fundamental concepts, experimental observations are reviewed and some fundamental scahng relations are discussed which characterize dislocation patterns under widely varying conditions. The emergence of dislocation patterns is related to the non-equilibrium dynamics of the system of interacting dislocation lines. Modelling approaches include discrete dislocation dynamics simulations as well as models based on ideas from non-equilibrium thermodynamics and statistical mechanics.
1. I N T R O D U C T I O N Plastic deformation of crystalline solids is intimately related to the motion and proliferation of linear lattice defects, i.e. dislocations. Preservation of the lattice structure during plastic deformation implies that plastic displacements can occur only by shifting adjacent lattice planes against each other by a lattice vector ('slip vector') b. If the slipped area is bounded, the boundary line corresponds to a lattice dislocation and the slip vector is the Burgers vector of this dislocation. For energetic reasons, in a given lattice structure slip occurs preferentially on the closest-packed lattice planes and the slip vector is usually a vector connecting nearest neighbors (i.e. b = \b\ is the interatomic spacing). A set of parallel lattice planes on which slip occurs ('slip planes') is characterized by their normal vector n; a slip plane normal plus slip vector define a slip system. In the following we will formally distinguish different slip systems by an index f3; each value of P corresponds to a particular pair (6, n). As crystalline solids deform plastically, the expansion of slipped areas leads to an increase in the dislocation line length within the crystal ('dislocation multiplication'). This proliferation of dislocations goes along with their spatial re-distribution and, more often than not, the formation of heterogeneous dislocation patterns (Figure 1). The characteristics of such patterns depend on material properties and deformation conditions. Two extreme cases are illustrated in Figure 1 (a) and (b). Although in both cases the material (monocrystalline Cu), stress state (uniaxial tension/compression) and crystal orientation
216
M. Zaiser
Figure 1. Dislocation patterns formed in different materials and under different deformation conditions, (a) FVactal dislocation cell structure formed in a Cu single crystal deformed in tension along the [100] axis; deformation was carried out at room temperature to a stress (resolved shear stress in each of the 8 active slip systems) of 75.6 MPs; transmission electron micrograph (TEM micrograph) after Mughrabi et. al. [1]. (b) TEM micrograph of the 'labyrinth' structure formed in a [100] oriented Cu single crystal after cyclic deformation (tension/compresssion) at a strain amplitude 7 = 1.8 x 10~^ to a saturation stress amplitude of 40 MPa, after Gong et. al. [2]; (c) Dislocation cell structure in a GaAs single crystals; the pattern has formed after plastic relaxation of thermal stresses during crystal growth; dislocation positions are mapped by etch pits, courtesy of P. Rudolph, (d) Dislocation cell structure in an Al polycrystal after rolling, TEM micrograph after Liu et. al. [3].
Dislocation patterns in crystalline solids - phenomenology and modelling
111
(stress axis along the [100] lattice direction) are the same, two different deformation modes (unidirectional versus cyclic straining) yield completely different morphologies of the dislocation pattern: while cyclic deformation leads to highly regular 'labyrinth' patterns of intersecting dislocation-dense walls with a well defined spacing, unidirectional deformation yields a very irregular pattern which is characterized by fractal scale-invariance rather than any characteristic length scale [4]. Such scale-invariance is, however, an exception rather than the rule - the majority of dislocation patterns in unidirectionally deformed crystals pertains to the cellular type illustrated in Figure 1 (c) and (d); these patterns, while irregular, exhibit a well-defined characteristic length scale (mean cell size A).
2. D I S L O C A T I O N D Y N A M I C S : F U N D A M E N T A L S 2.1. Forces and interactions in dislocation systems Dislocations move under the action of stresses. Specifically, the so-called Peach-Koehler force acting on a dislocation segment of unit length, Burgers vector h and tangent vector t is given by /PK - {ha) X t
(1)
where cr is the stress tensor at the location of the dislocation segment. From a dynamical point of view it is important to distinguish glide motion of a dislocation line within its slip plane (which corresponds to the plane spanned by h and t^ with normal vector n) from out-of-plane 'climb' motions. The latter cannot take place without inserting or removing material, and therefore dislocation climb is possible only at temperatures where longrange self-diffusion may occur. The glide component of the Peach-Koehler force can be written as /G = Tfe6(nxi)
.
(2)
Tg^J = sari is called the resolved shear stress in the slip system (6, n); s = h/h is the unit sUp vector. The key problem of dislocation dynamics is to evaluate the stresses at the positions of all dislocation line segments. The local stress at a given point in the crystal lattice is a sum of externally applied and internal stresses. As 'external' we classify any stresses acting from outside on the dislocation system - due to tractions applied to the surface of the dislocated body, temperature gradients giving rise to thermal stresses, etc. Such stresses act as external driving forces which may induce dislocation motion and plastic flow (see below). Dislocation interactions, on the other hand, are embodied within the internal stress field created by the dislocation system itself. Dislocations are sources of internal stresses. The stress field of a straight dislocation with Burgers vector h and tangent vector t is given by
'SA^ = T "i^n^-^f^ ,
(3)
where r is a vector of modulus |f| = r in the plane normal to t, and G is the shear modulus of the material. The dislocation stress field decays radially like b/r and may be strongly
218
M. Zaiser
anisotropic; its dependence on the azimuthal angle 0 is governed by the angle 0 between the Burgers vector h and line direction t^ which defines the type of the dislocation. For instance, dislocations with 6 parallel to Tare referred to a screw dislocations, dislocations with 6 perpendicular to Tedge dislocations, and dislocations with 9 = ±7r/3 or ±27r/3 are called 60° dislocations. Explicit expressions for the stress fields of edge, screw and general ('mixed') dislocations in isotropic media can be found in standard works (e.g. [5]). Here we only note that the stress field of a straight dislocation is antisymmetric in both b and For a quasi-two-dimensional system of straight parallel dislocations with common line direction ^, the internal stress at the point rin the plane perpendicular to the dislocation lines is simply obtained by summation over all dislocations, J
For a three-dimensional system of curved dislocation lines, the situation is slightly more complicated. A dislocation segment of tangent vector T, Burgers vector 6, and unit length situated at the origin creates at the point f = (r, 0, ^ ) a stress
.(M>-) = ^^M|LM) .
(5)
Explicit expressions for the stress fields of short straight segments of general orientation can be found in [6]. The internal stress field at r is obtained by summation over all segments, ^'^'(r) = E
[ or(6,,r(.),f - f(s))ds
.
(6)
Here, r{s) is the parametrization of a three-dimensionally curved dislocation Ci with Burgers vector fei, and t{s) is the corresponding bundle of tangent vectors. Again, the summation runs over all dislocation lines ^. 2.2. Dislocation motion and plastic flow Dislocations move under the action of Peach-Koehler forces. Moving dislocations are subject to electron and phonon drag leading to a friction force per unit length / F = X^ which is proportional to the dislocation velocity. Since the effective mass of a dislocation is small, dislocation glide often takes place in an over-damped manner and can therefore be described by 'Aristotelian' dynamics, v = h{r^lx
•
(7)
This description ceases to be correct when dislocation motion is controlled by thermal activation processes. Thermal activation of dislocations may be important for the overcoming of localized obstacles such as solute atoms, radiation- or deformation-induced ^For topological reasons, dislocations cannot terminate inside the crystal and the Burgers vector is constant along the dislocation line. Any dislocation must either form a closed loop or connect two triple nodes where J^j ^i = 0. In a finite crystal, one or both of these nodes may be replaced by a point at the surface, and in Etn infinite crystal by the point at infinity.
Dislocation patterns in crystalline solids - phenomenology and modelling
219
point defects and their agglomerates, or small precipitates. Thermal activation is also crucial for the motion of dislocations in crystals with high Peierls barriers: If a dislocation moves along its slip plane, its core configuration and hence its energy changes in a lattice-periodic manner. In certain materials the resulting energy barriers to dislocation motion ('Peierls barriers') are significant and at sufficiently low temperatures, dislocation motion is governed by their overcoming via a thermally activated double-kink mechanism. In cases where dislocation motion is controlled by thermal activation, the velocity (defined on a scale which is large in comparison with the spacing of the relevant obstacles) is given by an Arrhenius-type expression. H{T)
1/6 exp
(8)
kT
where z/ is a characteristic frequency of the order of magnitude of the Debye frequency. The stress dependence of the activation energy H{T) reflects the way how the energy profile of the relevant obstacles is tilted by the resolved shear stress acting on the dislocation. Whatever the stress dependence of its velocity, a gUding dislocation (dislocation segment) of tangent vector t moves in the direction n x f of the glide component of the Peach-Koehler force. For dynamic purposes it is often convenient to distinguish dislocations according to their direction of motion into different 'populations'. For a 2D dislocation system of straight parallel dislocations, t can take only two values which define two possible 'signs' s := sign(fe.f) of a dislocation. For each slip system f3 and sign s we may formally introduce a discrete dislocation density pj^^ by
p^D'{^ = j:^ssMAr~r^)
,
(9)
J
where the sum runs over all dislocations. The local shear strain rates on the different slip systems are related to the dislocation densities and dislocation velocities by Orowan's relation,
7^(0 = EV''(0K^xt>(r) .
(10)
s
Depending on the level of description (discrete vs. continuum), p^'^{r) in this equation is understood as the discrete density given by Eq. (9), or an appropriate coarse-grained version thereof. Analogous expressions can be formulated in 3D. The plastic strain rate tensor is obtained from the shear strain rates on the different slip systems via i{r)^J^M^^^(r)
(11)
where the projection tensors M^ = [s^ ^ ^^Jsym are the symmetrized tensor products of the respective unit slip vectors and slip plane normals. 2.3. Scaling relations for dislocation patterns Irrespective of their morphology, dislocation patterns in plastically deformed crystals obey some fairly universal scahng relations. We discuss these scaHng relations first for
220
M. Zaiser
the simplified case of a system of straight parallel dislocations which can be envisaged as point particles in a plane normal to their line direction. We assume that temperatures are sufficiently low such that dislocations move only by gUde, and that Peierls barriers are low such that glide may take place at arbitrarily small stresses. Under these circumstances a necessary and sufficient condition for a system of N dislocations to be in static equilibrium at an applied stress (Text is that the resolved shear stresses acting on all dislocations must be zero, i.e. ni = 0 ,
hi
iG\l...N]
.
(12)
Now assume that we increase the applied stress by a factor r/ and simultaneously decrease all dislocation spacings by the same factor. Prom Eqs. (3) and (12) it follows immediately that the new dislocation arrangement is also in static equilibrium. This simple scaling invariance also holds in the limit AT ~> cx) in an infinite crystal. In this case the re-scaling by a factor r; simply increases the dislocation density (number of dislocations per unit area) by a factor of r/^. Prom this scaling property the following conclusions derive: • If a given dislocation arrangement of average density p is stable under a certain applied stress (Text', then there exists a one-parameter manifold of dislocation arrangements of density p' with the same morphology that are stable under stresses (p'/p)VVe«. • If a dislocation arrangement exhibits inhomogeneous patterns with a characteristic length A (for examples see (b)-(d) in Pigure 1) and is stable under the applied stress o-ext, then morphologically equivalent arrangements with characteristic length A' are stable under stresses (X/X')(Tcxt• Assume that the applied stress is slowly increased until sustained dislocation motion occurs on a slip system (6, n). The corresponding critical resolved shear stress Tg^ is also called the flow stress of this slip system, and the critical resolved shear stress of the slip system(s) where long-range dislocation motion first occurs is simply called the flow stress r^ — min[Tg^]. Prom scaling it follows that, for dislocation patterns of similar morphology but different total density, flow stresses are proportional to the square root of dislocation density. This is often expressed in terms of Taylor's relation r' = aGby/p
,
(13)
where G is the shear modulus of the material, and the constant a depends weakly on the morphology of the dislocation arrangement (typically 0.2 < a < 0.4). In general, because of dislocation multiplication (cf. below) plastic deformation goes along with an increase in dislocation density, and this in turn implies an increasing flow stress (work hardening).
Dislocation patterns in crystalline solids - phenomenology and modelling
iu|
r-, 10'
^
I I I miif
111
I I I iini|
10'i ••*-.»
2, io°i •
Fe Cu Al
•
• o
g 10"
10"'
E 10'
)30
GaAs
iiiiiiH
^ 10"
^. oc
I iiiinn
I I I mm
I I limn
i
iiiiiilj
10° 10' 10' 10' Mean cell size Jl [^im] "I i i i i i i |
o
I I iiiiii|
I I limn
GaAs
•
Fe
•
CuMn
I I iiini|
10' I I iiiiij
,90)
,9'
cd
^10° c
g
1 ^°' i 10-^10"
I inn
III!
I I I linn
I I I nun
10^ 10^ 10' Mean cell size A [\xm]
10°
' ' ''""I
10^
Figure 2. Scaling of dislocation cell patterns in metals and semiconductors. Top figure: cell size vs. flow stress (resolved shear stress in the active slip system(s)). Bottom figure: cell size vs. mean dislocation spacing. Data for pure metals after Holt [7], data for CuMn after Neuhaus and Schwink [8], data for GaAs after Rudolph et. al. [9].
The above arguments can be generalized to three-dimensional systems of curved dislocation lines. For a 3D dislocation arrangement in static equilibrium, the internal stress field (6) must balance the externally applied stress everywhere along the dislocation lines. Again, by increasing the applied stress by a factor r/, and shrinking all distances including the length ds of a line element (cf, Eq. (6)) by the same factor, we obtain another static dislocation arrangement. This procedure increases the dislocation density (now defined as line length per unit volume) by a factor of 77^, and we recover the same scaling relations as above. The experimental data in Figure 2 illustrate the relations between dislocation densities, charsu^teristic lengths of dislocation cell patterns, and the flow stresses at which these patterns have formed. It is seen that the general scaling relations r^ a y/p oc 1/A are
222
M. Zaiser
fulfilled over a wide range of materials and deformation conditions, with proportionality factors that are remarkably insensitive to all peculiarities of the deformation process. Often, in the course of deformation the flow stress and dislocation density increase and the characteristic length of the dislocation pattern decreases in accordance with these scaling relations, while the morphology of the dislocation arrangement remains more or less unchanged. Such self-similar behavior of the dislocation pattern during hardening has been termed the 'law of similitude' [10]. At this point we have to address an apparent paradox: The scaling arguments we have formulated pertain to static dislocation arrangements while the formation of heterogeneous dislocation patterns necessarily involves the motion of dislocations. By looking at Eqs. (7) or (8) we see, however, that the dynamic behavior of dislocations is not invariant under the re-scaling procedure discussed above. For situations where the motion of dislocations is governed by drag forces or, more generally, the dislocations are highly mobile (e.g. at high temperatures), this paradox is readily resolved. In this case, numerical estimates show that at each given moment plastic deformation is carried by only a few dislocations (dislocation segments). These 'active' segments move rapidly in regions where the dislocation-induced internal stresses assist the externally applied stress. At the same time, the overwhelming majority of the dislocations is practically at rest. It has been estimated that, in a typical fee crystal deforming at a characteristic experimental strain rate of 10"^ s~\ only a fraction of 10~^ of all dislocations at each given moment appreciably contribute to the deformation [11,12]. Under these circumstances, scaling relations for static dislocation arrangements retain their validity even during plastic flow. The situation is more complicated in crystals where the motion of dislocations is limited by localized barriers that are not dislocation-related, such as Peierls barriers or localized obstacles. At low temperatures the overcoming of such obstacles may require substantial stresses (cf. Eq. (7); for appreciable dislocation motion to occur on typical experimental timescales, stresses must be high enough to reduce the activation enthalpy H{T) to about 25/cT). This may lead to a temperature-dependent flow stress contribution that is independent on dislocation density, and in such situations the scaling relationships of the 'law of similitude' do not apply.
3. D I S C R E T E DISLOCATION D Y N A M I C S ( D D D ) SIMULATIONS The most straightforward way of dealing with the dynamics of a many-dislocation system is to directly solve the equations of motion of the dislocation lines, keeping track of all the dislocation interactions. We first briefly discuss fully three-dimensional dislocation dynamics simulations which attempt to give a true representation of the evolution of the system of interacting dislocation lines, and then the less realistic but computationally much less intensive alternative of studying idealized quasi-two-dimensional systems consisting of parallel dislocations. 3.1. D D D simulation of 3-diinensional dislocation systems Several different schemes have been proposed for simulating the 3D dynamics of a system of interacting dislocations. These schemes mainly differ in the way how the dislocation
Dislocation patterns in crystalline solids - phenomenology and modelling
223
Figure 3. Dislocation pattern observed in a three-dimensional DDD simulation (simulation of an fee crystal deformed along a [100] axis to a total strain of 0.003; the image shows the dislocations contained within a 'TEM foil' of 5 //m thickness and [HI] orientation, after Madec et. al. [16].
lines are represented. Representations in terms of parametric splines have been adapted, using the spline parameters as generalized line coordinates [13]. Other simulations represent a dislocation line by a sequence of straight segments of variable orientation [14], using the nodes between segments as line coordinates. In the following we discuss in more detail the approach by Kubin, Devincre and co-workers [6,15-17] who use a representation of dislocation lines in terms of sequences of linear segments for which only a limited number of discrete orientations are allowed. Elastic interactions between dislocation segments are mediated by the corresponding segment stress fields which according to Eq. (5) decay in space like 1/r^. In this sense the computational problem is very similar to the molecular dynamics simulation of a system of particles with long-range (e.g. gravitational or electrostatic) interactions^. Once the stress acting on a dislocation segment is known, the Peach-Koehler force and the segment velocity can be calculated. In moving the segments, however, it is important to maintain connectivity of the dislocation line which for curved dislocations requires continuous adjustment of the segment length and/or the insertion and removal of segments. Additional problems arise if two dislocation segments get very close. Two dislocation segments of the same slip system but opposite sign may annihilate spontaneously if they ^Since piecewise linear representations of the dislocation lines give rise to artificial stress singularities at the 'corners', it may be necessary to apply corrections in order to correctly evaluate the interaction of adjacent segments.
224
M. Zaiser
approach each other below a critical distance (typically a few interatomic spacings)^. Another process may take place if two dislocation lines gliding on different slip planes intersect. If the Burgers vectors 61 and 62 and line configuration of the dislocations are such that the two lines attract each other, then upon contact they will form a new 'junction' segment with Burgers vector hi +62, thereby reducing the total elastic energy and dislocation line length. As a consequence, the intersection takes an H- rather than an X-shape and two triple nodes emerge"^. The junction segment initially runs along the direction of intersection of the slip planes of the two dislocations. If only discrete segment orientations are allowed, it is important that these include the orientation of potential junction segments. For instance, in fee crystals where the Burgers vectors are directed along [110] lattice directions and the slip planes are of [111] type, junction segments can have screw or 60° orientations and these orientations should be included in the discretization [17]. There are two reasons why junction formation is of crucial importance for the dynamical behavior of a dislocation system. On the one hand, junctions act as dislocation obsta^^les. A dislocation moving on its slip plane will continuously form junctions with 'forest' dislocations threading through this plane, and dislocation motion can only proceed if the stress acting on the dislocation is high enough to destroy these junctions. In metals deforming in multiple slip, this mechanism yields a major contribution to the flow stress. On the other hand, dislocation segments pinned at two anchoring points may bulge out and nucleate dislocation loops that expand during deformation. Expansion of dislocation loops increases the total dislocation line length and is thereby responsible for the increasing dislocation density in a deforming crystal. Through this mechanism junction formation (or, more general, the 'knitting' of a 3D dislocation network) is also crucial for dislocation multiplication and work hardening. Three-dimensional dislocation dynamics simulations correctly represent the processes of dislocation multiplication and junction formation. They account without additional assumptions for the multitude of conceivable bowing and intersection processes which may occur in the dislocation network. However, the neccessity to continuously re-evaluate the stresses acting on all dislocation segments tends to make such simulations computationally extremely intensive. Because of their high computational cost, present-day simulations of dislocation dynamics in bulk metals are restricted to comparatively small system sizes (typically about (10//m)^, though periodic boundary conditions may be imposed to mimic a bulk crystal), small strains (typically far less than 1% total deformation), and low dislocation densities (pL^ < 5 x 10^ where L is the characteristic linear dimension of the simulated volume). Under these circumstances, only the very first stages of dislocation cell patterning can be studied (Figure 3). Since dislocation densities are low, the cell size of the incipient dislocation cell patterns is of the same order as the size of the simulation cell (e.g.. Figure 3 shows a section of the simulation cell plus its first periodic images), and therefore not much information about the spatial morphology of the emergent patterns can be obtained.
^In case of two edge segments, annihilation may leave an agglomerate of point defects in the crystal. ^Only triple nodes are stable in a dislocation system; intersections which do not lead to junction formation imply that the two dislocations repel each other and, hence, no quadruple nodes are formed.
Dislocation patterns in crystalline solids - phenomenology and modelling
225
0.75 ixm
Figure 4. Dislocation pattern observed in a two-dimensional DDD simulation of deformation in symmetrical double slip; shear strain in each of the two sHp systems 7 = 0.073, resolved shear stress r = 2.6x 10~^G, total dislocation density 2 x lO^^m"^; after Benzerga et. al. [18].
3.2. D D D simulation of 2-dimensional dislocation systems Because of the tremendous reduction in degrees of freedom, two-dimensional dislocation systems can be simulated to much larger strains and dislocation densities. Since the basic scaling relations for dislocation patterns are similar in two and three dimensions, one may ask whether such simulations are a way of modelling the later stages of dislocation pattern formation and to the evolution of these patterns during sustained deformation. A 2D dislocation system is essentially a system of interacting point particles moving in a plane. For 2D dislocation dynamics to be representative of the behavior of a real, three-dimensional crystal, two conceptual problems must be addressed: (i) In a system of straight dislocations there is no mechanism for increasing the dislocation density (dislocation multiplication) which in 3D occurs by bulging of segments in the dislocation network and expansion of the ensuing dislocation loops. Since straight dislocations can neither bulge nor expand, one has instead to devise some phenomenological rule for throwing new dislocations into the system, (ii) Since dislocations cannot intersect in 2D, the formation/destruction of junctions and the corresponding work hardening have to be accounted for in a phenomenological manner. The problem is complicated by the fact that dislocation loop generation and destruction of junctions in a 3D dislocation system may be 'many-body' processes which in general simultaneously involve several dislocation lines. The approach adopted in the most recent 2D simulations is to model junction formation as a dislocation pair reaction: as soon as two dislocations of different slip systems approach beyond a certain collision radius, they form a sessile junction. This junction can be broken if a critical stress is exceeded which is related to the dislocation configuration in its surrounding. Source formation and operation are modelled in a similar spirit. This procedure makes it necessary to isolate and parameterize a limited set of dislocation mechanisms. It introduces into the models a
226
M. Zaiser
number of phenomenological parameters which have to be determined either by matching to 3D simulations or to experimental data [18]. The advantage of 2D simulations lies in the possibility to achieve strains and dislocation densities sufficiently high such that at least the early stages of dislocation pattern formation and evolution can be observed (Figure 4) and statistically significant information on the morphology of the dislocation patterns can be deduced. However, even if the problems relating to mapping 3D dislocation processes onto a 2D geometry can be solved, the approach is restricted to certain glide geometries since it is difficult to represent simultaneous dislocation glide on more than two intersecting slip planes in a 2D model.
4. C O N T I N U U M D I S L O C A T I O N D Y N A M I C S A P P R O A C H E S In view of the computational complexity of discrete dislocation dynamics simulations, repeated attempts have been made to arrive at some kind of coarse-grained description of dislocation dynamics. There are two fundamental problems: a) how to define coarsegrained dislocation densities in a manner which retains the relevant information about the kinetic properties of the discrete dislocation lines, and b) how to obtain the corresponding equations of motion. Two main approaches may be followed - an 'energetic' approach which starts out from the elastic energy associated with the dislocation system, and a 'dynamic' approach which determines equations of motion for dislocation densities in a phenomenological manner. 4.1. Linear irreversible thermodynamics and energy minimization Dislocations move under the action of forces, which implies that they move 'downhill' in an energy functional. Any static dislocation arrangement necessarily corresponds to a local minimum of the elastic energy^. It is therefore tempting to address dislocation patterning in terms of energy minimization arguments. An early attempt in this direction was made by Holt [7] who studied the stability of a spatially homogeneous dislocation system with conserved total dislocation density. In the following we present a slightly generalized version of this model which works analogous to the Cahn-Hilliard theory of spinodal decomposition [19]. The elastic energy of the internal stress field is given by E,^ = / ( T ( r ) C - V ( r ) d V
(14)
where C~^ is the tensor of elastic compliances and V is the crystal volume. In the following we consider a system of straight parallel dislocations. If we denote by
(15)
where the integration is carried over the plane normal to the dislocation. The dislocation stress field is given by Eq. (3), and accordingly the dislocation self-energy diverges with ^Note that the elastic energies associated with the dislocation system are usually so large that entropy effects can safely be neglected.
Dislocation patterns in crystalline solids - phenomenology and modelling
227
increasing crystal radius R like E^^^ = K^Gb\n{R/b) where K^ is a constant of the order of unity which depends on the type of the dislocation. The energy divergence which occurs according to Eqs. (3,15) as one approaches the dislocation core may be truncated at radius b since, on scales comparable to the atomic spacing, local linear elasticity ceases to be valid and Eq. (3) no longer applies. Far away from the dislocation, on the other hand, linear elasticity is perfectly valid and, hence, the logarithmic divergence of the self-energy is limited only by the finite crystal size. The interaction energy of two dislocations of slip systems f3 and /?' and signs s and s' at the respective positions r and r*' is given by E^nt""\r-
r') = ss' j a^{f- f")C-^(T^\r'
- r")d?r"
.
(16)
This interaction energy can be written as E^""^ = ss'K^^'{(j))Gb\n{R/\r - r'\) where K^^ {(j)) is a non-dimensional function that depends on the relative orientation of the two dislocations in the plane. Note that the dislocation interaction is strongly anisotropic; its angular average vanishes, J K^^\(f)d(f) — 0. Now the energy of a discrete dislocation system characterized by the discrete densities PD{^ as defined in Eq. (9) is readily written as
E = / ^-(r)^i,dV + E ^ - 7 / P'ii'i^Pn'inE^^'''(^-
^"'OdVdV .
(17)
To arrive at a continuum description we average over an ensemble of statistically equivalent dislocation systems. This leads to continuous densities p^'^{r) — (PD (r))- Averaging the product of discrete densities in the interaction term of Eq. (17) leads to pair densities p f | ' ' " " V , f ' ) - / ' ^ ( r ) / ' ' " ' ( r ' ) [ l +
(18)
jp[r')EUr-r'W
Here the average self- and interaction energies of a dislocation axe given by E^M = T.f"EL I3,s
EUr^= E ss'f-f-'E^,-'{r)d^^'-'(r^ /3l3',ss'
,
(19)
228
M. Zaiser
where /^'"^ = p^'^/p- In a random dislocation arrangement without correlations, the average interaction energies vanish and the energy density (18) diverges logarithmically. The presence of dislocation pair correlations re-normalizes this divergence [12]. To derive the temporal evolution of the dislocation pattern from the elastic energy functional we make the following assumptions: (i) only the total dislocation density changes whereas the 'composition' of the dislocation arrangement as expressed by the density fractions /^'* remains invariant, (ii) the total number of dislocations is preserved, i.e., the time evolution of p is given by the continuity equation dtp{r^ = ~VJ{r)
(20)
(iii) the dislocation flux J is proportional to the gradient of the average dislocation energy, J = —XP^E(f}/x where x is an effective friction coefficient. The problem is very similar to the linear irreversible thermodynamics treatment of a system with conserved order parameter as developed, for instance, in the Cahn-Hilliard theory of spinodal decomposition. Now we consider a homogeneous dislocation arrangement of total density po and study the time evolution of small space-dependent density fluctuations Sp{r). Retaining in the equation of evolution only terms of linear order in 6p^ we obtain Xdt6p(r) - poA J6p{r')EUr-
r')dh'
« poAp^'^<^/^(r) + VD^^^V6p{f)]
(21)
where
D^'^ ^ J EUr)d'r
, Z>(^) = | ^ 5 ^ , ( r l d V .
(22)
In the last step we have performed a Taylor expansion of p(f') around f^ — f, using the fact that the dislocation pair correlation functions and therefore also the average interaction energy £'mt(^— ^ 0 are short-ranged functions, cf. Eq. (19). Re-writing the evolution equation for Sp in Fourier space we get xdtSp(k) = /OoP[-/)(°^ + kD^^^k]6p{k)
(23)
Since dislocations arrange such as to reduce their total energy, the coefficients D^^^ as well as the components of the matrix D^^^ are negative. This implies a long-wavelength instability similar to the instability encountered in spinodal decomposition. The amplification of fluctuations is in general anisotropic; the direction(s) of maximum amplification corresponds to the Eigenvector(s) of JD^^^ with the largest Eigenvalue D^j^. The corresponding wavelength of dislocation density fluctuations is given by A = TT Note that, since the range of the dislocation pair correlations is proportional to the dislocation spacing, Z)^°^ oc l/po and D^^^ a l/(po)^- Hence the wavelength of maximum instability which characterizes the emergent patterns is proportional to (but in general much larger than) the dislocation spacing, in agreement with the principle of similitude. This 'energetic' approach to dislocation patterning has certain attrax^tive features, as the treatment of dislocation patterning within the general framework of linear irreversible thermodynamics leads to a simple continuum theory which correctly accounts for the
Dislocation patterns in crystalline solids - phenomenology and modelling
229
observed scaling relation A oc p~^/^, while the possibility of describing the entire dislocation population in terms of a single partial differential equation significantly reduces the computational complexity. Furthermore, the approach offers a simple explanation for dislocation patterning: the elastic energy of a dislocation is reduced if the dislocation density increases, and therefore the dislocation system segregates into regions of high and low dislocation density. On the other hand, the approach rises several problems: • The assumption that the dislocation flux follows the gradient of the elastic energy may not be warranted. The motion of dislocations is subject to dynamic constraints, e.g., at low temperatures their glide is confined to the sUp planes. Even at higher temperatures, where out-of-plane motions are possible, the effective mobilities for in-plane and out-of-plane motions are rather different.
• As dislocations of different slip systems may move in different slip planes, their fluxes have different directions.
• The approach cannot account for dislocation motions driven by an external stress. Under an externally applied stress, positive and negative dislocations may move in opposite directions, and the evolution of the dislocation arrangement cannot be described in terms of a single dislocation flux even if only dislocations of a single slip system are present. • During plastic deformation the dislocation density continually increases, and hence the assumption that the dislocation density behaves like a conserved order parameter is not justified. • It has also been criticized that the predicted proportionality of the pattern wavelength and the dislocation spacing is actually a consequence of the assumed scaling properties of the pair correlation functions. However, a recent study has to a certain extent justified the energy-based approach. By statistically averaging the equations of motion of a discrete dislocation system, Zaiser and co-workers arrived at evolution equations which under certain restrictive assumptions (dislocations of a single slip system, no external stress) are similar to Eq. (22). The dynamic constraints to dislocation motion (glide motion can occur on a single slip plane only) introduce additional anisotropics; for the case of a system of straight parallel edge dislocations it has been found that the instabiUty leads to the formation of periodic walls perpendicular to the dislocation glide direction [20]. Such walls are indeed observed in cycfically deformed metals (Figure 1 (b)). In prolonged cyclic deformation the dislocation system reaches a steady state where the dislocation density remains practically constant, and one may conjecture that periodic reversal of the external stress reduces the influence of directed dislocation fluxes on the dislocation patterning. In unidirectional deformation, on the other hand, evolution of the dislocation microstructure is characterized by directed dislocation fluxes and increasing dislocation densities, and it is difficult to see how these features can be accommodated within the 'energetic' approach.
230
M. Zaiser
4.2. S y n e r g e t i c m o d e l s A plastically deforming crystal is a driven system which by the continuous influx and dissipation of mechanical work is kept in an out-of-equilibrium state. This observation has prompted attempts to model dislocation patterning as a 'synergetic' pattern formation phenomenon, in analogy with spatial patterning phenomena observed in many driven far-from-equilibrium systems, e.g., the Belushov-Zhabotinskii reaction in chemical kinetics or the Taylor instability in hydrodynamics (for overviews of synergetic patterning, see [21,22]). In the theoretical description of these phenomena, spatial patterns emerge from symmetry-breaking instabilities in systems of nonlinear reaction-diffusiontransport equations. It has been proposed to adopt this framework to dislocation systems by distinguishing dislocation populations pk of different specification and writing down reaction-transport equations of the type [23-25] dtPk^VJk
= fk[[pi]) .
(24)
Here, the non-linear reaction terms fk model local interactions between dislocation populations and the flux terms describe long-range dislocation transport. Several phenomenological models of this type have been proposed in the literature, assuming different transport mechanisms to be operative. For instance, Kratochvil considered the flux of edge dislocation dipoles induced by their 'sweeping' by moving screw dislocations [26]. Walgraef and Aifantis [24] consider single-slip cyclic deformation and describe the net transport arising from the forward-backward motion of dislocations during a stress cycle by diffusion-like terms of the form Jk = \^Dpk. In its simplest one-dimensional form, the Walgraef Aifantis model reads ^
^
= ^ - ^ ^ + /(Pi,pJ
,
^ A 0 + p(Pi)-/(A,Pm) .
(25)
Here, p ^ and pi are the densities of mobile and immobile (dipole) dislocations, respectively, each population consisting of dislocations of both signs. The non-linear reaction term is assumed as f{puPm) = Cipi — C2PmP\ where Ci and C^ were introduced as phenomenological coefficients. The Walgraef-Aifantis model allows to obtain periodic solutions corresponding to the ladder-like dislocation arrangement in persistent slip bands (PSB, see Figure 5, right). An extension of the model to more than one dimension has been also used to investigate the competition between matrix and PSB patterns as observed experimentally (Figure 5, left) [24]. In either case, the model leads to estimates of the pattern wavelength in terms of the diffusionlike coefficients D^^ and Di entering the reaction-diffusion scheme (25). A general problem with the phenomenological synergetic approach is that, in the absence of a well-defined procedure for coarse-graining dislocation dynamics, the definition of the dislocation populations in Eq. (24), the determination of the functional form of the reaction terms, and the mathematical specification of the dislocation fluxes are all left to educated guess. With some mathematical training it is possible to devise nonUnear equations which produce a particular type of spatial patterns but, since the results are in a sense pre-determined by the phenomenological 'input', it is arguable whether this really
Dislocation patterns in crystalline solids - phenomenology and modelling
M
PSB M PSB
cT"*
25-r
1
1
1
1
231
1
1
j
Til
H
\ X)
1 ^H 0
54
CO
b 0 r 0
u
1 2
1 4
u
1 6
1 8
uu 1 10
1 12
\
1 1
Distance [M.m]
Figure 5. Left: dislocation structure in single-slip cyclic deformation; the ladder-like structure corresponds to a persistent slip band (PSB); the surrounding patchy structure is referred to as 'matrix' (M). Right: One-dimensional dislocation density pattern in a persistent slip band obtained from the Walgraef-Aifantis model; Courtesy of E.G. Aifantis.
solves the problem of dislocation patterning. Nevertheless the synergetic approax^h has been important from a conceptual point of view: If the dislocation system is envisaged as a driven out-of-equilibrium system, one cannot simply take for granted the appUcabiUty of energy minimization concepts or of linear irreversible thermodynamics, and the driven dynamics of the dislocation ensemble is put into the focus of interest.
5. STOCHASTIC APPROACHES 5.1. Discrete stochastic dislocation dynamics The discrete stochastic dislocation d5niamics approach proposed by Groma and coworkers [27,28] considers the stress-driven motion of discrete dislocations but uses a statistical description of the force created by the internal stress field. As discussed in detail in [12], in many situations the internal stress field associated with a dislocation arrangement can be split into a long-wavelength part ('mean-field stress') which varies in spax^e on the scale A of dislocation density variations and can be expressed as a functional of the dislocation densities [12,27], plus a short-wavelength part which fluctuates on the scale p""^/^ of the average local dislocation spacing. The probability distribution of this fluctuating part of the internal stress has been evaluated in [27], and the short-range nature of its spatial correlations has been demonstrated in [12]. The basic idea of a discrete stochastic dislocation dynamics simulation is to calculate explicitly only the mean-field stresses, while replacing the short-wavelength flucutations of the internal stress field by a statistically equivalent stochastic process. The meanfield stresses are functional of the dislocation densities which change slowly in space and
232
M. Zaiser
time. Since these stresses need not be updated in every time step and, moreover, can be evaluated from the dislocation densities by convolution using Fast Fourier TVansformation (FFT), a very considerable reduction in the computational cost of the simulation can in principle be achieved.
Figure 6. Simulated dislocation pattern using a stochastic dislocation dynamics code; left: dislocation density pattern, right: greyscale pattern of the excess Burgers vector density; after Groma [28]
The crucial problem with this method is to correctly represent the statistical properties of the fluctuating stresses acting on the dislocations. Groma et. al. use a strong approximation by representing the temporal statistics of the internal stresses at the dislocation positions in terms of the statistics of the internal stress fluctuations at a random point in space. This is feasible if (a) only a small fraction of the dislocations is mobile at each given moment and (b) these mobile dislocations move as individuals, each of them individually scanning the internal stress 'landscape' created by the majority of the dislocation arrangement which remains at rest. While assertion (a) is presumably valid, it is rather difficult to assess the validity of assumption (b). In fact, analytical arguments [11,12,29] as well as simulations [30,31] and experiment [30,32] indicate that dislocations move in a strongly correlated, avalanche-like manner where dynamic interactions among the moving dislocations may be of crucial importance. An adequate theoretical understanding of these collective effects and an appropriate mathematical description of the corresponding
Dislocation patterns in crystalline solids - phenomenology and modelling
233
fluctuation phenomena are still missing. In spite of these problems, simulations of deformation-induced dislocation patterns using the discrete stochastic dislocation dynamics approach yield interesting results. An example is given in Figure 6 which shows the dislocation pattern forming in a 2D dislocation system deforming in symmetrical double slip. The hierarchical nature of the dislocation pattern with long straight walls carrying substantial excess Burgers vector (and, hence, substantial lattice rotations) compares well with certain experimentally observed cellular microstructures (see e.g. Figure 1 (d)). 5.2. Continuum stochastic dislocation dynamics The continuum stochastic dislocation dynamics method formulated by Hahner [29] and elaborated by Hahner and Zaiser [4,11,12,33] approaches the problem of dislocation patterning on a much more phenomenological level. This approach starts out from phenomenological differential equations which describe dislocation density evolution and work hardening on a macroscopic scale. The complex dynamics of plastic flow on mesoscopic scales is then taken into account by considering the shear strain rates on the different slip systems [i.e. the dislocation fluxes, cf. Eq. (10)] as fluctuating functions of space and time. Since dislocation accumulation and dislocation reactions are driven by the dislocation fluxes, this impUes that dislocation density evolution is described in terms of a set of stochastic differential equations. For illustration, we discuss the simplest model of this type [34]. We assume deformation to occur in symmetrical double slip. Dislocations are stored in the crystal after travelling a mean ghde path L = B/y/p that is proportional to, but much larger than, the average dislocation spacing. The increase of the total dislocation density p is then given by
dtP^fvrt = §Vpm + s^]
(26)
where in the second step we have split the (local) strain rate in either of the sUp systems into the mean strain rate and a spatio-temporally fluctuating contribution. Fluctuations in the local strain rates are modelled by assuming that straining proceeds in the form of a shot noise where the strain rate is composed of discrete random 'events', J(r,t) = 'ZjS{t-U)g(r-ri)
(27)
i
The times ti and locations rj of the different 'slip events' are assumed as statistically independent random variables. The shape function g{f) and event amplitude 7 are chosen to reflect general scaling relations and phenomenological observations of collective dislocation motion: a) In agreement with the scaling relations underlying the 'law of similitude' we assume that the range of spatial correlations in plastic flow decreases in inverse proportion with the flow stress. This implies that 5' is a function of f/r^. b) The event amplitude 7 increases in proportion with the flow stress. This reflects the scaling relation 7 = pbL [cf. Orowan's relation, Eq. (10)] where L can be understood as the mean path travelled by a dislocation during a collective slip event. Since p oc (r^)^ and L oc l/r^ it follows that 7 oc r ^ c) Correlations in plastic flow are strongly anisotropic, their range in the direction of dislocation motion significantly exceeds their range normal to that direction.
234
M. Zaiser
Relevant information on dislocation patterning can be directly obtained by studying the stochastic differential equation (26). As discussed in detail in [34], the flow stress of an inhomogeneous dislocation arrangement can be written as r^ = aGb{y/p) where (...) denotes an average over the inhomogeneous microstructure. It then follows from Eq. (26) that the flow stress increases linearly with average strain, r^ = {aGB/2){y). Equation (26) is conveniently re-written by introducing the transformation of variables
I
1
2
yo_t>_oo-yo-g.^..^-^ 3
scaled dislocation density p l
Figure 7. Probability distributions of dislocation densities calculated from Eq. (29) for low (Q2 = 0.1, solid line) and high (Q^ = 2, dotted line) noise amplitudes. Data points: Results obtained from simulations of deformation in discrete random events with event sizes corresponding to the same effective noise amplitudes.
P ~ kV(<^G^)l^P5^ = [^l{'^GB{i))]i' ^ Furthermore we idealize the stochastic process <^7 = 7 ~ (7) by a Gaussian white noise process w(t). This leads to the stochastic differential equation
^fP=
sfp~p-Qp^
(28)
The 'noise amplitude' Q reflects the coarseness of plastic deformation. In scaled variables, Q is time-independent and proportional to 7/r. The Fokker-Planck equation corresponding to the stochastic differential equation (28) can be solved analytically; using Ito calculus its steady-state solution is found as
{^yfp-pj
(29)
Note that this transformatioii is again consistent with the scahng relations of the 'law of similitude'
Dislocation patterns in crystalline solids - phenomenology and modelling
235
where AT is a normalization constant. The shape of this probability distribution depends on the value of the 'noise amplitude' Q which acts as a control parameter of the system. Below a critical noise amplitude Q^ — 1/2 the probability density exhibits a maximum which, at small noise amplitudes, lies close to the average density. Above the critical noise amplitude this maximum disappears and gives way to a hyperbolic decay with an exponential cut-off. This noise-induced transition has been interpreted as a transition towards the formation of cell structures. This is illustrated in Figures 7 and 8.
Figure 8. Dislocation density patterns obtained from simulations of the continuum stochastic dislocation dynamics model discussed in the text; top: large noise amplitude, bottom: small noise amplitude. The probability distributions of dislocation densities corresponding to the top and bottom patterns are given by the open and full symbols in Figure 7, respectively.
Figure 7 shows scaled probability distributions (all densities have been normalized by their strain-dependent average values). The full line has been calculated from Eq. (29) for Q^ = 0.1, and the dotted line for Q^ = 2. The data points show the results of simulations assuming discrete 'deformation events' with amplitudes corresponding to the same noise strengths. The simulations have been carried out by assuming that the function g{f) in Eq. (27) decays exponentially as a function of IF— r^l with a decay length of one average
236
M Zaiser
dislocation spacing in the direction perpendicular to the glide direction and a decay length of 10 average dislocation spacings in the glide direction. Spatial patterns obtained from these simulations are shown in Figure 8. The top image obtained for the large noise amplitude shows a pattern of dislocation-dense walls enclosing dislocation-depleted cell interiors. The walls follow the direction of the slip planes and the overall morphology of the pattern is very similar to the patterns observed in twodimensional DDD simulations (Figure 4). For the small noise amplitude, on the other hand, the dislocation pattern is more or less homogeneous and exhibits no discernable features. Irrespective of the technical details, stochastic dislocation dynamics approaches lead to an interpretation of dislocation patterning that is very different from the energetic approaches outlined earlier. According to the stochastic interpretation, dislocation patterning is a far-from-equilibrium phenomenon driven by fluctuating dislocation fluxes. These fluxes occur only if an external driving is provided in terms of an applied stress or imposed macroscopic strain rate. Due to dislocation interactions, the dislocation fluxes on mesoscopic scales are heterogenous in space and time and this heterogeneity is ultimately responsible for the emergence of heterogeneous dislocation patterns.
6. C O N C L U S I O N S In spite of the fact that the spontaneous formation of dislocation patterns is an ubiquitous phenomenon in plastic deformation of crystalline solids, to date there exists no commonly accepted approach towards computational modelling and/or theoretical understanding of dislocation patterning. This observation is particularly astonishing in view of the fact that dislocation patterns obey simple scaling relations which can be observed over at least five decades in pattern 'wavelength\ Furthermore, the same scaling relations are observed to hold in widely differing materials (metals, ionic solids, semiconductors). In spite of this remarkable degree of universality in the phenomenology of dislocation patterns, a commonly accepted theory does not (yet) exist. This is partly due to the fact that three-dimensional dislocation dynamics simulation, which is the most straightforward approach towards computational modelling of dislocation patterns, is still limited by the substantial computational cost of simulating systems of long-range interacting, flexible and reactive lines. Two-dimensional simulations reduce the computational cost but increase the phenomenological 'input' required in the models and are restricted to a limited set of deformation geometries. Continuum dislocation dynamics approaches constitute an alternative, but suffer from the fact that there exists no systematic procedure for coarse-graining systems of flexible lines with long-range interactions. Stochastic models reduce the computational cost by using 'mean-field-plusfluctuation' approaches to the internal stresses and/or dislocation fluxes. In this case, the fundamental problem is that despite recent progress the fluctuation properties of driven many-dislocation systems are theoretically poorly understood, and therefore the models have to rely on crude approximations or heavy phenomenological input. Is there a way out of this impasse? Of course, the exponential growth of available computing power will ultimately beat the power-law increase of computational cost as
Dislocation patterns in crystalline solids - phenomenology and modelling
237
three-dimensional simulations are carried to larger system sizes, strains and dislocation densities. However, the experience of the last decade has shown that progress along this direction has not been as rapid than initially expected. In the opinion of the present author, an alternative strategy may be to further develop our understanding of the nonequiUbrium statistical mechanics of systems of interacting lines in order to obtain coarsegrained descriptions and/or fluctuation properties of dislocation systems. Ultimately, it should not be necessary to know the position of every single dislocation segment in order to understand how dislocation patterns emerge.
ACKNOWLEDGEMENTS The author expresses particular thanks Prof. Peter Rudolph for drawing his attention towards the formation of dislocation cell patterns in semiconductor crystal growth and for inspiring discussions on this subject. The present overview could not have been written without discussions, collaborations and controversies involving numerous other groups and individuals. These scientific interactions have been greatly helped by financial support of the European Commission through two TMR/RTN networks under Contracts No ERBFMRX-CT-960062 and HPRN-CT-2002-00198, which hereby is gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
H. Mughrabi, T. Ungar, S. Kienle and M. Wilkens, Phil. Mag. A 53 (1986) 793. B. Gong, Z. Wang and Z.G. Wang, Acta Mater. 45 (1997) 1365, 1379. Q. Liu, D. Juul Jensen and N. Hansen, Acta Mater. 46 (1998) 5819. M. Zaiser, K. Bay and P. Hahner, Acta Mater. 47 (1999) 2463. J.P. Hirth and J. Lothe, Theory of Dislocations, Wiley, New York, 1982. B. Devincre and M. Condat, Acta Metall. Mater. 40 (1992) 2629. D. Holt, J. Appl. Phys. 41 (1970) 3197. R. Neuhaus and Ch. Schwink, Phil. Mag. A 65 (1992) 1463. P. Rudolph, Journal of Crystal Growth, in the press. (Data presented at the 15th American Conference on Crystal Growth and Epitaxy, Keystone, July 20-24, 2003.) N. Hansen and D. Kuhlmann-Wilsdorf, Mater. Sci. Engng. 81 (1986) 141. M. Zaiser, Mater. Sci. Engng. A 309-310 (2001) 304. M. Zaiser and A. Seeger, in: Dislocations in Solids, Vol. 11, eds. F.R.N. Nabarro and M.S. Duesbery (North-Holland, Amsterdam, 2002) p. 1. N.M. Ghoniem, S.H. Tong and L.Z. Sun, Phys. Rev. B 61 (2000) 913. O. Politano and J.M. Salazar, Mater. Sci. Engng. 309 (2001) 261. B. Devincre and L.P. Kubin, Mater. Sci. Engng. A 234-236 (1997) 8. R. Madec, B. Devincre and L.P. Kubin, Scripta Mater. 47 (2002) 689. R. Madec, B. Devincre and L.P. Kubin, Phys. Rev. Letters 89 (2002) 255508. A.A. Benzerga, Y. Brechet, A. Needleman and E. Van der Giessen, Model. Simul. Mater, Sci. Engng. 12 (2004) 159. J.W. Cahn and J.E. Hilliard, J. Chem. Phys. 28 (1958) 258. M. Zaiser, M.-C. Miguel and I. Groma, Phys. Rev. B 64 (2001) 224102.
238
M. Zaiser
21. H. Haken, Synergetics (Springer, Berlin, 1978). 22. G. Nicolis and I. Prigogine, Self-Organisation in Non-Equilibrium Systems (Wiley, New York, 1987). 23. E.G. Aifantis, Mater. Sci. Engng. 81 (1986) 563. 24. D. Walgraef and E.G. Aifantis, Int. J. Engng. Sci. 23 (1985) 1351, 1359, 1365. 25. E.G. Aifantis, Int. J. Plasticity 3 (1987) 211. 26. J. Kratochvil, Scripta Metall. Mater. 26 (1993) 113. 27. I. Groma and B. BaJco, Phys. Rev. B 58 (1998) 2969. 28. I. Groma and B. Bako, Phys. Rev. Letters 84 (2000) 1487. 29. P. Hahner, Appl. Phys. A 62 (1996) 473. 30. M.-G. Miguel, A. Vespignani, S. Zapped, J. Weiss amd J. Grasso, Nature 410 (2001) 667. 31. M.-G. Miguel, A. Vespignani, M. Zaiser and S. Zapped, Phys. Rev. Letters 99 (2002) art. no. 165501 32. H. Neuhauser, in: Dislocations in Solids, Vol. 6, ed. F.R.N. Nabarro, North-Holland, Amsterdam, 1984, p. 319 33. P. Hahner, K. Bay and M. Zaiser, Phys. Rev. Letters 81 (1998) 2470. 34. M. Zaiser, Mater. Sci. Engng. A 249 (1998) 145.